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Contents
List of Tables List of Figures Preface Acknowledgments 1
Introduction 1.1 Hydrologic Frequency Analysis 1.2 General Aspects and Approaches 1.3 Other Models 1.4 Return Period, Probability, and Plotting Positions 1.5 Flood Frequency Models 1.6 Hydrologic Risk 1.7 Regionalization 1.8 Tests on Hydrologic Data 1.8.1 Test for Independence and Stationarity 1.8.2 Tests Homogeneity and Stationarity 1.8.3 Test for Outliers
2
Selection and Evaluation of Parent Distributions: Conventional Moments 2.1 Moments of Distributions and Their Sample Estimates 2.2 Moment Ratio Diagrams (MRDs) 2.3 Probability Plots 2.4 Selection of Distributions 2.4.1 Chi-Square and Psi Tests 2.4.2 Kolmogorov–Smirnov Test 2.5 Regional Homogeneity and Regionalization
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3
Selection and Evaluation of Parent Distributions; Probability Weighted Moments and L–Moments 3.1 Moments of Distributions and Their Sample Estimates 3.2 L-Moment Ratio Diagrams 3.3 Goodness-of-Fit Tests 3.3.1 Tests Based on L-Moments 3.3.2 Regional Homogeneity Tests 3.4 A Case Study 3.4.1 Data and Preliminary Analysis 3.4.2 Regional Homogeneity 3.4.3. Regional Quantile Estimates
4
Parameter and Quantile Estimation 4.1 Introduction 4.2 Parameter Estimation 4.2.1 Method of Moments (MOM) 4.2.2. Method of Maximum Likelihood (ML) 4.2.3 Method of Probability Weighted Moments (PWM) 4.3 Quantile Estimation 4.4 Confidence Intervals 4.4.1 Standard Error in the MOM 4.4.2 Standard Error in the MLM 4.4.3 Standard Error in the PWM
5
Normal and Related Distribution 5.1 Normal Distribution 5.1.1 Introduction 5.1.2 Parameter Estimation Method of Moments: Maximum Likelihood (ML) Method: PWM Method: 5.1.3 Quantile Estimates 5.1.4 Standard Error: Method of Moments: ML Method: PWM Method: 5.2 Two-Parameter Lognormal (LN[2]) Distribution 5.2.1 Introduction 5.2.2 Parameter Estimation Method of Moments: Maximum Likelihood (ML) Method: PWM Method:
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5.2.3 Quantile Estimates 5.2.4 Standard Error Method of Moments: ML Method: 5.3 Three-Parameter Lognormal (LN[3]) Distribution 5.3.1 Introduction 5.3.2 Parameter Estimation Method of Moments: The Maximum Likelihood (ML) Method: PWM Method: 5.3.3 Quantile Estimates 5.3.4 Standard Error Method of Moments: ML Method: 6
The Gamma Famil 6.1 Exponential Distribution 6.1.1 Introduction 6.1.2 Parameter Estimation Method of Moments: Maximum Likelihood (ML) Method: PWM Method: 6.1.3 Quantile Estimates 6.1.4 Standard Error Method of Moments: ML Method: PWM Method: 6.2 Two-Parameter Gamma (G[2]) Distribution 6.2.1 Introduction 6.2.2 Parameter Estimation Method of Moments: Maximum Likelihood (ML) Method PWM Method 6.2.3 Quantile Estimates 6.2.4 Standard Errors Method of Moments ML Method 6.3 Pearson (3) Distribution 6.3.1 Introduction 6.3.2 Parameter Estimation Method of Moments Maximum Likelihood (ML) Method PWM Method:
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6.3.3 Quantile Estimation 6.3.4 Standard Errors Method of Moments ML Method 6.4 Log-Pearson (3) Distribution 6.4.1 Introduction 6.4.2 Studies on Skewness Coefficients 6.4.3 Parameter Estimation Method of Moments Maximum Likelihood (ML) Method PWM Method 6.4.4 Quantile Estimation 6.4.5 Standard Error Method of Moments ML Method: 6.5 U.S. Water Resources Council Method (WRCM) 6.5.1 Introduction 6.5.2 Frequency Analysis Procedure by the WRCM 6.5.3 Outlier Tests 6.5.4 Confidence Limits 7
Extreme Value Distributions 7.1 Generalized Extreme Value (GEV) Distribution 7.1.1 Parameter Estimation Method of Moments: Maximum Likelihood (ML) Method PWM Method 7.1.2 Quantile Estimates 7.1.3 Standard Error Method of Moments ML Method PWM Method 7.2 The Extreme Value Type I EV(1) Distribution 7.2.1 Parameter Estimation Method of Moments Maximum Likelihood (ML) Method PWM Method 7.2.2 Quantile Estimates 7.2.3 Standard Error Method of Moments ML Method PWM Method 7.3 Weibull Distribution
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7.3.1 Parameter Estimation Method of Moments Maximum Likelihood (ML) Method PWM Method 7.3.2 Quantile Estimates 7.3.3 Standard Errors Method of Moments ML Method 8
The Wakeby Distribution 8.1 The Five-Parameter Wakeby Distribution (WAK(5)) 8.1.1 Introduction 8.1.2 Parameter Estimation 8.1.3 Quantile Estimation 8.1.4 Standard Error 8.2 The Four-Parameter Wakeby Distribution (WAK(4)) 8.2.1 Introduction 8.2.2 Parameter Estimation 8.2.3 Quantile Estimation 8.2.4. Standard Error 8.3 The Generalized Pareto Distribution 8.3.1 Introduction 8.3.2 Parameter Estimation Method of Moments Maximum Likelihood (ML) Method PWM Method 8.3.3 Quantile Estimation 8.3.4 Standard Error Method of Moments ML Method PWM Method
9
The Logistic Distribution 9.1 Logistic Distribution 9.1.1 Introduction 9.1.2 Parameter Estimation Method of Moments Maximum Likelihood (ML) Method PWM Method 9.1.3 Quantile Estimation 9.1.4 Standard Error Method of Moments ML Method
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PWM Method 9.2 Generalized Logistic Distribution 9.2.1 Introduction 9.2.2 Parameter Estimation Method of Moments Region I –10 < CS < 10; –1/3 < k < 1/3 Region II 0 < CS < 10; 1/3 < k < 1/2 Region III –10 < CS < 0; –1/2 < k < –1/3 The Maximum Likelihood (ML) Method PWM Method 9.2.3 Quantile Estimation 9.2.4 Standard Error Method of Moments ML Method 10
Computer Program 10.1 Introduction 10.2 Description of Program
References
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List of Tables
Commonly Used Plotting Position Formulas Annual Maximum Flows in Wabash River at Lafayette, IN Annual Maximum Flows in Wildcat Creek near Jerome, IN Annual Maximum Flows in Salt Creek near Harrodsburg, IN Wabash River Basin Stations Mean Flow and Moment Ratios for 93 Stations in the Wabash River Basin Flow Data for Wildcat Creek (Station 46) at Kokomo, IN Flow Data for Mill Creek (Station 93) near Manhattan, IN L-Moments and Ratios for the Wabash River Basin Stations Homogeneity Measures for the Wabash River Basin Goodness of Fit Measures (ZDIST) for the Wabash River Basin Regional Homogeneity Analysis Using Wiltshire’s Method Estimated Quantiles for Station 95 in the WFWR Region for Different At-Site and Regional Distributions Tippecanoe River Near Delphi, IN (Station 43) Data Parameter Estimates for Example 5.1.1 Quantile Estimates and their Standard Errors (in parentheses) for Example 5.1.3
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Parameter Estimates for Example 5.2.1 Quantile Estimates and their Standard Errors (in parentheses) for Example 5.2.3 Parameter Estimates for Example 5.3.1 Quantile Estimates and Their Standard Errors (in parentheses) for Example 5.3.3 Parameter Estimates for Example 6.1.1 Quantile Estimates and Standard Errors (in Parentheses) for Example 6.1.3 Parameter Estimates for Example 6.2.1 Coefficients of a Polynominal in γ for the Parameters G, 1/A, B and H3 Quantile Estimates and Standard Errors (in Parentheses) for Example 6.2.3 Data from Eel River at North Manchester, IN Parameter Estimates for Example 6.3.1 Quantile Estimates and Their Standard Errors (in Parentheses) for Example 6.3.3 Data from Wabash River at Logansport, IN Parameter Estimates for Example 6.4.1 Quantile Estimates and their Standard Errors (in Parentheses) for Example 6.4.2 KT Values for Pearson Type III Distribution (Positive Skew) KT Values for Pearson Type III Distribution (Negative Skew) KN Values for Outlier Test Results for Example 6.5.1 Annual Flood Peaks in Floyd River Floods and 90% Confidence Intervals Results without Outlier
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Values of a, b, c, f, g, and h in the Covariance Matrix of ML Estimates of GEV Parameters Annual Maximum Flows in White River near Nora, IN Parameter Estimates for Example 7.1.1 Elements of the Asymptotic Covariance Matrix of the PWM Estimators of the Parameters of the GEV Distribution (Hosking et al. 1985) Variances and Covariances of Parameter Estimates in Example 7.1.3 Quantile Estimates and their Standard Errors (in parentheses) for Example 7.1 Annual Maximum Flows in Sugar Creek at Crawfordsville, IN Parameter Estimates for Example 7.2.1 Quantile Estimates and Their Standard Errors (in Parentheses) for Example 7.2.2 Parameter Estimates for Example 7.3.1 Quantile Estimates and Their Standard Errors (in Parentheses) for Example 7.3.3 Quantile Estimates for Example 8.1.2 Quantile Estimates for Example 8.2.1 The Wabash River Flows at Mt. Carmel (Threshold = 50,000 cfs) Parameter Estimates for Example 8.3.1 Quantile Estimates and their Standard Errors (in Parentheses) for Example 8.3.2 Parameter Estimates for Example 9.1.1 Quantile Estimates and their Standard Errors (in Parentheses) for Example 9.1.3 Annual Maximum Flow in East Fork White River at Seymour, IN Parameter Estimates for Example 9.2.1 Quantile Estimates and their Standard Errors (in Parenthesis) for Example 9.2.3
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List of Figures
Sample Residual Plot by using the LMS Method. Cs – Ck Moment ratio diagram. Watershed boundaries for the data in Table 2.1.1 β1 – β2 Moment ratio diagram. Cs – Ck Moment ratio diagram for 93 Wabash River basin stations. β1 – β2 Moment ratio diagram for 93 Wabash River basin stations. Probability plots for stations 46 and 93 compared with the theoretical linear relationship for the exponential distribution. Probability plots for the data from stations 93 and 112 compared with the theoretical linear relationship for the exponential distribution. Location maps of gauging stations in the Wabash River basin. The LCv, LCs moment ratio diagram for 93 stations in the Wabash River basin. The LCs – LCk moment ratio diagram for 93 stations in the Wabash River basin. Homogeneous flood frequency regions in Indiana (USGS 1984). The LCv, LCs moment ratio diagram for the WEST region. The LCs, LCk moment ratio diagram for the WEST region.
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The LCv, LCs moment ratio diagram for the UPPER region. The LCs, LCk moment ratio diagram for the UPPER region. Observed and Estimated Flows and 95% Confidence Intervals for Tippecanoe River Data used in Examples 5.1.1 to 5.1.3. Observed and estimated flows and 95% confidence intervals for Wabash River data used in Examples 5.2.1 to 5.2.3. Observed and estimated flows and 95% confidence intervals for the Wabash River data used in Examples 5.3.1 to 5.3.3. Observed and estimated flows and 95% confidence intervals for the Wabash River data used in Examples 6.1.1 to 6.1.3. Observed and estimated flows and 95% confidence intervals for the Wabash River data used in Examples 6.2.1 to 6.2.3. Observed and estimated flows and 95% confidence intervals for the Eel River data used in Examples 6.3.1 to 6.3.3. Observed and estimated flows and 95% confidence intervals for the Wabash River data used in Examples 6.4.1 to 6.4.3. Location of stations with reference numbers Skewness coefficients of stations. Coefficient of map skew, Cm, for use with WRCM. Results for Example 6.5.1. Results for Example 6.5.2. Relationship between the skewness coefficient and k for the GEV distribution. Observed and estimated flows and 95% confidence intervals for the White River data used in Examples 7.1.1 to 7.1.3.
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Observed and estimated flows and 95% confidence intervals for the Sugar Creek data used in Examples 7.2.1 to 7.2.3. Observed and estimated flows and 95% confidence intervals for the Tippecanoe River data used in Examples 7.3.1 to 7.3.3. Observed and estimated flows for the Wabash River data used in Examples 8.1.1 and 8.1.2. Observed and estimated flows for the Wabash River data used in Examples 8.2.1 and 8.2.2. Observed and estimated flows and 95% confidence intervals for the Wabash River data used in Example 8.3.1 to 8.3.3. Observed and estimated flows and 95% confidence intervals for the Tippecanoe River data used in Examples 9.1.1 to 9.1.3. Cs vs. K relationship for the generalized logistic distribution. Observed and estimated flows and 95% confidence intervals for the East Fork White River data used in Examples 9.2.1 to 9.2.3.
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Preface
Frequency analysis of floods is a very active area of investigation in Statistical Hydrology. Various distributions, methods of estimation of parameters, problems related to regionalization, and other related topics are being investigated. Much of this material is in journals and reports, and usually in a form which is not easily accessible to students and practitioners. Consequently, the gap between research and practice is growing in this field. The main purpose of this book is to present many of these distributions and estimation procedures in a unified fashion so that they would be available to students and practitioners. An attempt is made to make the book self-contained. The details behind the procedures are given in full so that the reader can see the basis for the computations. For each of the distributions considered—three-parameter estimation methods, the method of moments, the maximum likelihood method, and the method of probability-weighted moments—are discussed. Where they are available, the standard errors of estimates are also discussed. Each procedure is illustrated with real data. Most of the computations discussed in the book have been reprogrammed for use with personal computers. Executable versions of these programs are available on a CD-ROM from the senior author. The main distribution families discussed in the book have been chosen because of their popularity within the hydrogic community. There are many other distributions which have been proposed. Often these are discussed in a few papers and are either not used or studied further. Similarly, parameter estimation methods outside the better known ones are also not discussed. We would be glad to hear from the readers about the material discussed in the book and related matters. After about five decades of work, the field is still evolving. Only by increased use of these distributions and methods can a consensus be reached about them. We hope the material in this book contributes to that goal.
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Acknowledgments
We would like to thank Professor R. L. Kashyap of the School of Electrical Engineering, Professors A. G. Altschaeffl, R. S. Govindaraju, M. Hondzo, G. D. Jeong, D. A. Lyn, E. D. Sutton, and E. C. Ting of the School of Civil Engineering, Purdue University for their friendship and support. Mrs. Dinah Hackerd has cheerfully and patiently typed several versions of this book over the past five years. Professors W. H. C. Maxwell, B. C. Yen, and M. H. Garcia of the Department of Civil Engineering, University of Illinois, provided excellent facilities for A. R. Rao to spend a sabbatical during the Spring Semester of 1998. Mr. Siddhartha S. Rao of Purdue University was of great help with the computer programs. Dean R. J. Schwartz, School of Engineering, Purdue University graciously considered that writing this book was a worthwhile effort and granted the necessary sabbatical leave during Spring Semester, 1998. We are greatful to all of these individuals.
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CHAPTER 1
Introduction
1.1
Hydrologic Frequency Analysis
The primary objective of frequency analysis is to relate the magnitude of extreme events to their frequency of occurrence through the use of probability distributions (Chow et al., 1988). Data observed over an extended period of time in a river system are analyzed in frequency analysis. The data are assumed to be independent and identically distributed. The flood data are considered to be stochastic and may even be assumed to be space and time independent. Further, it is assumed that the floods have not been affected by natural or manmade changes in the hydrological regime in the system. In practice, the true probability distribution of the data at a site or a region is unknown. The assumption that data in a given system arise from a single-parent distribution may be questionable when data from large watersheds are analyzed. In such cases, more than one type of rainfall or flow may contribute to extreme events in a region. However, for the analysis to be of practical use, simpler distributions are often used to characterize the relation between flood magnitudes and their frequencies. The performance of distributions is evaluated by using different statistical tests. Quite often, many assumptions made in flood frequency analysis may be invalid. At any rate these assumptions have been questioned and discussed extensively (Kleme s˘, l987a; Kleme s˘, l987b; Yevjevich, 1968).
1.2
General Aspects and Approaches
There have been many discussions of general aspects of frequency analysis. There are also several summaries and discussions of flood frequency analysis: Chow (1964), Yevjevich (1972), Haan (1977), Kite (1977), Singh (1987), Potter (1987), Bobeé and Ashkar (1991), McCuen
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(1993), and Stedinger et al. (1993). The literature in this field is vast and growing. Todd (1957) discussed the basic principles of frequency analysis of stream flow data and outlined computational procedures. Construction of probability papers for the exponential, bounded exponential and log Pearson (3) distributions were discussed by Burkhardt and Prakash (1976). Linsley (1986) discussed the accuracy of flood estimates. The gap between research and practice in flood research and strategies to close them were discussed by Pilgrim (1986). One of Pilgrim’s conclusions was that inadequate attention was given by researchers to problems which are important to practitioners. Flood frequencies have also been estimated by using observed or simulated rainfall data and valid watershed models. A technique of using rainfall data to obtain flood frequencies was tested as early as 1957. The TR-20 computer program was used by Danushkodi (1979) to simulate floods. Alexander (1963) discussed the method of storm transposition to estimate the frequency of rare floods. Fleming and Franz (1971) compared different methods of estimation of flood frequencies. They concluded that the Hydrocomp Simulation Program was most successful in reproducing flood frequency curves determined from historic streamflow records. The distribution of simulated floods matched that of observed floods. Two derived distribution techniques to derive flood frequency distributions were investigated by Moughamian et al. (1987). Both of them were found to perform poorly and it was concluded that fundamental improvements were needed before the derived distribution approach could be used with confidence for flood frequency analysis. Raines and Valdes (1993) evaluated two methods based on derived distributions. Neither of these methods gave consistently better results than those obtained by using the log Pearson (3) distribution. Bradley and Potter (1991) developed a new statistical procedure using approximate relationships between flood quantities which could be compared for different scenarios. Bradley and Potter (1992) presented a new approach for flood frequency analysis of flows simulated by using models. In this method, peak discharges are conditioned on runoff volumes. The effect of changes in temperature and precipitation on frequency of extreme runoff was studied by Krasouskaia (1993). Earlier work in flood frequency analysis covered a wide range. For example, Snyder (1958) developed an approach based on the rational method, the time of concentration and unit hydrograph interpretation, to compute the flood discharge probability. Benson (1959) investigated
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the effect of channel slope on flood frequencies and found it to be second in importance to drainage area. Alexander et al. (1969) discussed the problem of estimating the relationship between the magnitude and frequency of rare floods. The use of moment ratio diagrams was also discussed by them. The random occurrence of rare floods and the use of Poisson distribution in flood frequency analysis was discussed by Kirby (1969). The relationship between flood data and watershed characteristics in small basins in Pennsylvania was discussed by White and Reich (1970). A method of incorporating past water levels at a site into probability analysis was developed by Gerard and Karpuk (1979). The importance of visual interpretation of observed flood series was emphasized by Reich and Renard (1981). Envelope curves for extreme flood events in the U.S. were developed by Crippen (1982). The concept of robust flood frequency models was introduced by Kuczera (1982a), who found that regionalized estimates were preferable to estimates based on short record lengths and estimates which combined both site and regional information were preferable for larger record lengths. The effect of serial dependence on the reliability of the T-year event was investigated by Tasker (1983). Cox regression model was used by Smith and Karr (1986) for flood frequency analysis. These models allowed incorporation of time varying exogenous information into flood frequency analysis. An empirical Bayes approach to combine site-specific and regional information was developed by Kuczera (1982b). A Posterior distribution of the power normal distribution and of the T-year flood were derived by Kuczera (1983). Smith (1987) developed procedures to estimate recurrence intervals of large floods. Adamowski and Feluch (1990) developed a non-parametric flood frequency analysis method which can also use historical information. Flood data from 383 sites in the southwestern U.S. were analyzed by Vogel et al. (1993) to explore the suitability of different distributions to model flood frequencies. L-moment ratio diagrams were used by them for selection of distributions. Such a variety of activity—and the above discussion is by no means comprehensive—has brought about some strong criticism. Klemes˘ (1987a) has stated, “At best, much of flood frequency analysis is just a part of small sample theory in disguise, the term “flood” being used merely as a name for the numbers employed; at worst, it is a pretentious game draining resources both from hydrology and engineering research, and a cheap opportunity to satisfy the need of academics to publish papers and supply easy topics for graduate students who know little beyond elementary statistics, probability theory, and computer programming.”
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However, quite a bit of research related to parameter estimation, different probability distributions, and regionalization methods has been completed during the last two decades. Work related to these aspects has been quite focused and deserves a summary and exposition. Some of these distributions and techniques are discussed in this book.
1.3
Other Models
Apart from the methods of frequency analysis based on the distributions discussed in this book, several other methods and models have been proposed. A method of estimating the frequency of annual peak flows by using monthly peak flows was developed by Whisler and Smith (1957). A distribution-free model based on the theory of extreme values has been developed by Todorovic and Zelenhasic (1970). Todorovic and Rousselle (1971) developed a model by relaxing the assumption of identical distribution of flows. Instead, they assumed that only those exceedences occurring during a particular season to be identically distributed. A relation between peak discharge and maximum 24-hour flow was proposed by Watt (1971). The time of occurrence of extreme floods was considered by Todorovic and Woolhiser (1972). Gupta et al. (1976) derived a nonasymptotic expression for the joint distribution function of the largest flood and its time of occurrence. A model based on partial duration series, in which both the time of occurrence and the magnitude are time-dependent random variables, was developed by North (1980). Kavvas (1982a,b) developed a stochastic trigger model for flood peaks and applied it to data from Goksu-Karahacili River in Turkey. A cluster process of the Neyman-Scott type was proposed to model flood peaks by Cervantes et al. (1983). Annual flood probabilities were estimated by using a Fourier series method by Wu and Woo (1989). Basin scale in flood peak distribution was represented by using a lognormal model by Smith (1992). Gupta et al. (1994) used the multi-scaling theory of floods for regional quantile analysis. Boughton (1980) developed and fitted a three-parameter distribution to annual flood data from 78 catchments in eastern Australia. A nonlinear relationship was detected between the frequency factor and a double logarithmic function of the recurrence interval. A general double bounded probability density function was developed and methods of parameter estimation were also discussed. Phien and Jivajirajah (1984) proposed the SB distribution for flood frequency analysis.
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Bayesian methods have also been used in flood frequency analysis. A procedure to incorporate uncertainties in the data in the extrapolation from limited records and in model selection was presented in a Bayesian framework by Tang (1980). Russell (1982) presented a Bayesian method to estimate design floods when data are insufficient to carry out frequency analysis. As mentioned earlier, distributions, other than those used in this book, have been proposed or used for flood frequency analysis. Singh and Singh (1985a,b) derived the two-parameter gamma and LP(3) distributions by using the maximum entropy principle. Moharram et al. (1993) compared different estimation methods for the three-parameter generalized Pareto distribution and showed that the least squares method had the lowest root mean square error. The probability weighted moments method had the smallest bias when the shape parameter was smaller than zero. The log-logistic distribution was compared to the GEV, LN(3), and P(3) distributions by using data from Scotland by Ahmad et al. (1988). The log-logistic distribution was found to perform better than other distributions and hence was recommended for further analysis. Rossi et al. (1984) introduced the two-component extreme value distribution for flood frequency analysis. They also presented an estimation procedure which can be readily used for the problem of regional flood estimation. Further discussion of the properties of two-component extreme value distribution was provided by Beran et al. (1986). Although a large part of flood frequency analysis is based on parametric methods, nonparametric methods are receiving increased attention. Bardsley (1989) developed a nonparametric method in which historical paleoflood data may be included in the flood frequency analysis. Adamowski (1985) introduced the nonparametric kernel estimation method for flood frequency analysis. A variable kernel nonparametric estimation method, which can include paleoflood information, was developed by Guo (1991). Lall et al. (1993) discussed the selection of the kernel function and bandwidth in kernel flood frequency estimators. They concluded that variable-band widths with heavy-tailed kernels were the best for flood frequency analysis. A kernel estimator for the quantile function was developed by Moon and Lall (1994). There is a long tradition of transforming hydrologic data before analyzing them. Logarithmic transformation is very commonly used. Power (Shanks and Rao, 1976) and SMEMAX (Rasheed et al., 1982) transformations have also been used. Jain and Singh (1986) compared the SMEMAX transformation and its modified versions to the power transformation by using flood data. They concluded that the power
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transformation was superior in rendering the data approximately normal. Transformations introduce bias, and this aspect must be considered when transformed data are used. Wilson et al. (1990) investigated the bias introduced by logarithmic transformation of data by using the LN(2) distribution.
1.4
Return Period, Probability, and Plotting Positions
Flood peaks do not occur with any fixed pattern in time or magnitude. Time intervals between floods vary. The definition of return period is the average of these inter-event times between flood events (Cunnane, 1989). Large floods naturally have large return periods and vice versa. The definition of the return period may not involve any reference to probability. However, a relationship between the probability of occurance of a flood and its return period can be justified. A given flood q with a return period T may be exceeded once in T years. Hence the probability of exceedence is P(QT > q) = 1/T. The cumulative probability of non-exceedence, F(QT) is given by Eq. 1.4.1. 1 F ( Q T ) = P ( Q T ≤ q ) = 1 – P ( Q T > q ) = 1 – --T
(1.4.1)
Equation 1.4.1 is the basis for estimating the magnitude of a flood, QT, given its return period T. Substituting F(QT) = 1 – 1/T in a known statistical distribution function, one can solve for the magnitude of QT. Often, the data are plotted on probability paper to check whether they follow a particular distribution, to detect errors, and to check for outliers. Probability plots require an initial estimate of the probability of non-exceedence F = (F (QT)), which is called a “plotting position.” Plotting positions are also used to estimate parameters by using the probability weighted moments (PWM) method (Chapter 3). A plottingposition formula which is used in this book is that given by Hosking (1990): i – 0.35 F = ------------------, i = 1, 2, …, N N
(1.4.2)
where N is the sample size and i is the rank of the observations in ascending order. The formula in Eq. 1.4.2 is believed to give acceptable results for some common three-parameter distributions and is used in
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the PWM method (Hosking, 1990). Some other commonly used plotting-position formulas are given in Table 1.4.1 (Cunnane, 1989). Table 1.4.1. Commonly Used Plotting Position Formulas
Plotting Position
Formula T
F = 1 – 1/T
N+1 ------------m
i ------------N+1
Gringorton
N + 0.12 --------------------m – 0.44
i – 0.44 --------------------N + 0.12
Hazen
N ----------------m – 0.5
i – 0.5 --------------N
Blom
N + 0.25 ----------------------m – 0.375
i – 0.375 --------------------N + 0.25
Cunnane
N + 0.2 -----------------m – 0.4
i – 0.4 -----------------N + 0.2
California
N ---m
i–1 ---------N
Chegodayev
N + 0.4 -----------------m – 0.3
i – 0.3 -----------------N + 0.4
Adamowski
N + 0.5 -------------------m – 0.24
i – 0.26 -----------------N + 0.5
EWSD
N+1–α ----------------------m–α
i ----------------------N+1–α
Weibull
Note: i is the rank in ascending order = N – m + 1; m is the rank in descending order = N – i + 1; N is the number of observations. From Cunnane (1989).
Research on plotting positions has had a long history and the work is still continuing. Srikanthan and McMahon (1981) suggested that the value of α in the plotting position formula for the LP(3) distribution is 0.4. Adamowski (1981) proposed a new plotting position formula which is based on the mean square criterion. Reimius (1982) also proposed a new plotting position formula. Zhang (1982) derived a plotting position formula for data which may include extraordinarily large and small historical values. Xuewu
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et al. (1984) developed a plotting-position formula by Monte Carlo method for skewness coefficients between 0 and 2. Arnell et al. (1986) derived an unbiased plotting position formula for the generalized extreme value distribution. Hirsch and Stedinger (1987) analyzed plotting position formulas. The effect of misspecification of n, the length of the period during which any floods greater than the threshold value occurs and the uncertainty of the threshold value are discussed. Nguyen et al. (1989) proposed a new plotting-position formula for Pearson (3) distribution. The PWM is used to estimate the exact plotting positions. They also proposed a simple formula which represents a very good approximation to exact plotting positions. Guo (1990a) proposed an unbiased plotting position formula based on the same concept as the historically weighted moments. Guo (1990b) analyzed unbiased plotting-position formulas for the GEV distribution. He found the Cunnane formula to be superior to others. Wang (1991) explained the large bias produced by the existing plotting position formulas and developed a modified procedure. The effects of four plotting-position formulas on estimating the probability weighted moments of LN(3), LP(3), GEV, and Wakeby distributions were investigated by Haktanir and Bozduman (1995). They found that GEV and LP(3) distributions along with the Landwehr plotting-position formula were found to be the best combination.
1.5
Flood Frequency Models
In flood frequency analysis, a unique relationship between a flood magnitude and the corresponding recurrence interval T is sought. The task is to extract information from a flow record to estimate the relationship between Q and T. Three different models may be considered for this purpose (Cunnane, 1989). These models are (1) the annual maximum series (AM) model, (2) the partial duration series (PD) or peaks over a threshold (POT) model, and (3) the time series (TS) model. In the annual maximum (AM) flow series, only the peak flow in each year of record is considered. However, the use of an AM series may involve some loss of information. For example, the second or third peak within a year may be greater than the maximum flow in other years and yet they are ignored (Kite, 1977; Chow et al. 1988). This situation is avoided in the partial duration (PD) or the peaks over a threshold (POT) models where all peaks above a certain base value are considered. The base is usually selected low enough to include at least one
CH01 Page 9 Monday, February 4, 2002 11:26 AM
event in each year (Kite, 1977). The PD (or POT) model, however, is limited by the fact that observations may not be independent (Chow et al., 1988). According to Cunnane (1989), the AM model is statistically more efficient than the PD model when λ is small (λ < 1.65) where λ is the mean number of peaks per year included in the PD series. The return period TE for a PD (or POT) model is related to the return period T of an AM model by Eq. 1.5.1 (Chow, 1964). T T E = log ------------ T – 1
–1
(1.5.1)
The relative difference between TE and T is greatest for small values of T and converges to 0.5 as T increases. Rosbjerg (1977) demonstrated that when events follow a Poisson process, the relationship between return periods of annual maximum and partial duration series given in Eq. 1.5.1 is exactly satisfied. The discussion in this book is based on the annual maximum series (AM) model. An earlier discussion and comparison of annual maximum and partial duration series is found in Chow (1964). Cunnane (1973) developed a method to compare the statistical efficiency of the T-year flood by two different methods. He found that estimates based on annual exceedence series had larger variance than those based on annual maximum series for return periods greater than 10 years. Cunnane (1973) also demonstrated that the variance of estimates based on partial duration series was smaller than those based on annual maximum series if the partial duration series contained at least 1.65 N values where N is the number of years of record. The role of truncation levels in defining partial flood series was investigated by Ashkar and Rousselle (1983). Some of the problems involved in the practical application of partial duration series models were also discussed by them. The relative efficiencies of using the partial duration and annual maximum series in flood frequency analysis were evaluated by Tavares, Valdares, and Da Silva (1983) who also presented some general recommendations about truncation levels. Ashkar and Rousselle (1987) developed a procedure to choose a truncation level for selecting the partial duration series. This procedure is based on the equality of mean and variance of Poisson distribution. The distributions of the T-year flood and the risk of its being exceeded within a given period were derived and a risk based design technique was developed by Rasmussen and Rosbjerg (1989). The bias
CH01 Page 10 Monday, February 4, 2002 11:26 AM
and variance of quantile estimates from a partial duration series have been derived by Buishand (1990). In the time series (TS) model (Flood Studies Report, 1975) the flow hydrograph is considered to be a time series in which the flows are represented by a series of ordinates at equally spaced intervals of time. Time intervals of days, months, and years are commonly used in hydrological time series, but in flood frequency analysis, most commonly, only days are used. Ideally, if a hydrograph is considered to be a stochastic process in continuous time, properties of such a series can be deduced from those of the parent process. If Q(t) is the flow on day t, a time series model may be written as the sum of trend, seasonal, and stochastic components. Estimation of model formulation and parameters proceed together through the three components beginning with trend and ending with the stochastic component. Recently, methods are being developed to incorporate prerecord flood information into the observed data. Tasker and Thomas (1978) analyzed several procedures for treating prerecord flood information in flood frequency analyses by using Monte Carlo experiments. It was found that when sample skew is adjusted for historic information before it was weighted with a generalized skew coefficient, it is better to use historic record length in computing the weighing factor. Condie and Lee (1982) considered the problem of parameter estimation in the LN(3) distribution when observed and historic data are combined. Hosking and Wallis (1986a,b) used computer simulation experiments to assess whether the use of a single paleoflood estimate will increase the accuracy of estimation of extreme floods in flood frequency analysis. Stedinger and Cohn (1986) investigated flood quantile estimators which can use historical and paleoflood information in flood frequency analysis. Hosking and Wallis (1986a,b) assessed the value of historical information in flood frequency analysis by computer simulation. Wall et al. (1987) combined site-specific and historic data to estimate floods. The method was tested by using Pennsylvania data.
1.6
Hydrologic Risk
A relation between the magnitude, design period in years and the probability of not exceeding that magnitude in the design period was derived by Riggs (1961). A method based on Poisson distribution, to compute the probability of occurrence or exceedence of a flood of a specified magnitude at least once in a given time period, was devel-
CH01 Page 11 Monday, February 4, 2002 11:26 AM
oped by Hall and Howell (1963). Yen (1970) derived an expression for the risk of failure associated with a return period and the expected life of a project. The bias in computed flood risk was discussed by Hardison and Jennings (1972) who recommended that the accuracy of procedures used in flood frequency analysis be appraised in terms of standard errors. Stedinger (1983b) developed methods to estimate design floods with specified design probability. Both Bayesian and classical approaches were used by Stedinger (1983b). Chow and Takase (1977) derived probabilistic models for adopting hydrologic extremes as design criteria in water resources project planning. In one of these models a hydrologic event of a given recurrence interval is used, whereas the extreme event observed in the record is used in the other model. A general formula for calculating the probability of failure of water projects was developed by Schuzheng (1985) who also discussed its applications. The relationship between the recurrence interval, distribution of events, risk and the memoryless property of distribution of time to next episode were examined by Loaiciga and Marino (1991).
1.7
Regionalization
The availability of data is an important aspect in frequency analysis. The estimation of probability of occurrence of extreme floods is an extrapolation based on limited data. Thus the larger the database, the more accurate the estimates will be. From a statistical point of view, estimation from small samples may give unreasonable or physically unrealistic parameter estimates, especially for distributions with a large number of parameters (three or more). Large variations associated with small sample sizes cause the estimates to be unrealistic. In practice, however, data may be limited or in some cases may not be available for a site. In such cases, regional analysis is most useful. Regional analysis is based on the concept of regional homogeneity which assumes that annual maximum flow populations at several sites in a region are similar in statistical characteristics and are not dependent on catchment size (Cunnane, 1989). Although this assumption may not be strictly valid, it is convenient and effective. Regionalization serves two purposes. For sites where data are not available, the analysis is based on regional data (Cunnane, 1989). For sites with available data, the joint use of data measured at a site, called at-site data, and regional data from a number of stations in a region provides sufficient information to enable a probability distribu-
CH01 Page 12 Monday, February 4, 2002 11:26 AM
tion to be used with greater reliability. This type of analysis represents a substitution of space for time where data from different locations in a region are used to compensate for short records at a single site (National Research Council, 1988; Stedinger et al., 1993). Many types of regionalization procedures are available (Cunnane, 1988 and 1989). One of the simplest procedures which has been used for a long time is the index flood method. The key assumption in the index flood method is that the distribution of floods at different sites in a region is the same except for a scale or index flood parameter, which reflects rainfall and runoff characteristics of each region. The index flood may be the mean flood, although any location parameter of the frequency distribution may be used (Hosking and Wallis, 1991). In this case, ˆ T at a given site for a given return period regional quantile estimates Q T can be obtained as in Eq. 1.7.1, where qT is the quantile estimate from the regional distribution for the given return period, and µi is the mean flow at the site. QˆT = µ i q T
(1.7.1)
The regional distribution parameters are obtained by using the regional weighted average of dimensionless moments obtained by using the dimensionless rescaled data qij = Qij/ µˆ i . Another method of obtaining the regional distribution parameters is the station year approach (Cunnane, 1989) where all the data are pooled, after dividing them by the mean µ i at each site, and are treated as a single sample. The joint use of at-site and regional data is advisable, provided that a reasonably homogeneous flood region can be identified. This aspect is discussed in Sections 2.5 and 3.4. The data at a site may be used when the record at a station is exceptionally long, or when regional data are not available, or when a region is heterogeneous.
1.8
Tests on Hydrologic Data
Two basic assumptions in statistical flood frequency analysis are the independence and stationarity of the data series. In addition, the assumption that the data come from the same distribution (homogeneity) is made. The following tests, which are also discussed by Bobeé and Ashkar (1991), are commonly used to test for stationarity, homogeneity and independence of data. Other (similar) tests are found in Kite (1977).
CH01 Page 13 Monday, February 4, 2002 11:26 AM
1.8.1 Test for Independence and Stationarity Given a sample of size N, the Wald-Wolfowitz (1943) (W-W) test is used to test for the independence of a dataset and to test for the existence of trends in it. For a data set x1, x2,....xN the statistic R is calculated from Eq. 1.8.1. N–1
R =
∑ xi xi + 1 + x1 xN
(1.8.1)
i=1
When the elements of the sample are independent, R follows a normal distribution with mean and variance given by Eqs. 1.8.2 and 1.8.3, ( s1 – s2 ) R = -------------------N–1 2
(1.8.2)
( s 1 – 4s 1 s 2 + 4s 1 s 3 + s 2 – 2s 4 ) S2 – s4 2 - – R + ----------------------------------------------------------------------------var ( R ) = -------------N–1 ( N – 1)( N – 2) 2
4
2
2
(1.8.3)
where s r = N m r ′ and m r ′ is the rth moment of the sample about the origin. 1⁄2 The statistic u = ( R – R ) ⁄ ( var ( R ) ) is approximately normally distributed with mean zero and variance unity and is used to test the hypothesis of independence at significance level α , by comparing the statistic u with the standard normal variate uα/2 corresponding to a probability of exceedence α /2. The method is illustrated in Example 1.8.1. Program wwtest to perform the computations is discussed in Chapter 10.
EXAMPLE 1.8.1 The wwtest is used to analyze the annual maximum flow data from the Wabash River at Lafayette, Indiana. The data are given in Table 1.8.1. The following results were obtained from program wwtest. 11 11 Statistic R = 2.3861 × 10 ; mean value R = 2.35 × 10 ; Standard devi11 11 2.3861 × 10 – 2.35 × 10 9 ation σ R = 5.36 × 10 ; the statistic u = ---------------------------------------------------------------- = 9 5.36 × 10 0.7248.
CH01 Page 14 Monday, February 4, 2002 11:26 AM
The test value u = 0.7248 is less than the critical value at 5% significance level u0.025 = 1.96. Thus we can accept the hypothesis of independence and stationarity. The Wabash River data are concluded to be independent and stationary at the 5% significance level.
Table 1.8.1.
Year
1907 1908 1909 1910 1911 1912 1913 1914 1915 1916 1917 1918 1919 1920 1921 1922 1923 1924 1925
Annual Maximum Flows in Wabash River at Lafayette, IN
Flow (cfs)
41500 57000 44000 49000 31000 45900 19000 41100 37300 76000 33200 61200 76000 59800 44400 58400 53600 59800 63300
Year 1926 1927 1928 1929 1930 1931 1932 1933 1934 1935 1936 1937 1938 1939 1940 1941 1942 1943 1944 1945 1946 1947
Flow (cfs) 57700 64000 63500 38000 74600 13100 37600 67500 21700 37000 93500 58500 63300 74400 34200 14600 44200 13100 73300 46600 39400 41200
Year 1948 1949 1950 1951 1952 1953 1954 1955 1956 1957 1958 1959 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969
Flow (cfs) 41300 62000 90000 50600 41900 35000 16500 35300 30000 52600 99000 89000 39500 55400 46000 63000 58300 36500 14600 64900 68500 69100
Year 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991
Flow (cfs) 42600 31000 39400 40700 53400 36000 43900 23600 50500 49700 48100 44500 56400 60800 40400 80400 41600 14700 33300 40700 53300 77400
1.8.2 Tests Homogeneity and Stationarity In this test two samples of size p and q with p ≤ q are compared. The combined data set of size N = p + q is ranked in increasing order. The Mann-Whitney (1947) (M-W) test considers the quantities V and W in Eqs. 1.8.4 and 1.8.5.
CH01 Page 15 Monday, February 4, 2002 11:26 AM
( p( p + 1)) V = R – -------------------------2
(1.8.4)
W = pq – V
(1.8.5)
R is the sum of the ranks of the elements of the first sample (size p) in the combined series (size N ), and V and W are calculated from R, p, and q. V represents the number of times an item in sample 1 follows an item in sample 2 in the ranking. Similarly, W can be computed for sample 2 following sample 1. The M-W statistic U is defined by the smaller of V and W. When N > 20 and p, q > 3, and under the null hypothesis that the two samples came from the same population, U is pq approximately normally distributed with mean U = ------ and variance 2 var (U ), pq var ( U ) = ----------------------N (N – 1)
3
N –N ----------------- – ∑ T 12
(1.8.6)
where T = ( J – J ) ⁄ 12 and J is the number of observations tied at a given rank. T is summed over all groups of tied observations in both 1⁄2 samples of size p and q. The statistic u = ( U – U ) ⁄ [ var ( U ) ] is used to test the hypothesis of homogeneity at significance level α by comparing it with the standard normal variate for that significance level. Programs mwtest and mw2test are used to compute the statistics. By using the program mwtest, the results of homogeneity within a given data set is tested by splitting it into two subsets of sizes p and q. The program mw2test is used for comparing two different data sets. 3
EXAMPLE 1.8.2 a. Use the mwtest to check the homogeneity and stationarity of the annual maximum flow data from the Wabash River station at Lafayette, Indiana given in Table 1.8.1. The data set was divided into two sets each of length p = 42, q = 43. Value of test statistic: U = 794; mean value: U = 903 ; ∑ T = 0 ; standard deviation: σ U = 113.77; standardized test value
CH01 Page 16 Monday, February 4, 2002 11:26 AM
794 – 903 u = ------------------------ = – 0.96 . 113.77 Since |u| = 0.96 is less than the critical value u0.025 = 1.96, the Wabash River data can be considered to be homogeneous and stationary at 5% level of significance. b. Use the M-W test to check the applicability of the index flood method (Section 1.7) by using the flow data from the Wildcat Creek at Jerome and the data from the Wabash River at Lafayette and the Salt Creek near Harrodsburg. The flow data for Wildcat Creek and Salt Creek are given in Tables 1.8.2 and 1.8.3. Each data set is divided by its mean, then homogeneity is studied. For the Wabash River and Wildcat Creek data we have: p = 30; q = 85; v = 1263, w = 1287, R = 1728; ∑T = 0; Value of test statistic: U = 1263; mean value: U = 1275 ; standard 1263 – 1275 deviation: σ U = 157.0; standardized test value: U = ------------------------------ = 157.0 –0.0764. Since |u| = 0.0764 is less than u0.025 = 1.96, the two data sets can be considered homogeneous. For the Wildcat Creek and Salt Creek data we have: p = 30; q = 36; v = 717; w = 363; R = 1182;
∑T
= 0.
Value of test statistic: U = 363; mean value: U = 540 ; standard 363 – 540 deviation: σ U = 77.65; standardized test value: U = ------------------------ = 77.65 -2.28 Since |u| = 2.28 is greater than u0.025 = 1.96, the null hypothesis that the two data sets are homogeneous cannot be accepted at 5% significance level. The two data sets are concluded to come from different distributions.
1.8.3 Test for Outliers An outlier is an observation that deviates significantly from the bulk of the data, which may be due to errors in data collection, or recording, or due to natural causes. The presence of outliers in the data causes difficulties when fitting a distribution to the data. Low and high out-
CH01 Page 17 Monday, February 4, 2002 11:26 AM
Table 1.8.2.
Year 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 Table 1.8.3.
Annual Maximum Flows in Wildcat Creek near Jerome, IN
Flow (cfs) 3990 3390 4160 1500 632 2540 3150 2790 2180 1710
Year 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981
Flow (cfs) 2910 2240 2720 2270 3700 1260 2760 3290 6140 1180
Year 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991
Flow (cfs) 3060 2260 2050 4590 2450 798 2750 5130 2240 6890
Annual Maximum Flows in Salt Creek near Harrodsburg, IN
Year
Flow
Year
Flow
Year
Flow
Year
Flow
1956
5900
1965
3710
1974
1980
1983
2170
1957
9680
1966
2860
1975
2230
1984
2200
1958
7940
1967
2690
1976
2020
1985
2240
1959
17900
1968
2940
1977
2020
1986
2160
1960
22000
1969
2840
1978
2440
1987
1230
1961
20200
1970
1940
1979
2180
1988
2060
1962
5200
1971
2350
1980
2130
1989
2230
1963
9400
1972
2360
1981
1990
1990
2060
1964
8000
1973
3340
1982
2100
1991
2160
(cfs)
(cfs)
(cfs)
(cfs)
liers are both possible and have different effects on the analysis. The Grubbs and Beck (1972) test (G-B) may be used to detect outliers. In this test the quantities xH and xL are calculated by using Eqs. 1.8.7 and 1.8.8, x H = exp ( x + k N s )
(1.8.7)
x L = exp ( x – k N s )
(1.8.8)
CH01 Page 18 Monday, February 4, 2002 11:26 AM
where x and s are the mean and standard deviation of the natural logarithms of the sample, respectively, and kN is the G-B statistic tabulated for various sample sizes and significance levels by Grubbs and Beck (1972). At the 10% significance level, the following approximation proposed by Pilon et al. (1985) is used, where N is the sample size. kN = –3.62201 + 6.28446N 1/4 –2.49835N1/2 + 0.491436N 3/4 – 0.037911N
(1.8.9)
Sample values greater than xH are considered to be high outliers, while those less than xL are considered to be low outliers. The program gbtest in Chapter 10 may be used to calculate xH and xL. The U.S. Water Resources Council (1981) method of testing outliers has a very high threshold (McCormick and Rao, 1995). For example, in one of the stations, the mean value of 9796 cfs and a standard deviation of 5166 cfs were obtained for observed flows. The mean plus two standard deviations is 20128 cfs. There were two observed flows higher than 20128 cfs. In this case the high outlier threshold by the Water Resources Council Method is 48640 cfs, which corresponds to the mean plus seven and half times the standard deviation, which is extremely high. An alternative method to detect outliers is the least median square (LMS) method, which was investigated by McCormick and Rao (1995). In this method, the general equation used is of the form in Eq. 1.8.9, Y i = β0 + β1 X i + εi
(1.8.9)
In the above equation, Yi is the response variable; β 0 and β 1 are parameters to be estimated; Xi is the independent variable; and ε i is the deviation from the predicted value. The estimated value from the model is given by Eq. 1.8.10, Yˆ i = b 0 + b 1 X i
(1.8.10)
where Yˆ i is the estimated response variable; b0 and b1 are the estimates of the parameters β 0 and β 1 . The residual value, the difference between the predicted and observed values, is then defined as in Eq. 1.8.11. e i = Y i – Yˆ i
(1.8.11)
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In the LMS method, b0 and b1 in Eq. 1.8.10 are determined such that the median value of the squared residual is minimized: 2
Minimize [med( e i )] A robust method is one which will not overemphasize any particular portion of the data range, such as the higher values of the dependent variable. The LMS method is a robust method. A scale estimate is used by the LMS method to define how well the data are fitted by the straight line. The initial scale estimate, s0 is given by Eq. 1.8.12. 5 0 2 s = 1.486 1 + ------------ MED ( e i ) n – 2
(1.8.12)
Each observation is then assigned a weight, corresponding to whether it is within a reasonable range of the initial scale estimate. 0 w i = 1 if e i ⁄ s ≤ 2.5 0 otherwise
The final scale estimate used in the LMS method is then: n
∑ ( wi ei ) 2
σ* =
i=1 n
----------------------
(1.8.13)
∑ wi – 2 i=1
Outliers may be detected from a plot of ei vs. Yˆ i , also known as a residual plot. The criterion to test whether an observation is an outlier is whether it has a residual value greater than a multiple of the final scale estimate. In the following discussion, a range of 2.5 σ is used. The data may be analyzed by the LMS method by using the Gumbel distribution. The extreme value type I, or Gumbel distribution may be expressed as in Eq. 1.8.14. QT = b0 + b1 Y T
(1.8.14)
CH01 Page 20 Monday, February 4, 2002 11:26 AM
QT is the flow corresponding to the return period T, and YT is the reduced variate. T Y T = – ln ln ------------ T – 1
(1.8.15)
The return period T is estimated by using a suitable formula (Table 1.4.1) such as the California formula. The LMS solution is highly dependent on the spacing of the independent variable. Since the differences in YT values decrease as the rank of the value increases, there is a heavier concentration of data points in the lower YT range. This concentration can result in a solution that will best fit the lower to middle values of YT , while leaving higher values outside the 2.5 σ range. This problem was overcome by dropping data points corresponding to the lowest values of YT for each station until the number of values outside of the 2.5 σ range is approximately the same as the number of values above the upper threshold. This forces the median to account for the upper portion rather than the lower portion of the data. A typical result from a station with the highest 20 observations is given in Figure 1.8.1. The β 0, β 1 and σ * values for this case are 266, 162, and 12.5. A plot of ei vs Yˆ is given in Figure 1.8.1. The largest observed flow is an outlier in this case, as it is outside the ± 2.5 σ band. Any observations below the upper threshold which are outside the 2.5 σ band would not be classified as outliers, but simply poorly fitted. Further details of the method are found in McCormick and Rao (1995).
EXAMPLE 1.8.3 Identify outliers in the Wabash River and Wildcat Creek data given in Tables 1.8.1 and 1.8.2 by using G-B test. For the Wabash River data in Table 1.8.1 at 10% significance level we have: N = 85; kn = 2.9611; s = 0.4589; x = 10.77 xH = 185,370 cfs; xL = 12,236 cfs
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One observation, x = 190,000 cfs (Year 1913) is greater than xH and therefore it is considered a high outlier. No low outliers were detected. For the Wildcat Creek data given in Table 1.8.2 we have: N = 30; kn = 2.56398; s = 0.5291914; x = 7.842699. xH = 9,892.53 cfs; xL = 655.81 cfs
One observation, x = 632 cfs (Year 1966) is less than xL and therefore is considered a low outlier at 10% significance level. No high outliers were detected.
Figure 1.8.1.
Sample Residual Plot by using the LMS Method.
CHAPTER 2
Selection and Evaluation of Parent Distributions: Conventional Moments
2.1 Moments of Distributions and Their Sample Estimates Moments about the origin or about the mean are used to characterize probability distributions. Moments about the origin are the expected values of powers of a random variable. For a distribution with a probability density function f(x), the rth moment about the origin is given by Eq. 2.1.1. ∞
µ′ r =
∫x
r
f ( x ) dx,
µ′ 1 = µ = mean
(2.1.1)
–∞
The central moments µr are computed by Eq. 2.1.2. ∞
µr =
∫ ( x – µ′ )
f ( x ) dx,
r
1
µ1 = 0
(2.1.2)
–∞
It can be easily proved that the relationship between µ r ′ and µ r is given by Eqs. 2.1.3 and 2.1.4 (Kendall and Stewart, 1967). r
µr =
1 ∑ j µ′r – j( − µ′)
r
j
(2.1.3)
j=0
r
µ′ r =
∑ j µr – j ( − µ′1 ) r
j
(2.1.4)
j=0
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Sample moments m′ r and mr , on the other hand, are calculated by using Eqs. 2.1.5 and 2.1.6.
n
1 r m′ r = --- ∑ x i , m′ 1 = x = sample mean ni = 1
(2.1.5)
n
1 r m r = --- ∑ ( x i – x ) , m 1 = 0 ni = 1
(2.1.6)
Relationships as in Eqs. 2.1.3 and 2.1.4 hold for sample moments as well. These sample moments are often biased and may be corrected (Cunnane, 1989). For example, some of the corrected central moments are given below:
n
1 2 ˆ 2 = ------------ ∑ ( x i – x ) m n – 1i = 1
(2.1.7)
n
n 3 ˆ 3 = ---------------------------------- ∑ ( x i – x ) m (n – 1)(n – 2) i = 1
2
(2.1.8)
n
n 4 ˆ 4 = --------------------------------------------------- ∑ ( x i – x ) m (n – 1)(n – 2)(n – 3) i = 1
(2.1.9)
However, in small samples the bias may be larger than can be properly corrected by using simple expressions in n. The conventional moment ratios are defined as below. 1⁄2
Coefficient of variation C v = z = µ 2 /µ′ 1
(2.1.10)
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3⁄2
Coefficient of skewness C s = γ 1 = µ 3 /µ 2
(2.1.11)
Coefficient of kurtosis C k = γ 2 = µ 4 /µ 2
(2.1.12)
2
Sample moment ratios are calculated by substituting estimates mr in Eqs. 2.1.5 to 2.1.9 to their counterparts µr in Eqs. 2.1.10 to 2.1.12. Further correction of bias is required for the sample moment ratios, which depends on the sample size, skewness of parent population, and the form of the parent distribution (Wallis et al., 1974). Moments of distributions and their properties have been discussed extensively in the literature. Distribution functions for the mean, standard deviation, and skewness coefficients were computed by Wallis et al. (1974) by using the Monte Carlo method. Small samples from normal, Gumbel type-I, log normal, P(3), Weibull, and Pareto distributions were used in their study. Kirby (1974) showed that sample skewness coefficient and coefficient of variation of positive data, the maximum standardized variate, and the standardized range have bounds which depend only on the sample size. A least squares procedure which allows the construction of confidence regions and tolerance limits for distribution functions has been developed. Some statistics of historical and simulated flood sequences were examined in real and log space by Landwehr et al. (1978), who bring out the difficulties involved in inferring the properties of floods in real space from those in log space. A consequence of this study is that the construction and use of skew maps in log space may not be very useful. Gingras et al. (1994) found that unimodal distributions reflected a single flood generating mechanism whereas multimodal densities reflected two or more mechanisms. Different methods of estimation of skewness coefficient are compared with respect to bias, variance, mean square error, and robustness by Yevjevich and Obeysekara (1984). They proposed a new skewness estimator which uses the estimated covariance between subsample mean and variance.
EXAMPLE 2.1.1 Calculate the mean, coefficient of variation, coefficient of skewness, and coefficient of kurtosis at each station for the data from 93 stations in the Wabash River basin in Indiana. The station information
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is given in Table 2.1.1 and the watershed boundaries are shown in Figure 2.1.1. The computed moments are given in Table 2.1.2.
2.2
Moment Ratio Diagrams (MRDs)
For a given distribution, conventional moments can be expressed as functions of the parameters of distributions. It follows that the higher order moments can be expressed as functions of lower order moments. For two-parameter distributions, the moment µ3 can be expressed as 3⁄2 a unique function of µ2. For example, Cs = µ3 / µ 2 can be expressed 1⁄2 2 as a unique function of C v = µ 2 / µ′ 1 . Similarly, C k = µ 4 / µ 2 of a three-parameter distribution can be expressed as a unique function of Cs. The Cs – Ck moment ratio relationship for some popular threeparameter distributions are shown in Figure 2.2.1. The abbreviations used in Figure 2.2.1 are:
Figure 2.2.1.
Cs – Ck Moment ratio diagram.
GEV: Generalized extreme value distribution GLOG: Generalized logistic distribution
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Figure 2.1.1.
Watershed boundaries for the data in Table 2.1.1.
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Table 2.1.1. Wabash River Basin Stations St. No.
USGS No.
Station Name
Area mi2
Record Extent
N
Upper Wabash River Basin: 19 03322500 Wabash River Near New Corydon
262.00 1952–1988 (37)
20 03322900 Wabash River at Linn Grove
453.00 1964–1991 (28)
21 03323500 Wabash River at Huntington
721.00 1951–1991 (41)
22 03324000 Little River Near Huntington
263.00 1944–1991 (48)
23 03324200 Salamonie River at Portland
85.60 1960–1991 (32)
24 03324300 Salamonie River Near Warren
425.00 1958–1991 (34)
25 03324500 Salamonie River at Dora
557.00 1924–1991 (68)
26 03325000 Wabash River at Wabash
1768.00 1914–1991 (68)
27
03325311 Little Mississinewa River at Union City
28 03325500 Mississinewa River Near Ridgeville 29 03326070 Big Lick Creek Near Hartford City 30 03326500 Mississinewa River at Marion 31 03327000 Mississinewa River at Peoria 32 03327500 Wabash River at Peru
9.67 1983–1991 (09) 133.00 1947–1991 (45) 29.20 1972–1991 (18) 682.00 1923–1991 (68) 808.00 1952–1991 (40) 2686.00 1943–1991 (49)
33 03327520 Pipe Creek Near Bunker Hill
159.00 1969–1991 (23)
34 03328000 Eel River at North Manchester
417.00 1924–1991 (68)
35 03328430 Weesau Creek Near Deedsville
8.87 1971–1991 (21)
36 03328500 Eel River Near Logansport 37 03329000 Wabash River at Logansport 38 03329400 Rattlesnake Creek Near Patton
789.00 1943–1991 (49) 3779.00 1969–1991 (68) 6.83 1969–1991 (23)
39 03329700 Deer Creek Near Delphi
274.00 1943–1991 (49)
40 03330500 Tippecanoe River at Oswego
113.00 1950–1991 (42)
41
03331110 Walnut Creek Near Warsaw
19.60 1970–1991 (22)
42 03331500 Tippecanoe River Near Ora
856.00 1947–1991 (45)
43 03333000 Tipecanoe River Near Delphi
1865.00 1940–1987 (48)
Middle Wabash River Basin: 44 03333450 Wildcat Creek Near Jerome 45 03333600 Kokomo Creek Near Kokomo
146.00 1962–1991 (30) 24.70 1960–1991 (32)
46 03333700 Wildcat Creek at Kokomo
242.00 1956–1991 (36)
47 03334500 South Fork Wildcat Creek near Lafayette
243.00 1943–1991 (49)
48 03335000 Wildcat Creek Near Lafayette 49 03335500 Wabash River at Lafayette
794.00 1955–1991 (37) 7267.00 1907–1991 (85)
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Table 2.1.1. St. No.
Wabash River Basin Stations (continued)
USGS No.
Station Name
50 03335690 Mud Pine Creek Near Oxford 51 03335700 Big Pine CR NR Williamsport 52 03336000 Wabash River at Covington 53 03339108 East Fork Coal Creek Near Hillsboro 54 03339500 Sugar Creek at Crawfordsville 55 03340500 Wabash River at Montezuma
Area mi2
Record Extent
N
39.40 1971–1991 (21) 323.00 1956–1987 (32) 8218.00 1927–1991 (65) 33.40 1969–1991 (23) 509.00 1939–1991 (53) 1118.00 1925–1991 (67)
56 03340800 Big Raccoon Creek Near Fincastle
139.00 1958–1991 (34)
57 03340900 Big Raccoon Creek Near Ferndale
222.00 1958–1991 (34)
58 03341300 Big Raccoon Creek at Coxville
448.00 1958–1988 (31)
White River West Fork Basin: 67 03347000 White River at Muncie
241.00 1926–1991 (66)
68 03347500 Buck Creek Near Muncie
35.50 1955–1991 (37)
69 03348000 White River at Anderson
406.00 1911–1991 (81)
70 03348020 Killbuck Creek Near Gaston
25.50 1968–1991 (24)
71 03348350 Pipe Creek at Frankton
113.00 1969–1991 (23)
72 03349000 White River at Noblesville
858.00 1947–1991 (45)
73 03350700 Stony Creek Near Noblesville 74 03351000 White River Near Nora 75 03351310 Crooked Creek at Indianapolis 76 03351400 Sugar Creek Near Middletown 77 03351500 Fall Creek Near Fortville 78 03352500 Fall Creek at Millersville 79 03353000 White River at Indianapolis
50.80 1968–1991 (24) 1219.00 1930–1991 (62) 17.90 1970–1991 (22) 5.80 1969–1989 (21) 169.00 1942–1991 (50) 298.00 1930–1991 (62) 1635.00 1912–1991 (80)
80 03353120 Pleasant Run at Arlington Ave at Indianapolis
7.58 1960–1991 (32)
81 03353180 Bean Creek at Indianapolis
4.40 1971–1991 (21)
82 03353200 Eagle Creek at Zionsville
103.00 1958–1991 (34)
83 03353500 Eagle Creek at Indianapolis
174.00 1939–1991 (53)
84 03353600 LittleEagle Creek at Speedway
23.90 1960–1991 (32)
85 03353620 Lick Creek at Indianpolis
15.60 1971–1991 (21)
86 03353700 West Fork White Lick Creek at Danville
28.80 1957–1991 (35)
87 03353800 White Lick Creek near Mooresville 88 03354000 White River Near Centerton
212.00 1957–1991 (35) 2444.00 1947–1991 (45)
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Table 2.1.1. St. No.
Wabash River Basin Stations (continued)
USGS No.
Station Name
89 03354500 Beanblossom Creek at Beanblossom 90 03357350 Plum Creek near Bainbridge
Area mi2
Record Extent
N
14.60 1952–1991 (40) 3.00 1970–1991 (22)
91 03357500 Big Walnut Creek Near Reelsville
326.00 1950–1991 (42)
92 03358000 Mill Creek Near Cataract
245.00 1950–1991 (42)
93 03359000 Mill Creek Near Manhattan
294.00 1940–1991 (52)
94 03360000 Eel River at Bowling Green
830.00 1931–1991 (61)
95 03360500 White River at Newberry
4688.00 1908–1991 (84)
Middle Wabash River Basin: 96 03361000 Big Blue River at Carthage
184.00 1952–1991 (41)
97 03361500 Big Blue River at Shelbyville
421.00 1944–1991 (48)
98 03361650 Sugar Creek at New Palestine
93.90 1968–1991 (24)
99 03361850 Buck Creek at Acton
78.80 1968–1991 (24)
100 03362000 Youngs Creek Near Edinburgh
107.00 1943–1991 (49)
101 03362500 Sugar Creek Near Ediburgh
474.00 1943–1991 (49)
102 03363000 Driftwood River Near Edinburgh 103 03363500 Flatrock River at St. Paul 104 03363900 Flatrock River at Columbus 105 03364000 East Fork White River at Columbus 106 03364200 Haw Creek Near Clifford 107 03364500 Clifty Creek at Hartsville 108 03365000 Sand Creek Near Brewersville 109 03365500 East Fork White River at Seymour 110 03366200 Harberts Creek Near Madison 111 03366500 Muscatatuck River Near Deputy
1060.00 1942–1991 (50) 303.00 1931–1991 (61) 534.00 1968–1991 (24) 1707.00 1947–1991 (45) 47.50 1968–1991 (24) 91.40 1948–1991 (44) 155.00 1948–1986 (39) 2341.00 1924–1991 (68) 9.31 1969–1991 (23) 293.00 1948–1991 (44)
112 03368000 Brush Creek Near Nebraska
11.40 1956–1991 (36)
113 03369000 Vernon Fork Muscatatuck River Near Butler
85.90 1942–1991 (50)
114 03369500 Vernon Fork Muscatatuck River at Vernon 115 03371500 East Fork White River Near Bedford
198.00 1940–1991 (52) 3861.00 1940–1991 (52)
116 03371500 Back Creek at Leesville
24.10 1971–1991 (21)
117 03371520 Stephens Creek Near Bloomington
10.90 1971–1991 (21)
118 03372500 Salt Creek Near Harrodsburg 119 03373500 East Fork White River at Shoals
432.00 1956–1991 (36) 4927.00 1904–1991 (87)
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Table 2.1.2. Mean Flow and Moment Ratios for 93 Stations in the Wabash River Basin
Station No. N Upper Wabash River 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43
37 28 41 48 32 34 68 68 9 45 18 68 40 49 23 68 21 49 68 23 49 42 22 45 48
Mean Flow
Cv
Cs
Ck
4137 5175 5969 3543 2439 7227 6703 18422 248 4311 846 11699 7535 21992 2581 4357 267 8278 33544 254 4910 406 193 4602 12665
0.3809 0.3618 0.4444 0.2962 0.2666 0.3527 0.4721 0.5239 0.6052 0.5811 0.4719 0.4480 0.7761 0.5675 0.4059 0.4094 0.4419 0.4060 0.4582 0.5470 0.6806 0.4012 0.6492 0.4062 0.3719
0.4082 –0.1511 1.3166 0.3304 –0.5949 0.3981 0.9617 1.1706 1.4070 1.6871 1.3816 0.4622 1.7592 1.6981 0.6389 0.5522 0.7505 1.0984 1.1923 1.1336 1.9872 1.3101 1.5277 0.7083 0.1194
4.0620 3.6050 5.1097 2.9195 3.2761 3.0936 3.8599 4.3144 6.7335 7.0080 5.3188 2.8207 6.4368 6.2692 3.6501 3.1356 3.1154 3.9266 5.0002 1.8534 7.8512 5.3053 5.4810 2.7986 2.7081
2891 521 4257 5723 10814 52621 1823 5764 52700 1627 10638 65078
0.4978 0.4143 0.4710 0.6512 0.5097 0.4789 0.6689 0.4314 0.4965 0.3264 0.5355 0.4657
1.0160 0.2447 0.1867 1.4616 0.7549 2.3008 1.7011 0.3276 1.0592 0.1032 0.8310 0.9054
4.5288 3.5391 2.4899 5.1311 3.4762 13.2124 7.2812 3.6210 4.7599 3.4300 3.4576 5.5285
Middle Wabash River 44 45 46 47 48 49 50 51 52 53 54 55
30 32 36 49 37 85 21 32 65 23 53 67
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Table 2.1.2. Mean Flow and Moment Ratios for 93 Stations in the Wabash River Basin (continued) Station No. 56 57 58
N 34 34 31
Mean Flow 5776 1775 9331
Cv 0.6634 0.5856 0.8193
Cs 1.2754 3.2419 2.4792
Ck 4.0785 17.2119 11.3150
5403 834 7726 476 2132 11258 964 14171 1435 507 3175 4479 20019 1038 392 4800 6472 1395 1156 1775 9010 25666 1798 388 9501 5672 3011 13691 39404
0.5232 0.4321 0.6184 0.6419 0.5504 0.5144 0.4365 0.5152 0.7733 0.5736 0.5577 0.5757 0.5335 0.4858 0.4374 0.5197 0.6796 0.5278 0.5087 0.6213 0.4482 0.3929 0.7580 0.5049 0.5848 0.4587 0.4528 0.4914 0.5103
0.7955 1.0441 1.3669 1.2736 1.0877 1.1875 0.7780 0.7549 2.5396 0.8141 1.6412 1.1099 1.5989 1.0810 0.7269 0.9569 2.6755 1.1021 0.6417 2.6889 0.7106 0.4887 2.9829 1.3044 1.5491 0.8828 2.1834 1.0592 1.5674
3.4726 4.2850 5.9890 3.5798 4.1913 4.2234 4.3012 3.0376 11.1789 3.6314 5.9983 4.5368 8.0404 4.7820 3.2000 4.6539 14.7910 4.2210 2.9964 13.5705 3.1714 3.2104 14.8323 5.0451 6.9844 3.4841 9.3136 4.2609 7.2921
4254 7368 1435 2532
0.5176 0.4357 0.2710 0.5024
1.6606 0.7756 –0.0738 2.3800
7.8570 3.2621 2.4625 10.2050
West Fork White River 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95
66 37 81 24 23 45 24 62 22 21 50 62 80 32 21 34 53 32 21 35 35 45 40 22 42 42 52 61 84
East Fork White River 96 97 98 99
41 48 24 24
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Table 2.1.2. Mean Flow and Moment Ratios for 93 Stations in the Wabash River Basin (continued) Station No. 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119
N 49 49 50 61 24 45 24 44 39 68 23 44 36 50 52 52 21 21 36 87
Mean Flow 3860 9109 16581 7383 9495 24610 1931 4024 7869 32714 1066 16194 2049 7238 14863 38615 3128 1335 4690 40569
Cv 0.6260 0.5550 0.5244 0.5807 0.4450 0.5009 0.2339 0.5518 0.4851 0.5062 0.3364 0.5404 0.6719 0.5702 0.6480 0.4528 1.0043 0.8374 1.1066 0.5347
Cs 0.9972 1.4600 1.0684 0.9299 1.1287 0.7380 –1.5976 1.2271 1.0338 0.4905 1.1373 1.8406 4.5024 2.4588 2.1710 0.2860 3.3181 2.7585 2.4065 2.3305
Ck 3.5364 5.8272 3.6915 3.3933 4.2314 2.8769 7.6442 4.8776 4.7096 2.9919 5.5884 8.7935 26.3001 12.2314 9.7230 2.5616 14.8579 11.7482 8.2069 13.0289
LOGN: Three-parameter log normal distribution P-III: Pearson type III distribution GPAR: Generalized Pareto distribution Another version of moment ratio diagram (MRD) is the relation 2 between β1 = C s and β2 = Ck. This relationship is commonly denoted as MRD (β1, β2) and is shown in Fig. 2.2.2 (Bobeé and Ashkar, 1989, 1991). It can be shown that β2 – β1 – 1 ≥ 0 for any statistical distribution. Consequently there is an “impossible region” in the MRD corresponding to (β2 – β1 – 1 < 0). This MRD is similar to the Cs – Ck diagram in Figure 2.2.1 except that in the MRD (β1, β2), the value of Cs is squared. Distributions of four or more parameters occupy an area on the β1, β2 (or Cs, Ck) diagram rather than being single curves. Consequently, they provide greater flexibility in representing data than twoor three-parameter distributions. On the other hand, two-parameter distributions appear as single points in Figure 2.2.2 and provide no flexibility with respect to shape. Given a data set, pairs of
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Figure 2.2.2.
β1 – β2 Moment ratio diagram.
Cˆ s , ˆC k ( or βˆ 1 , βˆ 2 ) are plotted on the C s – C k ( or β 1 – β 2 ) diagram. The location of the sample estimate with respect to the distributions gives an indication of the suitability of a distribution to the data. However, if the sample size is small, the bias in the values of higher moments ( m 3 , m 4 ; ˆC s , ˆC k ) may be large enough to give misleading results. The bias correction depends on the sample size, parent skewness and the form of the parent distribution (Cunnane, 1989, App. 3). If more than one data set is available for a region, Cˆ s , ˆC k ( or βˆ 1 , βˆ 2 ) for each station as well as their regional average are plotted. Such a plot gives a better idea of the appropriate distribution which can be used in that region, provided that the region is sufficiently homogeneous. Homogeneity of a region is tested by using the methods discussed in Section 2.5. The Cs – Ck relationships can be obtained from the material discussed in the following chapters. Alternatively, the following approximations can be used for constructing the Cs – Ck (or β1 – β2) moment ratio diagrams which are quite accurate for Ck < 40. 1. Uniform:
Cs = 0.0
,
Ck = 1.8
2. Exponential:
Cs = 2.0
,
Ck = 9.0
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3. Gumble (EVI):
Cs = 1.1396
,
Ck = 5.4002
4. Logistic:
Cs = 0.0
,
Ck = 4.2
5. Normal:
Cs = 0.0
,
Ck = 3.0
6. Lognormal: 2
Ck = 3 + 0.025653 Cs + 1.720551 C s + 3
4
5
6
0.041755 C s + 0.046052 C s – 0.00478 C s + 0.000196 C s 7. Generalized Logistic: 2
4
6
Ck = 4.2 + 2.400505 C s + 0.244133 C s – 0.00933 C s 8
+ 0.002322 C s
8. Generalized Extreme Value (GEV): 2
3
Ck = 2.695079 + 0.185768 Cs + 1.753401 C s + 0.110735 C s 4
5
6
7
+ 0.037691 C s + 0.0036 C s + 0.00219 C s + 0.000663 C s 8
+ 0.000056 C s 9. Gamma and Pearson III:
2
Ck = 3 + 1.5 C s
10. Generalized Pareto: 2
3
Ck = 1.8 + 0.292003 Cs + 1.34141 C s + 0.090727 C s + 4
5
6
7
0.022421 C s + 0.00400 C s + 0.000681 C s + 0.000089 C s + 8
0.000005 C s
11. Weibull: Same as GEV but with Cs replaced by –Cs.
EXAMPLE 2.2.1 Using the results obtained in Example 2.1.1, compute the Cs and Ck values for the 93 Indiana stations and plot them along with the theoretical Cs – Ck relationships for different three-parameter distributions. Discuss the results.
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The Cs – Ck MRD is shown in Figure 2.2.3 and the β1 – β2 MRD is shown
Figure 2.2.3.
Cs – Ck Moment ratio diagram for 93 Wabash River basin stations.
in Figure 2.2.4. It is observed from the two MRD’s that the stations cluster around the Pearson(3) distribution P(3) and therefore P(3) distribution may be expected to give the best regional fit for the data. However, for some data from individual stations the Weibull distribution and the lognormal distribution are also possible candidates.
2.3
Probability Plots
Probability plots are used to visually evaluate the agreement between distributions and observed data. Observed data are plotted against the values estimated from the fitted distribution. If the fitted distribution is the exact parent distribution, this relationship should appear as a straight line through the origin with a 45° slope. The suitability of a distribution may then be judged by the correspondence between quantile estimates and those given by the straight line. This analysis can be made only after estimating the parameters and fitting the distribution. Alternatively, in the case of two-parameter distributions, a similar procedure (Stedinger et al., 1993) may be applied before the parame-
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Figure 2.2.4.
β1 – β2 Moment ratio diagram for 93 Wabash River basin stations.
ters are estimated. For a given two-parameter distribution, the distribution function can be written in the form of Eq. 2.3.1, Q–α F = G -------------- β
(2.3.1)
where α and β are the distribution location and scale parameters, respectively, and thus the estimated quantile may be obtained as in Eq. 2.3.2. ˆ = α + βG –1 ( F ) Q
(2.3.2)
ˆ and G –1 ( F ) . Equation 2.3.2 represents a linear relationship between Q ˆ then the relationship between If the estimate of the observed Q is Q –1 the observed Q and G (F) should be linear if the fitted distribution is the exact parent distribution. For two-parameter distributions, G–1(F) depends only on F, which can be estimated by using a suitable plotting position formula (Section 1.4). By plotting the observed data qi against G–1(Fi), those distributions which give a straight line relationship on
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the probability plot can be selected. This is usually done by preparing in advance a special plotting paper on which the F,Q relationship would appear as a straight line for each distribution. This paper can be used to directly plot the values of Qi against the plotting position Fi. For three-parameter distributions, the function G–1(F) depends on the skewness coefficient Cs and probability plots can be used only for a specific value of Cs assumed in advance. Probability plots, although helpful in choosing between alternative distributions, suffer from a distinct possibility of error due to large variations in the plotting behavior of samples drawn from a given distribution (Cunnane 1989).
EXAMPLE 2.3.1 Use the probability plot method to investigate the suitability of the Exponential distribution for two stations in the Wabash River basin. The cumulative probability function of the Exponential distribution is given by F = 1– e –( x – ε ) /α and the quantile function is therefore Q = x = ε − α log (1 – F). Since T = 1/(1 – F), Q = ε + α log T. Therefore, the relationship between Q and log T should be a straight line. The flow data for stations 46 (Wildcat Creek at Kokomo) and 93 (Mill Creek near Manhattan) (see Tables 2.1.1 and 2.1.2) are given in Tables 2.3.1 and 2.3.2. The Q,T relationships are plotted on a log scale in Figure 2.3.1 using a plotting position formula F = (i – 0.35)/n, or equivalently T = n/(n – i + 0.35) where i is the rank in ascending order. Also shown in Figure 2.3.1 are the theoretical straight-line relationships for both stations. As seen from Figure 2.3.1, the Q,T relationship for station 93 is fairly close to the theoretical straight line indicating that the Exponential distribution may be suitable for the data from this station. On the other hand, the Q,T relationship at station 46 deviates from the theoretical straight line indicating that the Exponential distribution is not suitable for the data at this station. The results obtained from the probability plot is in agreement with the results that would be obtained by using MRDs. The Exponential distribution plots as Cs = 2 and Ck = 9, station 93 plots as Cs = 2.18 and Ck = 9.31, while station 58 plots as Cs = 0.19 and Ck = 2.49.
2.4
Selection of Distributions
The choice of distributions to be used in flood frequency analysis has been a topic of interest for a long time. Hazen (1914) looked at this question in the context of storage design for municipal water supply. Probability distributions that best fit distributions of annual precipi-
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tation and runoff series were analyzed by Markovic (1965). A method of selection of a distribution which best fits the data was proposed by Gupta (1970). Gupta’s method is based on the coefficient of determination. A nonparametric test is used to detect differences between distributions. Table 2.3.1.
1956 1957 1958 1959 1960 1961 1962 1963 1964 1965 1966 1967
Table 2.3.2
1940 1941 1942 1943 1944 1945 1946 1947 1948 1949 1950 1951 1952
Flow Data for Wildcat Creek (Station 46) at Kokomo, IN
2060 4860 6920 8100 1410 3290 5320 2760 6900 1870 846 3670
1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979
5080 4420 3160 2530 4970 3600 6590 4050 5740 1400 4180 4940
1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991
7150 1740 4380 4030 2770 7240 3720 1020 4240 6070 4180 8070
Flow Data for Mill Creek (Station 93) near Manhattan, IN
4020 3270 5340 5800 5960 4000 3200 5000 4800 5000 8960 3000 3290
1953 1954 1955 1956 1957 1958 1959 1960 1961 1962 1963 1964 1965
3690 1540 4040 3180 2940 2600 2440 2740 2750 2860 2920 2980 2930
1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978
2130 2250 2710 2780 2730 2430 2110 2440 2290 2520 2000 1650 2260
1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991
2410 2060 2050 2050 1930 1990 2030 2190 1830 1750 1870 1840 3060
McCuen and Rawls (1979) discussed criteria to evaluate the usefulness of hydrologic analysis. They also presented a classification system for categorizing the available procedures of flood frequency
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Figure 2.3.1. Probability plots for stations 46 and 93 compared with the theoretical linear relationship for the exponential distribution.
analysis. A literature review and recommendation on reporting flood frequency analysis procedures were also presented by them. McCuen (1979a) presented definitions of statistical terms used in flood frequency analysis and an interpretation of these terms. Campbell and Sidel (1984) compared the Type I Extreme Value EV1(2), two-parameter log normal (LN (2)), three-parameter log Nor-
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mal (LN (3)), and log Pearson type III (LP(3)) distributions for fitting flood data from Oregon. They selected the LP(3) distribution as the best. Akaike’s information criterion was proposed for the choice of distributions by Turkman (1985). Vogel (1986) proposed a new probability plot correlation test for the EV1(2) distribution. He also discussed the probability plot correlation coefficient test for the normal and log normal distribution hypotheses. This work was extended by Vogel and McMartin (1991) who developed a probability plot correlation coefficient hypothesis test for the Pearson type III (P(3)) distribution. They also presented a new estimator of the skewness coefficient. Nine distributions were used with data from 45 unregulated streams in Turkey by Haktanir (1992) who concluded that LN(2) and EV1(2) distributions were superior to other distributions. Bobeé et al. (1993) reviewed commonly used procedures for flood frequency estimation. They gave some reasons for the state of confusion in these comparative studies and presented broad lines of a comparison strategy. A total of 1819 site-years of data from 19 stations in the world were analyzed by Onoz and Bayazit (1995). They used seven distributions and found that the GEV distribution was superior to other distributions. A conclusion, based on the results of the above cited studies, is that most of the methods available for selection of distributions from small samples are not sensitive enough to discriminate among distributions. Studies along the lines suggested by Bobeé et al. (1993) may help in resolving the outstanding issues in this important topic. However, probability distributions for flood frequency analysis have been selected by using chi-square, Kolmogorov-Smirnoff tests and Akaike’s Information Criterion (AIC). Turkman (1985) used the AIC to choose the most likely among the EV1(2), Frechet and Weibull distributions. Moon et al. (1993) compared several tail probability estimators and found that the variable kernel method performed consistently well in terms of bias and root mean square error. These procedures can be used to test distributions separately, but not as discriminatory tests for choosing between one distribution and another (Cunnane, 1989). Typical tests that are used are the chi-square test and the KolmogorovSmirnov test (Kite, 1977).
2.4.1 Chi-Square and Psi Tests In the chi-square test, data are first divided into k class intervals. The
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statistic χ in Eq. 2.4.1 is distributed asymptotically as chi-square with k – 1 degrees of freedom. 2
( O –E ) 2
k
j j ∑ -----------------Ej
χ = 2
(2.4.1)
j=1
In Eq. 2.4.1 Oj is the observed number of events in the class interval j, Ej is the number of events that would be expected from the theoretical distribution and k is an arbitrary number of classes to which the observed data are divided. If the class intervals are chosen such that each interval corresponds to an equal probability, then Ej = n/k where n is the sample size and k is the number of class intervals, and Eq. 2.4.1 reduces to Eq. 2.4.2 (Mann and Wald, 1942) k
k 2 2 χ = --- ∑ O j – n nj=1
(2.4.2)
Class intervals can be computed by using the inverse of the distribution function corresponding to different values of probability F similar to estimating quantiles. These aspects are discussed in Section 4.3. The psi test (Tribus, 1969) is similar in structure to the chi-square test. The statistic ψ used in this test is given by Eq. 2.4.3 where N is the number of observations and k is the number of class intervals. Equation 2.4.3 can be manipulated to yield Eq. 2.4.4, under the assumption that the sum of the errors is small. k
∑ εj j=1
k
=
∑ (E j– O j) j=1
k
O ψ = 10N ∑ O j log 10 ------j Ej j=1
(2.4.3)
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k
N ∑ (E j– O j) j=1 ψ = ----------------------------------2E j 2
(2.4.4)
The term on the right-hand side of Eq. 2.4.4 is very similar to the corresponding term in Eq. 2.4.1, with the important difference that the number of observations N is involved in Eq. 2.4.4. When the ratio Oj/Ej is nearly unity, there is close correspondence between the results given by chi-square and psi tests. The parameter ψ is chi-square distributed with k – 1 – m degrees of freedom where m is the number of parameters estimated. Further discussion of the psi test is found in Arora and Rao (1985).
2.4.2 Kolmogorov–Smirnov Test A statistic based on the deviations of the sample distribution function FN(x) from the completely specified continuous hypothetical distribution function Fo(x) is used in this test. The test statistic D is defined in Eq. 2.4.5. D N = max F N ( x )–F o ( x )
(2.4.5)
The values of FN(x) are estimated as N j ⁄ N where Nj is the cumulative number of sample events at class limit j. Fo(x) is then 1/k, 2/k, ... etc., where k is the number of class intervals. Class limits are obtained the same way as in the chi-square test. The value of DN must be less than a tabulated value of DN at the required confidence level (Kolmogorov, 1933; also Hogg and Tanis, 1988, Table VIII) for the distribution to be accepted. Other goodness-of-fit measures are available such as the leastsquares test (Kite, 1977) and the probability plot correlation coefficient test (Filliben, 1975). Goodness-of-fit tests have very low statistical power (Cunnane, 1989). Since the parameters of the tested distributions are estimated from samples, it follows that several candidate distributions may be considered to be similar. Consequently there is a very high probability that real differences will not be detected by these tests. In a study in which the LP(3), Wakeby, and mixed double exponential distributions were fitted to annual maximum flow data from 25
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rivers from around the world, Arora and Rao (1985) tested the goodness-of-fit by using chi-square and psi tests. The LP(3) distribution provided a good fit by the chi-square test for 13 of the 25 data sets, marginal fit for 7 data sets and the fits were unacceptable for 5 data sets. The psi test results showed that 9 data sets fitted well, 3 were marginal, and 13 were unacceptable. For the mixed double-exponential distribution, the chi-square test indicated that 9 data sets provided good fit, 8 marginal, and 9 sets were poor. The psi test indicated that 6 data sets fitted well, 10 marginal, and 9 sets were poor. The Wakeby distribution chi square test indicated that 11 sets fitted the data well, six were marginal, and the fits were unacceptable for 8 of them. The psi test indicated good fit for 10 data sets, marginal fit for 6 data sets, and unacceptable fit for 9 data sets. The results of fitting distributions to data is thus diverse. Even in light of these weak tests, distributions are not acceptable in many cases. Consequently a single distribution is not acceptable for all the data. The same conclusion presents itself with other tests based on L-moments which are discussed later in this book.
EXAMPLE 2.4.1 The chi-square test as well as the Kolmogorov-Smirnov test are applied to data from two stations 93 (Mill Creek near Manhattan) and 112 (Brush Creek near Nebraska) to check the goodness of fit assuming an exponential distribution for the data. Each of the data sets is divided into eight classes (k = 8). Class intervals corresponding to equal probabilities of Ej = 1/8 = 0.125 are calculated from the quantile function Q = ε – α log (1-F). The parameter estimates obtained by the method of moments (MOM) are: 1⁄2 αˆ = m 2 and εˆ = m′ 1 – αˆ
Station 93: N = 52, m′ 1 = 3012,
αˆ = 1364,
1⁄2
= 1364, m2 εˆ = 1648
Q = 1648 – 1364 log (1-F).
8 2 χ = ------ (374) – 52 = 5.53 52
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Class 1 2 3 4 5 6 7 8
Class Fo Limit 0.125 1830 0.250 2040 0.375 2288 0.500 2593 0.625 2985 0.750 3538 0.875 4483 1.000 ∞
nj = ΣΟi 3 10 18 24 35 41 45 52
Oi 3 7 8 6 11 6 4 7 ΣΟi2 = 374
χ 0.90 (8–1) = 12.02 > χ 2
Fn = nj/n 0.0577 0.1923 0.3462 0.4615 0.6731 0.7885 0.8654 1.0000
|Fn – Fo| 0.0673 0.0577 0.0288 0.0385 0.0481 0.0385 0.0096 0.0000 Dn = 0.0673
2
According to the chi-square test we can accept the null hypothesis that the data are exponentially distributed at 10% significance level. Also DN = 0.0673 while at 10% significance level we have Dcritical = 0.1692. Thus we can accept the null hypothesis that the data are exponentially distributed using the Kolmogorov-Smirnov test at 10% significance level. Station 112: N = 36,
m′ 1 = 2049, m 2
αˆ = 1377,
εˆ = 672
1⁄2
= 1377
Q = 672 – 1377 log (1 – F)
8 2 χ = ------ (250) – 36 = 19.55 36 χ
2 0.90
(7) = 12.02 < χ
2 0.99
(7) = 18.48 < χ
2
According to the chi-square test we reject the null hypothesis that the data are exponentially distributed at both 10% and 1% significance levels.
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Class 1 2 3 4 5 6 7 8
Fo 0.125 0.250 0.375 0.500 0.625 0.750 0.875 1.000
Class Limit 856 1069 2288 2593 2985 3538 4483 ∞
Oi 0 2 5 7 9 9 3 1 ΣΟi2 = 250
nj = ΣΟi Fn = nj/n 0 0 2 0.0556 7 0.1944 14 0.3889 23 0.6389 32 0.8889 35 0.9722 36 1.000
|Fn – Fo| 0.1250 0.1944 0.1806 0.1111 0.0139 0.1389 0.0972 0.0000 Dn = 0.1944
Also, Dn = 0.1944 while at 20% and 10% significance levels the critical values of Dn are 0.178 and 0.206, respectively. Thus we reject the hypothesis that the data are exponentially distributed using the KolmogorovSmirnov test at the 20% level but not at the 10% significance level. The quantile plots for the two stations in this example are shown in Figure 2.4.1. As expected, the plot for station 93 is much closer to a straight line relationship than the plot for station 112.
EXAMPLE 2.4.2 For the annual maximum flow data from Susquehanna River near Harrisburg, Pennsylvania for the period 1891 to 1967 (N = 71) the observed frequency and the expected frequency from the LP(3) distributions are given below. Determine, by using the ψ-test whether the LP(3) distribution is acceptable for these data.
(Ej – Oj)2/2Ej
Class
Oj
Ej
1
0.141
0.183
0.0048
2
0.535
0.479
0.0033
3
0.239
0.225
0.00044
4
0.042
0.070
0.0056
5
0.014
0.028
0.007
6
0.028
0.014
0.007 Σ = 0.0562
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Figure 2.4.1. Probability plots for the data from stations 93 and 112 compared with the theoretical linear relationship for the exponential distribution.
The sum of (Ej – Oj)2/2Ej is 0.0562. As N = 71, ψ value is 3.99. χ ( k – 1 – m ), ( 6 – 1 – 3 ) is 5.99. The null hypothesis that these data are distributed as LP(3) distribution is accepted. 2 0.95
2.5
Regional Homogeneity and Regionalization
Regional estimates of flood flow statistics are investigated in the context of regionalization and regional homogeneity studies. Kleme s˘ (1988) has discussed the methods and merits of regional flood frequency analysis. Cunnane (1988) has reviewed the regional frequency analysis methods. Matalas et al. (1975) demonstrated that the relationship between the mean and standard deviation of regional estimates of skewness for historical flood sequences is incompatible with the corresponding relationship for several well known distributions
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such as EV1(2), P(3), LN(2) and others. Wallis et al. (1977) investigated the differences in relationship between the mean and standard deviation of regional estimates of skewness for observed flood sequences and the corresponding theoretical relationships. These differences were explained in terms of spatial and temporal distribution of skewness values. Tasker (1978) demonstrated that the method of estimation of weighted average skew by using the weighing factor recommended by the Hydrology Committee of the Water Resources Council resulted in a poorer estimate of the population skew coefficient than using the sample skew coefficient alone. The optimal estimate of skew coefficient involves a factor which is a function of the number of observations, standard deviation of skew coefficients and population skew coefficient. Mosley (1981) performed a cluster analysis of data characterizing flood hydrology of selected New Zealand catchments. Catchments of similar hydrologic characteristics were delineated. Waylen and Woo (1981) used discriminant analysis to separate the Fraser River catchment in British Columbia into regions. The mean and standard deviation of floods, within each region, were related to the physiographic and climatic variables. Tasker (1982) compared methods of regionalization by using split sample data. Cluster analysis methods were used to define “homogeneous” hydrologic regions. Kuczera (1983) used the empirical Bayes procedure, with the normal probability model, to combine site and regional information. Acreman and Sinclair (1986) used the NORMIX multivariate clustering algorithm to classify drainage basins. The method was tested by using data from 168 basins in Scotland. Boes et al. (1989) developed methods to estimate annual flood quantiles based on a regional Weibull model. Estimation techniques such as the MOM, PWM, and MLM were compared. Bhaskar and O’Connor (1989) compared cluster analysis and the method of residuals for flood regionalization. The results were quite dissimilar. Burn (1989) used cluster analysis to delineate groups of stations into a region for regional flood frequency analysis. Garros-Berthet (1994) applied a station–year approach to rainfall–runoff data from Indonesia, Malaysia, and the Philippines to define regional distributions of maximum floods. Cluster analysis was used to define groups of homogeneous stations. Fill and Stedinger (1995) compared the relative performances of Dalrymple’s test, a normalized quantile test based on L-moment parameters and a method of moment CV–test. Dalrymple’s test was shown to be erroneous. The
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L-moment test is shown to be always more powerful than the other two tests. Regional frequency analysis methods are based on the assumption that the standardized variable qt = QT/µi at each station (i) has the same distribution at every site in the region under consideration (see Section 1.7). In particular Cv(q) and Cs(q), the coefficient of variation and the coefficient of skewness of q, are considered to be constant across the region (Cunnane, 1989). Departures from this assumption may lead to biased quantile estimates at some sites. Sites with Cv and Cs nearest to the regional average may not suffer from such bias, but large, biased quantile estimates are expected for sites whose Cv and Cs deviate from average. Good results may be obtained by regionalization, especially in cases of short records, provided that the degree of heterogeneity is not great. In such cases, the large number of sites contributing to parameter estimation compensates for regional heterogeneity. A method of assigning homogeneous regions is geographical similarity in soil types, climate and topography. However, geographically similar regions may not be similar from the flood frequency point of view (Cunnane, 1989). On the other hand, two sites in different regions may prove to be similar with respect to flood frequency, despite the fact that they are geographically different. Another approach (Wiltshire, 1986a,b) is to initially divide the entire group of catchments into two or more groups based on one or more chosen basin characteristic such as large and small, or wet and dry. The internal homogeneity and mutual heterogeneity of these groups are then expressed in terms of a flow statistic such as Cv. The process is then repeated by altering the partition points until an acceptable set of regions has been identified. Considerable effort has been expended on developing and testing methods for estimating flood frequencies at ungaged locations. Regionalization of watershed and flow characteristics has played an important role in these studies. An early effort by Bodhaine (1960) dealt with determining the magnitude and frequency of floods in the Pacific Northwest. Alexander (1972) investigated the estimate K in the relationship between flood flow Q and catchment area A, Q = CAK where C is a constant. Schrader et al. (1981) used the LN(3) distribution in a regionalization study. Data from 132 watersheds in Pennsylvania were used by them. Waylen and Woo (1981) used the Gumbel distribution in a regionalization study of the flood data from the Fraser River basin in Canada. They used discriminant analysis to separate basins into
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regions. The mean and standard deviation of flood data were related to physiographic and climatic variables. This study was continued in Waylen and Woo (1984). The effect of using mean flow for regionalization studies was investigated by Stedinger (1983a). He found that when the mean flood is used for regionalization, the resulting curves may not represent the true flood distribution. Stedinger found that the use of logarithms of flood values and unbiased moment or probability weighed estimators overcame this problem. The correlation among flood flows in a region places severe limits on the accuracy of moments. The annual peak flows from 57 watersheds in Pennsylvania were analyzed to test whether they were correlated by Wall and Englot (1985). Out of 57 data series, 55 were found to be uncorrelated. Wiltshire (1986a,b) proposed a procedure to classify basins into distinct homogeneous groups for regional flood frequency analysis. Model error estimators for weighted and generalized least square (GLS) regression were investigated by Stedinger and Tasker (1986). These estimators were shown to be more accurate than ordinary least squares estimators. GLS estimators performed well even when normality assumptions were violated. The search for distributions which are insensitive to regional variation is another aspect that has received attention. The three-parameter GEV distribution was shown by Lattenmeir et al. (1987) to be relatively insensitive to moderate regional variation in the coefficient of variation. It was also shown to be quite insensitive to variation in the skew coefficient. Jin and Stedinger (1989) developed generalized maximum likelihood estimators which use both at-site historical information and systematic flood records. Two methods of estimation of regional probabilities are discussed by Gottschalk (1989). For independent data, Gottschalk used the order statistics for estimating regional probabilities, whereas for correlated data, he defined an equivalent number of regional series and used the theory of independent data. A regional flood frequency model based on a large quantile was developed by Smith (1989). Exceedences of the specified quantile are modelled by a generalized Pareto distribution. Gabrielle and Arnell (1991) developed a regional flood frequency estimation method based on two component extreme value and generalized extreme value distributions. They demonstrated that dividing a region to subregions to estimate some parameters can lead to more precise quantile estimates. The GEV distribution was used by Farquharson et al. (1992) to develop
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regional frequency curves from 162 stations in Africa which are located in arid and semi-arid areas. Zrinji and Bunn (1994) used the region of influence approach to regionalize an area. The advantages of this method were discussed by using data from Newfoundland, Canada. L-moment tests were used by Gingras et al. (1994) to delineate homogeneous regions in Ontario and Quebec. The use of L-moments to test regional homogeneity (Hosking and Wallis, 1991) has received considerable attention recently. Acreman and Sinclair (1986) proposed a method of identifying homogeneous regions by using a clustering algorithm depending on the catchment characteristics and then using a likelihood ratio test to check whether an estimated GEV distribution for a region differs significantly from that of another region. The Wiltshire (1986a,b) Cv-based test involves the statistic S in Eq. 2.5.1. ( C vj – C vo ) ∑ --------------------------Uj j=1 N
S =
2
(2.5.1)
In Eq. 2.5.1, N is the number of sites in the region, Cvj is the coefficient of variation at site j and Cvo given by Eqs. 2.5.2 and 2.5.3: N
∑ C vj / U j j=1 N
C vo = -------------------------
(2.5.2)
∑ 1/U j
j=1
and V U j = ---nj
(2.5.3)
In Eq. 2.5.3 nj is the record length at site j and V is the regional variance defined in Eq. 2.5.4. 1 V = ---N
N
∑ n jv j
(2.5.4)
j=1
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In Eq. 2.5.4, vj is given by Eq. 2.5.5: 2
nj
v j = (n j - 1)
∑
C
(i)
vn – 1
i=1
-
nj
∑C v l=1
(l) n–1
⁄ nj ⁄ nj
(2.5.5)
(k)
C vn – 1 in Eq. 2.5.5 is the Cv computed from a sample of size (nj – 1) with the kth observation removed. The statistic S in Eq. 2.5.1 has the 2 2 form of a χ statistic. S is expected to be χ distributed with (N – 1) degrees of freedom. If the value of S exceeds the critical value of 2 χ ( N – 1 ) at a particular significance level, then the hypothesis that the region is homogeneous is rejected and the region is regarded as heterogeneous. However, this test is likely to be effective only for large regions having large record lengths. Further developments using distribution-based tests are given by Wiltshire (1986a,b).
EXAMPLE 2.5.1 Use the Wiltshire (1986) method to investigate the homogeneity within two regions in the Wabash River basin. The two regions are the West Fork White River region (29 stations) and the upper Wabash River region (25 stations). The Cv values for these stations are given in Table 2.1.2. The program discussed in Chapter 10 is used for calculating the test statistic S for each region. The following results are obtained: West Fork White River Region: n = 29, v = 0.3535, S = 24.42 Since χ 0.90 ( 29 – 1 ) = 37.92 > s , we cannot reject the null hypothesis of homogeneity at 10% significance level. 2
Upper Wabash River Region: n = 25, v = 0.1799, S = 78.92 Since χ 20.90 ( 25 – 1 ) = 33.20 < χ 20.99 ( 25 – 1 ) = 42.98 < S , we reject the null hypothesis of homogeneity at both 10% and 1% significance levels.
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CHAPTER 3
Selection and Evaluation of Parent Distributions: Probability Weighted Moments and L-Moments
3.1 Moments of Distributions and Their Sample Estimates Probability weighted moments (PWMs) are defined by Greenwood et al. (1979) as 1
M p, r, s = E [ x F ( 1 – F ) ] = p
r
s
∫ [ x(F )] F (1 – F ) p
r
s
dF
(3.1.1)
0
In particular, the following two moments M1,0,s and M1,r,0 are often considered: 1
M 1, 0, s = α s =
∫ x(F )(1 – F )
s
dF
(3.1.2)
0
1
M 1, r, 0 = β r =
∫ x ( F )F
r
dF
(3.1.3)
0
where p, r, and s are real numbers. When r and s are equal to zero and p is a non-negative number, Mp,o,o represents the conventional moment of order p about the origin, µ′p . When p = 1 and either r or s is equal to zero, then M1,r,0 = βr and M1,0,s = αs are linear in x and of sufficient generality for parameter estimation (Hosking, 1986a). As x takes only the power of one, simpler relationships are obtained between the parameters of the distributions and probability weighted
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moments than the corresponding relationships in conventional moments. For an ordered sample x1 ≤ … ≤ xN, N > r, N > s unbiased sample PWMs are given by Eqs. 3.1.4 and 3.1.5 (Hosking, 1986a). N
1 ˆ 1, 0, s = --- N – i x i / N – 1 a S = αˆ s = M s s N i∑ =1
(3.1.4)
N
1 ˆ 1, r, 0 = --- i – 1 x i / N – 1 b r = βˆ r = M r r N i∑ =1
(3.1.5)
Special cases of these estimators include the sample mean –1 x = N ∑ x i = a 0 = b 0 and the extreme data values x1 = N aN–1 and xN = N bN–1. Alternatively, consistent but biased estimators of PWMs may be obtained by using the plotting position Fi = (i – 0.35)/N. There is no theoretical reason to prefer plotting position estimators to the unbiased estimators. However, practical experience shows that plotting position estimators sometimes yield better estimates of parameters and quantiles. The plotting position estimates for PWMs are given by Eqs. 3.1.6 and 3.1.7. N
1 ˆ 1, 0, s = --- ( 1– F i ) s x i a S = αˆ s = M N i∑ =1
(3.1.6)
N
ˆ r = Mˆ 1, r, 0 = ---1- ∑ F i r x i br = β Ni = 1
(3.1.7)
The PWMs αs and βr are related as in Eq. 3.1.8. s
αs =
∑ i ( – 1 ) s
r
i
βi , βr =
i=0
In particular: α0 = β0 α1 = β0 – β1 α2 = β0 – 2β1 + β2 α3 = β0 – 3β1 + 3β2 – β3
∑ i r
( –1 ) αi i
(3.1.8)
i=0
, , , ,
β0 = a0 β1 = α0 – α1 β2 = α0 – 2α1 + α2 β3 = α0 – 3α1 + 3α2 – α3
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The same relationships hold for the sample PWM estimates as and br . Hosking (1986 and 1990) introduced the L-moments, which are linear functions of PWMs. The L-moments are more convenient than PWMs because they can be directly interpreted as measures of scale and shape of probability distributions. In this respect they are analogous to conventional moments. L-moments are defined by Hosking in terms of the PWMs α and β as r
λr + 1 = ( – 1 )
r
∑
r
∑ p r, k β k
p r, k α k = *
*
k=0
(3.1.9)
k=0
where p r, k = ( – 1 ) *
r–k
r r + k . k k
(3.1.10)
In particular, λ1 = λ2 = λ3 = λ4 =
α0 α0 – 2α1 α0 – 6α1 + 6α2 α0 – 12α1 + 30α2 – 20α3
= = = =
β0 2β1 – β0 6β2 – 6β1 + β0 20β3 – 30β2 + 12β1 – β0
Sample L-moments (lr) are calculated by replacing α and β in Eq. 3.1.9 by their sample estimates a and b in Eqs. 3.1.4 and 3.1.5 or from Eqs. 3.1.6 and 3.1.7. L-moment ratios, which are analogous to conventional moment ratios are defined by Hosking (1990) as in Eqs. 3.1.11 and 3.1.12, τ = λ2 / λ1
(3.1.11)
τr = λr / λ2 , r ≥ 3
(3.1.12)
where λ1 is a measure of location, τ is a measure of scale and dispersion (LCv), τ3 is a measure of skewness (LCs), and τ4 is a measure of kurtosis (LCk). Sample L-moment ratios (t and tr) are calculated by replacing λr in Eqs. 3.1.11 and 3.1.12 by their sample estimates lr . It can be shown (Hosking, 1990) that for r greater than or equal to 3, the absolute value of τr is less than one. Furthermore, if x ≥ 0 almost surely, then τ, the LCv of x satisfies 0 < τ < 1. This boundedness of L-moment
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ratios is an advantage (Hosking, 1990) because it is easier to interpret a measure such as τ3, which is constrained to lie within the interval (–1,1), than conventional skewness coefficient which can take arbitrarily large values. L-moment and nonparametric methods were coupled to underlying distributions which are nonunimodal by Gingras and Adamowski (1992). The advantages of L-moments compared to product moments are brought out by Vogel and Fennessey (1993).
EXAMPLE 3.1.1 Calculate the L-moments and moment ratios for the 93 Wabash River basin stations listed in Table 2.1.1. For the 93 Wabash River basin stations in Table 2.1.1, the calculated L-moments and moment ratios are as given in Table 3.1.1. The results in Table 3.1.1 correspond both to the biased estimators computed by using a plotting position of Fi = (i – 0.35)/N and the unbiased estimators. A location map of these stations is shown in Figure 3.1.1.
3.2
L-Moment Ratio Diagrams
Analogous to the conventional MRDs, the L-moment ratio diagrams are based on relationships between the L-moment ratios. A diagram based on LCs (τ3) versus LCk (τ4), such as that in Figure 3.4.2, can be used similar to the conventional MRDs to identify appropriate distributions. For a given region, the sample L-moment ratios t3 and t4 for each station as well as their regional average are plotted on the Lmoment ratio diagram. A suitable parent distribution is that which averages the scattered data and around which the data spread consistently. However, just as with conventional MRDs, a certain degree of regional homogeneity (Section 3.4) must be satisfied in order to obtain a suitable regional parent distribution. As mentioned before, Lmoment ratio diagrams are based on unbiased sample quantities in contrast to Cs and Ck which have to be corrected for bias. It was shown by Hosking (1990) that Cs and Ck values from several samples drawn from three different distributions lie close to a single line on the graph and overlap each other offering little hope of identifying the population distribution. In contrast, the sample L-moment ratios plot as fairly well separated groups and permit better discrimination between the
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Figure 3.1.1.
Location maps of gauging stations in the Wabash River basin.
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Table 3.1.1. L-Moments and Ratios for the Wabash River Basin Stations
l1
t
Biased t3
t4
t
t3
t4
4137 5175 5969 3543 2439 7227 6703 18422 248 4311 846 11699 7535 21992 2581 4357 267 8278 33544 254 4910 406 193 4602 12665
0.22 0.21 0.24 0.17 0.16 0.20 0.26 0.28 0.33 0.30 0.26 0.26 0.39 0.29 0.23 0.23 0.25 0.22 0.25 0.32 0.34 0.21 0.34 0.23 0.21
0.05 –0.03 0.27 0.08 –0.08 0.10 0.20 0.24 0.26 0.28 0.29 0.11 0.32 0.34 0.15 0.14 0.20 0.25 0.20 0.09 0.36 0.25 0.36 0.19 0.03
0.19 0.20 0.20 0.13 0.15 0.17 0.15 0.15 0.24 0.23 0.23 0.12 0.28 0.19 0.19 0.16 0.15 0.21 0.17 –0.02 0.25 0.24 0.17 0.12 0.13
0.21 0.20 0.24 0.17 0.15 0.20 0.26 0.28 0.33 0.30 0.25 0.25 0.39 0.34 0.23 0.23 0.25 0.22 0.25 0.32 0.34 0.21 0.34 0.23 0.21
0.03 –0.07 0.27 0.07 –0.12 0.09 0.19 0.24 0.26 0.28 0.30 0.10 0.32 0.35 0.14 0.13 0.19 0.25 0.20 0.06 0.36 0.25 0.37 0.18 0.02
0.18 0.19 0.19 0.10 0.13 0.15 0.14 0.15 0.31 0.23 0.21 0.11 0.29 0.18 0.17 0.16 0.12 0.20 0.17 –0.08 0.25 0.24 0.15 0.10 0.11
0.27 0.24 0.27 0.34 0.29 0.24 0.35 0.25
0.19 0.06 0.06 0.30 0.17 0.20 0.28 0.04
0.23 0.19 0.10 0.22 0.17 0.24 0.21 0.13
0.27 0.23 0.27 0.34 0.28 0.24 0.35 0.25
0.18 0.04 0.04 0.30 0.16 0.20 0.27 0.01
0.23 0.18 0.09 0.23 0.16 0.24 0.22 0.12
Stat. No. N Upper Wabash 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43
37 28 41 48 32 34 68 68 9 45 18 68 40 49 23 68 21 49 68 23 49 42 22 45 48
Unbiased
Middle Wabash 44 45 46 47 48 49 50 51
30 32 36 49 37 85 21 32
2891 521 4257 5723 10814 52621 1823 5764
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Table 3.1.1. L-Moments and Ratios for the Wabash River Basin Stations (continued) Stat. No. 52 53
N 65 23
l1 52700 1627
t
Biased t3
0.27 0.19
54
53
10638
55
67
65078
56
34
57 58
Unbiased
0.19 0.06
t4 0.18 0.21
t 0.27 0.19
t3 0.18 0.03
t4 0.18 0.20
0.30
0.19
0.16
0.30
0.19
0.15
0.26
0.10
0.18
0.25
0.09
0.17
5776
0.35
0.33
0.18
0.35
0.33
0.17
34
1775
0.26
0.32
0.28
0.26
0.33
0.28
31
9331
0.39
0.36
0.26
0.39
0.37
0.27
West Fork White River 67
66
5403
0.29
0.17
0.13
0.29
0.16
0.12
68
37
834
0.24
0.20
0.21
0.23
0.20
0.21
69
81
7726
0.33
0.23
0.14
0.33
0.23
0.14
70
24
476
0.33
0.36
0.20
0.33
0.37
0.19
71
23
2132
0.30
0.24
0.24
0.30
0.24
0.22
72
45
11258
0.28
0.26
0.20
0.28
0.26
0.19
73
24
964
0.24
0.17
0.22
0.24
0.16
0.22
74
62
14171
0.29
0.18
0.15
0.29
0.18
0.14
75
22
1435
0.36
0.41
0.25
0.37
0.43
0.25
76
21
507
0.32
0.18
0.18
0.32
0.16
0.18
77
50
3175
0.29
0.30
0.22
0.29
0.30
0.21
78
62
4479
0.31
0.21
0.19
0.31
0.21
0.19
79
80
20019
0.28
0.21
0.18
0.28
0.21
0.18
80
32
1038
0.27
0.20
0.16
0.27
0.19
0.15
81
21
392
0.25
0.20
0.15
0.25
0.19
0.11
82
34
4800
0.29
0.17
0.20
0.29
0.16
0.20
83
53
6472
0.33
0.25
0.21
0.33
0.25
0.21
84
32
1395
0.29
0.24
0.17
0.29
0.23
0.16
85
21
1156
0.29
0.18
0.09
0.29
0.16
0.06
86
35
1775
0.30
0.31
0.26
0.30
0.31
0.14
87
35
9010
0.25
0.17
0.11
0.25
0.16
0.08
88
45
25666
0.22
0.11
0.16
0.22
0.10
0.14
89
40
1798
0.34
0.34
0.30
0.35
0.34
0.31
90
22
388
0.28
0.27
0.21
0.27
0.27
0.20
91
42
9501
0.31
0.24
0.21
0.31
0.24
0.20
92
42
5672
0.25
0.21
0.17
0.25
0.21
0.16
93
52
3011
0.22
0.36
0.25
0.22
0.37
0.24
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Table 3.1.1. L-Moments and Ratios for the Wabash River Basin Stations (continued) Stat. No. 94
N 61
l1 13691
t
Biased t3
0.27
95
84
39404
Unbiased
0.21
t4 0.17
t 0.27
t3 0.21
t4 0.17
0.27
0.22
0.17
0.27
0.22
0.17
0.27 0.24 0.16 0.24 0.34 0.29 0.29 0.32 0.24 0.28 0.13 0.30 0.27 0.29 0.19 0.27 0.25 0.28 0.32 0.26 0.39 0.37 0.46 0.27
0.22 0.18 –0.00 0.36 0.23 0.26 0.25 0.21 0.25 0.19 –0.13 0.24 0.18 0.11 0.16 0.25 0.44 0.25 0.31 0.09 0.57 0.45 0.62 0.23
0.22 0.15 0.09 0.37 013 0.20 0.17 0.14 0.17 0.12 0.28 0.17 0.15 0.13 0.18 0.30 0.40 0.25 0.25 0.11 0.46 0.36 0.36 0.24
0.27 0.24 0.16 0.24 0.34 0.29 0.29 0.32 0.24 0.28 0.12 0.30 0.27 0.29 0.18 0.27 0.25 0.28 0.32 0.26 0.40 0.38 0.46 0.27
0.22 0.17 –0.04 0.37 0.23 0.26 0.24 0.20 0.25 0.18 –0.19 0.23 0.18 0.10 0.15 0.25 0.46 0.25 0.31 0.08 0.61 0.48 0.64 0.22
0.22 0.13 0.03 0.40 0.12 0.20 0.16 0.13 0.14 0.11 0.27 0.16 0.14 0.12 0.14 0.31 0.41 0.26 0.25 0.10 0.52 0.40 0.38 0.24
East Fork White River 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119
41 48 24 24 49 49 50 61 24 45 24 44 39 68 23 44 36 50 52 52 21 21 36 87
4254 7368 1435 2532 3860 9109 16581 7383 9495 24610 1931 4024 7869 32714 1066 16194 2049 7238 14863 38615 3128 1335 4690 40569
distributions. Thus, the identification of a parent distribution can be achieved much more easily by using L-moment ratio diagrams than conventional MRDs, especially for skewed distributions. Some useful relationships for constructing the L-moment ratio diagram for some common distribution are given by Hosking (1990 and 1991a,b): 1. 2. 3.
Uniform: τ3 = 0 , τ4 = 0 Exponential: τ3 = 1/3 , τ4 = 1/6 Gumbel (EV1(2)): τ3 = 0.1699 , τ4 = 0.1504
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Logistic: τ3 = 0 , τ4 = 1/6 Normal: τ3 = 0 , τ4 = 0.1226 Generalized Pareto:
4. 5. 6.
τ4 = τ3 (1 + 5τ3)/(5 + τ3) or τ4 = 0.20196 τ3 + 0.95924 τ 3 – 0.20096 τ 3 + 0.04061 τ 3 2
7.
3
4
Generalized Logistic: τ4 = (1 + 5 τ 3 )/6 2
or τ4 = 0.16667 + 0.83333 τ 3 2
8.
Generalized Extreme Value: τ4 = 0.10701 + 0.11090 τ3 + 0.84838 τ 3 – 0.06669 τ 3 2
3
+ 0.00567 τ 3 – 0.04208 τ 3 + 0.03763 τ 3 4
9.
5
6
Gamma and Pearson III: τ4 = 0.1224 + 0.30115 τ 3 + 0.95812 τ 3 – 0.57488 τ 3 + 0.19383 τ 3 2
4
6
8
10. Lognormal (two and three parameters): t4 = 0.12282 + 0.77518 τ 3 + 0.12279 τ 3 – 0.13638 τ 3 + 0.11368 τ 3 2
4
6
8
11. Wakeby lower bound: τ4 = –0.07347 + 0.14443 τ3 + 1.03879 τ 3 – 0.14602 τ 3 + 0.03357 τ 3 2
3
4
12. Overall lower bound: τ4 = –0.25 + 1.25 τ 3 2
3.3 3.3.1
Tests Based on L-Moments Goodness-of-Fit Tests
Hosking and Wallis (1991) give a goodness-of-fit measure based on t 4 , the regional average of the sample L-kurtosis, mainly for three-param-
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eter distributions. Since all three-parameter distributions fitted to the data will have the same t3 on the LCs vs LCk diagram, the quality of fit can be judged by the difference between regional average t 4 and the value of τ4DIST for the fitted distribution. The statistic ZDIST given below: Z
DIST
= ( t 4 – τ4
DIST
)/σ 4
is a goodness-of-fit measure, where σ4 is the standard deviation of t 4 . The value of σ4 can be obtained by simulation after fitting a Kappa distribution to the observations (Hosking, 1988). A fit is declared adequate if ZDIST is sufficiently close to zero, a reasonable criterion being |ZDIST| ≤ 1.64. For small samples (N ≤ 20) or large L-skewness (τ3 ≥ 4) a correction of t 4 is required: instead of using t 4 , t 4 – β4 is used where β4 is the bias in the regional average L-kurtosis for regions with the same number of sites and the same record lengths as the observed data. β4 can also be obtained by simulations required to obtain σ4. The calculations for ZDIST can be made by the FORTRAN computer program “Fortran Routines for Use with the Method of L-moments” developed by Hosking (1991a).
3.3.2
Regional Homogeneity Tests
Hosking and Wallis (1991) give two statistics which are used to test regional homogeneity. The first statistic is a discordancy measure, intended to identify those sites that are grossly discordant with the group as a whole. The discordancy measure D estimates how far a (i) T (i) (i) given site is from the center of the group. If u i = [ t , t 3 , t 4 ] is the vector containing the t, t3 and t4 values for site (i), then the group average for NS sites is given by Eq. 3.3.1. NS
1 u = ------- ∑ u i NS i = 1
(3.3.1)
The sample covariance matrix is given by Eq. 3.3.2. NS
S = ( NS– 1 )
–1
∑ ( ui – u ) ( ui – u )
T
(3.3.2)
i=1
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The discordancy measure is defined by Eq. 3.3.3. T –1 1 D i = --- ( u i – u ) S ( u i – u ) 3
(3.3.3)
A site (i) is declared to be unusual if Di is large. A suitable criterion to classify a station as discordant is that Di should be greater than or equal to 3. The second statistic is a heterogeneity measure, intended to estimate the degree of heterogeneity in a group of sites and to assess whether they might reasonably be treated as homogeneous. Specifically, the heterogeneity measure compares the between-site variations in sample L-moments for the group of sites with that expected for a homogeneous region. Three measures of variability V1, V2, and V3 are available. 1. Based on LCv(t), the weighted standard deviation of (t) is Eq. 3.3.4, NS
V1 =
∑ N i(t
(i)
NS
∑ Ni
2
– t) /
i=1
(3.3.4)
i=1
where, NS in Eq. 3.3.4 is the number of sites, Ni is the record length at each site and t is the average value of t(i) given by Eq. 3.3.5. NS
NS
i=1
i=1
(i) t = ∑ N i t / ∑ N i
2.
(3.3.5)
Based on LCv and LCs, the weighted average distance from the site to the group weighted mean on a t vs. t3 graph is computed. NS
V2 =
∑ N i {( t
(i)
2
(i)
2 1⁄2
– t) +(t 3 – t 3 ) }
NS
/
i=1
3.
∑ Ni
(3.3.6)
i=1
Based on L-skewness (t3) and L-kurtosis (t4), the weighted average distance from the site to the group weighted mean on a t3 vs. t4 graph is computed in Eq. 3.3.7. NS
V3 =
(i)
(i)
∑ N i ( t 3 – t 3) +(t 4 -t 4 ) } i=1
2
2
1⁄2
NS
/
∑ Ni
(3.3.7)
i=1
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To evaluate the heterogeneity measures, a Kappa distribution (Hosking, 1988) is fitted to the group average L-moments 1, t, t 3 , b 4 . Simulations of a large number of regions, Nsim, from this Kappa distribution are performed. The regions are assumed to be homogeneous and the data are assumed to have no cross-correlation or serial correlation. The sites are assumed to have the same record lengths as their realworld counterparts. For each simulated region, Vi (where Vi is any of the three measures V1, V2, or V3 defined above) is calculated. From the simulated data the mean µv and standard deviation σv of the Nsim values of Vi are determined. The heterogeneity measure is defined in Eq. 3.3.8. H i = ( V i – µ v )/ σ v
(3.3.8)
A region is declared to be heterogeneous if Hi is sufficiently large. Hosking and Wallis (1991b) suggest that a region be regarded as acceptably homogeneous if Hi is less than 1, possibly heterogeneous if it is between 1 and 2, and definitely heterogeneous if Hi is greater than 2. Hosking and Wallis (1991) observed that statistics H2 and H3 based on the measures V2 and V3 lack the power to discriminate between homogeneous and heterogeneous regions and that H1 based on V1 had much better discriminating power. Therefore the H1 statistic based on V1 is recommended as a principal indicator of heterogeneity. If a Kappa distribution cannot be fitted ( t 4 is too large relative to t 3 ) , the generalized logistic distribution, a special case of the Kappa distribution, is used for simulation. Also, H1 was found to be a better indicator of heterogeneity in large regions, but has a tendency to give false indications of homogeneity for small regions.
3.4
A Case Study
In order to illustrate the use of concepts and tests discussed above the Wabash river basin data are presented as a case study in regionalization.
3.4.1
Data and Preliminary Analysis
Probability weighted moments as well as L-moments were calculated for each of the stations by using the unbiased estimators in Eqs. 3.1.4
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and 3.1.5. To obtain dimensionless moments, the L-moments are divided at each site by the mean at that site (µi). The L-moment ratios as well as their regional averages are calculated. The LCv – LCs moment ratio diagram for different stations is shown in Figure 3.4.1. The LCs – LCk moment ratio diagram for these stations as well as the theoretical LCs – LCk relationships for several three-parameter distributions are shown in Figure 3.4.2.
3.4.2
Regional Homogeneity
As one might expect for such a large region, the results in Figures 3.4.1 and 3.4.2 suggest that the region as a whole is heterogeneous. Data points are widely scattered in Figures 3.4.1 and 3.4.2. The points on the LCs – LCk diagram (Figure 3.4.2) extend over the region covered by several three-parameter distributions instead of clustering around a particular distribution. This conclusion is also supported by the results obtained from the regional homogeneity tests. The unbiased estimators of PWMs in Eqs. 3.1.4 and 3.1.5 are used in these tests. Values of the site discordancy measure Di, the heterogeneity measure H, and the goodness-of-fit measure (ZDIST) were computed for the whole region by using the FORTRAN computer program developed by Hosking (1991b). Of 93 sites, 12 were found to be discordant with the region as a whole. The heterogeneity measures H1, H2 and H3 computed before and after removing the discordant stations are given in the first row of Table 3.4.1. These results indicate heterogeneity with respect to both LCv values and the average distance within the LCv , LCs diagram (Figure 3.4.1), and possible heterogeneity with respect to average distance within the LCs, LCk diagram (Figure 3.4.2). Hosking (1991b) recommends the use of H1 to check homogeneity since both H2 and H3 can give false indication of homogeneity, especially in regions with a small number of sites. A value of H1 = 5.99 in Table 3.4.1, which is obtained before the discordant sites are removed, indicates a very high degree of heterogeneity. In addition, the value of H2 is 2.69, supporting this conclusion. The situation improves when discordant stations are removed; but H1 is 3.15 even after removing the discordant stations, suggesting that the region is heterogeneous. However, since the region is homogeneous with respect to H3, the parameter estimates may be obtained by using the regional average L-skewness along with the atsite mean and LCv similar to the GEV(2) method of Lettenmaier et al. (1987).
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Figure 3.4.1. river basin.
The LCv – LCs moment ratio diagram for 93 stations in the Wabash
Figure 3.4.2. river basin.
The LCs – LCk moment ratio diagram for 93 stations in the Wabash
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Table 3.4.1. Homogeneity Measures for the Wabash River Basin
Discordant Stations Removed
All stations Region NS ALL UWAB MWAB WFWR EFWR
93 25 15 29 24
H1
H2
H3
NS1
H1
H2
H3
5.99c 5.20c 1.85b 0.17a 3.38c
2.69c 4.03c 0.89a –1.56a 1.79b
1.25b 2.51c 0.10a –2.41a 2.21c
81 24 15 28 22
3.15c 4.97c 1.85b –0.28a 1.24b
0.10a 3.58c 0.89a –2.05a 0.28a
–1.40a 2.06c 0.10a –2.74a 1.10b
a = homogeneous, b = possibly heterogeneous, c = heterogeneous.
The main conclusion from the above analysis is that the Wabash River basin is statistically heterogeneous. For the region as a whole, the first row in Table 3.4.2 gives the goodness-of-fit measure for different distributions. The generalized extreme value distribution (GEV) seems to be the best choice if the region is to be considered as a single unit, although the ZDIST value is quite large for large portions of the basin. The stations were divided into four subregions according to their geographical location: the upper Wabash basin (UWAB, 25 stations), the middle Wabash basin (MWAB, 15 stations), the east fork White River basin (EFWR, 24 stations), and the west fork White River basin (WFWR, 29 stations). These subregions are shown in Figure 2.1.1. In Table 3.4.1 the heterogeneity measures are given for the four subbasins before and after removing the discordant sites. The results in Table 3.4.1 indicate that the upper Wabash basin is heterogeneous and may have to be further subdivided in order to obtain reliable homogeneous regions. Both the middle Wabash and east fork White River basin are possibly heterogeneous. The west fork White River basin, in spite of being the largest (29 stations), is the most homogeneous, with negative values of H indicating less spread than expected in a region of such size. The goodness-of-fit measure ZDIST in Table 3.4.2 indicates that the GEV distribution may be a good regional fit for the west fork White river and upper Wabash regions. The LN(3) distribution is suitable for the west fork White river, while the GLOG seems to be suitable for the middle Wabash river and east fork White river regions. However, the only reliable conclusion applies to the west fork White river region (GEV and LN(3)) since non-homogeneous regions, by definition, do not have a single parent distribution.
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Table 3.4.2. Goodness-of-Fit Measures (ZDIST) for the Wabash River Basin
Region ALL UWAB MWAB WFWR EFWR
GLOG 2.93 1.75 –0.18* 2.04 0.34*
GEV –2.10 –1.24* –2.24 –0.66* –1.76
LN(3) –3.33 –1.72 –2.57 –1.52* –2.51
P(3) –5.87 –2.89 –3.38 –3.18 –3.90
GEV: generalized extreme value; GLOG: generalized logistic; LN(3): three-parameter log normal, P(3): Pearson type III distributions. *The distribution may be accepted as a regional distribution (|ZDIST| < 1.64).
Other methods based on ordinary moments for assessing regional homogeneity have been proposed. The results from the L-moment method were compared to the method proposed by Wiltshire (1986a, b). In Wiltshire’s method, the statistic S is calculated by using the coefficient of variation (Cv) values from different stations within a region (see Section 2.5). This is equivalent to H1 in our analysis. The 2 statistic S is expected to be distributed as χ ( n – 1 ) where n is the number of sites in the region. The results from this method are given in Table 3.4.3. Considering the fact that both methods may give misleading results about homogeneity in small regions, it can be concluded that the results from Wiltshire’s method are in agreement with those from the L-moments analysis. Table 3.4.3. Regional Homogeneity Analysis Using Wiltshire’s Method
Region ALL UWAB MWAB WFWR EFWR
NS 93 25 15 19 24
S χ 0.90 ( n – 1 ) χ 0.95 ( NS – 1 ) 155.97 109.76 115.39 85.27 33.20 36.42 17.57 21.06 23.68 24.42 37.92 41.34 39.88 32.01 35.17 2
2
Comments heterogeneous heterogeneous homogeneous homogeneous heterogeneous
In a study to develop techniques for estimating magnitude and frequency of floods in Indiana streams, the USGS (USGS, 1984) divided the state into the seven regions shown in Figure 3.4.3. USGS used a regression analysis technique to arrive at this classification. Region 6 covers the middle Wabash River and part of upper Wabash regions of the present study. Both the east fork and west fork White River watershed are included in region 3 in Figure 3.4.3. As we have
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Figure 3.4.3.
Homogeneous flood frequency regions in Indiana (USGS 1984).
seen in this study, these regions are not homogeneous. The fact that regression equations developed to estimate flows are quite different for each of these regions also reflects the lack of homogeneity in the USGS classification of watersheds.
3.4.3
Regional Quantile Estimates
Regional estimates are obtained by using the estimation equations given in Section 3.1. For regional estimates, the regional average
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values l 1 , l 2 and t 3 are used to obtain dimensionless regional frequency distributions. For a given return period T, the regional growth factors, qT and qTi, respectively, are calculated and the regional quantile estimates are computed by using Eq. 1.7.1. To study the validity of regional analysis, the west fork White River was selected as a representative homogeneous region, and the upper Wabash region as a representative heterogeneous region. The LCv – LCs and the LCs – LCk moment ratio diagrams for the west fork White River region are shown in Figures 3.4.4 and 3.4.5, respectively. Similar diagrams for the upper Wabash region are shown in Figures 3.4.6 and 3.4.7. Comparing Figures 3.4.4 and 3.4.6 it can be seen that the west fork White River region is more homogeneous in terms of the LCv, LCs diagram. The spread around the mean value in Figure 3.4.4 is much smaller than in Figure 3.4.6. Similarly, the scatter of stations around the mean in Figure 3.4.5 is less than in Figure 3.4.7. Thus, the assumption that flows at different stations have the same parent distribution is more acceptable for the WFWR region than for the UWAB region due to the closeness of the distribution parameters for the WFWR basin. The quantile estimates for station 95 in the WFWR basin for different regional distributions are given in Table 3.4.4. The GEV and LN3 distributions give the best regional estimates. Further details of this study are found in Rao and Hamed (1997).
Observed (approx.)
Return Period (years)
Table 3.4.4. Estimated Quantiles for Station 95 in the WFWR Region for Different At-Site and Regional Distributions
10 65,640 20 76,846 50 92,259 100 112,752 150 122,608 200 127,536
At-site GEV
LN(3)
64,879 77,448+ 94,855 108,795– 117,298– 123,497–
65,280– 77,586 94,150+ 107,068 114,816 120,407
Regional P(3)
GLOG
GEV
LN(3)
65,789+ 63,278 66,368+ 66,848 77,545 76,242– 79,912+ 80,109 98,836 92,613+ 96,122 98,063+ 103,804 113,853+ 114,123+ 112,138– 110,297 125,517+ 123,504+ 120,606– 114,887 134,441+ 130,368+ 126,729–
P(3)
GLOG
67,443 80,082 96,334+ 108,433 115,463 120,437
64,676– 78,598+ 100,076 119,348 132,074 141,836
Underlined are closest to observed; + and – indicate over or under estimated, respectively.
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Figure 3.4.4.
The LCv – LCs moment ratio diagram for the WEST region.
Figure 3.4.5. The LCs – LCk moment ratio diagram for the WEST region.
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Figure 3.4.6.
The LCv – LCs moment ratio diagram for the UPPER region.
Figure 3.4.7. The LCs – LCk moment ratio diagram for the UPPER region.
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CHAPTER 4
Parameter and Quantile Estimation
4.1
Introduction
After a distribution or a number of distributions are selected to fit the data, their parameters must be estimated. Some of the common methods of parameter estimation are discussed in Section 4.2. The estimated parameters are used to calculate quantile estimates for different return periods or, conversely, to calculate the return period for a given flood magnitude. This is achieved by using the distribution function, in which the parameters of the distribution are replaced by their estimates and the relationship between return periods (T ) and probability of non-exceedence (F ) in the form F = 1 – 1/T is used. There are two types of error associated with quantile estimates. The first type arises from the assumption that the observed data follow a particular distribution. This error can be checked by goodness-of-fit tests as discussed in previous chapters (Sections 2.4 and 3.3). The second type is the error inherent in parameters estimated from small samples. This error can be reduced by using a method which gives minimum variance parameter estimates, which in turn would result in the smallest variance in quantile estimates. It is then possible to construct confidence intervals for the estimated quantiles by using information about the sampling variance of the parameter estimates (Section 4.4).
4.2
Parameter Estimation
A number of methods can be used for parameter estimation. These include the method of moments (MOM), the maximum likelihood method (MLM), the probability weighted moments method (PWM), the least squares method (LS), maximum entropy (ENT), mixed moments (MIX), the generalized method of moments (GMM), and incomplete means method (ICM). Three of the more commonly used methods are
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considered here, namely, the method of moments (MOM), the maximum likelihood method (MLM) and the probability weighted moments method (PWM). The maximum likelihood method (MLM) is considered the most efficient method since it provides the smallest sampling variance of the estimated parameters, and hence of the estimated quantiles, compared to other methods. However, for some particular cases, such as the Pearson type III distribution, the optimality of the ML method is only asymptotic and small sample estimates may lead to estimates of inferior quality (Bobeé and Ashkar, 1991). Also, the ML method has the disadvantage of frequently giving biased estimates, but these biases can be corrected. Furthermore, it may not be possible to get ML estimates with small samples, especially if the number of parameters is large. The ML method requires higher computational efforts, but with the increased use of high-speed personal computers, this is no longer a significant problem. The method of moments (MOM) is a natural and relatively easy parameter estimation method. However, MOM estimates are usually inferior in quality and generally are not as efficient as the ML estimates, especially for distributions with large number of parameters (three or more), because higher order moments are more likely to be highly biased in relatively small samples. The PWM method (Greenwood et al., 1979; Hosking, 1986a) gives parameter estimates comparable to the ML estimates, yet in some cases the estimation procedures are much less complicated and the computations are simpler. Parameter estimates from small samples using PWM are sometimes more accurate than the ML estimates (Landwehr et al., 1979c). Also, in some cases, such as the symmetric lambda and Weibull distributions, explicit expressions for the parameters can be obtained by using PWM, which is not the case with the ML or MOM methods. Kebaili-Bergaoui (1994) showed that the ML and ME methods of parameter estimation for Weibull, P(3), Galton, and Gumbel distributions are a particular case of generalized method of moments.
4.2.1 Method of Moments (MOM) Estimates of the parameters of a probability distribution function are obtained in the MOM by equating the moments of the sample with the moments of the probability distribution function. For a distribution with k parameters, α1, α2, …, αk which are to be estimated, the first
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k sample moments are set equal to the corresponding population moments that are given in terms of unknown parameters. These k equations are then solved simultaneously for the unknown parameters α1, α2, …, αk. 4.2.2. Method of Maximum Likelihood (MLM) Estimation by the ML method involves the choice of parameter estimates that produce a maximum probability of occurrence of the observations. For a distribution with a probability density function (pdf) given by f(x) and parameters α1, α2, …,αk, the likelihood function is defined as the joint pdf of the observations conditional on given values of the parameters α1, α2, …, αk in the form: n
L ( α 1 , α 2 , …, α k ) =
∏ f ( x i ; α1 , α2 , …, αk )
(4.2.1)
i=1
The values of α1, α2, …, αk that maximize the likelihood function are computed by partial differentiation with respect to α1, α2, …,αk and setting these partial derivatives equal to zero as in Eq. 4.2.2. The resulting set of equations are then solved simultaneously to obtain the values of α1, α2, …,αk, ∂ L ( α 1 , α 2 , …, α k ) ----------------------------------------------- = 0; ∂ αi
i = 1, 2, …, k
(4.2.2)
In many cases it is easier to maximize the natural logarithm of the likelihood function by using ∂ ln L ( α 1 , α 2 , …, α k ) ------------------------------------------------------ = 0; ∂ αi
i = 1, 2, …, k
(4.2.3)
4.2.3 Method of Probability Weighted Moments (PWM) Parameter estimates are obtained in this method, as in the case of MOM, by equating moments of the distributions with the corresponding sample moments. For a distribution with k parameters, φ1, φ2, …, φk, which are to be estimated, the first k sample moments are set equal
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to the corresponding population moments. The resulting equations are then solved simultaneously for the unknown parameters φ1, φ2, …, φk. Details of the estimation procedures for each of the distributions considered in this book by using the above three estimation methods are given in subsequent chapters.
4.3
Quantile Estimation
After the parameters of a distribution are estimated, quantile estimates (xT) which correspond to different return periods may be computed. As discussed before, the return period is related to the probability of non-exceedence (F ) by the relation, F = 1 – 1/T
(4.3.1)
where F = F(xT) is the probability of having a flood of magnitude xT or smaller. The problem thus reduces to evaluating xT for a given value of F. In practice, two types of distribution functions are encountered. The first type is that which can be expressed in the inverse form x T = φ ( F ) . In this case, xT is evaluated in a straight forward manner by replacing F by its value from Eq. 4.3.1. Examples of this type are the extreme value, Wakeby and logistic distributions. The second type of distribution cannot be expressed directly in the inverse from x T = φ ( F ) . In this case numerical methods are used to evaluate xT corresponding to a given value of F. This is done either by using numerical relationships between F and xT or by using tables which are prepared for this purpose. Chow (1964) proposed a general form for calculating xT as in Eq. 4.3.2, x T = u 1′ + K T µ 2
(4.3.2)
In Eq. 4.3.2, KT is the frequency factor which is a function of the return period and of the parameters of the distribution. Expressions for KT can be derived by using the same technique for the two types of distributions. In fact, from Eq. 4.3.2, it can be seen that KT is just the variable xT standardized by using the moments u 1′ and µ 2 of the distribution. It should be noted, however, that in Eq. 4.3.3, u 1′ and µ2, the distribution moments, are calculated by using their relationship with the estimated parameters and will be equal to the sample moments only when the MOM is used for parameter estimation.
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x T – µ 1′ K T = ----------------µ2
4.4
(4.3.3)
Confidence Intervals
It is clear that a point estimate of a certain quantile corresponding to a return period may be of no real significance unless there is an indication of the accuracy of the estimate. There have been several studies related to prediction accuracy. Normal and EV1(2) distributions were used by Nash and Amorocho (1966) to derive expressions for standard errors of sample estimates of flood magnitudes corresponding to specified return periods. These expressions are functions of sample size and parameters of normal and EV1(2) distributions. Recurrence intervals between exceedences of river levels were investigated by McGilchrist et al. (1969) by using exponential and gamma distributions. The variability of return periods in the presence of persistence was investigated by Lloyd (1970). The extreme value theory was extended to dependent observations by Tawn (1988). Risk analysis was extended to time dependent flood models by Nachtnabel and Konecny (1990). Return periods corresponding to mean annual flows were examined by Rao (1981). Victorov (1971) demonstrated that the effect of period of record on flood prediction can be very strong in the sense that floods estimated by using different 10-year records varied as much as 595%. A measure of the variability of the estimated value is the standard error of estimate sT which is defined as (Cunnane, 1989) in Eq. 4.4.1, sT =
E { xˆ T – E ( xˆ T ) }
2
(4.4.1)
The standard error of estimate accounts for the error due to small samples, but not the error due to the choice of inappropriate distribution. The standard error of estimate depends in general on the method of parameter estimation. Consequently, each method gives a different standard error of estimate. As mentioned before, the most efficient method is that which gives the smallest standard error of estimate. It can be shown by using the asymptotic theory that the distribution of 2 xˆ T is asymptotically normal with mean x T and variance s T as n → ∞ . Thus an approximate (1 – α) confidence interval for xT is given by xˆ T + t a/2 s T
(4.4.2)
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where t is the standard normal variate. Hebson and Cunnane (1987) found that xˆ T has a skewed distribution when it is estimated from small samples and by at-site estimation methods, but that they are very close to normal when estimated by combined at-site/regional methods. According to Cunnane (1989), guidelines for flood frequency analysis in the United States (USWRC, 1981) contain methods for computing confidence intervals for skewed distributions using noncentral t-distribution. Some of the earlier work related to confidence intervals has been summarized in Yevjevich (1964). Gladwell and Lin (1969) developed a graphical procedure to estimate confidence limits. This procedure is based on the known theoretical distribution of ordered data. Shane and Lynn (1969) assumed the number of flood peak occurrences to be Poisson distributed and the magnitudes to be exponentially distributed. Confidence limits are related to the measures of risk. Bobeé and Morin (1973) used the distribution functions of the order statistics for the Pearson (3) distribution to derive the confidence intervals associated with it. Kite (1975) used data generation experiments to derive distributions of extreme events generated from probability distributions commonly used in hydrology. These distributions were shown to be statistically indistinguishable from the normal distribution. By using this characteristic, confidence limits, which are based on mean and standard deviation of extreme event distribution are derived by Kite (1975). Hoshi and Burges (1981a) have given an approximation technique to compute the derivative of a standard gamma quantile with respect to the distribution shape parameter b. This derivative is needed to estimate the sampling variance of a specified quantile.
4.4.1 Standard Error in the MOM The standard error of estimate for a given three-parameter distribution with parameters α, β, and γ can be calculated by using the formula ∂x 2 ∂x 2 ∂x 2 2 s T = ------- var α + ------ var β + ------ var γ ∂α ∂β ∂γ ∂x ∂x ∂x ∂x + 2 ------- ------ cov ( α, β ) + 2 ------- ------ cov ( α, γ ) ∂α ∂β ∂α ∂γ
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∂x ∂x + 2 ------ ------ cov ( β,γ ) ∂β ∂γ
(4.4.3)
where the partial derivatives can be calculated from the relation x T = φ ( F ) or x T = µ 1′ +K T µ 2 as explained before. The variances and covariances of parameters are calculated by using the relationship between the parameters and the moments of the distribution µ 1′ , µ 2 and µ 3 , which are estimated by the sample moments m 1′ , m 2 and m 3 . For a given parameter, say α = α ( m 1′ , m 2 , m 3′ ) , we have ∂α 2 ∂α 2 ∂α 2 var α = ----------- var m 1′ + --------- var m 2 + --------- var m 3 ∂m 1′ ∂m 2 ∂m 3 ∂α ∂α ∂α ∂α + 2 ----------- --------- cov ( m 1′ , m 2 )+2 ----------- --------- cov ( m 1 , m 2 ) ∂m 1′ ∂m 2 ∂m 1′ ∂m 3 ∂α ∂α +2 --------- --------- cov ( m 1 , m 3 ) ∂m 2 ∂m 3
(4.4.4)
where the partial derivatives are obtained from the relationships between α with m 1′ , m 2 , m 3 used in parameter estimation. The variances and covariances of the moments m 1′ , m 2 , m 3 are given by Kendall and Stewart (1963) as in Eqs. 4.4.5 to 4.4.10. var m 1′ = µ 2 /N
(4.4.5)
1 2 var m 2 = ---- [ µ 4 – µ 2 ] N
(4.4.6)
1 2 3 var m 3 = ---- [ µ ( 6 – µ 3 – 6µ 4 µ 2 + 9 µ 2 ] N
(4.4.7)
cov ( m 1′ , m 2 ) = µ 3 /N
(4.4.8)
1 2 cov ( m 1′ , m 3 ) = ---- [ µ 4 – 3µ 2 ] N
(4.4.9)
1 cov ( m 2 , m 3 ) = ---- [ µ 5 – 4µ 3 µ 2 ] N
(4.4.10)
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Alternatively (Kite, 1977), by substituting the above relationships into Eq. 4.4.3 we get a simplified expression for evaluating the standard error as in Eq. 4.4.11. 2
K ∂K µ 2 2 s T = -----2 {1+ K T γ 1 + ------T [ γ 2 – 1 ]+ ---------T- MP[2γ 2 – 3γ 1 – 6+ K T ( γ 3 – 6γ 1 γ 2 ⁄ 4 – 10γ 1 ⁄ 4 )] N 4 ∂γ 1 ∂K 2 2 2 + ---------T- [γ 4 – 3γ 3 γ 1 – 6γ 2 + 9γ 1 γ 2 /4+ 35γ 1 /4 +9 ]} ∂γ 1
(4.4.11)
where γ 1 = µ 3 /µ 2
(4.4.12)
γ 2 = µ 4 /µ 2 2
(4.4.13)
γ 3 = µ 5 /µ 2
(4.4.14)
γ 4 = µ 6 /µ 2
(4.4.15)
3/2
5/2
3
and KT is the frequency factor. The expression for sT in Eq. 4.4.16 sT = δ
µ -----2 n
(4.4.16)
may be used where δ does not depend on µ2 or n. If the frequency factor K T does not depend on γ 1 (two-parameter distributions), then ( ∂K T ) ⁄ ( ∂γ 1 ) = 0 and the expression for δ simplifies to Eq. 4.4.17. 1/2
2 K δ = 1 + K T γ 1 + ------T [ γ 2 – 1 ] 4
(4.4.17)
4.4.2 Standard Error in the MLM For a particular distribution, when the parameters α, β, and γ are estimated by the MLM, the T-year quantile is given by Eq. 4.4.18.
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xˆ T = f ( α, β, γ , T )
(4.4.18)
In Eq. 4.4.18, T is constant for a given xT . The standard error in this case is given by Eq. 4.4.19 ∂x 2 ∂x 2 ∂x 2 2 s T = ------- var α + ------ var β + ------ var γ ∂α ∂β ∂γ ∂x + 2 ------- ∂α
∂x ----- cov ( α,β )+ 2 ∂β
∂x + 2 ------ ∂β
∂x ----- cov ( β,γ ) ∂γ
∂x ----- ∂α
∂x ----- cov ( α,γ ) ∂γ (4.4.19)
The partial derivatives in Eq. 4.4.19 can be calculated from the relation x T = φ ( F ) or xT = µ′ 1 + K T µ 2 . Kendall and Stewart (1967) have shown that the variance-covariance matrix in Eq. 4.4.20 is the inverse of the symmetric matrix in Eq. 4.4.21 (the expected information matrix) where L is the likelihood function. var α cov ( α,β ) cov ( α,γ ) var β cov ( β,γ ) var γ ∂ log L – ---------------2 ∂α 2
(4.4.20)
2 ∂ log L ∂ log L – ----------------- – ----------------∂α∂β ∂α∂γ 2
2 ∂ log L ∂ log L – ---------------- ----------------2 ∂β∂γ ∂β 2
E
(4.4.21)
∂ log L – ---------------2 ∂γ 2
where E represents the expected value.
4.4.3 Standard Error in the PWM The standard error of estimate may be calculated by Eq. 4.4.22 when the parameter set (α, β, γ) is estimated by PWM method
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∂x 2 ∂x 2 ∂x 2 ∂x ∂x 2 s T = ------- varα + ------ varβ + ------ varγ + 2 ------- ------ cov ( γ ,β ) ∂α ∂β ∂β ∂α ∂β ∂x ∂x + 2 ------- ------ cov ( α,γ ) + 2 ∂α ∂γ
∂x ∂x ----- ------ cov ( β,γ ) ∂β ∂γ
(4.4.22)
The partial derivatives in Eq. 4.4.22 are obtained from the inverse form x = x (F) for distributions which are expressible in this form. For other distributions, approximate values of these partial derivatives are obtained by using the frequency factor relationship x = µ1 + K T µ2 . The covariance matrix of α, β, and γ are obtained by using the covariance matrix for the PWMs βr or αr , where r takes the values 1, T 2, and 3. Hosking (1986a) shows the vector b = ( b 0, b 1 ,b 2 ) has asympT totically a multivariate Normal distribution with mean β = ( β 1 ,β 2 ,β 3 ) and covariance matrix N–1V. The elements of V = ( v ij ) are obtained by Eqs. 4.4.23 to 4.4.25. v rr = 2
∫ ∫ {F ( x)}
r
{ F ( y ) } F ( x ) { 1 – F ( y ) } dx dy r
(4.4.23)
x
1 v rs = lim N cov ( b r ,b s ) = --- ( g rs + g sr ) 2
(4.4.24)
N→∞
g rs = 2
∫ ∫ {F ( x)}
r
{ F ( y ) } ⋅ F ( x ) { 1 – F ( y ) } dx dy s
(4.4.25)
x
ˆ For a set of parameters Θ = (α, β, γ)T, let their PWM estimates be Θ T ˆ = f ( b ) be the relationship between the parameters = ( αˆ βˆ γˆ ) and let Θ and the PWMs br . The matrix G = (gij) where g ij = ∂ f i / ∂b j is defined. ˆ has a multivariate Normal distribution with Then, asymptotically, Θ mean f(β) = θ and covariance matrix n–1 GVGT. The expressions for evaluating sT in the case of PWM estimates are complicated and are available only for some of the distributions discussed in the following chapters.
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CHAPTER 5
Normal and Related Distributions
5.1
Normal Distribution
5.1.1 Introduction The normal distribution is used in frequency analysis for fitting empirical distributions to hydrological data, and in simulation of data. As many statistical parameters are approximately normally distributed, the normal distribution is often used for statistical inferences. An early application of the normal distribution to hydrologic variables was by Hazen (1914), who introduced the normal probability paper for analysis of hydrologic data. Markovi c´ (1965) found that the normal distribution could be used to fit the distribution of annual rainfall and runoff data. Slack et al. (1975) demonstrated that, in the absence of information about the distribution of floods and economic losses associated with the design of flood reduction measures, the use of the normal distribution is better than other distributions such as extreme value, lognormal or Weibull. The probability density function of a normally distributed variable x is given by 1 f ( x ) = ----------------e σ 2π
1 2 – --------2- ( x – µ ) 2σ
(5.1.1)
where µ and σ are the parameters of the distribution. The variable x can take any value in the range (– ∞, ∞). The standard normal variate u is a normal variable with a mean equal to zero and standard deviation equal to one. The probability density function of u is given by 1 –u2 /2 f ( u ) = ------------e 2π
(5.1.2)
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which can be numerically approximated by Eq. 5.1.3 (Abramowitz and Stegun, 1965) 10 – 1
f ( u ) = ( b 0 + b 2 u + b 4 u + b 8 u + b 10 u ) + ε ( u ) 2
4
8
(5.1.3)
In Eq. 5.1.3, 0 ≤ u < ∞ and b0 = 2.5052367b6
= 0.1306469
b2 = 1.2831204b8
= – 0.0202490
b4 = 0.2264718b10
= 0.0039132
The error ε(u) is less than 2.3 × 10–4. f(u) is an even function so that f(u) = f(–u). The cumulative distribution F(u) which is the area under the probability density function is given by, u
F (u) =
1 –t 2 /2 ---------- e dt 2π –∞
∫
(5.1.4)
The distribution function can be numerically approximated (Abramowitz and Stegun, 1965) as in Eq. 5.1.5. F ( u ) = 1 – f ( u ) ( b 1 q + b2 q + b3 q + b4 q + b5 q ) + ε ( u ) 2
3
4
5
(5.1.5)
In Eq. 5.1.5 1 q = ---------------- , p = 0.2316419 and 0 ≤ u < ∞ 1 + pu b1 = 0.319381530 b2 = –0.356563782 b3 = 1.781477937 b4 = –1.821255978 b5 = 1.330274429 The error ε(u) in Eq. 5.1.5 is less than 7.5 × 10–8, and F(–u) = 1 – F(u).
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5.1.2 Parameter Estimation Method of Moments: Since there are two parameters to be evaluated (α1 = µ and α2 = σ), the first two moments of the normal distribution are calculated as follows:
1 f ( x ) = -------------------- e α 2 2π ˙
∞
1 u′ 1 = x . -------------------- e α 2 2π –∞
∫
2 1 – --------2- ( x – α 1 ) 2α 2
2 1 – --------2- ( x – α 1 ) 2α 2
(5.1.6)
(5.1.7)
dx
x–α substitute z = --------------1 in Eq. 5.1.7. Then x = zα 2 + α 1, dx = α 2 dz α2 ∞
–z
2
1 ------2 µ′ 1 = ( zα 2 + α 1 ) ----------e dz 2π –∞
∫
∞
∞
∫
∫
α1 α 2 –z2 ⁄ 2 µ′ 1 = ---------ze dz + ---------2π 2π –∞
(5.1.8)
2
e
–z ⁄ 2
dz
(5.1.9)
–∞
The first integral is equal to zero while the second integral is equal to 2π α1 . ∴ µ′ 1 = ---------2π
2π = α 1
(5.1.10)
The second moment is computed in a simpler form if central moments are considered, ∞
1 µ 2 = ( x – µ′ 1 ) ------------------e α 2 2π –∞
∫
2
1 2 – --------2- ( x – α 1 ) 2α 2
dx
(5.1.11)
But from 5.1.10, µ′ 1 = α 1
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–1 2 --------- ( x – α 1 )
∞
2 2α 1 ∴ µ 2 = ( x – α 1 ) ------------------e 2 α 2 2π –∞
∫
2
dx
(5.1.12)
–1 ⁄ 2
Substituting y = ( x – α 1 ) / 2α 2 and dx = ( α 2 / 2 )y dy into Eq. 5.1.12 after changing limits of integration, and noting that y is an even function and that the integral is double that from zero to infinity we get: 2
2
2
2α µ 2 = ---------2 π
∞
∫y
1 ⁄ 2 –y
(5.1.13)
e dy
0
1 π The integral in Eq. 5.1.13 is equal to Γ ( 3 ⁄ 2 ) = --- Γ ( 1 ⁄ 2 ) = ------2 2 2
2α π 2 ∴ µ 2 = ---------2 ------- = α 2 π 2
(5.1.14)
From Eqs. 5.1.10 and 5.1.14 and replacing µ′ 1 and µ 2 by their sample estimates m′ 1 and m 2 the parameter estimates by the method of moments are αˆ 1 = m′ 1
(5.1.10a)
2 αˆ 2 = m 2
(5.1.14a)
and
Maximum Likelihood (ML) Method: The likelihood function of a sample of size N from a normal distribution is given by L(α1, α2). N
–1 --------n 2α 22
1 L ( α 1 , α 2 ) = ------------------ e α 2π 2
∑(x – α ) i
2
1
i=1
(5.1.15)
Taking the natural logarithm of Eq. 5.1.15 we get
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log L ( α 1 , α 2 ) = – N log α 2 – N log
1 2π – ---------2 2α 2
N
∑ ( x i – α1 ) . (5.1.16) 2
i=1
To obtain the ML estimates the derivatives of Eq. 5.1.16 are set equal to zero. ∂ logL ( α 1 , α 2 ) 1 ------------------------------------ = ---------2 ∂α 1 2α 2
N
∑ 2 ( xi – α1 )
= 0
(5.1.17)
i=1
Equation 5.1.17 gives 5.1.18. 1 αˆ 1 = ---N
N
∑ xi
(5.1.18)
= m′ 1
i=1
Similarly, for the parameter α2, ∂ log L ( α 1 , α 2 ) 2 N -------------------------------------- = – ----- + ---------3 ∂α 2 α 2 2α 2
1 2 ∴ αˆ 2 = ---N
N
∑ ( xi – α1 ) = 0 2
(5.1.19)
i=1
N
∑ ( xi – α1 )
2
= m2
(5.1.20)
i=1
From 5.1.19 and 5.1.20 we see that α1 and α 2 can be estimated by the sample mean and variance. Consequently, for the normal distribution, both the method of moments and the maximum likelihood method give the same results. 2
PWM Method: Since the normal distribution function cannot be explicitly expressed in terms of x, the evaluation of the probability weighted moments becomes a complicated procedure. However, the resulting expressions are simple. Hosking (1990) gives the following properties of the normal distribution λ1 = β0 = α1
(5.1.21)
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α λ 2 = 2β 1 – β 0 = ------2π
(5.1.22)
λ τ 3 = -----3 = 0 λ2
(5.1.23)
λ 30 –1 ( 2 – 9) τ 4 = -----4 = ------ tan π λ2
= 0.1226
(5.1.24)
The PWM parameter estimates are then given in terms of sample moments l1 and l2 as αˆ 1 = l 1 αˆ 2 =
π l2
(5.1.25)
(5.1.26)
EXAMPLE 5.1.1 Estimate the parameters of a normal distribution to be fitted to the annual maximum flow data of the Tippecanoe River near Delphi given in Table 5.1.1. Use all the three methods discussed above.
Method of Moments: From Table 2.1.2
n = 48, m′ 1 = 12665 , Cv = 0.3719, Cs = 0.1194 The observations have a small skew coefficient which suggests that the Normal distribution may be suitable for frequency analysis at this station. From Eqs. 5.1.10 and 5.1.14 we get
αˆ 1 = m′ 1 = 12665
(1)
1⁄2 αˆ 2 = m 2 = C v µ′1 = 4710
(2)
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Table 5.1.1 Tippecanoe River Near Delphi, Indiana (Station 43) Data
Year
Ann. Max. Flow
1940 1941 1942 1943 1944 1945 1946 1947 1948 1949 1950 1951 1952 1953
6290 2700 13100 16900 14600 9600 7740 8490 8130 12000 17200 15000 12400 6960
Year 1954 1955 1956 1957 1958 1959 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970
Ann. Max. Flow 6500 5840 10400 18800 21400 22600 14200 11000 12800 15700 4740 6950 11800 12100 20600 14600 14600
Year 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987
Ann. Max. Flow 8900 10600 14200 14100 14100 12500 7530 13400 17600 13400 19200 16900 15500 14500 21900 10400 7460
ML Method: It has been shown that the ML solution is identical to the MOM solution. The parameter estimates, quantile estimates, and standard errors are the same as those obtained by MOM.
PWM Method: From Table 3.1.1 for the given data (station no. 43) we have
n = 48, l1 = 12665, t = 0.215 From Eqs. 5.1.25 and 5.1.26 we get
αˆ 1 = l i = 12665 αˆ 2 =
π ( 0.215 × 12665 ) = 4826
The parameter estimates are summarized in Table 5.1.2.
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Table 5.1.2
Parameter Estimates for Example 5.1.1
Parameter MOM MLM PWM
α1 12665 12665 12665
α2 4710 4710 4826
5.1.3 Quantile Estimates For a given return period T, the corresponding probability of nonexceedence is F = 1 – 1/T. It is easy to calculate the standard normal variate u corresponding to a probability F of non-exceedence. Abramowitz and Stegun (1965) give this value of u as, 2
C0 + C1W + C2W u = W – ----------------------------------------------------------3 + ε ( P ) 2 1 + d1W + d2W + d3W
(5.1.27)
where C0 = 2.515517
d1 = 1.432788
C1 = 0.802853
d2 = 0.189269
C2 = 0.010328
d3 = 0.001308 W =
– 2 log ( P ) for P < 0.5
(5.1.28)
where P = 1 – F, the probability of exceedence, and the error ε(P) is less than 4.5 × 10–4. For P > 0.5, P is replaced by 1 – P and the required u is the computed u but with an opposite sign. The required quantile estimate is then calculated as xˆ T = αˆ 1 + uαˆ 2
(5.1.29)
which is in the form of Eq. 4.3.2 in which the frequency for the KT is replaced with the standard normal variate u.
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EXAMPLE 5.1.2 Estimate the 100-year-flood for the Tippecanoe River data by using the parameters estimated in Example 5.1.1.
Quantile Estimates Based on MOM and MLM Parameters: From Eq. 5.1.29 we get xˆ T = 12665 + 4710 u where u depends on the return period T. For example if T = 100 then, probability of non-exceedence F = 1 – 1/T = 1 – 0.01 = 0.99, probability of exceedence P = 1 – F = 1 – 0.99 = 0.01. From Eq. 5.1.28 w =
– 2 log ( 0.01 ) = 3.034854
From Eq. 5.1.27 u = 2.326785 , ∴xˆ 100 = 12665 + 4710 × 2.326785 cfs. ∴xˆ 100 = 23624 cfs. Quantile Estimates Based on PWM Parameters: From Eq. 5.1.29 we get xˆ t = 12665 + 4826 u For T = 100 we have u = 2.326785. xˆ 100 = 12665 + 4826 × 2.326785 cfs. xˆ 100 = 23894 cfs.
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5.1.4 Standard Error: Method of Moments: Following Kite (1977), it is noted in Eq. 4.4.11 that for a two-parameter distribution we have ∂K T ---------- = 0 ∂γ 1 and 1⁄2
2 KT δ = 1 + γ 1 K T + [ γ 2 – 1 ] -------4
(5.1.30)
But for the normal distribution we also have: µ2 = α2
(5.1.31)
µ3 = 0 , γ 1 = 0
(5.1.32)
µ 4 = 3α 2 , γ 2 = 3
(5.1.33)
2
4
Furthermore, for the normal distribution KT is equal to u where u is the standard normal variate. Substituting Eqs. 5.1.31 to 5.1.33 in Eq. 5.1.30 we get, 2 1⁄2
u δ = 1 + ----2
(5.1.34)
and the standard error sT is sT = [ 1 + u / 2 ] 2
1⁄2
α ------2n
(5.1.35)
ML Method: From Kite (1977) we have
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var ( α 1 ) = α 2 /n
(5.1.36)
var ( α 2 ) = 2 α 2 /n
(5.1.37)
cov ( α 1 , α 2 ) = 0
(5.1.38)
2
2
4
2
We have X = α1 + u α2 . 2
Thus ∂X --------- = 1 ∂α 1
(5.1.39)
1 ∂X ---------2 = u --------2α 2 ∂α 2
(5.1.40)
sT is estimated from Eq. 4.4.19, which in this case reduces to Eq. 5.1.41. ∂X 2 ∂x 2 2 2 s T = --------- var ( α 1 ) + ---------2 var ( α 2 ) ∂α 1 ∂α 2 ∂x +2 --------- ∂α 1
∂x -------- cov ( α 1 , α 22 ) ∂α 2
(5.1.41)
2
Substituting Eq. 5.1.36 to 5.1.40 into 5.1.41 we get Eq. 5.1.42. 2 α α 2 1 2 s T = -----2 + u ---------2 ⋅ 2 -----2 n n 4α 2
(5.1.42)
2 u α 2 ∴ s T = 1 + ----- -----2 2 n
(5.1.43)
4
2
Equation 5.1.43 is the same as Eq. 5.1.35 which was obtained for the MOM.
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PWM Method: According to Hosking (1986a), the variances and covariances for the parameters α1 and α2 are, α var (α1) = -----2 n 2
(5.1.44)
α var (α2) = 0.5113 -----2 n
(5.1.45)
cov (α 1, α2) = 0
(5.1.46)
∂x --------- = 1 ∂α 1
(5.1.47)
∂x --------- = u ∂α 2
(5.1.48)
2
We have from Eq. 5.1.29,
The standard error is given by Eq. 4.4.22 which in this case reduces to Eq. 5.1.49 ∂x 2 ∂x ∂x 2 ∂x 2 s T = --------- var ( α 1 ) + --------- var ( α 2 ) + 2 --------- --------- cov ( α 1 , α 2 ) (5.1.49) ∂α 1 ∂α 2 ∂α 1 ∂α 2 Substituting Eqs. 5.1.44 to 5.1.48 in Eq. 5.1.49 gives: 2 α 2 s T = ( 1 + 0.5113u ) -----2 n 2
(5.1.50)
The standard error for the PWM method is slightly greater than that obtained by either the MOM or the ML method.
EXAMPLE 5.1.3 Estimate the standard errors for floods of different recurrence intervals for the data used in Examples 5.1.1 and 5.1.2. Summarize the results.
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Standard Error Based on MOM and MLM Parameters: From Eq. 5.1.35 we get,
s T = [ 1 + u /2 ] 2
1⁄2
4710 ------------48
For example, if T = 100 then u = 2.326785 and
s T = [ 1 + 2.326785 /2 ] 2
1⁄2
4710 ------------- = 1309 cfs 48
Standard Error Based on PWM Method Parameters From Eq. 5.1.50,
4826 sT = [1 + 0.5113 u2]1/2 -----------48 For example if T = 100 then u = 2.326785 and 2 1⁄2
s T = [ 1 + 0.5113 × 2.326785 ]
4826 ------------- = 1352 cfs 48
The quantile estimates and standard errors for different return periods T estimated by using the MOM, ML, and PWM methods are given in Table 5.1.3. These results are obtained by the computer programs discussed in Chapter 10. In Figure 5.1.1, observed and estimated flows are shown, as well as the 95% confidence interval for the estimates, given by the MO, ML, and PW methods. Table 5.1.3 Quantile Estimates and their Standard Errors (in parentheses) for Example 5.1.3
T 10 20 50 100 200
P (%) 10 5 2 1 0.5
MOM 18701 (917) 20412 (1043) 22338 (1199) 23622 (1308) 24797 (1412)
MLM 18701 (917) 20412 (1043) 22338 (1199) 23622 (1308) 24797 (1412)
PWM 18849 (945) 206039 (1075) 22576 (1237) 23891 (1352) 25095 (1460)
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Figure 5.1.1. Observed and estimated flows and 95% confidence intervals for Tippecanoe river data used in Examples 5.1.1 to 5.1.3.
5.2
Two-Parameter Lognormal (LN(2)) Distribution
5.2.1 Introduction The probability density function of a logarithmic normally distributed variable x with two parameters (LN(2)) is given by
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[ – log x – u y ] 2 1 - f ( x ) = --------------------- exp ------------------------------2 2σ y xσ y 2π
(5.2.1)
where µy and σy are the mean and standard deviation of the natural logarithms of x. If the variable log(x) is standardized as in Eq. 5.2.2, log ( x ) – µ u = ---------------------------y σy
(5.2.2)
the standard normal variate u is obtained with pdf given in Eq. 5.1.2.
5.2.2 Parameter Estimation Method of Moments: Chow (1954, 1959) worked extensively with the lognormal distribution. He developed a graphical method to determine the frequency factor in LN(2) distribution. The first two moments of the LN(2) distribution are given by Kite (1977) as Eqs. 5.2.3 and 5.2.4. µ′ 1 = e σ
2
µy + σy / 2
2
µ 2 = ( e y – 1 ) ⋅ ( µ′ 1 )
(5.2.3) 2
(5.2.4)
Taking natural logarithms of Eqs. 5.2.3 and 5.2.4 we get Eqs. 5.2.5 and 5.2.6. log µ′ 1 = µ y + σ y / 2 2
(5.2.5) σ
2
log µ 2 = 2 log µ′ 1 + log ( e y – 1 )
(5.2.6)
Solving Eqs. 5.2.5 and 5.2.6 and replacing µ′ 1 and µ 2 by their sample estimates m′ 1 and m2 we get Eqs. 5.2.7 and 5.2.8. m2 2 σˆ y = log -------------- + 1 ( m′ ) 2 1
(5.2.7)
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2 µˆ y = log m′ 1 – σˆ y / 2
(5.2.8)
Equations 5.2.7 and 5.2.8 give the relationships between the param2 eters σ y and µy and the sample mean m′ 1 and sample variance m2. An alternative method is to calculate the variable y = log (x), compute 2 µy and σ y of the transformed variable, and proceed as described in Section 5.1 for the normal distribution, bearing in mind that the variable in this case is log (x) and not x itself. A problem arises when this technique is applied to data which include zeros. In such cases, the solution is either adding 1.0 to all data, or replacing zero data with 1.0, or even ignoring zero observations (Kilmartin and Peterson, 1972). Maximum Likelihood (ML) Method: The likelihood function of a sample of size N from a log-normal distribution is given by Eq. 5.2.9 1 1 1 N ------------------ exp – ---------2 L = ----------N σ 2π 2σ y ∏ xi y
N
∑ [ log x i – µ y ] i=1
2
(5.2.9)
i=1
Taking the natural logarithm of L in Eq. 5.2.9 we get Eq. 5.2.10. N
log L =
∑ log i=1
1 N 1 N ---- – ---- log ( 2π ) – ---- log σ 2y – ---------2 2 xi 2 2σ y
N
∑ [ log x i – µ y ]
2
(5.2.10)
i=1
Differentiating the log likelihood function with respect to µy and σ y and equating to zero gives Eqs. 5.2.11 and 5.2.12.
2
1 µˆ y = ---N
1 2 σˆ y = ---N
N
∑ log x i
= m' 1 y
(5.2.11)
i=1
N
∑ ( log x i – µˆ y )
2
= m2 y
(5.2.12)
i=1
These estimates correspond to the method of moments estimates when the transformed observations y = log(x) are normally distributed.
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PWM Method: Complex manipulations are required to derive the probability weighted moments for the lognormal distribution. The following Eqs. 5.2.13 and 5.2.14 are given by Hosking (1990) where erf (x) is the error function and F(·) is the normal distribution function which can be evaluated by using Eq. 5.1.5. λ1 = e λ2 = e
2
µy + σy / 2
2
µy + σy ⁄ 2
(5.2.13)
erf ( σ y / 2 )
(5.2.14)
where x 2 2 –u erf ( x ) = ------- e du = 2F ( x 2 ) – 1 π 0
∫
(5.2.15)
Equations 5.2.13 and 5.2.14 can be solved for µy and σ y replacing λ1 and λ2 by the sample L-moments l1 and l2 to give: 2
l –1 –1 σˆ y = 2er f ---2 = 2 er f ( t ) l 1
(5.2.16)
2 σˆ µˆ y = log l 1 – -----y 2
(5.2.17)
erf–1(t) is evaluated by using the relationship in Eq. 5.2.15 as u ⁄ 2 , where u is the standard normal variate obtained from Eq. 5.1.27 corresponding to F = ( t + 1 ) /2 .
EXAMPLE 5.2.1 Estimate the parameters of a two-parameter LN distribution to be fitted to the Wabash River annual maximum flow data at Lafayette, IN (Table 1.8.1). Use the three methods discussed above.
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Method of Moments: The direct method is used here. The indirect method which uses the logarithms of flow is identical to the ML method which will be discussed next. From Table 2.1.2 we have
N = 85, m′1 = 52621, cv = 0.4789 From Eqs. 5.2.7 and 5.2.8 we get, 2 σˆ y = { log [ ( 0.4789 ) + 1 ] }
1⁄2
= 0.4544
( 0.4544 ) µˆ y = log ( 52621 ) – ----------------------- = 10.7676 2 2
ML Method: From the data in Table 1.8.1 we can calculate the first two moments of the logarithms of the data to get
m' 1 y = 10.7711 , m 2 y = 0.2081 . From Eqs. 5.2.11 and 5.2.12 we get,
µˆ y = m' 1 y = 10.7711 1⁄2 σˆ y = m 2 y = 0.4562
PWM Method: From Table 3.1.1 for the given station (No. 49) we have, N = 85, l1 = 52621, t = 0.2370. From Eq. 5.2.16 we have, –1 σˆ y = 2er f ( 0.2370 )
0.2370 + 1 F = ------------------------- = 0.6185 2 P = 1 – F = 1 – 0.6185 = 0.3815
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From Eq. 5.1.28 w =
– 2 log P = 1.3883 .
From Eq. 5.1.27 u = 0.3015.
erf–1 (0.2370) = u/ 2 = 0.2132 From Eq. 5.2.16 –1 σˆ y = 2er f ( 0.2377 ) = 0.4264
and from Eq. 5.2.17, 2
0.4264 µˆ y = log ( 52621 ) – ------------------ = 10.7800 2 The parameters estimated by the three methods are summarized in Table 5.2.1. Table 5.2.1
Parameter Estimates for Example 5.2.1
Parameter MOM ML PWM
µˆ y 10.7676 10.7711 10.7800
σˆ y 0.4544 0.4562 0.4264
5.2.3 Quantile Estimates For a given return period T, the probability of non-exceedence is F = 1 – 1 ⁄ T . First the standard normal variate (u) is calculated by using Eqs. 5.1.27 and 5.1.28. The quantile estimate is then obtained as log xˆ T = µˆ y + u σˆ y thus xˆ T = e
µˆ y + uσˆ y
(5.2.18) (5.2.19)
If the frequency factor is used instead, KT is given by (Kite, 1977):
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2
σˆ u – σˆ /2
e y y –1 K T = ---------------------------2 1⁄2 σˆ y (e – 1)
(5.2.20)
xT is given by Eq. 5.2.21, xˆ T = µˆ + K T σˆ
(5.2.21)
where µˆ and σˆ here are the mean and standard deviation of the original observations.
EXAMPLE 5.2.2 Estimate the 100-year flood in the Wabash River at Lafayette, Indiana, by using the three methods discussed in this section. Quantile Estimate by the Method of Moments: From Eqs. 5.2.18 and 5.2.19
log xˆ T
= 10.7676 + 0.4544 u; xˆ T = exp [ 10.7676 + 0.4544u ]
For example, if T = 100 then u = 2.326785
xˆ 100 = exp [ 10.7676 + 0.4544 × 2.326785 ] = 135744 cfs xˆ 100 = 136611 cfs Quantile Estimate Based on the Maximum Likelihood Method: From Eqs. 5.2.18 and 5.2.19 we get
log xˆ T = 10.7711 + 0.4562u ; xˆ T = exp [ 10.7711 + 0.4562 u ] For example, if T = 100 then u = 2.326785,
xˆ 100 = exp [ 10.7711 + 0.4562 × 2.326785 ] = 136460 cfs Quantile Estimate Based on the PWM Method: From Eqs. 5.2.18 and 5.2.19
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log xˆ T = 10.7800 + 0.4264 u ; xˆ T = exp [ 10.7800 + 0.4264 u ] For example, if T = 100 then u = 2.326785 and xˆ 100 = exp [ 10.7800 + 0.4264 × 2.326785 ] = 129591 cfs
5.2.4 Standard Error Method of Moments: There are two methods for calculating the standard error by the method of moments. The first method involves the evaluation of the standard error for the logarithmic transformation of the observations. The resulting standard error sTL is in logarithmic units, s TL = [ 1 + u /2 ] 2
1⁄2
σ ------yn
(5.2.22)
where σy is the standard deviation of yi = log xi . In this method, confidence limits are first calculated in logarithmic units and then converted into original units. The second method involves the direct application of the standard error formula to the observations. The values of γ1 and γ2 are given in Eqs. 5.2.23 and 5.2.24, (Kite, 1977). 3σ
2
σ
2
µ3 e y – 3e y + 2 γ 1 = -------= --------------------------------2 3⁄2 3⁄2 σ µ2 (e y – 1) 2 2 2 σ /2 8 σ /2 6 σ /2 4 µ γ 2 = -----42 = ( e y ) + 2 ( e y ) + 3 ( e y ) – 3 µ2
(5.2.23)
(5.2.24)
The coefficient of variation z and σy are related as in Eq. 5.2.25. σ
2
z = (e y – 1)
1⁄2
(5.2.25)
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Substituting Eq. 5.2.25 in Eqs. 5.2.23, 5.2.24, and 5.1.30 we get Eq. 1⁄2 5.2.26 where z is the coefficient of variation z = µ 2 /µ′ 1 and KT is the frequency factor given by Eq. 5.2.20. 2 1⁄2
KT 3 8 6 4 2 δ y = 1 + ( z + 3z ) K T + ( z + 6z + 15z + 16z + 2 ) -------4
(5.2.26)
The standard error is thus given by σ s T = δ y ------n
(5.2.27)
where σ is the standard deviation of the observations and is equal to 1⁄2 m2 . ML Method: Following Kite (1977) we have, as before xˆ T = e
µ y + uσ y
(5.2.28)
Therefore, µ + uσ ∂X ---------T = e y y ∂µ y
(5.2.29)
µ + uσ
∂xˆ ue y y --------T2 = ------------------2σ y ∂σ y
(5.2.30)
ω is defined as in Eq. 5.2.31.
ω = e
µ y + uσ y
(5.2.31)
Since the standard error is given by ∂x 2 ∂x 2 ∂x ∂x T 2 2 2 s T = -------T- var ( µ y ) + --------T2 var ( σ y ) + 2 -------T- --------cov ( µy, σy ) , 2 ∂µ y ∂σ ∂µ y ∂σ y y
(5.2.32)
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substituting Eq. 5.2.31 into Eq. 5.2.32 and simplifying, we get uω uω 2 2 2 2 s T = ω var ( µ y ) + ----------2- var ( σ y ) + --------- cov ( µ y , σ y ) σ y 4σ y 2
2
2
(5.2.33)
The variances and covariances are given by σ var ( µ y ) = -----y n
(5.2.34)
var ( σ y ) = 2σ y ⁄ N
(5.2.35)
cov ( µ y , σ y ) = 0
(5.2.36)
2
2
4
2
Substituting Eqs. 5.2.34 to 5.2.36 in 5.2.33 we get 1 2 2 ( µ + uσ ) 2 2 s T = ---- σ y e y y [ 1 + u /2 ] N
(5.2.37)
But we have as before e
µy + uσ y
= xT
(5.2.38)
and xT = µ ( 1 + K T z )
(5.2.39)
1⁄2
where z is the coefficient of variation µ 2 ⁄ µ′ 1 and KT is the frequency factor given by Eq. 5.2.20. Substituting Eqs. 5.2.38 and 5.2.39 in Eq. 5.2.37 we get Eq. 5.2.40, 2 2 2 2 σ [ log ( z + 1 ) ] ( 1 + K T z ) ( 1 + u /2 ) 2 - s T = ----- --------------------------------------------------------------------------------2 n z
(5.2.40)
which is in the form of Eq. 4.4.16 with 1⁄2 1⁄2 1 2 2 δ z = --- ( 1 + K T z ) ( 1 + u /2 ) [ log ( z + 1 ) ] . z
(5.2.41)
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EXAMPLE 5.2.3 Estimate the standard error of the 100-year flood computed by using the parameters estimated by the MOM and the MLM. Compute the magnitude of floods for other recurrence intervals. Summarize the results. Standard Error by using the MOM Parameters: From Eq. 5.2.25 we have σˆ
2
z = (e y – 1)
1⁄2
= (e
( 0.4544 )
2
– 1)
1⁄2
= 0.4789
which is also the coefficient of variation of the observations. Also from Eq. 5.2.20,
KT = (e
2
0.4544u – ( 0.4544 ) /2
– 1 )/ ( e
2
( 0.4544 ) ⁄ 2
– 1)
1⁄2
and the standard error can be calculated by substituting in Eq. 5.2.26. For example if T = 100, KT = 3.3320, and from Eq. 5.2.26, δy = 4.9279. Hence, from Eq. 5.2.27 knowing 1⁄2
σ = m2
= C v m′ 1 = 25198
25198 S T = 4.9279 ---------------- = 13469 cfs 85 Standard Error by using the ML Parameters: From Eq. 5.2.37 we have
s T = [ 1 + u /2 ] 2
1⁄2
µ + uσ
σye y y -------------------n
10.7711 + 2.326785 × 0.4562
0.4562e s T = 1.9253 × -------------------------------------------------------------- = 12732 cfs 85 The quantile estimates and standard errors were estimated for different return periods by using the MOM and MLM methods. Quantile estimates for the PWM method are also computed for these recurrence intervals.
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These results calculated by the computer program discussed in Chapter 10 are shown in Table 5.2.2. The observed and estimated flows as well as the 95% confidence interval for the estimates given by the MOM, MLM, and PWM are shown in Figure 5.2.1. Table 5.2.2 Quantile Estimates and their Standard Errors (in parentheses) for Example 5.2.3
T 10 20 50 100 200
5.3
P (%) 10 5 2 1 0.5
MOM 84961 (6510) 100210 (8532) 120670 (11298) 136580 (13469) 152975 (15715)
MLM 85461 (5708) 100868 (7656) 121555 (10606) 137653 (13113) 154247 (15860)
PWM 82984 96888 115343 129560 144101
Three-Parameter Lognormal (LN(3)) Distribution
5.3.1 Introduction The three-parameter lognormal distribution is similar to the LN(2) distribution except that x is shifted by an amount (a) which represents a lower bound. The normally distributed variable becomes log (x – a) with the pdf, 1 1 2 f ( x ) = --------------------------------- exp – ---------2 [ log ( x – a ) – µ y ] ( x – a )σ y 2π 2σ y
(5.3.1)
where µy and σ y are the location and scale parameters, which correspond to the mean and variance of the logarithm of the shifted variable (x – a). The standardized variable u is obtained as in Eq. 5.3.2. 2
log ( x – a ) – µ u = ------------------------------------y σy
(5.3.2)
Sangal and Biswas (1970) suggested a procedure to estimate the parameters of the LN(3) distribution in which only the mean, median
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Figure 5.2.1. Observed and estimated flows and 95% confidence intervals for Wabash River data used in Examples 5.2.1 to 5.2.3.
and standard deviation of the data are used. The mathematical properties of the LN(3) distribution were discussed by Burges et al. (1975). They also compared two methods of estimation of the third parameter ‘a’ of the LN(3) distribution. Singh and Singh (1988) used the maximum entropy method to estimate the parameters of the LN(3) distribution. They compared their method to the ML method of parameter estimation. Stevens (1992) developed ML estimators of the LN(3) distribution in which historical data can also be included. He demonstrated by using Monte
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Carlo simulation that inclusion of historical data reduced the bias and variance of extreme flows. Hoshi et al. (1989) investigated different quantile estimation procedures for the lognormal distribution. They compared the average bias and the root mean square errors associated with these procedures.
5.3.2 Parameter Estimation Method of Moments: The first two moments of the LN(3) are given by (Kite, 1977), m¢ 1 = a + e s
2
m y + s y /2
2
m 2 = ( e y – 1 )e
(5.3.3) 2
2m y + s y
(5.3.4)
The coefficient of variation of (x – a), z2, is given by (Kite, 1977) as in Eq. 5.3.5, 2§3
1–w z 2 = -----------------1§3 w
(5.3.5)
where w is defined by Eq. 5.3.6. 1§2
– g 1 + (g 1 + 4) w = ---------------------------------------2 2
(5.3.6)
g1 in Eq. 5.3.6 is the skewness coefficient of the original variable x. The values of the parameters after replacing m¢ 1 , m 2 and g1 by their sample estimates m¢ 1 , m 2 and g1 and using Eqs. 5.3.5 and 5.3.6 to obtain z2 are given by: 2 sˆ y = [ log ( z 2 + 1 ) ]
1§2
1 2 mˆ y = ( log ( m 2 § z 2 ) ) – --- log ( z 2 + 1 ) 2
(5.3.7)
(5.3.8)
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aˆ = m¢ 1 – m 2 § z 2
(5.3.9)
An equivalent procedure is given in Eqs. 5.3.10 to 5.3.13 (Chow 1954, Hoshi et al., 1989). 2 1§2
G = g1 ( 4 + g1 ) 1§3
Ï Ï ¸ 1 2 sˆ y = Ì log Ì 1 + --- ( g 1 + G ) ý 2 Ó Ó þ
(5.3.10) 1§3
Ï ¸ 1 2 + Ì 1 + --- ( g 1 – G ) ý 2 Ó þ
1§2
¸ –1 ý þ
m2 1 mˆ y = --- log ---------------------------2 2 sˆ y sˆ 2 e (e y – 1)
(5.3.11)
(5.3.12)
2
aˆ = m¢ 1 – e
mˆ y + sˆ y /2
(5.3.13)
The Maximum Likelihood (ML) Method: The log-likelihood function of the LN3 distribution is given in Eq. 5.3.14. N
N N 1 2 log L = – Â log ( x i – a ) – ---- log ( 2p ) – ---- ( log s y ) – ---------2 2 2 2s y i=1
N
 [ log ( x i – a ) – m y ]
2
i=1
(5.3.14) Differentiating the log-likelihood function with respect to a, my, s y and equating the resulting expressions to zero we get (Kite, 1977) Eqs. 5.3.15 to 5.3.17. 2
N
1 m y = ---- Â log ( x i – a ) Ni = 1
(5.3.15)
N
1 2 2 s y = ---- Â [ log ( x i – a ) – m y ] Ni = 1 N
 ( xi – a ) ( my – sy ) = –1
i=1
2
N
log ( x – a )
i  ------------------------( xi – a )
(5.3.16)
(5.3.17)
i=1
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By substituting µy and σ y from Eqs. 5.3.15 and 5.3.16 into 5.3.13, an equation only in a is obtained which is solved numerically. The numerical solution can be obtained by using any iterative procedure. Newton’s method is discussed herein. Our objective function is taken as F(a) in Eq. 5.3.18. 2
N
N
F (a) =
∑ ( xi – a )
–1
( µy – σy ) – 2
∑ ( xi – a )
–1
log ( x i – a )
(5.3.18)
i=1
i=1
An initial value of ao is needed. This initial value may be the MOM estimate. However, one should be careful since a defined by MOM may be greater than the minimum observed value. In such a case, a value which is smaller than the minimum observed value may be chosen. Using the initial value ao we calculate µyo, σyo from Eqs. 5.3.15 and 5.3.16. These values are substituted in Eq. 5.3.18 to get F(ao) which is specified to be zero. The initial value ao is then updated as follows a n + 1 = a n – F ( a n )/F′ ( a n )
(5.3.19)
where F′ ( a ) is obtained by differentiating Eq. 5.3.18 to get Eq. 5.3.20. N
N
dF ( a ) 2 –2 –2 F′ ( a ) = --------------- = ( µ y – σ y ) ∑ ( x i – a ) – ∑ ( x i – a ) log ( x i – a ) da i=1 i=1
+
∑ ( xi – a )
2
–2
dµ dσ –1 + --------y – ---------y ∑ ( x i – a ) da da
(5.3.20)
The derivatives of µy and σ y are obtained by differentiating Eqs. 5.3.15 and 5.3.16 to get, 2
dµ 1 --------y = – --da n 2
n
∑ ( xi – a )
–1
,
(5.3.21)
i=1
n
dσ y dµ 2 –1 --------- = --- ∑ [ log ( x i – a ) – µ y ] – ( x i – a ) – --------y . ni = 1 da da
(5.3.22)
The iterations are repeated until F(a) is sufficiently close to zero. It should be noted, however, that in some cases a solution may not exist for the ML equations.
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PWM Method: The expressions for the moments of the LN(3) distribution are given by Hosking (1990) as follows: λ1 = a + e λ2 = e
2
µ y + σ y /2
2
µ y + σ y /2
(5.3.23)
erf ( σ y /2 )
(5.3.24)
and 6 τ 3 = ------π
σ y /2
∫
0
2
erf ( x/ 3 )e ---------------------------------- dx erf ( σ y /2 ) –x
(5.3.25)
An approximate solution for the parameter estimates is given by Hosking (1990) as in Eqs. 5.3.26 to 5.3.29, 3 5 σˆ y = 0.999281 z – 0.006118 z + 0.000127 z
(5.3.26)
2 σˆ σˆ µˆ y = log l 2 /erf -----y – -----y 2 2
(5.3.27)
2
aˆ = l 1 – e
µˆ y + σˆ y /2
(5.3.28)
where
z =
8 –1 1 + t 3 --- Φ ------------ 2 3
(5.3.29)
Φ ( · ) = F ( · ) is the standard Normal distribution function given by –1 –1 Eq. 5.1.4. Φ (· ) = F (· ) = u can be obtained by Eq. 5.1.27.
EXAMPLE 5.3.1 Estimate the parameters of a three-parameter lognormal distribution for the annual maximum flood data of Wabash River at
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W. Lafayette, Indiana given in Table 1.8.1. Use the three methods discussed in this section. Method of Moments: From Table 2.1.2 we have n = 85, m′ 1 = 52621, Cv = 0.4789, Cs = 2.3008 From Eq. 5.3.6
w = [ – 2.3008 + ( 2.3008 + 4 ) 2
1⁄2
]/2 = 0.3738
From Eq. 5.3.5 2⁄3
1 – 0.3738 z 2 = ------------------------------ = 0.6678 1⁄3 0.3738 From Eqs. 5.3.7, 5.3.8, and 5.3.9
0.4789 × 52621 aˆ = 52621 – ------------------------------------- = 14885 0.6678 2 σˆ y = [ log ( 0.6678 + 1 ) ]
1⁄2
= 0.6073
0.4789 × 52621 1 2 µˆ y = log ------------------------------------- – --- log ( 0.6678 + 1 ) = 10.3539 2 0.6678 ML Method: Applying the Newton method outlined in Section 5.3.1b we get the following estimates
aˆ = –16122; µˆ y = 11.0827; σˆ y = 0.3279 To verify the solution we first calculate n
∑ ( xi – a )
–1
= 1.3789 × 10
–3
i=1 n
∑ ( xi – a )
–1
log ( x i – a ) = 1.5133 × 10
–2
i=1
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Then we substitute in Eq. 5.3.18 to get, –3
F ( a ) = [ 11.0827 – ( 0.3279 ) ] × 1.3789 × 10 – 1.5133 × 10 2
–2
= 7 × 10 ≈ 0 –2
Thus Eqs. 5.3.15 to 5.3.17 are verified. PWM Method: From Table 3.1.1 we have n = 85, l1 = 52621, t = 0.2370, and t3 = 0.2045. We have
1 + t3 1 + 0.2045 ------------- = ------------------------- = 0.6023 2 2 To obtain Φ
–1
[0.6023] we calculate P = 1 – 0.6023 = 0.3977.
Substituting this value into Eqs. 5.1.28 and 5.1.27 we get,
W =
– 2 log ( 0.3977 ) = 1.3580 , Φ [ 0.6023 ] = 0.2592 –1
Substituting the above value in Eq. 5.3.29 we get,
z =
8 --- × 0.2592 = 0.4232 3
ˆ y = 0.4225 . From Eq. 5.3.26 we get σ σˆ y In Eq. 5.3.27, we must first evaluate erf ----- 2
σˆ y = 0.2113. Since erf(x) = 2 Φ ( x 2 ) – 1 ----2 erf (0.2113) = 2Φ ( 0.2113 2 ) – 1
To obtain Φ ( 0.2113 2 ) = Φ ( 0.2988 ) , we use Eqs. 5.1.3 and 5.1.5. Φ ( 0.2988 ) = F ( 0.2988 ) = 0.6174
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erf (0.2113) = 2 × 0.6174 – 1 = 0.2348. From Eq. 5.3.27 we get
0.2370 × 52621 0.4225 µˆ γ = log ------------------------------------- – ------------------ ∴ µˆ γ = 10.7906 2 0.2349 2
Substituting µy and σy in Eq. 5.3.28 we get, 2
0.4225 aˆ = 52621 – exp 10.7906 + ------------------ = – 474 . 2 The parameter estimates obtained in this example are given in Table 5.3.1. Table 5.3.1 Parameter Estimates for Example 5.3.1
Parameter MOM MLM PWM
a 14885 –16122 –474
µy 10.3539 11.0827 10.7906
σy 0.6073 0.3279 0.4225
5.3.3 Quantile Estimates For a given return period T, the corresponding probability of nonexceedence is F = 1 – 1 ⁄ T . First the standard normal variate u is calculated using Eqs. 5.1.27 and 5.1.28. The quantile estimate can be obtained as ln ( xˆ T – a ) = µ y + uσ y .
(5.3.30)
Thus, xˆ T = a + e
µ y + uσ y
.
(5.3.31)
If the form of the frequency factor KT given by Kite (1977) is used instead
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2
2
( uσ – σ /2 )
e y y –1 K T = -----------------------------2 1⁄2 σ (e y – 1)
(5.3.32)
xˆ T = µˆ + K T σˆ
(5.3.33)
and
where µ and σ2 are the mean and variance of the original variable x.
EXAMPLE 5.3.2 Estimate the 100-year flood for the Wabash River at Lafayette, Indiana by using the LN(3) distribution. Use the parameters estimated by the three methods discussed in Example 5.3.1. Quantile Estimates Based on MOM Parameters: From Eq. 5.3.31,
xˆ T = 14885 + exp ( 10.3539 + 0.6073u ) For example if T = 100, then u = 2.326785 and
xˆ 100 = 14885 + exp (10.3539 + 0.6073 × 2.326785) = 143806 cfs Quantile Estimates Based on ML Parameters: From Eq. 5.3.31
xˆ T = – 16122 + exp [ 11.0827 + 0.3379u ] For example if T = 100, then u = 2.326785 and
xˆ 100 = –16122 + exp [11.0827 + 0.3279 × 2.326785] = 123354 cfs Quantile Estimates Based on PWM Parameters: From Eq. 5.3.31
xˆ T = – 474 + exp [ 10.7906 + 0.4225u ]
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For example if T = 100 then u = 2.326785 and
xˆ 100 = – 474 + exp [ 10.7906 × 0.4225 + 2.326785 ] = 129315 cfs
5.3.4 Standard Error Method of Moments: Two methods can be used for evaluating the standard error. The first one involves the calculation of the standard error of the transformed variable y = log (xT – a). The standard error in this method is given in logarithmic units by σ 2 2 s T = -----y ( 1 + u /2 ) n 2
(5.3.34)
Again, confidence limits may be calculated in logarithmic units and then converted into original linear units. The second method is the direct method (Kite, 1977). In the equation for hydrologic frequency analysis, xT = µ + K T σ
(5.3.35)
KT is the frequency factor given by Eq. 5.3.32. The value of sT may be calculated by ∂x 2 ∂x 2 ∂x 2 2 s T = ---------T- var ( µ′ 1 ) + --------T var ( µ 2 ) + --------T var ( µ 3 ) ∂µ′ 1 ∂µ 2 ∂µ 3 ∂x ∂x ∂x ∂x + 2 --------T ---------T- cov ( µ′ 1 ,µ 2 ) + 2 --------T ---------T- cov ( µ′ 1 , µ 3 ) ∂µ 1 ∂µ′ 2 ∂µ 1 ∂µ′ 3 ∂x ∂x + 2 --------T --------T cov ( µ 2 , µ 3 ) ∂µ 2 ∂µ 3
(5.3.36)
Considering Eq. 5.3.35 and introducing the skewness coefficient γ1 we get ∂x ---------T- = 1 ∂µ′1
(5.3.37)
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∂x ∂K 1 --------T = ------ K – 3γ 1 -------∂γ 1 ∂µ 2 2σ
(5.3.38)
∂x 1 ∂K --------T = -----2 -------∂µ 3 σ ∂γ 1
(5.3.39)
Also from Kendall and Stuart (1963), it is known that, var ( µ′ 1 ) = µ 2 ⁄ N
(5.3.40)
1 2 var ( µ 2 ) = ---- [ µ 4 – 2µ 2 ] N
(5.3.41)
1 2 3 var ( µ 3 ) = ---- [ µ 6 – µ 3 – 6µ 4 µ 2 + 9µ 2 ] N
(5.3.42)
cov ( µ′ 1 , µ 2 ) = µ 3 ⁄ N
(5.3.43 )
1 2 cov ( µ′ 1 , µ 3 ) = ---- [ µ 4 – 3µ 2 ] N
(5.3.44)
1 cov ( µ 2 , µ 3 ) = ---- [ µ 5 – 4µ 3 µ 2 ] N
(5.3.45)
The central moments µr are given by (Kite, 1977). µ2 = e µ3 = e µ4 = e µ5 = e µ6 = e
2
2
2σ y + 4µ y
2
(e
(e
2
2
2
2
10σ y
2
15σ y
σ
σ
2
2
4σ y
+2
e
6σ y
10σ y
2
(5.3.47)
2 3σ y
2
– 5e
(5.3.46)
(e y – 1)
2
+ 3e 2
+ 10e
2
– 6e
2
(e y – 1)
3σ y ⁄ 2 + 3µ y
(e y + 1) (e
5σ y /2 + 5µ y
3σ y + 6µ y
σ
2
σ y + 2µ y
3σ y
6σ y
σ
– 3
(5.3.48)
2
– 10e y + 4 )
2
+ 15e
2σ y
2
– 20e
3σ y
σ
(5.3.49)
2
+ 15 e y – 5 )
(5.3.50)
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∂K ∂z ∂ ω ∂K T ---------- = ---------T- · -------2 -------∂z 2 ∂ ω ∂γ 1 ∂γ 1
(5.3.51)
where 2z 2 1⁄2 ∂K ------- = ------------2 [ exp { [ log ( 1 + z 22 ) ] u – [ log ( 1 + z 22 ) ] ⁄ 2 } ] ∂z 2 1 + z2 u 1 1 1 1 --------------------------------------------- – ---- – -------3 + -------3 + ------- 2 1⁄2 z 2z 2 2 2z 2 2z 2 2z 2 [ log ( 1 + z 2 ) ]
(5.3.52)
∂z 2 –1 1 - 1 + --------------- = -----------2⁄3 2⁄3 ∂ω 3ω ω
(5.3.53)
γ1 ∂ω 1 -------- = – --- + --------------------------2 ∂γ 1 2 2 ( γ 1 + 4 )1 ⁄ 2
(5.3.54)
Substituting Eqs. 5.3.37 to 5.3.51 into 5.3.36 an expression is obtained for sT. ML Method: The standard error is given by Eq. 5.3.55. ∂x 2 ∂x 2 ∂x ∂x ∂x 2 2 2 2 s T = ------ var ( a ) + -----------2 var ( σ y ) + -------- var ( µ y ) + 2 ------ ---------2 cov ( a 2 σ y ) ∂a ∂a ∂σ ∂µ y (2σ y y ∂x ∂x ∂x ∂x 2 + 2 ------ -------- cov ( a, µ y ) + 2 ---------2 -------- cov ( σ y , µ y ) ∂a ∂µ y ∂σ ∂µ y y
(5.3.55) We have (Kite, 1977), xT = a + e
µ y + uσ y
∂x ------ = 1 ∂a
(5.3.56)
(5.3.57)
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µy + uσ
If w = e
µ y + uσ y
y ∂x ue --------2 = ------------------2σ y ∂σ y
(5.3.58)
µ + uσ ∂x -------- = e y y ∂µ y
(5.3.59)
is substituted into Eq. 5.3.55 it results in Eq. 5.3.60.
2
2
uw uw 2 2 2 2 s T = var ( a ) + ----------2- var ( σ y ) + w var ( µ y ) + ------- cov ( a, σ y ) σ y 4σ y 2 uw 2 + 2w cov ( a, µ y ) + --------- cov ( σ y , µ y ) σy
(5.3.60)
The variances and covariances of the parameters (Kite, 1977) are, 1 var ( a ) = -----------2ND
(5.3.61)
σ y σ y + 1 2 ( σ2y – µ y ) σ2y – 2µ y --------------- e var ( µ y ) = -------–e ND 2σ 2y
(5.3.62)
2
2
2 2 2(σ – µ ) σ – 2µ σy 2 2 var ( σ y ) = -------[ ( σ y + 1 )e y y – e y y ] ND
(5.3.63)
– 1 σ2 /2-µ cov ( a, µ y ) = ------------ e y y 2ND
(5.3.64)
– 1 2 σ2 /2-µ 2 cov ( a, σ y ) = -------- σ y e y y ND
(5.3.65)
– 1 2 σ2 –2 µ 2 cov ( µ y, σ y ) = -------- σ y e y y ND
(5.3.66)
2
D is the determinant of the matrix I.
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( s y + 1 )e 2
n I = -----2 sy
e
e
2
2 (sy – my)
2
e
sy § 2 – my
( –e
2
sy § 2 – my
)
2 sy § 2 – my
0
0
1 § 2sy
2
–e
sy § 2 – my
(5.3.67)
1
2
2 s y + 1 2 ( s2y – m y ) ( 2s y + 1 ) s2y –2m y -e D = --------------e – ---------------------2 2 2s y 2s y 2
(5.3.68)
Substituting these expressions in Eq. 5.3.60 we get an expression for estimating the standard error sT .
EXAMPLE 5.3.3 Estimate the standard errors of the 100-year flood computed in Example 5.3.2, based on the Method of Moments and the Maximum Likelihood Method. Estimate the standard errors of floods of different recurrence intervals. Standard Error Based on MOM Estimates: From Example 5.3.1 we have g1 = 2.3008 and w = 0.3738 Consider the case when T = 100, then u = 2.326785. We have from Eq. 5.3.32 2
( 2.326785 ¥ 0.6073 – 0.6073 /2 )
e –1 K 100 = ----------------------------------------------------------- = 3.6185 2 1§2 0.6073 (e – 1) From Eqs. 5.3.51 to 5.3.54
2.3008 ∂w -------- = – 0.5 + ----------------------------------------- = – 0.1226 1§2 2 ∂g 1 2 ( 2.3008 + 4 ) ∂z 2 –1 1 ------- = -----------------------------1 + -------------------------- = – 1.8803 2§3 2§3 ∂w 3 ( 0.3738 ) ( 0.3738 ) ∂K ------- = 1.2706 ∂z 2
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∂K T ---------- = – 0.1226 × – 1.8803 × 1.2706 = 0.2929 ∂γ 1 From Eqs. 5.3.37 to 5.3.39
∂x ---------T- = 1 , ∂µ′ 1 ∂x 1 –5 --------T = ----------------------------------------------- [ 3.6185 – 3 × 2.3008 × 0.2929 ] = 3.1650 × 10 ∂µ 2 2 × 0.4789 × 52621 ∂x 1 – 10 --------T = --------------------------------------------2 × 0.2929 = 4.6135 × 10 ∂µ 3 ( 0.4789 × 52621 ) From Eqs. 5.3.46 to 5.3.50 8 13 18 µˆ 2 = 6.3496 × 10 ; µˆ 3 = 3.6812 × 10 ; µˆ 4 = 5.5176 × 10 ;
24 29 µˆ 5 = 1.0724 × 10 ; µˆ 6 = 3.0609 × 10
From Eqs. 5.3.40 to 5.3.45
6.3496 × 10 6 var ( µ′ 1 ) = ------------------------------ = 7.4701 × 10 85 8
1 18 8 2 16 var ( µ 2 ) = ------ [ 5.5176 × 10 – ( 6.3496 × 10 ) ] = 6.0170 × 10 85 1 3 29 13 27 var ( µ 3 ) = ------ [ 3.0609 × 10 – 3.6812 × 10 – 6µ 4 µ 2 + 9µ 2 ] = 3.3649 × 10 85 cov ( µ′ 1 ,µ 2 ) = µ 3 /n = 4.3308 × 10
11
cov ( µ′ 1 ,µ 3 ) = [ µ 4 – 3µ 2 ]/n = 5.0684 × 10 2
16
1 22 cov ( µ 2 ,µ 3 ) = --- [ µ 5 – 4µ 3 µ 2 ] = 1.1516 × 10 n
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Substituting the above values in Eq. 5.3.36 we get –5 2
s T = 1 × 7.4701 × 10 + ( 3.1650 × 10 ) 6.0170 × 10 + ( 4.6135 × 10 6
2
16
3.3649 × 10 + 2 × 1 × 3.1650 × 10 × 4.3308 × 10 27
–5
+ 2 × 1 × 4.6135 × 10
– 10
× 4.6135 × 10
– 10
× 1.1516 × 10
)
11
× 5.0684 × 10 + 2 × 3.1650 × 10 16
– 10 2
–5
22
s T = 1.944 × 10 , s T = 34560 cfs . 2
9
Standard Error based on ML Estimates: Consider the case where T = 100, then u = 2.326785. From Eq. 5.3.68 we have
( 0.3279 ) + 1 2 D = -------------------------------exp [ { 2 ( 0.3279 ) – 11.0827 } ] 2 2 ( 0.3279 ) 2
2 ( 0.3279 ) + 1 - exp [ ( 0.3279 ) 2 – 2 × 11.0827 ] – ----------------------------------2 2 ( 0.3279 ) 2
D = 2.2278 × 10
– 11
From Eqs. 5.3.61 to 5.3.66,
1 2 8 var ( aˆ ) = -------------------------- = 2.6404 × 10 ; var ( µˆ y ) = 0.0708 ; var ( σˆ y ) = 0.0035 ; 2 × 85 × D 2 2 cov ( aˆ , µˆ y ) = – 4284.1 ; cov ( aˆ , σ y ) = 921.2609 ; cov ( µˆ y , σˆ y ) = – 0.0149 .
Also
w = e
µ y + uσ y
= 139460 .
Substituting these values into Eq. 5.3.60 we get,
s T = 1.4786 × 10 . s T = 12160 cfs . 2
8
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The quantile estimates and standard errors for different return periods T using the MOM and MLM methods and also the quantile estimates for the PWM are given in Table 5.3.2. These estimates are calculated by the computer program discussed in Chapter 10. The observed and estimated flows as well as the 95% confidence interval for the estimates given by the MOM, MLM, and PWM are shown in Figure 5.3.1. Table 5.3.2 Quantile Estimates and Their Standard Errors (in parentheses) for Example 5.3.3
T 10 20 50 100 200
P (%) 10 5 2 1 0.5
MOM 83222 (7485) 100090 (10101) 124103 (21219) 143763 (34560) 164842 (51782)
MLM 82886 (4881) 95412 (6658) 111415 (9577) 123341 (12159) 135229 (15036)
PWM 82977 96820 115167 129282 143705
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Figure 5.3.1. Observed and estimated flows and 95% confidence intervals for the Wabash River data used in Examples 5.3.1 to 5.3.3.
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CHAPTER 6
The Gamma Family
6.1
Exponential Distribution
6.1.1 Introduction The exponential distribution is a special case of the Gamma family of distributions which include the Pearson (3), one- and two-parameter Gamma, log-Pearson (3), and the generalized Gamma distributions. The pdf of the Pearson (3) distribution is given by ( x – ε)
– ---------------1 β–1 α f ( x ) = --------------------( x – ε) e β α · Γ(β)
(6.1.1)
where Γ ( · ) is the gamma function defined by Eq. 6.1.2, ∞
∫
Γ ( y + 1 ) = t e dt, y + 1 > 0 y –t
(6.1.2)
Γ ( y + 1 ) = yΓ ( y ), y > 0
(6.1.3)
Γ ( y ) = Γ ( y + 1 )/y, y<1
(6.1.4)
Γ ( n ) = ( n – 1 )!, n is a positive integer
(6.1.5)
0
with the following properties
The gamma function can be evaluated numerically by using the approximation in Eq. 6.1.6 (Abramowitz and Stegun, 1965). Γ ( y + 1 ) = 1 + b 1 y + b 2 y + …. + b 8 y + ε ( y ) , 2
8
(6.1.6)
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In Eq. 6.1.6, 0 ≤ y ≤ 1 and b1 = –0.57719 1652
b5 = –0.75670 4078
b2 = 0.98820 5891
b6 = 0.48219 9394
b3 = –0.89705 6937
b7 = –0.19352 7818
b4 = 0.91820 6857
b8 = 0.03586 8343
The absolute error in the approximation is ε ( y ) ≤ 3 × 10 . The above approximation is used to evaluate Γ ( y ) for 0 ≤ y ≤ 1 . For other values of y, Γ ( y ) is expanded using Eq. 6.1.3 or Eq. 6.1.4. For example, –7
Γ ( 3.15 ) = 2.15 Γ ( 2.15 ) = 2.15 × 1.15 Γ ( 1.15 ) , with y = 0.15, and Γ ( 0.25 ) Γ ( 1.25 ) Γ ( – 0.75 ) = ------------------- = ------------------------------ with y = 0.25 – 0.75 – 0.75 × 0.25 Special values of gamma include Γ ( 2 ) = Γ ( 1 ) = 1 and Γ ( 1 ⁄ 2 ) = π . The exponential distribution is obtained from Eq. 6.1.1 by setting β = 1. x – ε
α 1 – ---------f ( x ) = --- e α
(6.1.7)
The variable x can take any value in the range ε < x < ∞, where ε is a lower bound. The probability distribution function of x is given by
F( x) = 1 – e
x–ε – ----------- α
(6.1.8)
6.1.2 Parameter Estimation Method of Moments: The first two moments of the exponential distribution are computed as follows.
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∞
x – ε
α x – ---------dx µ′1 = --- e α
∫
(6.1.9)
ε
x – ε ∞ ∞ – ---------α dx . | + e ε ε
x–ε = -x exp – ----------- α
∫
∴ µ′ 1 = ε + α ∞
µ′ 2 =
∫ ε
µ′ 2 = – x e 2
x–ε – ----------- α
2
(6.1.11)
x – ε
α x – ------------- e dx α
∞
|
(6.1.10)
∞
+2
ε
∫
xe
(6.1.12)
x–ε – ----------- α
dx
(6.1.13)
ε
The second integral in Eq. 6.1.13 is 2α µ′ 1 = 2α (ε + α), and thus µ′ 2 = ε + 2αε + 2α 2
2
µ 2 = µ′ 2 – ( µ′ 1 ) = α 2
(6.1.14) 2
(6.1.15)
Similarly µ 3 = 2α
3
(6.1.16)
µ 4 = 9α
4
(6.1.17)
Parameter estimates in this case are obtained by replacing µ′ 1 and µ2 by the sample first and second moments, m′ 1 and m2, respectively to give αˆ and εˆ .
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2 αˆ = m 2
(6.1.18)
εˆ = m′ 1 – αˆ
(6.1.19)
Maximum Likelihood (ML) Method: The likelihood function of a sample of size n from an exponential distribution is given by Eq. 6.1.20. N
1 L = -----N- e α
1 – --α
∑ ( x – ε) i
i=1
1 log L = – N log α – --α
(6.1.20)
N
∑ ( xi – ε )
(6.1.21)
i=1
Differentiating the log-likelihood equation and setting the derivative equal to zero we get ∂ log L N 1 --------------- = – ---- + -----2 ∂α α α 1 ∴ αˆ = ---N
N
∑ ( xi – ε )
= 0
(6.1.22)
–ε
(6.1.23)
i=1
n
∑ ( x i – ε ) = m′1 i=1
L is an increasing function of ε, and hence the ML estimate of ε is the largest value that ε can attain. However, as x is bounded, ε < x < ∞, the value of ε cannot be larger than the minimum observed value of x = xmin. If ε is greater than xmin, the likelihood function would be equal to zero. Thus, εˆ = min ( x i ) = x 1
(6.1.24)
The statistic x1 is a biased estimator of ε since x1 is always greater than ε and therefore E(x1) > ε. In order to obtain a better estimator, the distribution of x1 is considered. The probability density function of x1 is given by Eq. 6.1.25 (Hogg and Tanis, 1988).
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f ( x1 ) = N [ 1 – F ( x ) ]
N–1
f ( x)
(6.1.25)
x – ε
α N – N ---------f ( x 1 ) = ---- e α
(6.1.26)
Equation 6.1.26 is the pdf of an exponential distribution with mean 2 2 ε + α ⁄ N and variance α ⁄ N . Therefore we have µ′ 1 ( x 1 ) = ε + α/N
(6.1.27)
µ′ 1 ( x ) = ε + α
(6.1.28)
If µ′ 1 (x) is estimated by m′1 and µ 1 (x1) by x1 (the only observation) we get α and ε as in Eqs. 6.1.29 and 6.1.30 (Cohen and Helm (1972), Flood Studies Report (1975)). N ( m′1 – x 1 ) αˆ = --------------------------(N – 1)
(6.1.29)
N x 1 – m′1 εˆ = ----------------------(N – 1)
(6.1.30)
The new estimators are functions of m′1 and x1 which are sufficient statistics for ε and α. The new estimators are unbiased and are of minimum variance. PWM Method: The inverse form of Eq. 6.1.8 is given in Eq. 6.1.31 x = ε – α ln ( 1 – F )
(6.1.31)
The probability weighted moments are calculated as 1
α0 =
∫ 0
1
x dF =
∫ [ ε –α ln ( 1 – F ) ] dF
(6.1.32)
0
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∴ α0 = ε + α 1
α1 =
(6.1.33)
1
∫
x ( 1 – F ) dF =
0
∫
[ ε – α ln ( 1 – F ) ] ( 1 – F ) dF
(6.1.34)
0
ε α α 1 = --- + --2 4
(6.1.35)
The L-moments are then calculated as λ1 = α0 = ε + α
(6.1.36)
λ 2 = α 0 – 2α 1 = α/2
(6.1.37)
The PWM estimates are obtained by substituting l1 and l2 into the corresponding moments λ1 and λ2 to obtain Eqs. 6.1.38 and 6.1.39. αˆ = 2l 2
(6.1.38)
εˆ = l 1 – 2l 2
(6.1.39)
EXAMPLE 6.1.1 Estimate the parameters of the exponential distribution for annual maximum Wabash River flood data, at Lafayette, Indiana. The data are given in Table 1.8.1. Method of Moments: From Table 2.1.2 we have for the Wabash River at Lafayette (station 49) the following N = 85, m′1 = 52621, C v = 0.4789 . 1⁄2
m2
= m′1 C v = 52621 × 0.4789 = 25200
From Eqs. 6.1.18 and 6.1.19 we get, 1⁄2 αˆ = m 2 = 25200 .
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1⁄2 εˆ = m′1 – m 2 = 27421
ML Method: From Table 2.1.2 and Table 1.8.1 N = 85, m′ 1 = 52621, X min = 13100. From Eqs. 6.1.29 and 6.1.30 we get
85 ( 52621 – 13100 ) αˆ = ----------------------------------------------- = 39991 ( 85 – 1 ) ( 85 × 13100 – 52621 ) εˆ = ----------------------------------------------------- = 12629 ( 85 – 1 ) PWM Method: From Table 3.1.1 we have N = 85, l1 = 52621, t = 0.2369 From Eqs. 6.1.38 and 6.1.39 we get
αˆ = 2l 2 = 2 ×0.2369 × 52621 = 24932 εˆ = l 1 – 2 l 2 = 52621 – 24932 = 27689 The parameter estimates obtained in this example are given in Table 6.1.1. Table 6.1.1 Parameter Estimates for Example 6.1.1
Parameter MOM MLM PWM
ε 27421 12629 27689
α 25200 39991 24932
6.1.3 Quantile Estimates The exponential distribution can be expressed in the inverse form given by Eq. 6.1.40.
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x = ε – α log ( 1 – F )
(6.1.40)
1 The T-year flood xˆ T is calculated by substituting F = 1 – --- into Eq. T 6.1.40 giving Eq. 6.1.41. 1 xˆ T = εˆ – αˆ log --- T
(6.1.41)
xˆ T = εˆ + αˆ log ( T ) .
(6.1.42)
Rearranging Eq. 6.1.42 we can write xˆ T = ( εˆ + αˆ ) + αˆ [ log ( T ) – 1 ] .
(6.1.43)
Comparing Eq. 6.1.43 with Eq. 4.3.2 and considering Eqs. 6.1.11 and 6.1.15 we get Eq. 6.1.44. K T = log ( T ) – 1
(6.1.44)
EXAMPLE 6.1.2 Estimate the 100-year flood for the Wabash River annual maximum flows at Lafayette by using the parameters estimated in Example 6.1.1. Method of Moments: From Eq. 6.1.42 we have, xˆ T = 27421 + 25200 log (T). For T = 100,
xˆ 100 = 27421 + 25200 log (100). ∴ xˆ 100 = 143471 cfs ML Method: From Eq. 6.1.42 we have, xˆ T = 12629 + 39991 log (T). For T = 100,
xˆ 100 = 12629 + 39991 log (100). ∴ xˆ 100 = 196794 cfs PWM Method:
From Eq. 6.1.42 we have, xˆ T = 27689 + 24932 log (T).
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For T = 100, xˆ 100 = 27689 + 24932 log (100). ∴ xˆ 100 = 142505 cfs
6.1.4 Standard Error Method of Moments: The frequency factor KT given by Eq. 6.1.44 does not depend on γ1, ∂K T ---------- = 0 ∂γ 1 and the value of δ is given by Eq. 4.4.17. Therefore, for the exponential distribution, 3
2α γ 1 = -------- = 2 3 α
(6.1.45)
4
9α - = 9 γ 2 = -------4 α
(6.1.46)
Substituting γ1 and γ2 from Eqs. 6.1.45 and 6.1.46 into Eq. 4.4.17 we get Eq. 6.1.47. 2 1⁄2
δ = [ 1 + 2K T + 2K T ]
(6.1.47)
The standard error is given by Eq. 6.1.48. α 2 2 s T = ----- [ 1 + 2K + 2K T ] N 2
(6.1.48)
The variances and covariances of the parameters in this case are given by Eqs. 6.1.49 to 6.1.51. 2
2α var αˆ = --------N
(6.1.49)
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α var εˆ = ----N
2
(6.1.50)
–α cov ( εˆ , αˆ ) = --------N 2
(6.1.51)
ML Method: Due to the fact that the range of the variable x in the exponential distribution depends on the parameter ε, the matrix form cannot be used to compute variances and covariances of parameters except for the case in which ε is known. With the exception of this case, the standard expected value techniques are used. When the value of ε is known, then the variance of α is obtained as shown below (Kendall and Stuart, 1963). ∂ log L var αˆ = – --------------- ∂α 2 αˆ = α 2
–1
2 ∂ log L N 2 ---------------- = -----2 – -----3 Σ ( x i – ε ) 2 ∂α α α
(6.1.52)
(6.1.53)
Substituting Σ(xi – ε) = Nα from Eq. 6.1.23, Eq. 6.1.53 is simplified as Eq. 6.1.54. 2 2 ∂ log L N N ---------------- = -----2 – -----3 ( Nα ) = – -----2 2 α ∂α α α
(6.1.54)
Substituting Eq. 6.1.54, in Eq. 6.1.52 we get α var αˆ = ----N 2
(6.1.55)
In this case the standard error is given by α 2 2 s T = ----- ( K + 1 ) N 2
(6.1.56)
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When both ε and α are unknown, var ( ε ) = var ( x 1 ) = α /N and can be neglected since it is of order (N–2). By using the minimum variance unbiased estimators, Cohen and Helm (1972) and also Sarhan (1954) give the variances and covariances of the parameters as in Eqs. 6.1.57 to 6.1.59. 2
2
α var αˆ = ------------N–1
(6.1.57)
α var εˆ = ----------------------N (N – 1)
(6.1.58)
–α cov ( εˆ , αˆ ) = ----------------------N (N – 1)
(6.1.59)
2
2
2
The standard error sT in this case is obtained by substituting Eqs. 6.1.57 to 6.1.59 in Eq. 4.4.22. 2
2 N 2K α 2 s T = ----- 1 + 2K T + --------------TN N–1
(6.1.60)
PWM Method: For the exponential distribution, variances and covariances of the parameters are given by (Hosking, 1986a) Eqs. 6.1.61 to 6.1.63. 2
4α var αˆ = --------3N
(6.1.61)
α var εˆ = ------3N
(6.1.62)
2
α cov ( εˆ , αˆ ) = ------3N 2
(6.1.63)
Comparing these variances with the variances of ML estimates in Eqs. 6.1.57 to 6.1.59, it can be seen that the PWM estimators are inferior to the ML estimators. The standard error can be calculated by substi-
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tuting Eqs. 6.1.61 to 6.1.63 into the general form in Eq. 4.4.22 to get Eq. 6.1.64. 4 2 α 2 s T = ----- 1 + 2K + --- K T N 3 2
(6.1.64)
EXAMPLE 6.1.3 Estimate the standard errors of the 100-year flood by using the parameters estimated by the three different methods in Example 6.1.1. Also compute the magnitudes of floods for different recurrence intervals and the corresponding standard errors. Standard Error Based on MOM Parameters: From Eq. 6.1.44, KT = log (100) – 1 = 3.60517, T = 100. sT is estimated from Eq. 6.1.48.
( 25200 ) 2 2 s T = --------------------- [ 1 + 2 × 3.60517 + 2 × 3.60517 ]; s T = 15986 cfs 85 2
Standard Error Based on the ML Parameters: From Eq. 6.1.44 we have, KT = log (100) – 1 = 3.60517, T = 100. From Eq. 6.1.60 we get, 2 85 ( 39991 ) 2 s T = --------------------- 1 + 2 × 3.60517 + ------ × 3.60517 ; s T = 20048 cfs 85 84 2
Standard Error Based on PWM Parameters: From Eq. 6.1.44 we have, KT = log (100) – 1 = 3.60517 From Eq. 6.1.64 we get, 2 4 ( 24932 ) 2 s T = --------------------- 1 + 2 × 3.60517 + --- × 3.60517 ; s T = 13667 cfs 85 3 2
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The Quantile estimates for different return periods T as well as the standard error using the ML, MOM, and PWM methods are given in Table 6.1.2. These results are obtained by the computer program discussed in Chapter 10. The observed and estimated flows as well as the 95% confidence interval for the estimates given by the MOM, MLM, and PWM are shown in Figure 6.1.1.
Figure 6.1.1. Observed and estimated flows and 95% confidence intervals for the Wabash River data used in Examples 6.1.1 to 6.1.3.
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Table 6.1.2 Quantile Estimates and Standard Errors (in parentheses) for Example 6.1.3
T 10 20 50 100 200
6.2
P (%) 10 5 2 1 0.5
MOM 85444 (7230) 102910 (9838) 125999 (13329) 143465 (15984) 160931 (18647)
MLM 104713 (10006) 132433 (13028) 169077 (17025) 196798 (20048) 224518 (23072)
PWM 85106 (6552) 102393 (8682) 125244 (11517) 142530 (13670) 159817 (15826)
Two-Parameter Gamma (G(2)) Distribution
6.2.1 Introduction The two-parameter gamma (G[2]) distribution is a special case of the Pearson (3) distribution in which the parameter ε in Eq. 6.1.1 is equal to zero. The pdf of the G(2) distribution is thus 1 β – 1 – ( x/α ) f ( x ) = ----------------- x e β α Γ(β)
(6.2.1)
The variable x in this case has a lower bound of zero instead of ε, 0 < x < ∞. The distribution function of x cannot be obtained in closed form. It is given in Eq. 6.2.2. x
1 F ( x ) = ----------------β α Γ(β)
∫x
β – 1 – ( x/α )
e
dx
(6.2.2)
0
If y = x/α is substituted into Eq. 6.2.2 we get y
1 F ( y ) = -----------Γ(β)
∫
y
β – 1 –y
e dy
(6.2.3)
0
From Abramowitz and Stegun (1958) F(y) can be written as, F ( y ) = F ( χ |v ) 2
(6.2.4)
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where F ( χ |v ) is the chi-square distribution with degrees of freedom 2 v = 2β and χ = 2y. Kendall and Stuart (1963) show that the expression for u1, given in Eq. 6.2.5, 2
2 1⁄3
χ u 1 = ----- v
9v 1 ⁄ 2 2 + ------ – 1 ------ 2 9v
(6.2.5)
is approximately N(0,1) when v > 30. Therefore, for a given x, F(x) 2 can be calculated by first calculating y = x/α and χ = 2y and then substituting these values in Eq. 6.2.5 to get u1. Given u1, the value of F(x) can be obtained from Eq. 5.1.5. 6.2.2 Parameter Estimation Method of Moments: The first and second moments of the G(2) distribution are computed by substituting y = x/α in Eq. 6.2.1, 1 β – 1 –y F ( y ) = ------------ y e . Γ(β) 1 ∴ µ′ 1 = -----------Γ(β)
(6.2.6)
∞
∫
( αy )y
β – 1 –y
e dy
(6.2.7)
0
α µ′ 1 = -----------Γ(β)
∞
∫
β –y
y e dy
(6.2.8)
0
Recognizing that ∞
∫
β – dy
y e
= Γ(β + 1) = β Γ(β) ,
(6.2.9)
0
the expression for µ′ 1 is given by Eq. 6.2.10. α µ′ 1 = ------------ β Γ ( β ) = αβ . Γ(β)
(6.2.10)
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The second moment about the mean is given by, 1 µ 2 = -----------Γ(β)
∞
∫
( αy – αβ ) y 2
β – 1 –y
e
(6.2.11)
dy
0
∞
∞
∫
∫
2 α β + 1 –y = ------------ y e dy – 2β Γ(β) 0
0
2 y e dy + β β –y
∞
∫
y
e dy
β – 1 –y
0
α 2 ∴ µ 2 = ------------ [ Γ ( β + 2 ) – 2βΓ ( β + 1 ) + β Γ ( β ) ] Γ(β)
(6.2.12)
2
µ2 = α
2
Γ(β + 2) βΓ ( β + 1 ) 2 --------------------- – 2 ------------------------- + β Γ(β) Γ(β)
µ 2 = α 2 [ ( β + 1 ) β – 2β + β ] = α β 2
2
2
(6.2.13)
(6.2.14)
(6.2.15)
Similarly, it can be shown that the skewness coefficient Cs is given by Eq. 6.2.16. α 2 C s = ------ ------α β
(6.2.16)
Parameter estimates are obtained by substituting the expressions for µ′ 1 , µ 2 to the corresponding sample moments m′1 , m 2 with the result, 2
m αˆ = -------m′1
(6.2.17)
( m′1 ) βˆ = -------------m2 2
(6.2.18)
Maximum Likelihood (ML) Method The likelihood function of a sample of size N from the G(2) distribution is given by Eq. 6.2.19.
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N
1 L = ----------------β α Γ(β)
N
N
β–1
∏ xi
–1 -----α
∑x
i
i=1
e
(6.2.19)
i=1
The log-likelihood function is given by Eq. 6.2.20. N
log L = – N β log α – N log Γ ( β ) + ( β – 1 )
1 ∑ log x i – --αi=1
N
∑ xi
(6.2.20)
i=1
Differentiating Eq. 6.2.20 with respect to α and β and setting the derivatives equal to zero gives Eqs. 6.2.21 and 6.2.22. ∂ ln L Nβ 1 ------------ = – ------- + -----2 ∂α α α
N
∑ xi
= 0
(6.2.21)
i=1
N
Γ′ ( β ) ∂ ln L ------------ = N log α – N ------------- + ∑ log x i Γ(β) i = 1 ∂β
(6.2.22)
Bobée and Ashkar (1991) proposed the following procedure to solve Eqs. 6.2.21 and 6.2.22. By taking the natural logarithm of Eq. 6.2.21 and subtracting Eq. 6.2.22, Eq. 6.2.23 is obtained. E(β) = U
(6.2.23)
U = log A – log G
(6.2.24)
where
1 A = ---N
N
∑ xi
= Arithmetic Mean = m′1
(6.2.25)
i=1
G = ( x 1 x 2 …x N )
1 ---N
= Geometric Mean
(6.2.26)
An approximate solution for β is given by Eqs. 6.2.27 and 6.2.28. For 0 ≤ U ≤ 0.5772 ,
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1 2 βˆ = ---- [ 0.5000876 + 0.1648852 U – 0.054427U ] U
(6.2.27)
For 0.5772 ≤ U ≤ 17.0 2
8.898919 + 9.059950 U + 0.9775373 U βˆ = -------------------------------------------------------------------------------------------------2 U ( 17.7928 + 11.968477 U + U )
(6.2.28)
The maximum error of approximation in Eq. 6.2.27 is 0.0088% and in Eq. 6.2.28 is 0.0054%. The value of αˆ is obtained from Eq. 6.1.21 as in Eq. 6.2.29. αˆ = A/βˆ
(6.2.29)
PWM Method The L-moments of the G(2) distribution are given by Eqs. 6.2.30 and 6.2.31 (Hosking, 1990). Parameter estimates are obtained by replacing λ1 and λ2 by their corresponding sample estimates l1 and l2. λ 1 = αβ
(6.2.30)
1 Γ(β + 1 ⁄ 2) λ 2 = ------- α ---------------------------Γ(β) π
(6.2.31)
The solution for αˆ and βˆ (Hosking, 1990) is obtained as follows, where t = l 2 /l 1 . 2 If 0 < t < 1 ⁄ 2 , then z = πt and βˆ is given by Eq. 6.2.32. 2 3 βˆ 1 = ( 1 – 0.3080z ) / ( z – 0.05812z + 0.01765z )
(6.2.32)
If 1 ⁄ 2 ≤ t < 1 , then z = 1 – t and β is given by Eq. 6.2.33. 2 2 βˆ = ( 0.7213z – 0.5947z ) / ( 1 – 2.1817z + 1.2113 z )
(6.2.33)
αˆ = l 1 / βˆ
(6.2.34)
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EXAMPLE 6.2.1 Estimate the parameters of the G(2) distribution for the annual maximum Wabash River flood data at Lafayette, Indiana given in Table 1.8.1. Method of Moments: From Table 2.1.2 we have for the Wabash River at Lafayette (station 49) the following
N = 85, m′1 = 52621, C v = 0.4789 m 2 = ( C v m′1 ) = 6.3505 × 10 2
8
6.3505 × 10 αˆ = ------------------------------ = 12068 52621 8
( 52621 ) βˆ = ------------------------------8 = 4.3602 6.3505 × 10 2
ML Method: From Eq. 6.2.25 we have
A = m' 1 = 52621, log ( A ) = 10.8709 From Eq. 6.2.25 we have,
1 log ( G ) = ---N
N
∑ log x i
= µ y where y = log xi
i=1
From example 5.2.1, µy = 10.7711 = log (G). From Eq. 6.2.24, U = 10.8709 – 10.7711 = 0.0998 Using Eq. 6.2.27, since U < 0.5772, 1 2 βˆ = ---------------- [ 0.5000876 + 0.1648852 × 0.0998 – 0.054427 × ( 0.0998 ) ] = 5.1732 0.0998
Substituting into Eq. 6.2.29,
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52621 αˆ = ---------------- = 10172 . 5.1732 PWM Method: From Table 3.1.1 we have n = 85, l1 = 52621,t = 0.2370 Since t = 0.2370< 0.5, z = π t = 0.1765 2
From Eq. 6.2.32, βˆ = 5.4131. Substituting in Eq. 6.2.34,
52621 αˆ = ---------------- = 9721 5.4131 The parameter estimates computed in this example are given in Table 6.2.1. Table 6.2.1
Parameter Estimates for Example 6.2.1
Parameter MOM MLM PWM
αˆ 12068 10172 9721
βˆ 4.3602 5.1732 5.4131
6.2.3 Quantile Estimates Kite (1977) shows that the frequency factor KT for the two-parameter Gamma distribution is given by Eq. 6.2.35, χ C 2 K T = -------------s – ----4 Cs 2
(6.2.35)
where Cs is the skewness coefficient of the data. 3⁄2
C S = µ3 / µ2
(6.2.36)
χ in Eq. 6.2.35 is obtained from the inverse of Eq. 6.2.5 as 2
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2 2 χ = v 1 – ------ + u 9ν
2 -----9ν
3
(6.2.37)
In Eq. 6.2.37, u is the standard normal variate corresponding to a probability of non-exceedence of P = 1 – 1/T. Several expressions for calculating KT directly from CS have been listed by Bobée and Ashkar (1991). Of these the following three are noteworthy. 1. Wilson-Hilferty Transformation (Wilson and Hilferty, 1931) The Wilson-Hilferty approximation is quite accurate for CS ≤ 1.0 and maybe sufficiently accurate for CS as high as 2. 3
C 2 C K T = ------ ------S u – ------S + 1 – 1 , C S > 0 6 CS 6
(6.2.38)
2. Modified Wilson-Hilferty Transformation (Kirby, 1972) 3
K T = A [ max { H , 1 – ( G/6 ) + ( G/6 ) u }) – B ] 2
(6.2.39)
for 0.25 ≤ Cs ≤ 9.75 where A, B, and G are functions of only CS and H = [ B – ( 2/C s ) / A ]
1⁄3
(6.2.40)
Kirby (1973) gives a table for values of G, A, and B. To facilitate the calculation of A, B, and G and their derivatives, Hoshi and Burges (1981a) give polynominal approximations for G, 1/A, B, and H 3 in the form 2 5 Zˆ = a 0 + a 1 γ + a 2 γ + … + a 5 γ
(6.2.41)
where Zˆ is the estimate of parameter Z and γ is the coefficient of skew Cs. The values of coefficients a0,....a5, are given in Table 6.2.2.
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Table 6.2.2 Coefficients of a Polynominal in γ for the Parameters G, 1/A, B, and H3
a0 a1 a2 a3 a4 a5 *MRE%
G –0.385205 E 0.100426 E 0.651207 E –0.149166 E 0.163945 E –0.583804 E 0.99
–2 +1 –2 –1 –2 –4
1/A 0.199447 E 0.484890 E 0.230935 E –0.152435 E 0.160597 E –0.558690 E 0.41
–2 +0 –1 –1 –2 –4
*Maximum relative error of parameter estimate = 100
B 0.990562 E 0.319647 E –0.274231 E 0.777405 E –0.571184 E 0.142077 E 0.30
( Z – Zˆ )/Z
+0 –1 –1 –2 –3 –4
H3 –0.365164 E 0.175924 E –0.121835 E 0.782600 E –0.777686 E 0.257310 E 4.38
– – – – – –
2 1 1 2 3 4
in the range (0.25 ≤ γ ≤ 9.75).
Hoshi and Burges (1981a)
3. Cornish-Fisher Transformation (Fisher and Cornish, 1960) 2
3
4 2 C u 3 – 7u C u2 – 1 C 6u + 14u – 32 K T = u + ------S -------------- + -----4-S ----------------- – -----5-S -------------------------------------- 2 3 2 9 2 405
4
5
C ( 9u 5 + 256u 3 – 433u ) C ( 12u 6 – 143u 4 – 923u 2 + 1472 ) + -----7-S ---------------------------------------------------- + -----8-S --------------------------------------------------------------------------4860 25515 2 2 6
C S ( 3753u 7 + 4353u 5 – 289517u 3 – 289717u ) -----------------------------------------------------------------------------------------------------– -----10 9185400 2
(6.2.42)
for C s ≤ 2 In all cases, for negative values of C S , K T = - K′T ( – C S ) where K′T is the frequency factor with the exceedence probability, 1 P = 1 – F = --T The T-year quantile is then calculated by using Eq. 6.2.43, xˆ T = αˆ βˆ + K T
2 αˆ βˆ
(6.2.43)
where KT is calculated by using an appropriate formula from those given above.
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EXAMPLE 6.2.2 Estimate the 100-year flood for the Wabash River annual maximum flows at Lafayette using the parameters estimated in Example 6.2.1. Quantile Estimate Based on MOM Parameters: From Eq. 6.2.43 we have xT = 52621 + 25199 KT, where KT can be obtained by using any of the equations in Section 6.2.3. For example if T = 100, then u = 2.326785 (see Example 5.1.2). For the fitted distribution we have from Eq. 6.2.16
α 2 2 C S = ------ ------- = -------------------- = 0.9578 α β 4.3602 Using the Wilson-Hilferty transformation in Eq. 6.2.38 we get
˙ 3 0.9578 2 0.9578 K 100 = ---------------- ---------------- 2.326785 – ---------------- + 1 – 1 . ∴ K 100 = 3.0033 0.9578 6 6 ∴xˆ 100 = 52621 + 25199 × 3.0033 = 128301 cfs Quantile Estimate Based on ML Parameters: From Eq. 6.2.43 we have, XT = 52621 + 23136 KT, where KT can be obtained by using any of the equations in Section 6.2.3 as appropriate. For example, if T = 100 then u = 2.326785. For the fitted distribution we have from Eq. 6.2.16,
α 2 2 C S = ------ ------- = -------------------- = 0.8793 α β 5.1732 Using the Wilson-Hilferty transformation in Eq. 6.2.38, 3
K 100
2 = ---------------0.8793
0.8793 0.8793 - 2.326785 – ---------------- + 1 – 1 -------------- 6 6
K 100 = 2.9510. ∴ Xˆ 100 = 52621 + 23136 × 2.9510 = 120895 cfs
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Quantile Estimates Based on the PWM Parameters: From Eq. 6.2.45, XT = 52621 + 22617 KT, where KT can be calculated by using any of the equations given in Section 6.2.3 as appropriate. For example if T = 100 then u = 2.326785. For the fitted distribution we have from Eq. 6.2.16,
α 2 2 C S = ------ ------- = -------------------- = 0.8596 α β 5.4131 Using the Wilson-Hilferty transformation in Eq. 6.2.38, 3
0.8596 2 0.8596 K 100 = ---------------- ---------------- 2.326785 – ---------------- + 1 – 1 = 2.9378 0.8596 6 6 Xˆ 100 = 52621 + 22617 × 2.9378 = 119065 cfs
6.2.4 Standard Errors Method of Moments Bobée (1973) has shown that the standard error of xˆ T is given by Eq. 6.2.44. 2 ∂K 2 1 σ 2 2 2 S T = ----- ( 1 + K T C V ) + --- K T + 2C V ---------T- ( 1 + C V ) ∂C S 2 N
(6.2.44)
In Eq. 6.2.44, CV is the coefficient of variation, σ2 = α2β is the variance of the observations, KT is the frequency factor and ( ∂K T ) ⁄ ( ∂C S ) is calculated from the appropriate formula for KT by partial differentiation with respect to CS. ML Method According to Bobée and DesGroseilliers (1985) the standard error in the ML method is given by Eq. 6.2.45. σ 2 s T = ----- δ T N 2
(6.2.45)
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δT in Eq. 6.2.45 is given by Eq. 6.2.46. 1 δ T = ------βη
1 + εK 2 K 2 1 ( βψ' – 1 ) ------------------T- + ------T + -----2 4β β β
∂K T 2 εK T ∂K T --------- + ----------- --------- ∂C S β β ∂C S
(6.2.46)
KT is the frequency factor in Eq. 6.2.46. σ2 = α2β is the variance of the fitted distribution.
(6.2.47)
α ε = ------ ( sign of α = + 1 or –1 ) α
(6.2.48)
1 η = ψ′ – --β
(6.2.49)
d log Γ ( β ) Γ′ ( β ) ψ = ψ ( β ) = ----------------------- = ------------dβ Γ(β)
(6.2.50)
d log Γ ( β ) ψ′ = ψ′ ( β ) = ------------------------2 dβ
(6.2.51)
2
From Abramowitz and Stegun (1965) we have Eq. 6.2.52. 1 1 log Γ ( β ) = β – --- log ( β ) – β + --- log ( 2π ) 2 2 1 1 1 1 1 + --------- – --------------3 + -----------------5 – -----------------7 + -----------------9 12β 360β 1260β 1680β 1188β
(6.2.52)
Differentiating Eq. 6.2.52 twice we get Eqs. 6.2.53 and 6.2.54. 1 1 1 1 1 1 ψ ( β ) = log ( β ) – ------ – -----------2 + --------------4 – --------------6 + --------------8 – --------------2β 12β 120β 252β 240β 132β 10 and ψ ( 1 ) = – γ = ( – 0.577215 )
(6.2.53)
(Euler’s number)
1 1 1 1 1 1 10 ψ′ ( β ) = --- + --------2 + --------3 – -----------5 + -----------7 – -----------9 + --------------β 2β 6β 30β 42β 30β 132β 11
(6.2.54)
and ψ′ ( 1 ) = π /6 . 2
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EXAMPLE 6.2.3 Estimate the standard errors of the 100-year flood computed by the G(2) distribution. Use the parameters estimated by the different methods in Example 6.2.1. Also compute the floods for different recurrence intervals listed in Example 4.1.3 and the corresponding standard errors. Standard Error Based on MOM Estimates: Consider the case with T = 100, K100 = 3.0033, Cs = 0.9578, Cv = 0.4789, u = 2.326785. In order to evaluate ( ∂K T ) ⁄ ( ∂C S ) we differentiate Eq. 6.2.38 which is used to evaluate KT. 3
2
C C ∂K T – 2 C S 2 C ---------- = ------2 ------ u – ------S + 1 – 1 + ------ 3 ------S u – ------S + 1 6 ∂C S C S 6 6 6 C S
u 2C S --6- – -------36
∂K T ---------- = – 3.1356 + 3.7970 = 0.6614 ∂C S Substituting in Eq. 6.2.45 we get, 2 αˆ βˆ 1 2 2 2 2 s T = --------- ( 1 + 3.0033 × 0.4789 ) + --- ( 3.0033 + 2 × 0.4789 × 0.6614 ) ( 1 + 0.4789 ) 85 2
2
8
s T = 1.0515 × 10 ; s T = 10254 cfs
Standard Error Based on ML Estimates: Consider the case where T = 100. To evaluate ∂K T ⁄ ∂C S we differentiate Eq. 6.2.38 which is used to evaluate KT, Cs = 0.8793, u = 2.326785
∂K –2 ---------T- = ------2 ∂C S C S
3
2
CS u 2C C C 2 C - u – ------S + 1 – 1 + ------ 3 ------S u – ------S + 1 --- – ---------S ---- 6 6 6 6 C S 6 36
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∂K T ---------- = – 3.3561 + 4.0269 = 0.6708 ∂C S From Eq. 6.2.47, σ = 23136. From Eq. 6.2.48, ε = +1; KT = 2.9510. From Eq. 6.2.54 we get for β = 5.1732, ψ′ ( β ) = 0.2132 .
1 5.1732
From Eq. 6.2.49, η = 0.2132 – ---------------- = 0.0199 . Substituting the above in Eq. 6.2.46, δT = 8.8418 Substituting the above in Eq. 6.2.45,
( 23136 ) 2 7 s T = --------------------- × 8.8418 = 5.5680 × 10 ; s T = 7462 cfs 85 2
The Quantile estimates for different return periods T by the ML, MOM, and PWM methods as well as the standard errors using the ML, MOM are given in Table 6.2.3. The results are obtained by using the Computer Program discussed in Chapter 10. The observed and estimated flows as well as the 95% confidence interval for the estimates given by the MOM, ML, and PWM methods are shown in Figure 6.2.1. Table 6.2.3 Quantile Estimates and Standard Errors (in parentheses) for Example 6.2.3
T 10
P (%) 10
20
5
50
2
100
1
200
0.5
MOM 86381 (5341) 99719 (6780) 116208 (8728) 128092 (10216) 139604 (11709)
MLM 83585 (3480) 95527 (4692) 110201 (6277) 120725 (7462) 130888 (8637)
PWM 82881 94481 108710 118905 128739
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Figure 6.2.1 Observed and estimated flows and 95% confidence intervals for the Wabash River data used in Examples 6.2.1 to 6.2.3.
6.3
Pearson(3) Distribution
6.3.1 Introduction The probability density function of the Pearson (3) distribution is given by Eq. 6.3.1
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x – γ
α 1 x – γ β – 1 – ---------f ( x ) = ---------------- ----------- e αΓ ( β ) α
(6.3.1)
The variable x in a Pearson (3) distribution can take values in the range γ < x < ∞. Generally α can be positive or negative, but for negative values of α the distribution becomes upper bounded and is therefore not suitable for analyzing maximum events. Equation 6.3.3 is obtained by substituting Eq. 6.3.2 into Eq. 6.3.1. x–γ y = ----------- α
(6.3.2)
1 β – 1 –y f ( y ) = ------------ y e Γ(β)
(6.3.3)
The distribution functions of x and y are given by Eqs. 6.3.4 and 6.3.5.
x
1 F ( x ) = ---------------αΓ ( β )
∫ γ
1 F ( y ) = -----------Γ(β)
x – γ --------- α
x–γ β – 1 – ----------- α
e
dx
(6.3.4)
x–γ ----------α
∫
y
β – 1 –y
e
dy
(6.3.5)
0
The value of F(x) for a given x can be obtained analytically by using the same method given for the Gamma (2) distribution, but in this case the value of y is calculated from Eq. 6.3.2. Matalas and Wallis (1973) compared the moment and maximum likelihood estimates computed by using Pearson type (3) variates. The ML estimates were less biased and not as variable as the comparable moment estimates. The differences between the estimates became quite pronounced for small samples. General expressions are derived by Bobeé (1973) for the sample error of T-year events which are estimated by using Pearson (3) distribution. These are compared to sample errors estimated by using
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simpler assumptions and the advantages of using the general expressions are demonstrated. In a similar vein, Condie (1977) has derived an expression to estimate the asymptotic standard error of estimate of the T-year event. Buckett and Oliver (1977) compared the MOM and MLM for P(3) distribution and recommended the MLM. The skewness coefficient of P(3) distribution was investigated by Lall and Beard (1982). The PWM was applied to the P(3) distribution by Song and Ding (1988), who derived expressions relating the parameters of the P(3) distribution and PWM. The PWM was found to be as efficient as the MOM. Ding et al. (1989) derived expressions for the PWMs of normal, lognormal and P(3) distributions so that their parameters can be estimated. Chowdhury and Stedinger (1991) proposed a method to construct approximate confidence intervals for quantiles of a P(3) distribution when the coefficient of skewness is estimated by using different methods. They established the superiority of their confidence interval. A parameter estimation method for the P(3) distribution based on order statistics was evaluated by Durrans (1992a,b). This method is shown to yield parameters and 100-year quantile estimates which have lower biases and variances than those of the method of moments. Shaligram and Lele (1978) analyzed flood flows from sixteen streams in India by using Pearson (3) distribution. Singh and Singh (1985a) derived the P(3) distribution by using the principle of maximum entropy. 6.3.2 Parameter Estimation
Method of Moments: The first moment of the Pearson (3) distribution is given by Eq. 6.3.6. 1 µ′1 = ---------------αΓ ( β )
∞
∫ α
(x – γ )
x – γ β – 1 – --------------α x ----------- e dx α
(6.3.6)
Equation 6.3.7 is obtained by substituting y from Eq. 6.3.2 into Eq. 6.3.6. 1 µ′1 = -----------Γ(β)
∞
∫
( αy + γ )y
β – 1 –y
e dy
(6.3.7)
0
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1 µ′ 1 = -----------Γ(β)
∞
∫
∞ β –y
αy e dy + γ
0
∫y
β – 1 –y
e dy
(6.3.8)
0
1 µ′ 1 = ------------ [ αΓ ( β + 1 ) + γΓ ( β ) ] Γ(β)
(6.3.9)
∴ µ′ 1 = αβ + γ
(6.3.10)
Similarly the second and third moments as well as the coefficient of skew γ1 are obtained by using Eqs. 6.3.11 to 6.3.13. µ2 = α β
(6.3.11)
µ 3 = 2α β
(6.3.12)
2
3
3⁄2
γ 1 = µ 3 /µ 2
2 Substituting for µ 3 and µ 2 , γ 1 = ------- . β
(6.3.13)
(6.3.14)
Parameter estimates are obtained by replacing µ′1 , µ2 and µ3 by their corresponding sample values, or equivalently, µ′1 , µ2 and γ1 by their sample estimates m′1 , m 2 and C S 2 βˆ = ( 2/C S )
αˆ =
( m 2 /βˆ )
γˆ = m' 1 – m 2 βˆ
(6.3.15)
(6.3.16)
(6.3.17)
Maximum Likelihood (ML) Method: The likelihood function for a sample of size N of the Pearson (3) distribution is given by Eq. 6.3.18.
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N
1 L = ---------------αΓ ( β )
N
N
xi – γ ∏ -----------α
β–1 –
xi – γ
- ∑ -----------α i=1
e
(6.3.18)
i=1
The log-likelihood function is given by Eq. 6.3.19. N
log L = – N log α – N log Γ ( β ) + ( β – 1 )
∑ log ( x i – γ ) – N ( β – 1 ) log α i=1
N
1 – --α
∑ ( xi – γ )
(6.3.19)
t=1
Differentiating Eq. 6.3.19 with respect to α, β, and γ and equating the results to zero results in Eqs. 6.3.20 to 6.3.22. N
Nβ 1 ------- – -----2 α α
∑ ( xi – γ ) N
– Nψ ( β ) +
= 0
(6.3.20)
i=1
xi – γ
∑ log -----------α
= 0
(6.3.21)
i=1
N ---- – ( β – 1 ) α
N
∑ -----------xi – γ 1
= 0
(6.3.22)
i=1
where ψ(β) = ∂log Γ (β)/∂β = Γ′ (β)/ Γ (β) and is given in Eq. 6.2.53. These three equations (6.3.20 to 6.3.22) are solved simultaneously ˆ and γˆ . An iterative to obtain the maximum likelihood estimates αˆ , β, numerical solution to these equations is proposed by Matlas and Wallis (1973). However, they noticed that a solution may not always exist, especially for very small sample skew (γ1) values. Also, when β < 1 there is no solution and thus the coefficient of skew γ1 = 2/ β must not exceed the value of 2. If γ1 > 2 then the method of conditional ML discussed by Bobée and Ashkar (1991) must be used. In this method, the value of γˆ is set to the minimum observation (α > 0) and the rest of the procedure is similar to the G(2) distribution discussed in Section 6.2. For negative values of Cs (α < 0) the distribution becomes upper bounded and is therefore not suitable for frequency analysis of maximum events.
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PWM Method: Similar to the Normal distribution, the P(3) distribution cannot be expressed in an inverse form X = X(F ) and therefore the expressions for probability weighed moments and L-moments are complicated. The PWMs given by Hosking (1986a) are given in Eqs. 6.3.23 to 6.3.25, λ 1 = γ + αβ
(6.3.23)
Γ(β + 1 ⁄ 2) 1 λ 2 = ------- α ---------------------------Γ(β) π
(6.3.24)
τ 3 = 6I 1 ⁄ 3 ( β, 2β ) – 3
(6.3.25)
where Ix(p,q) is the incomplete beta function. A simplified approximate solution for Eq. 6.3.25 is given by Hosking (1991a) as follows For t3 ≥ 1/3, let tm = 1 – t3 and ( 0.36067t m – 0.5967t m + 0.25361t m ) βˆ = -------------------------------------------------------------------------------------------------2 3 ( 1 – 2.78861t m + 2.56096t m – 0.77045t m ) 2
3
(6.3.26)
2
For t3 < 1/3, let tm = 3πt 3 and ( 1 + 0.2906t m ) βˆ = ---------------------------------------------------------------2 3 ( t m + 0.1882t m + 0.0442t m )
(6.3.27)
substituting these into Eqs. 6.3.23 and 6.3.24 we get
αˆ =
Γ ( βˆ ) π l 2 ---------------------------Γ ( βˆ + 1 ⁄ 2 ) γˆ = l 1 – αˆ βˆ
(6.3.28)
(6.3.29)
ˆ in Eq. 6.3.28 is It should be noted that for negative values of t 3 ,α of negative sign.
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EXAMPLE 6.3.1 Estimate the parameters of the Pearson (3) distribution for the annual maximum Eel River (at N. Manchester, IN) flood data given in Table 6.3.1. Table 6.3.1. Data from Eel River at North Manchester, IN
Year Ann. Max. Flow Year Ann. Max. Flow Year Ann. Max. Flow 1924 3590 1947 3210 1970 2630 1925 5480 1948 4220 1971 3680 1926 3960 1949 5900 1972 2600 1927 3030 1950 6700 1973 3120 1928 6080 1951 4780 1974 3790 1929 5880 1952 4240 1975 3300 1930 4120 1953 2400 1976 3830 1931 550 1954 1930 1977 2840 1932 1990 1955 4080 1978 4560 1933 3880 1956 3170 1979 4100 1934 1600 1957 3660 1980 4560 1935 4580 1958 3960 1981 5560 1936 7500 1959 7050 1982 8180 1937 5680 1960 2820 1983 4990 1938 7320 1961 2820 1984 3670 1939 4880 1962 3880 1985 8240 1940 3100 1963 3600 1986 3590 1941 1000 1964 5400 1987 2210 1942 2969 1965 3520 1988 4360 1943 5890 1966 2700 1989 4060 1944 5620 1967 7750 1990 6320 1945 3990 1968 7940 1991 8740 1946 3770 1969 5220 Method of Moments: From Table 2.1.2 we have for these data:
N = 68, m′ 1 = 4357.79, c v = 0.4094 and C S = 0.5522 From Eqs. 6.3.15 to 6.3.17, 2 βˆ = ( 2/0.5522 ) = 13.1179
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αˆ =
( 0.4094 × 4357.79 ) / 13.1179 = 492.584 2
2 γˆ = 4357.79 – ( 0.4094 × 4357.79 ) × 13.1179 = – 2103.9
ML Method: Solving Eq. 6.3.20 with Eq. 6.3.22 we get
1 αˆ = ---N
N
N
∑ ( x i – γ ) – ------------------------N 1 i=1 ∑ ----------------( xi – γ )
(1)
i=1
1 βˆ = -------Nα
N
∑ ( xi – γ )
(2)
i=1
ˆ and βˆ from Eqs. For a given initial value of γ we can evaluate α 1 and 2 above and the objective is to satisfy Eq. 6.3.21 which can be put in the following form: N
F =
∑ log ( xi – γ ) – N log α – N
ψ(β) = 0
(3)
i=1
This problem is solved using a Newton iteration method to improve the initial value of γ as,
γ n + 1 = γ n – F/F′
(4)
where F′ is the derivative of F in Eq. 3 given by N
F′ =
1
∂β N ∂α ------- – Nψ′ ( β ) -----∂γ ∂γ
– ---∑ ----------------( xi – γ ) α
i=1
(5)
where N
N ∑ 1/ ( x i – γ ) ∂α i=1 ------- = – 1 + ---------------------------------------N 2 ∂γ 1 ∑ ----------------( xi – γ ) i=1 2
(6)
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and 1 ∂β –1 ------ = ------ – ----------2 ∂γ α Nα
N
∂α
∑ ( x i – γ ) -----∂γ
(7)
i=1
The parameters obtained by this procedure are as follows:
αˆ = 473.3643, βˆ = 13.9205, γˆ = – 2230.7578 To verify this solution we calculate the following: N
N
∑ ( xi – γ ) = 448217.3078 ; ∑ 1/ ( x i – γ ) = 1.11143117 × 10 i=1
–2
i=1
N
∑ log ( x i – γ )
= 595.48972 ; ψ ( β ) = 2.597698
i=1
68 × 13.9205 1 ------------------------------- – --------------------------2 × 448217.3078 = – 5 × 10 –4 ≈ 0 (6.3.23) 473.364 ( 473.364 ) – 68 × 2.597698 + 595.48972 – 68 log ( 473.3643 ) = – 2 × 10 ≈ 0 (6.3.24) –2
68 –2 –5 ------------------- – ( 13.9205 – 1 ) ×1.11143 × 10 = 5 × 10 ≈ 0 473.364
(6.3.25)
therefore the ML equations are satisfied. PWM Method: From Table 3.1.1 we have the following data: N = 68, l1 = 4357.79, t = 0.2295 and t3 = 0.1407 Since t3 = 0.1407 is less than 1/3, from Eq. 6.3.27 we get tm = 3 π (0.1407)2 = 0.1866.
( 1 + 0.2906 × 0.1866 ) βˆ = ----------------------------------------------------------------------------------------------------------------------------- = 5.4499 2 3 [ 0.1866 + 0.1882 × ( 0.1866 ) + 0.0442 × ( 0.1866 ) ] From Eq. 6.1.3:
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Γ (5.4499) = 4.4499 × 3.4499 × 2.4499 × 1.4499 × Γ (1.4499), Γ (5.9499) = 4.9499 × 3.9499 × 2.9499 × 1.9499 × Γ (1.9499). From Eq. 6.1.6 we get:
Γ (1.4499) = 0.885662, So Γ (5.4499) = 48.296, Γ (1.9499) = 0.79842, So Γ (5.9499) = 110.194. From Eqs. 6.3.28 and 6.3.29:
αˆ =
48.296 π ( 0.2295 × 4357.79 ) ------------------- = 776.921 110.194
γˆ = 4357.79 – 776.921 × 5.4499 = 123.648 The parameter estimates obtained in this example are given in Table 6.3.2. Table 6.3.2 Parameter Estimates for Example 6.3.1
Parameter
α
β
γ
MOM
492.584
13.1179
–2103.9
MLM
473.364
13.9205
–2230.8
PWM
776.921
5.4499
123.65
6.3.3 Quantile Estimation Quantile estimation is carried out as in the Gamma (2) case, using the same frequency factor KT , but the quantile xT in this case is given by
x T = αˆ βˆ + γˆ + K T
2 αˆ βˆ
(6.3.30)
where KT is the frequency factor corresponding to a return period of T years, and can be evaluated by using the formulas given in Section 6.2.3.
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EXAMPLE 6.3.2 Estimate the 100-year flood for the Eel River data in Table 6.3.1 by using the parameters estimated in Example 6.3.1. Quantile Estimates Based on MOM Parameters: From Eq. 6.3.30, xˆ T = 4357.79 + 1784.07 K T . For T = 100, u = 2.326785 and Cs = 2 ⁄
βˆ = 0.5522 .
By using the Wilson-Hilferty transformation in Eq. 6.2.38: 3
K 100
2 = ---------------0.5522
0.5522 0.5522 - 2.326785 – ---------------- + 1 – 1 = 2.725887 -------------- 6 6
∴ xˆ 100 = 4357.79 + 1784.07 × 2.725887 = 9221 cfs Quantile Estimates Based on ML Parameters: From Eq. 6.3.30, xˆ T = 4357.79 + 1766.13 K T . For
example
if
T
=
100,
then
u
=
2.326785
and
C S = 2 ⁄ 13.9205 = 0.5361 . By using the Wilson-Hilferty transformation in Eq. 6.2.38: 3
K 100
0.5361 2 0.5361 = ---------------- ---------------- 2.326785 – ---------------- + 1 – 1 = 2.7145 0.5361 6 6
therefore xˆ 100 = 4357.79 + 1766013 × 2.7145 = 9152 cfs . Quantile Estimate Based on PWM Parameters: From Eq. 6.3.30 we get, xˆ T = 4357.79 + 1813.72 K T For example if T = 100 then u = 2.326785 and Cs =
2 ⁄ βˆ = 0.856714 By using the Wilson-Hilferty transformation in Eq. 6.2.38: 3
0.856714 2 0.856714 K 100 = ---------------------- ---------------------- 2.326785 – ---------------------- + 1 – 1 = 2.93586 0.856714 6 6
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Xˆ 100 = 4357.79 + 1813.72 × 2.93586 = 9683 cfs Other quantile estimates are given in Table 6.3.3.
6.3.4 Standard Errors Method of Moments The standard error is calculated by using Eq. 4.4.11. To further simplify the expressions it is noted that γi, i = 2, 3, 4 are given by Eqs. 6.3.31 to 6.3.33 (Kite, 1977; Bobée 1973). γ 2 = 3 ( 1 + γ 1 /2 )
(6.3.31)
γ 3 = γ 1 ( 10 + 3γ 1 )
(6.3.32)
2
2
γ 4 = 5 ( 3 + 13 γ 1 /2 + 3γ 1 /2 ) 2
4
(6.3.33)
Substituting Eqs. 6.3.31 to 6.3.33 into Eq. 4.4.11 results in Eq. 6.3.34. 2 µ ∂K K 2 2 3 s T = -----2 1 + K T γ 1 + ------T ( 3γ 1 /4 + 1 ) + 3K T ---------T- ( γ 1 + γ 1 /4 ) N ∂γ 1 2 ∂K 2 2 4 + 3 -------- ( 2 + 3γ 1 + 5γ 1 /8 ) ∂γ 1
(6.3.34) In Eq. 6.3.34, KT is the frequency factor, µ2 is the second central 2 moment and is estimated by α βˆ , γ 1 is the coefficient of skew and is estimated by 2 ⁄ βˆ and ∂K T ⁄ ∂γ 1 is evaluated by partial differentiation of the formula used to evaluate KT. ML Method The standard error in this method is calculated by using Eq. 4.4.19. Eqs. 6.3.35 to 6.3.41 are from Kite (1977),
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ψ′ ( β ) 1 ----------------- – -------------------2 (β – 2) (β – 1)
1 var ( α ) = -------------2 Nα D
(6.3.35)
2 var ( β ) = ------------------------------4 NDα ( β – 2 )
(6.3.36)
βψ′ ( β ) – 1 var ( γ ) = -------------------------2 Nα D
(6.3.37)
( –1 ) cov ( α, β ) = -------------3 Nα D
1 1 ----------------- – ----------------(β – 2) (β – 1)
(6.3.38)
1 cov ( α, γ ) = -------------2 Nα D
1 ----------------- – ψ′ ( β ) (β – 1)
(6.3.39)
β ----------------- – 1 (β – 1)
(6.3.40)
–1 cov ( β, γ ) = -------------3 Nα D where 1 D = -----------------------4 ( β – 2 )α
2β – 3 2ψ′ ( β ) – -------------------2 (β – 1)
(6.3.41)
The partial derivatives are obtained from Eq. 6.3.30 as ∂x -------T- = β + K T ∂α K ∂x T -------- = α + ------T 2 ∂β
α β -----α
(6.3.42)
2 α ∂K 2 α /β – ---------- ---------Tβ ∂C S
(6.3.43)
∂x -------T- = 1 ∂γ
(6.3.44)
( 2K T ) ⁄ ( ∂C S ) is obtained from the formula used to evaluate KT. Substituting the above equations into Eq. 4.4.19 the required estimate of sT is obtained.
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EXAMPLE 6.3.3 Estimate the standard errors of the 100-year flood for the Eel River data in Table 6.3.1 by using the parameters estimated in Example 6.3.1. Standard error by the MOM estimates: Consider the case where T = 100. We have K100 = 2.725887 3
Ï CS Ê ¸ C - u – ------Sˆ + 1 ý – 1 Ì ----Ë ¯ 6 Ó6 þ
∂K –2 ---------T- = ------2 ∂g 1 CS
2
ÏC ¸ Ï u 2C ¸ C 3 Ì ------S Ê u – ------Sˆ + 1 ý Ì --- – ---------S ý Ë ¯ 6 6 Ó þ Ó 6 36 þ
2 + -----CS
∂K T ---------- = – 4.936413 + 5.640619 = 0.704206 ∂g 1 Substituting these in Eq. 6.3.34 we get,
( 1784.07 ) 2 s T = -------------------------68 2
2 ¸ ( 2.725887 ) Ï 3 ( 0.5522 ) 2 1 + 2.725887 ¥ 0.5522 + ----------------------------- Ì -------------------------- + 1 ý 2 4 Ó þ
+ 3 ¥ 2.72588 ¥ 0.704206 { 0.5522 + ( 0.5522 ) /4 } 3
+ 3 ( 0.704206 ) [ 2 + 3 ( 0.5522 ) + 5 ( 0.5522 ) /8 ] 2
2
4
s T = 698148, s T = 834 cfs 2
Standard error by the ML estimates:
Consider the case where T = 100, K100 = 2.7145. ∂K 100 – 2 ------------- = ------2 ∂C S CS
3
Ï CS Ê ¸ C - u – ------Sˆ + 1 ý – 1 Ì ----Ë ¯ 6 Ó6 þ 2
ÏC ¸ Ï u 2C ¸ C 2 + ------ 3 Ì ------S Ê u – ------Sˆ + 1 ý Ì --- – ---------S ý CS Ó 6 Ë 6¯ þ Ó 6 36 þ
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∂K 100 ------------- = – 5.063926 + 5.769517 = 0.705591 ∂C S From Eq. 6.2.54 we get ψ′ ( β ) = 0.074479 From Eqs. 6.3.35 to 6.3.41: 1 D = -----------------------------------------------------------------4 ( 13.9205 – 2 ) ( 473.3643 )
2 × 13.9205 – 3 – 16 2 × 0.074479 – ------------------------------------= 2.5969 × 10 2 ( 13.9205 – 1 )
1 var ( α ) = ---------------------------------------------------------------------------------2 – 16 68 × ( 473.3643 ) × 2.5969 × 10
0.074479 1 --------------------------------- – -----------------------------------2 = 65145.67 ( 13.9205 – 2 ) ( 13.9205 – 1 )
2 var ( β ) = ---------------------------------------------------------------------------------------------------------------------- = 189.229 – 16 4 ( 473.3643 ) ( 13.9205 – 2 ) 68 × 2.59693 × 10 13.9205 × . 074479 – 1 var ( γ ) = ---------------------------------------------------------------------------------= 9296370.23 2 – 16 68 × ( 473.3643 ) × 2.5969 × 10 –1 1 1 --------------------------------- – --------------------------------- = – 3466.36 cov ( α, β ) = ---------------------------------------------------------------------------------3 – 16 ( 13.9205 – 2 ) ( 13.9205 – 1 ) 68 × ( 473.3643 ) × 2.5969 × 10 1 1 cov ( α, γ ) = ------------------------------------------------------------------------------------------------------------------ – 0.074479 = 737288.46 2 – 16 ( 13.9205 – 1 ) 68 × ( 473.3643 ) × 2.5969 × 10 –1 13.9205 cov ( β, γ ) = ------------------------------------------------------------------------------------------------------------------ – 1 = – 41320.81 3 – 16 ( 13.9205 – 1 ) 68 × ( 473.3643 ) × 2.5969 × 10
Also from Eqs. 6.3.42 to 6.3.44: ∂ xT -------- = 13.9205 + 2.7145 ∂α ∂ xT 2.7145 -------- = 473.3643 + ---------------2 ∂β
13.9205 = 24.0484
2
( 473.3643 ) ( 473.3643 ) ----------------------------- – --------------------------------- × 0.705591 = 621.569 13.9205 13.9205 2
∂x --------T = 1 ∂γ
Substituting the values obtained above into Eq. 4.4.19:
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s T = ( 24.0484 ) 2
2
×65145.67 + ( 621.569 ) × 189.229
+ 1 × 9296370.23 – 2 +2
2
2×4.0484 ×621.569× 3466.36
2×4.0484 ×1× 737288.46– 2
6×21.569 ×1× 41320.81
s T = 545275.85, s T = 738 cfs 2
Standard errors for other recurrence intervals are given in Table 6.3.3. The observed and estimated flows and 95% confidence intervals are shown in Figure 6.3.1. Table 6.3.3 Quantile Estimates and Their Standard Errors (in parentheses) for Example 6.3.3
T 20
P (%) 5.00
40
2.50
60
1.67
80
1.25
100
1.00
120
0.83
140
0.71
160
0.63
180
0.56
200
0.50
MOM 7545 (490) 8296 (638) 8711 (715) 8998 (780) 9215 (830) 9390 (872) 9536 (908) 9662 (940) 9772 (967) 9869 (992)
MLM 7508 (463) 8246 (576) 8654 (645) 8934 (696) 9148 (736) 9319 (769) 9463 (797) 9586 (822) 9693 (844) 9789 (863)
PWM 7715 8585 9073 9412 9671 9881 10057 10209 10341 10459
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Figure 6.3.1 Observed and estimated flows and 95% confidence intervals for the Eel River data used in Examples 6.3.1 to 6.3.3.
6.4
Log-Pearson (3) Distribution
6.4.1 Introduction If the variable log x is assumed to have a Pearson (3) distribution then the distribution of the variable x is a log-Pearson (3) (LP(3)) distribu-
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tion. The probability density function of a LP(3) distributed random variable is given by
log ( x ) – γ -----------------------α
1 f ( x ) = ------------------αxΓ ( β )
log ( x ) – γ β – 1 – -----------------------α
(6.4.1)
e
The pdf of the log-Pearson (3) distribution may take many different shapes (Bobée and Ashkar, 1991). For flood frequency analysis, only values of β greater than one and 1/α greater than zero are of interest. Negative coefficients of skew correspond to negative α values which are not acceptable as the distribution would then have an upper bound. The distribution function of the log-Pearson (3) distribution is given by Eq. 6.4.2. x
1 F ( x ) = ---------------αΓ ( β )
∫ 0
1 --x
log ( x ) – γ -----------------------α
log ( x ) – γ β – 1 – -----------------------α
e
dx
(6.4.2)
log ( x ) – γ If y = ------------------------ is substituted into Eq. 6.4.2 we get Eq. 6.4.3. α y
1 F ( x ) = -----------Γ(β)
∫
y
β – 1 –y
e dy
(6.4.3)
0
Equation 6.4.3 can be approximated by Eq. 6.2.4 where ν = 2β, χ = 2y and y = ( log ( x ) – γ ) ⁄ α . Based on Eq. 6.4.1, moments of the distribution can be derived (Bobée and Ashkar, 1991) as in Eq. 6.4.4. 2
γr
e µ′ r = ----------------------β ( 1 – rα )
(6.4.4)
It should be noted that higher order moments of order r do not exist if the value of α is greater than 1/r (Bobée and Ashkar, 1991). The moment ratios for the LP(3) distribution can be calculated by using Eqs. 6.4.5 to 6.4.7 (Bobée and Ashkar, 1991), ( 1 – α/k ) 2 C v = ----------------------- 1 – 2α/k
β
1⁄2
– 1
(6.4.5)
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–β 2α –β α –β α –3β 1 – 3α ------- -3 1 – ------- 1 – --- + 2 1 – --- k k k k C s = --------------------------------------------------------------------------------------------------------------------------–β – 2β 3 ⁄ 2 2α α 1 – ------- – 1 – --- k k –β
–β
–β
–β
(6.4.6)
3α α 2α α α C = 1 – 4α ------- – 4 1 – ------- 1 – --- + 6 1 – ------- 1 – --- – 3 1 – --- k k k k k k k -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------–β – 2β 2 α 1 – 2α ------- – 1 – --- k k – 2β
– 4β
(6.4.7)
where k depends on the logarithmic base used, k = 1 for natural logarithm and k = 0.434 for logarithm of base 10. The recommendation by a federal interagency group that LP(3) distribution be used by the U.S. government agencies resulted in considerable research on this distribution. As the LP(3) distribution depends on the skewness coefficients and their characteristics, they have also been intensively investigated. Benson (1968) discusses the result of the study by the federal interagency group. Matalas and Benson (1968) investigated the standard error of skewness coefficient. They recommended the use of the standard error of skewness coefficient derived by Fisher for samples from a normal distribution. Kirby (1973) demonstrated that the original Wilson-Hilferty transformation cannot preserve the mean, variance, skew and lower bound of the Pearson (3) variate, which it is intended to approximate, if the skewness coefficient is greater than 3. He proposed a modification of the Wilson-Hilferty transformation which preserves these moments and the lower bound for skewness coefficients larger than 3. There have been a number of studies related to estimation of parameters of LP(3) distributions. Bobeé (1975) introduced a new method of estimation of parameters which uses the moments of the original data. This new method is shown to be better than the method proposed by the Water Resources Council for floods with large return periods. Bobeé and Robitaille (1977) investigated different methods of fitting P(3) and LP(3) distributions by using several long-term records. P(3) distributions were found to fit the data better than LP(3) distributions. If LP(3) distribution is used, it is better to use the method of moments with observed data than log transformed data. The bias in the skewness coefficient should be corrected in either method. Nozdryn-Plotnicki and Watt (1979) used 37 long-term series from Canadian flow gaging stations in a study of parameter estimation for LP(3) distribution. They found that the parameter estimates are highly biased and had large variances although no significant bias was found in the quantile estimates.
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The bounds of LP(3) and its different forms are presented on the basis of the variance and skewness coefficient of a dimensionless variate with mean unity by Rao (1980). The bias in skew estimate is seen to affect the quantiles differently for different return periods. A method was developed by Hoshi and Burges (1981b) to estimate the parameters and quantiles of an LP(3) distribution, who recommend that the parameters of the LP(3) distribution should be fitted by using untransformed data. Rao (1980) compared two- and three-parameter distributions for flood frequency analysis. Phien and Hira (1983) presented four additional methods to estimate the parameters of the LP(3) distribution. A method based on the first two moments of the original data and the variance of the logtransformed variates is claimed to be superior to the ML method. Phien and Hsu (1985) developed a method to estimate the variance of the Tyear event by using the LP(3) distributions. Wallis and Wood (1985) investigated the relative accuracy of the Water Resources Council procedures by using a Monte Carlo simulation study to demonstrate their inadequacy. The quantile estimates obtained by these procedures and the LP(3) distribution were poorer than estimates computed by using an index flood approach with GEV or Wakeby distribution. They recommended re-evaluation of WRC Bulletin 17B guidelines. Ashkar and Bobeé (1987) compared four different versions of the method of moments used to fit the LP(3) distribution by using simulated data. Because the variances, covariances, and correlation coefficients computed by using the first-order asymptotic approximation may have considerable error, they recommended caution in using them. Ashkar and Bobeé (1988) developed confidence intervals for P(3) and LP(3) distributions. When the skewness coefficient was unknown, the use of Weibull distribution gave accurate confidence intervals. Arora and Singh (1989) compared different methods of parameter estimation of LP(3) distribution by the Monte Carlo simulation method. The U.S. Water Resources Council Method was found to perform poorly. The methods based on moments in real space and on mixed moments were found to be superior to other methods tested. Ashkar et al. (1992) demonstrated that an important cause of separation effect was the spatial mixing of skewness values within a region. They recommended that separation of skewness should not be used as a criterion for choosing the type of distribution to be used in flood frequency analysis. The asymptotic standard error of estimate of the T-year flood was obtained by using the equation for variance of estimate of a function by Pilon and Adamowski (1993). Their results
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indicated that historical information contributed significantly to the accuracy of estimation of quantiles, even when the measurement errors were large. Boughton (1976) used the LP(3) distribution in a study of flood estimation from short records. McMahon and Srikanthan (1981) investigated whether LP(3) distribution was applicable to Australian flood data and concluded that they were. They investigated the effects of sample size, distribution parameters and dependence on annual flood estimates. Boughton and Renard (1984) used log-Boughton and LP(3) distributions to investigate the flood frequency characteristics of 18 watersheds in southeastern Arizona. A generalized envelope for 100year flood for watersheds varying from 0.01 to 4000 mi2 was derived. Thomas (1985) reviewed Bulletins 17, 17A, and 17B of the Water Resources Council, giving special emphasis to techniques in Bulletin 17B. Singh and Singh (1988) used the principle of maximum entropy to derive a method of parameter estimation for the LP(3) distribution. These estimates were comparable to the estimates by MOM and MLM.
6.4.2 Studies on Skewness Coefficients There have been a number of studies related to the skewness coefficient, especially because of the importance given to the LP(3) distribution. McCuen (1979b) estimated the standard error of estimate of the skew map in Bulletin 17 of the Water Resources Council. This standard error is shown to be relatively large and comparable to the standard error of estimates obtained from annual flood series. Jackson (1981) discussed some problems in the application of WRC flood frequency guidelines which arose when they were applied to data from Susquehanna River basin. Tung and Mays (1981a) developed a method to estimate generalized values of mean, standard deviation, and skewness coefficient by using a weighted sum of the sample and regional statistics. Tung and Mays (1981b) presented a new procedure to combine regional and at-site skew coefficients. Tasker and Stedinger (1986) present a method based on weighted least squares regression to estimate generalized skew coefficients. The weights were determined by separating the residual variance into those due to modeling and sampling errors. A general formula is derived by Bobeé and Ashkar (1988) for the variance of the T-year event, based on the LP(3) distribution and the generalized method of moments.
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A generalized skew map for Louisiana streams was developed by Naghavi and Yu (1991). This state map was compared to the generalized skew map of the U.S. Water Resources Council. The skew map for Louisiana was found to be better. Spatial trends in the mean, standard deviation and skewness coefficient of log transformed annual flood peaks are delineated by Lichty and Karlinger (1995). A climate factor which quantifies the effects of long term climatic data is used in this study. A bias correction method based on the Bayesian method was developed for an order statistics based estimator of skewness coefficient by Durrans (1994a). The order statistic based estimator is shown to be algebraically bounded. Further comments on this paper are found in Durrans (1994b). We will next discuss the estimation of parameters of LP(3) distribution by the methods of moments, maximum likelihood, and probability weighted moments.
6.4.3 Parameter Estimation Method of Moments There are two ways of using the method of moments to estimate the parameters of the LP(3) distribution. The first method, called the indirect method of moments, is to take the natural logarithms of the random variable x and then apply a P(3) parameter estimation method to the transformed variable z = ln x. The second method is the direct application of the method of moments to the variable x. For the first, second, and third moments, if natural logarithms of Eq. 6.4.4 are taken we get Eqs. 6.4.8 to 6.4.10. log µ′1 = γ – β log ( 1 – α )
(6.4.8)
log µ′2 = 2γ – β log ( 1 – 2α )
(6.4.9)
log µ′3 = 3γ – β log ( 1 – 3α )
(6.4.10)
Manipulating Eqs. 6.4.8 to 6.4.10 gives Eq. 6.4.11 (Kite, 1977). 3 log µ′3 – 3 log µ′1 log [ ( 1 – α ) / ( 1 – 3α ) ] --------------------------------------- = ------------------------------------------------------2 log µ′2 – 2 log µ′1 log [ ( 1 – α ) / ( 1 – 2α ) ]
(6.4.11)
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The left-hand side of Eq. 6.4.11 can be calculated from the sample moments m′1 , m′2 , m′3 as log m′3 – 3 log m′1 B = ----------------------------------------log m′2 – 3 log m′1
(6.4.12)
Eq. 6.4.11 is then solved numerically for α. Alternatively following Kite (1977), if we define A = 1/α – 3
(6.4.13)
C = 1/ ( B – 3 )
(6.4.14)
then for 3.5 < B < 6.0, 2
A = – 0.23019 + 1.65262 C + 0.20911 C – 0.04557 C
3
(6.4.15)
and for 3.0 < B ≤ 3.5, A = – 0.47157 + 1.999 55 C .
(6.4.16)
α, β, and γ are estimated by Eqs. 6.4.8 and 6.4.9. αˆ = 1/ ( A + 3 )
(6.4.17)
log m′ 2 – 2 log m′ 1 log ( 1 + C v ) - = ------------------------------------------------------------------βˆ = ------------------------------------------------------------------2 2 [ log ( 1 – α ) – log ( 1 – 2α ) ] [ log ( 1 – α ) – log ( 1 – 2α ) ] 2
γˆ = log m′ 1 + β log ( 1 – α )
(6.4.18)
(6.4.19)
Maximum Likelihood (ML) Method The likelihood function of a sample of n observations from an LP(3) distribution is given by Eq. 6.4.20. N
1 –N [ αΓ ( β ) ] L = ----------N
∏ xi
N
∏
i=1
xi – γ log -----------α
1 – --(β – 1) α
e
∑ ( log x – γ ) i
i=1
(6.4.20)
i=1
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The log-likelihood function is given by Eq. 6.4.21. N
N
i=1
i=1
log L = – N β log α – N log Γ ( β ) – ∑ log x i + ( β – 1 ) ∑ log ( log x i – γ ) N
(6.4.21)
1 – --- ∑ ( log x i – γ ) α i=1
Differentiating Eq. 6.4.21 with respect to α, β, and γ and equating the results to zero results in Eqs. 6.4.22 to 6.4.24. N
∑ ( log x i – γ )
= nαβ
(6.4.22)
i=1 N
Nψ ( β ) =
∑ log [ ( log x i – γ ) ⁄ α ]
(6.4.23)
i=1 N
N = α ( β – 1 ) ∑ 1/ ( log x i – γ )
(6.4.24)
i=1
In Eq. 6.4.23 ψ ( β ) = Γ′ β /Γ( ( β) ) and is given by Eq. 6.2.53. These three equations (6.4.22 to 6.4.24) are solved numerically to find α, β, and γ. Comparing Eqs. 6.4.22 to 6.4.24 with Eqs. 6.3.2, to 6.3.22, it is seen that the parameter estimation procedure for the log Pearson (3) distribution using ML method is the same as the ML procedure for the P(3) distribution using the logarithms of the observations. Therefore all the conditions mentioned in Section 6.3.2 apply for this case also. PWM Method The direct application of the PWM method to the LP(3) distribution has not been reported. However, the PWM method can be applied to the transformed observation zi = log (xi) where zi is now distributed as a Pearson (3) variable and the procedure in Section 6.3.2 can be used for parameter estimation.
EXAMPLE 6.4.1 Estimate the parameters of the LP(3) distribution for the annual maximum flows of Wabash River at Logansport, Indiana given in Table 6.4.1. Use the direct and indirect MOM, MLM and PWM.
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Direct Method of Moments: Using the data from Table 6.4.1 we have
N = 68, m′1 = 33544.1, m′2 = 1.36141 ×10 ,m′3 = 6.58416 × 10 9
C v = 0.458166, C S = 1.19231, C k = 5.00025, m 2 = 2.36199 × 10
13
8
From Eqs. 6.4.12 and 6.4.14,
B = 2.92007, C = 1/ ( 2.92007 – 3 ) = – 12.5112 . Table 6.4.1 Data from Wabash River at Logansport, IN
Ann. Ann. Year Max. Flow Year Max. Flow 1942 29800 1943 89800 1944 49000 1945 31000 1946 26600 1947 31000 1948 30800 1949 40000 1924 45400 1950 70700 1925 52900 1951 33500 1926 41900 1952 34800 1927 44300 1953 28100 1928 40300 1954 13200 1929 29300 1955 27200 1930 61400 1956 25400 1931 11100 1957 39500 1932 30300 1958 52500 1933 46800 1959 69000 1934 16100 1960 28200 1935 26400 1961 35400 1936 63700 1962 38100 1937 48800 1963 37000 1938 48800 1964 40800 1939 57600 1965 27300 1940 30000 1966 14000 1941 14400 1967 41900
Ann. Year Max. Flow 1968 35600 1969 33100 1970 18000 1971 17800 1972 19900 1973 17800 1974 23900 1975 17900 1976 22300 1977 16700 1978 25800 1979 22800 1980 19800 1981 21300 1982 34100 1983 27400 1984 22200 1985 40300 1986 22900 1987 11400 1988 22800 1989 27100 1990 29600 1991 36400
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We use Eq. 6.4.16 to get,
A = – 0.47157 + 1.99955 × – 12.5112 = – 25.4883 and substituting A in Eq. 6.4.17,
αˆ = 1/ ( – 25.4883 + 3 ) = – 0.0444675 ˆ = –0.044380. The exact solution of Eq. 6.4.11 gives α From Eqs. 6.4.18 and 6.4.19, βˆ = 105.430, γˆ = 14.9987. Indirect Method of Moments: After taking logarithms of the observations,
N = 68, m′ 1 = 10.3229, m 2 = 0.200941 C v = 0.0434242, C S = – 0.0676579, C k = 2.93755 Using the same method used in Example 6.3.1, from Eqs. 6.3.15 to 6.3.17, 2 βˆ = ( 2/-0.0676579 ) = 873.8210
αˆ = – ( 0.200941/873.8210 ) = – 0.015164 γˆ = m' 1 – αβ = 10.3229 + 0.015164 × 873.821 = 23.5738 ML Method: After taking the logarithms of the observations, a P(3) distribution is fitted to the transformed data following the same procedure as in Example 6.3.1. Using the Newton method to solve Eqs. 6.4.22 to 6.4.24,
αˆ = – 0.017727, βˆ = 630.152, γˆ = 21.4936 To verify these estimates we calculate: n
ψ ( β ) = 6.44517; S 1 =
∑ ( log x i – γ )
= – 759.6036
i=1
n
S2 =
∑ 1/ ( log x i – γ ) = –6.09706 ; S 3 = i=1
n
log x i – γ
- ∑ log -------------------α
= 438.271 .
i=1
Substituting the above values into Equations 6.4.22 to 6.4.24 we get
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–759.6036 + 68 × 0.017727 × 630.152 ≈ 0
(6.4.22)
68 × 6.44517 – 438.271 ≈ 0
(6.4.23)
68 + 0.017727 (630.152 – 1) × (–6.09706) ≈ 0
(6.4.24)
The equations are satisfied, and the calculated parameters are the ML estimates. PWM Method: After taking the natural logarithms of the observations, consider the unbiased L-moments. l1 = 10.3229, l2 = 0.255077, t3 = –0.0178563 Since t3 = –0.0178563 < 1/3 we use Eq. 6.3.28:
t m = 3π × ( 0.0178563 ) = 0.00300505 2
1 + 0.2906 × 0.00300505 βˆ = ----------------------------------------------------------------------------------------------------------------------------------------------------------------2 3 [ 0.00300505 + 0.1882 × ( 0.00300505 ) + 0.0442 × ( 0.00300505 ) ] βˆ = 332.875 From Eq. 6.3.28 and 6.3.29:
Γ ( 332.875 ) αˆ = – π ×0.255077 × ---------------------------- = – 0.02479 Γ ( 333.375 ) γˆ = 10.3229 + 0.02479 × 332.875 = 18.5747 The parameter estimates computed in this example are summarized in Table 6.4.2. Table 6.4.2 Parameter Estimates for Example 6.4.1
Parameter MOM (Direct) MOM (Indirect) MLM PWM
α –0.0445 –0.01516 –0.01773 –0.02479
β 105.430 873.821 630.152 332.875
γ 14.999 23.574 21.4936 18.5747
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6.4.4 Quantile Estimation The T-year quantile is obtained by Eq. 6.4.25. z T = log x T = µ z + K T σ z
(6.4.25)
where µ z = γˆ + αˆ βˆ
(6.4.26)
σ z = αˆ βˆ
(6.4.27)
c s = 2/ βˆ
(6.4.28)
In Eq. 6.4.25 KT is the frequency factor corresponding to T-year return period and is obtained by any of the methods mentioned in Section 6.2.3. The value of xT is thus calculated by using Eq. 6.4.29. z
xT = e T
(6.4.29)
EXAMPLE 6.4.2 Estimate the 100-year flood for the Wabash River at Logansport data used in Example 6.4.1 by using the parameters estimated in Example 6.4.1. Quantile Estimates Based on MOM (Direct Method) Parameters: From Eqs. 6.4.25 to 6.4.29, µz = 14.9987 – 0.04438 × 105.430 = 10.3197.
σ z = 0.04438
105.430 = 0.45569; C S = 2/ 105.430 = 0.61928
And thus
xˆ T = exp ( 10.3197 + 0.45569 K T ) For example, if T = 100 we have u = 2.326785. By using the Wilson-
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Hilferty transformation in Eq. 6.2.38, KT = 2.1823.
∴ xˆ 100 = exp ( 10.3197 + 0.45569 × 2.1823 ) = 81974 cfs Quantile Estimate Based on MOM (Indirect) Parameters: From Eqs. 6.4.26 to 6.4.28:
µ z = 23.5738 – 0.015164 × 873.8210 = 10.32318 σ z = 0.015164
873.8210 = 0.44825; C S = 2/ 873.8210 = 0.06766
Thus,
xˆ T = exp ( 10.32318 + 0.44825 K T ) For example if T = 100, u = 2.326785 and K100 = 2.2765.
xˆ 100 = exp ( 10.32318 + 0.44825 × 2.2765 ) = 84425 cfs Quantile Estimates Based on ML Parameters: From Eqs. 6.4.26 to 6.4.28:
µ z = 21.4936 – 0.01773 × 630.152 = 10.3225 σ z = 0.01773 630.152 = 0.445073 C S = 2/ 630.152 = 0.07967 Thus:
xˆ T = exp ( 10.3225 + 0.445229 K T ) For example if T = 100 then u = 2.326785 and using the WilsonHilferty transformation KT = 2.2676
xˆ 100 = exp ( 10.3229 + 0.445229 × 2.4566 ) = 83450 cfs
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Quantile Estimates Based on PWM Parameters: From Eqs. 6.4.26 to 6.4.28:
µ z = 18.5747 – 0.02479 × 332.875 = 10.3229 σ z = 0.02479 332.875 = 0.45229;C S = 2/ 332.875 = 0.10962 . Thus:
xˆ T = exp ( 10.3229 + 0.45229 K T ) For example if T = 100 then u = 2.326785, using the Wilson-Hilferty transformation K 100 = 2.24546
xˆ 100 = exp ( 10.3229 + 0.45229 × 2.24546 ) = 83994 cfs Quantile estimates for this and other recurrence intervals are given in Table 6.4.3.
6.4.5 Standard Error Method of Moments An approximate standard error in this case can be obtained in logarithmic units using the equations in Section 6.3.4 and may then be converted to original units. An exact method to estimate the standard error is given by Hoshi and Burges (1981b). The standard error of xT is given by Eq. 6.4.30 which is derived from Eq. 6.4.29 ∂x 2 2 2 s T = -------T- var ( z T ) = x T var ( z T ) ∂z T
(6.4.30)
In Eq. 6.4.30, var (zT) is computed by Eq. 4.4.3 replacing x by z. To obtain the partial derivatives we consider Eq. 6.4.25 to get equations identical to Eqs. 6.3.42 to 6.3.44 but for the derivatives of zT instead of xT. The variances and covariances of the parameters α, β, and γ are obtained by Eq. 6.4.31.
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–1
S 11 var ( γ ) 0 cov ( γ , α ) cov ( γ , β ) = 0 0 var ( α ) cov ( α, β ) 0 var ( β ) 0
S 12 S 13 S 14 S 15 S 16
var ( m′1 )
S 22 S 23 S 24 S 25 S 26
cov ( m′1 , C v )
S 32 S 33 S 34 S 35 S 36
cov ( m′1 , C S )
0
0 S 44 S 45 S 46
var ( C V )
0
0 S 54 S 55 S 56
cov ( C v, C S )
0
0 S 64 S 65 S 66
var ( C s )
(6.4.31)
In Eq. 6.4.31, m′1 , Cv , and Cs are the sample mean, coefficient of variation and coefficient of skew. Bobée (1973) has given the following relationships for the variances and covariances of m′1 , Cv , and Cs. var ( m′ 1 ) = µ 2 /n 2
C var ( C V ) = ------V N
(6.4.32)
Ck – 1 2 -------------- + Cv – Cs Cv 4
(6.4.33)
9 1 2 2 2 2 var ( C s ) = ---- C 2 – C S + 9 – 6C K + --- ( C S C K – C S ) – 3 C S C 1 + 12 C S 4 N (6.4.34) 1 µ C cov ( m′1 , C V ) = ---- ------2- ------S – C v N µ′1 2 1⁄2
(6.4.35)
µ2 C - C K – 3 1 + ------s cov ( m′ 1, C s ) = ------- N 2 2
C cov ( C V , C S ) = ------V n
2
C 3 C 3 2 1 C 3C – --- ------S C K + --- ------S – C K + 3 + --- C S + --- ------1 – 2 ------S Cv 4 CV 2 4 CV 2 Cv
(6.4.36)
(6.4.37)
The values of Cv , Cs, Ck, C1, and C2 are given in Eqs. 6.4.51 to 6.4.55. The values of Sij in Eq. 6.4.31 are as follows ∂f 2 ∂f ∂f ∂f ∂f S 11 = ------ , S 12 = 2 ------ · ------- , S 13 = 2 ------ · ----- ∂γ ∂γ ∂α ∂γ ∂β ∂f 2 ∂f 2 ∂f ∂f S 14 = ------- , S 15 = 2 ------- · ------, S 16 = ------ ∂α ∂β ∂α ∂β
(6.4.38)
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∂f ∂g ∂f ∂g ∂f ∂g S 22 = ------ · ------- , S 23 = ------ · ------, S 24 = ------- · ------∂γ ∂α ∂γ ∂β ∂α ∂α ∂f ∂g ∂f ∂g ∂f ∂g S 25 = ------- · ------ + ------ · ------- , S 26 = ------ · -----∂α ∂β ∂β ∂α ∂β ∂β
(6.4.39)
∂f ∂h ∂f ∂h ∂f ∂h S 32 = ------ · ------- , S 33 = ------ · ------, S 34 = ------- · ------∂γ ∂α ∂γ ∂β ∂α ∂α ∂f ∂h ∂f ∂h ∂f ∂h S 35 = ------- · ------ + ------ · -------, S 36 = ------ · -----∂α ∂β ∂β ∂α ∂β ∂β
(6.4.40)
∂g 2 ∂g ∂g ∂g 2 S 44 = ------- , S 45 = 2 ------- · ------, S 46 = ------ ∂α ∂β ∂α ∂β
(6.4.41)
∂g ∂h ∂g ∂h ∂g ∂h ∂g ∂h S 54 = ------- · -------, S 55 = ------- ------ + ------ -------, S 56 = ------ · -----∂α ∂α ∂α ∂β ∂β ∂α ∂β ∂β
(6.4.42)
∂h 2 ∂h 2 ∂h ∂h S 64 = ------- , S 65 = 2 ------- ------, S 66 = ------ ∂α ∂β ∂α ∂β
(6.4.43)
∂f ∂f ∂f –1 where ------ = m′1 , ------- = β ( 1 – α ) m′1 , ------ = – log ( 1 – α ) m′1 ∂α ∂β ∂γ 1 ∂A –1 ∂B –1 ∂g 1 ∂A –1 ∂g ------- = C v --- ------- A – ------- B , ------ = C v --- ------- A 2 ∂α 2 ∂β ∂α ∂β ∂α
(6.4.44)
∂b –1 ------ B ∂β
(6.4.45)
∂C –1 3 ∂A –1 ∂h ∂C –1 3 ∂A –1 ∂h ------- = C s ------- C – --- ------- A , ------ = C s ------- C – --- ------- A ∂β ∂α 2 ∂α ∂β 2 ∂β ∂α
(6.4.46)
–β
and A = ( 1 – 2α ) – ( 1 – α ) B = (1 – α) –β
-
– 2β
(6.4.47)
–β
(6.4.48) –β
C = ( 1 – 3α ) – 3 [ ( 1 – α ) ( 1 – 2α ) ] + 2 ( 1 – α ) f = m′ 1 = exp ( γ ) ( 1 – α )
–β
– 3β
(6.4.49) (6.4.50)
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g = Cv = A
1⁄2
h = C s = C/ A
(6.4.51)
/B 3⁄2
(6.4.52)
–β
Also, if a r = ( 1 – rα ) , then 2
4
a 4 – 4a 3 a 1 + 6a 2 a 1 – 3a 1 C k = ---------------------------------------------------------2 A 2
(6.3.53)
2 3
5
a 5 – 5a 4 a 1 + 10a 3 a 1 – 10a a 1 + 4a 1 C 1 = -----------------------------------------------------------------------------------5⁄2 A 2
3
4
(6.4.54)
6
a 6 – 6a 5 a 1 + 15 a 4 a 1 – 20a 3 a 1 + 15a 2 a 1 – 5a 1 C 2 = ----------------------------------------------------------------------------------------------------------3 A
(6.4.55)
∂A –β – 1 – 2β – 1 – (1 – α) ] ------- = 2β [ ( 1 – 2α ) ∂α
(6.4.56)
∂A –β – 2β ------ = – ( 1 – 2α ) log ( 1 – 2α ) + 2 ( 1 – α ) log ( 1 – α ) ∂β
(6.4.57)
∂B ∂B ------- B –1 = β ( 1 – α ) –1, ------B –1 = – log ( 1 – α ) ∂β ∂α
(6.4.58)
with
∂C –β–1 –β–1 –3 β – 1 ------- = 3β [ ( 1 – 3α ) + ( 4α – 3 ) { ( 1 – α ) ( 1 – 2α ) } + 2(1 – α) ] ∂α ∂C –β –β ------- = – ( 1 – 3α ) log ( 1 – 3α ) + 3 [ ( 1 – α ) ( 1 – 2α ) ] ∂β – 6(1 – α)
– 3β
(6.4.59)
(6.4.60)
log ( 1 – α )
ML Method: As seen in Section 6.4.2b, fitting a LP(3) distribution to the data using the ML method is the same as fitting a P(3) distribution to the logarithms of the data. Therefore, we can obtain the standard error in the
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log-space using the equations in Section 6.3.4b. The standard error in the real space can then be obtained from Eq. 6.4.30.
EXAMPLE 6.4.3 Estimate the standard errors of the 100-year flood for the Wabash River (at Logansport, Indiana) data in Table 6.4.1 by using the parameters estimated in Example 6.4.1. Standard Errors Based on MOM (Direct Method): From Eqs. 6.4.54 and 6.4.55 we get C1 = 18.97278, C2 = 92.5468. We find the variances and covariances of the moments by using Eqs. 6.4.32 to 6.4.37. 8
6
var ( m′ 1 ) = 2.36199 × 10 /68 = 3.47351 × 10 ; var ( C V ) = 0.002371 .
var ( C S ) = 0.455083 ;cov ( m′ 1, C V ) = 14.289805 . cov ( m′ 1, C S ) = 64.375952; Cov ( C V , C S ) = 0.020356 From Eqs. 6.4.47 to 6.4.52 we get the following estimates. A B
= =
2.2158 × 10–5 0.010274
C f g h
= = = =
1.24366 × 10–7 33544.12 0.458166 1.192314
From Eqs. 6.4.56 to 6.4.60 and 6.4.44 to 6.4.46 we get
∂A ∂A –6 ------- = 0.003423, ------ = – 1.69 × 10 ∂β ∂α ∂B ∂B ------- B –1 = 100.95, ------B –1 = – 0.043423 ∂α ∂β ∂C – 5 ∂C –8 ------- = 2.518 ×10 , ------- = – 1.333 × 10 ∂β ∂α
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∂f 4 ∂f 6 ∂f 3 ------ = 3.3544 ¥ 10 , ------- = 3.386 ¥ 10 , ------ = – 1.4566 ¥ 10 ∂b ∂a ∂g ∂g ∂g ------- = – 10.8666, ------ = 0.0023864 ∂b ∂a ∂h ∂h ------- = – 34.8377, ------ = 0.0088823 ∂b ∂a From Eqs. 6.4.31 and 6.4.38 to 6.4.43 we get: var ( g ) 108.247 cov ( g , a ) 1.07274 cov ( g , b ) = 4986.36 var ( a ) 0.0106974 cov ( a, b ) 49.5722 var ( b ) 230067
The partial derivatives in Eq. 4.4.3 can be obtained for the case with T = 100,
∂K T /∂C S = 0.743242 ∂z T -------- = b – K T ∂a K ∂z -------T- = a + ------T 2 ∂b
b = 83.02277
2 2 ∂z a ∂K a ----- – ---------- ---------T- = – 0.0400487 ; -------T- = 1 b ∂C S ∂g b
var ( Z 100 ) = ( 83.0228 ) ¥ 0.0106977 + ( – 0.0400487 ) ¥ 230067 2
2
+ 1 ¥ 108.247 + 2 ¥ 83.0228 ¥ ( – 0.0400487 ) ¥ ( 49.5722 ) + 2 ¥ 83.0228 ¥ 1 ¥ 1.07274 + 2 ¥ ( – 0.0400487 ) ¥ 1 ¥ 4986.36 = 0.0199 Substituting these in Eq. 6.4.30:
s T = ( 81974 ) ¥ 0.0199 = 1.33728 ¥ 10 ; s T = 11564 cfs 2
2
8
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Standard Error Based on MOM (Indirect): Applying the same method used in Example 6.3.1 we find first from Eq. 6.2.38: ∂K 100 –2 ------------- = ------2 ∂C s C8
3
CS C 2 - u – ------S + 1 -1 + ----- ---- 6 6 C S
2
C u 2C C 3 ------S u – ------S + 1 --- – ---------S 6 6 6 36
∂K 100 ------------ = 0.738549 ∂C S Substituting the above in Eq. 6.3.34, var (z100) = 0.0198213. Substituting these in Eq. 6.4.30:
s T = ( 83450 ) × 0.0198213 = 1.380336 × 10 ; s T = 11749 cfs 2
2
8
Standard Error Based on ML Method: Consider the case where T = 100 ∂K 100 –2 ------------- = ------2 ∂C S CS
3
CS C 2 - u – ------S +1 – 1 + ----- ---- 6 C 6 S
2
C u 2C C 3 ------S u – ------S + 1 --- – ---------S 6 6 6 36
∂K 100 ------------ = 0.739072 ∂C s From Eq. 6.2.54:
ψ′ ( β ) = ψ′ ( 630.152 ) = 0.00158818 From Eqs. 6.3.35 to 6.3.41:
D = 2.15782 × 10 ; var ( α ) –5
var ( β ) = 2.1974 × 10 ; 7
= 0.00436349
var ( γ ) = 1721.72
cov ( α, β ) = 309.568; cov ( α, γ ) = 2.773803 cov ( β, γ ) = 1.94456 × 10
5
The partial derivatives are obtained from Eqs. 6.3.42 to 6.3.44.
∂x -------T- = 573.228 ∂α
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∂x T -------- = – 0.016905 ∂β ∂x -------T- = 1 ∂γ Substituting the above values in Eq. 4.4.19:
var ( z T ) = ( 573.228 ) × 0.00436349 + ( – 0.016905 ) 2
+ 1 × 1721.72 + 2
2
×2.1974 × 10
7
573.228 × ×( – 0.016905 ) × 309.568
+ 2 573.228 × ×1 × 2.73803 + 2 5 × 10 = 0.018244
(×– 0.016905 ) ×1 × 1.94456
From Eq. 6.4.30:
s T = ( 83450 ) × 0.018244 = 1.27049 × 10 ; 2
2
8
s T = 11272 cfs
The quantile estimates obtained by different methods and the standard errors where they are available are given in Table 6.4.3. The results are obtained by the computer program discussed in Chapter 10. The quantile plots of the MOM, ML, and PWM methods for these data are shown in Fig. 6.4.1. Table 6.4.3 Quantile Estimates and their Standard Errors (in parentheses) for Example 6.4.2
T P (%) 10 10 20
5
50
2
100
1
200
0.5
MOM (Direct) 53821.42 (3755.01) 62519.85 (5216.86) 73672.21 (8334.61) 81977.8 (11565.58) 90233.58 (15503.10)
MLM 53598.69 (3832.9) 62608.87 (5435.89) 74437.15 (8395.38) 83449.88 (11271.71) 92581.89 (14699.03)
PWM 54013.07 63101.87 74981.73 83994.4 93092.08
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Figure 6.4.1 Observed and estimated flows and 95% confidence intervals for the Wabash River data used in Examples 6.4.1 to 6.4.3.
6.5
U.S. Water Resources Council Method (WRCM)
6.5.1 Introduction The LP(3) distribution was recommended by the U.S. Water Resources Council for flood frequency analysis related to federal projects (U.S. Water Resources Council, 1967, 1976, 1977, and 1981, Benson, 1968). This recommendation was an attempt to provide a uniform approach
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to flood frequency analysis. As mentioned earlier, research on LP(3) distribution has brought out problems related to and disadvantages of using this distribution. Still, it is being widely used in the U.S. One of the important problems related to the use of the LP(3) distribution is the variability in skewness coefficients. Data from several stations in Indiana were investigated by McCormick and Rao (1995). The locations of the stations used in their study are shown in Figure 6.5.1. The skewness coefficients, Cs, for different stations are shown in Figure 6.5.2. In Figure 6.5.2, two statistics are given for each station. The statistic for the observed data is given at the top, while the statistic for the log-transformed data is given at the bottom. For example, at station 1, the value of Cs, is 0.887 for the observed data and –0.571 for the log-transformed data. Results in Figure 6.5.2 clearly bring out the difficulty of developing a map of skewness coefficients Cm. Nearby stations with no significant changes in topography have noticeably different skewness coefficients. For example, stations 1, 2, and 3, which are very close to each other, have values of Cs of 0.887, 0.825 and 0.248 for the observed data and –0.571, –0.267 and –0.556 for the log-transformed data. Similarly, stations 5 and 6 have values of Cs of 1.320 and –0.313 for the observed data, and –0.011 and –2.338 for the log-transformed data; stations 9, and 11 have Cs values of 0.226 and 1.018 for the observed data and –0.859 and 0.020 for the log-transformed data; stations 21, 22, and 24 have values of Cs of 0.705, 0.766, and 1.079 for the observed data and –0.544, –1.238 and –0.038 for the log-transformed data. Because of the proximity of these stations their skewness coefficients are expected to be close to each other. Similar skewness coefficients in an area imply that a skewness coefficient contour can be drawn for that area. Such a contour map is shown in Figure 6.5.3, according to which the skewness coefficient for most of Indiana is –0.4. However, there is a wide variability in skewness coefficients. Consequently, this kind of map shown in Figure 6.5.3 does not appear to be justified. However, the information in Figure 6.5.3 plays an important role in WRCM. 6.5.2 Frequency Analysis Procedure by the WRCM The annual maximum flood data are log (to the base 10) transformed. The mean, ( y ) , standard deviation ( s y ) and the skewness coefficient ( c s ) of the log-transformed data are computed. Cs is computed by Eq. 6.5.1, where N is the number of observations and yi are the logtransformed flood values.
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Figure 6.5.1.
Locations of stations with reference number.
N
N ∑ ( yi – y ) i=1 C s = ------------------------------------------3 ( N – 1 ) ( N – 2 )s y 3
(6.5.1)
The station skewness coefficient given by Eq. 6.5.1 is highly variable (Figure 6.5.2). Rather than using such an estimate, the Water
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Figure 6.5.2.
Skewness coefficients of stations.
Resources Council recommended the use of a weighted estimate Cw given by Eq. 6.5.2, where Cm is a “map skewness.” The map skewness Cw =
α C s + ( 1 – α )C m
(6.5.2)
is read off from Figure 6.5.3 for the location of interest. If the number of observations available for analysis is less than 25, then the map
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Figure 6.5.3.
Coefficient of map skew, Cm, for use with WRCM.
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skewness is used in the frequency analysis. If it is greater than 100, station skewness is used. If N is between 25 and 100 both station and map skewness values are used. The weight α is defined by Eq. 6.5.3. V (Cm) α = ------------------------------------V (Cm) + V (Cs)
(6.5.3)
In Eq. 6.5.3, V(Cm) is the variance of the map skewness Cm and is equal to 0.3025. The variance of the skewness coefficient, V(Cs), for LP(3) distribution has been shown by Wallis et al. (1974) to be:
V ( C s ) = 10
N A – B log 10 ------ 10
(6.5.4)
where, A = –0.33 + 0.08 |Cs| if |Cs|≤ 0.9 A = –0.52 + 0.3 |Cs| if |Cs| > 0.9 B = 0.94 – 0.26 |Cs| if |Cs| ≤ 1.5
(6.5.5)
B = 0.55 if |Cs| > 1.50 |Cs|is the absolute value of the station skew. The frequency factor KT is computed by using the approximation given by Kite (1977), in Eq. 6.5.6 1 3 2 2 2 3 4 5 K T = z – ( z – 1 )k + --- ( z – 6z ) k – ( z – 1 )k + zk + ( 1 ⁄ 3 )k (6.5.6) 3 where k = Cw/6. In Eq. 6.5.6, z is the standard normal variable corresponding to the return period T. Table 6.5.1 may also be used to estimate KT. The logarithm of the flood magnitude corresponding to recurrence interval T is given by Eq. 6.5.7. Y T = Y + K T sy Q T = 10
YT
(6.5.7) (6.5.8)
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Table 6.5.1. KT Values for Pearson Type III Distribution (Positive Skew)
Skew Coefficient Cs or Cw 3.0 2.9 2.8 2.7 2.6 2.5 2.4 2.3 2.2 2.1 2.0 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0
2
5
0.50 –0.396 –0.390 –0.384 –0.376 –0.368 –0.360 –0.351 –0.341 –0.330 –0.319 –0.307 –0.294 –0.282 –0.268 –0.254 –0.240 –0.225 –0.210 –0.195 –0.180 –0.164 –0.148 –0.132 –0.116 –0.099 –0.083 –0.066 –0.050 –0.033 –0.017 0
0.20 0.420 0.440 0.460 0.479 0.499 0.518 0.537 0.555 0.574 0.592 0.609 0.627 0.643 0.660 0.675 0.690 0.705 0.719 0.732 0.745 0.758 0.769 0.780 0.790 0.800 0.808 0.816 0.824 0.830 0.836 0.842
Return Period in Years 10 25 50 Exceedence Probability 0.10 0.04 0.02 1.180 2.278 3.152 1.195 2.277 3.134 1.210 2.275 3.114 1.224 2.272 3.093 1.238 2.267 3.071 1.250 2.262 3.048 1.262 2.256 3.023 1.274 2.248 2.997 1.284 2.240 2.970 1.294 2.230 2.942 1.302 2.219 2.912 1.310 2.207 2.881 1.318 2.193 2.848 1.324 2.179 2.815 1.329 2.163 2.780 1.333 2.146 2.743 1.337 2.128 2.706 1.339 2.108 2.666 1.340 2.087 2.626 1.341 2.066 2.585 1.340 2.043 2.542 1.339 2.018 2.498 1.336 1.993 2.453 1.333 1.967 2.407 1.328 1.939 2.359 1.323 1.910 2.311 1.317 1.880 2.261 1.309 1.849 2.211 1.301 1.818 2.159 1.292 1.785 2.107 1.282 1.751 2.054
100
200
0.01 4.051 4.013 3.973 3.932 3.889 3.845 3.800 3.753 3.705 3.656 3.605 3.553 3.499 3.444 3.388 3.330 3.271 3.211 3.149 3.087 3.022 2.957 2.891 2.824 2.755 2.686 2.615 2.544 2.472 2.400 2.326
0.005 4.970 4.909 4.847 4.783 4.718 4.652 4.584 4.515 4.444 4.372 4.298 4.223 4.147 4.069 3.990 3.910 3.828 3.745 3.661 3.575 3.489 3.401 3.312 3.223 3.132 3.041 2.949 2.856 2.763 2.670 2.576
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Table 6.5.1. (cont.)
Skew Coefficient Cs or Cw –0.1 –0.2 –0.3 –0.4 –0.5 –0.6 –0.7 –0.8 –0.9 –1.0 –1.1 –1.2 –1.3 –1.4 –1.5 –1.6 –1.7 –1.8 –1.9 –2.0 –2.1 –2.2 –2.3 –2.4 –2.5 –2.6 –2.7 –2.8 –2.9 –3.0
KT Values for Pearson Type III Distribution (Negative Skew)
2
5
0.50 0.017 0.033 0.050 0.066 0.083 0.099 0.116 0.132 0.148 0.164 0.180 0.195 0.210 0.225 0.240 0.254 0.268 0.282 0.294 0.307 0.319 0.330 0.341 0.351 0.360 0.368 0.376 0.384 0.390 0.396
0.20 0.846 0.850 0.853 0.855 0.856 0.857 0.857 0.856 0.854 0.852 0.848 0.844 0.838 0.832 0.825 0.817 0.808 0.799 0.788 0.777 0.765 0.752 0.739 0.725 0.711 0.696 0.681 0.666 0.651 0.636
Return Period in Years 10 25 50 Exceedence Probability 0.10 0.04 0.02 1.270 1.716 2.000 1.258 1.680 1.945 1.245 1.643 1.890 1.231 1.606 1.834 1.216 1.567 1.777 1.200 1.528 1.720 1.183 1.488 1.663 1.166 1.448 1.606 1.147 1.407 1.549 1.128 1.366 1.492 1.107 1.324 1.435 1.086 1.282 1.379 1.064 1.240 1.324 1.041 1.198 1.270 1.018 1.157 1.217 0.994 1.116 1.166 0.970 1.075 1.116 0.945 1.035 1.069 0.920 0.996 1.023 0.895 0.959 0.980 0.869 0.923 0939 0.844 0.888 0.900 0.819 0.855 0.864 0.795 0.823 0.830 0.771 0.793 0.798 0.747 0.764 0.768 0.724 0.738 0.740 0.702 0.712 0.714 0.681 0.683 0.689 0.666 0.666 0.666
100
200
0.01 2.252 2.178 2.104 2.029 1.955 1.880 1.806 1.733 1.660 1.588 1.518 1.449 1.383 1.318 1.256 1.197 1.140 1.087 1.037 0.990 0.946 0.905 0.867 0.832 0.799 0.769 0.740 0.714 0.690 0.667
0.005 2.482 2.388 2.294 2.201 2.108 2.016 1.926 1.837 1.749 1.664 1.581 1.501 1.424 1.351 1.282 2.216 1.155 1.097 1.044 0.995 0.949 0.907 0.869 0.833 0.800 0.769 0.740 0.714 0.690 0.667
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6.5.3 Outlier Tests As outliers affect the statistics, especially higher order statistics, WRCM recommends testing for outliers and eliminating them. A discussion of outliers is found in Section 1.8.3. The Water Resources Council (1981) recommends that if the station skewness coefficient is greater than +0.4, tests for high outliers be performed first. If the station skewness coefficient is smaller than –0.4, tests for low outliers should be performed first. If the station skewness is within ±10.4, tests for both high and low outliers should be performed. The threshold for high and low outliers, yH are given by Eqs. 6.5.9 and 6.5.10. yH = y + K N sY
(6.5.9)
yL = y – K N sY
(6.5.10)
The factor KN is a function of the sample size N and is given in Table 6.5.2. Table 6.5.2 KN Values for Outlier Test
Sample Size N 10 11 12 13 14 15 16 17 18 19 20 21 22 23
KN 2.036 2.088 2.134 2.175 2.213 2.247 2.279 2.309 2.335 2.361 2.385 2.408 2.429 2.448
Sample Size N 24 25 26 27 28 29 30 31 32 33 34 35 36 37
KN 2.467 2.486 2.502 2.519 2.534 2.549 2.563 2.577 2.591 2.604 2.616 2.618 2.639 2.650
Sample Size N 38 39 40 41 42 43 44 45 46 47 48 49 50 55
KN 2.661 2.671 2.682 2.692 2.700 2.710 2.719 2.727 2.736 2.744 2.753 2.760 2.768 2.804
Sample Size N 60 65 70 75 80 85 90 95 100 110 120 130 140
KN 2.837 2.866 2.893 2.917 2.940 2.961 2.981 3.000 3.017 3.049 3.078 3.104 3.129
Source: U.S. Water Resources Council, 1981.
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If a high outlier is found, WRCM recommends that it be compared to available historic flood data. If the high outlier is the maximum over a long period, the outlier is treated as historic flood data and excluded from analysis. In the absence of historic flood data, the outlier is retained as a part of observed record. If a low outlier is detected, it is deleted from the record and a conditional probability adjustment applied as described by U.S. Water Resources Council (1981).
6.5.4 Confidence Limits Confidence intervals are presented with the flood estimates. The confidence level β and the significance level α are related by Eq. 6.5.11. Usually the confidence level β is 1–β α = -----------2
(6.5.11)
specified. The confidence limits are a function of the recurrence interval T and the confidence level β. The upper limit UT,β and the lower limit LT,β are given by Eqs. 6.5.2 and 6.5.13 where KU and KL are the upper and lower confidence limit factors, which are also functions of T and β. UT
β,
= y + K U sY
(6.5.12)
LT
β,
= y + K Lsy
(6.5.13)
The confidence factors KU and KL are computed as follows. 1. 2.
The standard normal variate zα with exceedence probability α is estimated first. The frequency factor KT, corresponding to the recurrence interval T is estimated by Eqs. 6.5.6 or Table 6.5.1. The parameters p and q are computed by using Eqs. 6.5.14 and 6.5.15. 2
zα p = 1 = -------------------2( N – 1)
(6.5.14)
2
z 2 q = K T – ----α N
(6.5.15)
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3.
The confidence limit factors KU and K2 are computed by using Eqs. 6.5.16 and 6.5.17. 2
K T + K T – pq K U = -----------------------------------p
(6.5.16)
2
K T – K T – pq K L = ----------------------------------p 4.
(6.5.17)
The confidence limits are computed by using Eqs. 6.5.12 and 6.5.13.
EXAMPLE 6.5.1 Estimate the 100-year flood for the annual maximum flows of Wabash River at Logansport, Indiana given in Table 6.4.1. Check for outliers. Estimate the 90% confidence interval. a. Compute the statistics of logarithms of annual maximum flows.
y = 4.4832 sy = 0.1947 cs = –0.0677 b. As N is between 25 and 100, read off the map skewness and compute the weighted skewness coefficient. The map skewness coefficient is –0.4; N = 68.
A = – 0.33 + 0.08 – 0.0677 = 0.3245
(From Eq. 6.5.5)
B = 0.94 – 0.26 – 0.0677 = 0.9223
(From Eq. 6.5.5)
V ( C S ) = A – 10
N A – B log 10 x ------ 10
= 0.0808
(From Eq. 6.5.4)
Substituting these in Eq. 6.5.2,
0.3025 × – 0.0677 + 0.0808 + ( – 0.4 ) c w = ---------------------------------------------------------------------------------------- = 0.1382 0.3025 + 0.0808
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c. Compute K100 by Eq 6.5.6 or by Table 6.5.1 with c ω = –0.1382. K100 = 2.2245 log Q100 = 4.4832 + 2.2245 × 0.1947 = 4.9159 Q100 = 82393 cfs d. Develop 90% confidence intervals.
1 – 0.9 α = ---------------- = 0.05 2
(From Eq. 6.5.11)
From normal distribution tables, z = 1.6449 2
1.6449 p = 1 – ----------------------- = 0.9798 2 ( 68 – 1 )
(From Eq. 6.5.14)
2
1.6449 2 q = 2.2245 – ------------------ = 4.9088 68
(From Eq. 6.5.15)
From Eqs. 6.5.16 and 6.5.17,
2.2245 + 2.2245 – 0.9798 × 4.9088 K U = ------------------------------------------------------------------------------------------- = 2.6507 0.9798 2
2.2245 – 2.2245 – 0.9798 × 4.9088 K L = ------------------------------------------------------------------------------------------ = 1.8900 4.9088 2
From Eqs. 6.5.12 and 6.5.13
U 100, 0.9 = 4.4832 + 2.6507 × 0.1947 Q U = 10
U 100, 0.9
= 9.984 × 10 cfs 4
L 100, 0.9 = 4.4832 + 1.89 × 0.1947 Q L = 10
U 100, 0.9
= 70, 994 cfs
e. Check for outliers. From Table 6.5.2, KN = 2.8822
∴ y H = 4.4832 + 2.8822 × 0.1947 y L = 4.4832 – 2.8822 × 0.1947
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Q H = 10 Q L = 10
yH
yL
= 1.1063 × 10 cfs 5
= 8.3577 × 10 cfs 3
There are no outliers for these data. The results for other return periods are given in Table 6.5.3 and plotted in Figure 6.5.4. Table 6.5.3
T 3 10 20 100 200
Figure 6.5.4.
Results for Example 6.5.1
UT,0.9 42904 61343 76774 99842 111891
QT 38748 53656 62460 82476 91104
LT,0.9 35345 48043 55222 70994 77609
Results for Example 6.5.1.
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EXAMPLE 6.5.2 The annual maximum flood data for Floyd River is shown in Table 6.5.4 (U.S. Water Resources Council, 1977). Check the data for outliers. If any outliers are detected, remove the same and recompute the floods for recurrence intervals of 5, 10, 25, 50, 100, and 200 years. Table 6.5.4. Annual Flood Peaks in Floyd River
Year 1929 1930 1931 1932 1933 1934 1935 1936 1937 1938 1939 1940 1941 1942 1943
Flood Peaks
1460 4050 3570 2060 1300 1390 1720 6280 1360
Year 1944 1945 1946 1947 1948 1949 1950 1951 1952 1953 1954 1955 1956 1957 1958
Flood Peaks 7440 5320 1400 3240 2710 4520 4840 8320 13900 71500 6250 2260 318 1330 970
Year 1959 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973
Flood Peaks 1920 15100 2870 20600 3810 726 7500 7170 2000 829 17300 4740 13400 2940 5660
(U.S. Water Resources Council, 1977)
a. Compute the statistics of logarithms of the data.
y = 3.5553 sy = 0.464 Cs = 0.3566 The map skewness coefficient is –0.2. The weighted skewness coefficient by Eq. 6.5.2 is 0.1659. b. From Table 6.5.2 the KN value if 2.6710. yH = 3.5553 + 2.6710 × 0.464 yL = 3.5553 = 2.6710 × 0.464 QH = 6.2395 × 104
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QL = 207 The 715000 cfs flood is a high outlier. c. Eliminate the high outlier and recompute the statistics following the procedure in USWRC (1981).
y = 3.5374 sy = 0.4383 Cs = 0.1647 The weighted skewness coefficient by Eq. 6.5.4 is 0.0771. d. The floods corresponding to the different recurrence intervals along with their 90% confidence intervals are given in Table 6.5.5, and the resulting plot is shown in Figure 6.5.5a. Table 6.5.5. Floods and 90% Confidence Intervals
T 5 10 20 100 200
UT,0.90 8509 22462 36872 97347 140549
QT 6147 14381 21881 45158 66594
LT,0.90 4621 10200 14838 30113 39144
The results are plotted in Figure 6.5.5. The results of frequency analysis after removing the outlier are given in Table 6.5.6. The resulting plot is shown in Figure 6.5.5. Table 6.5.6.
Results without Outlier
T UT,0.90 5 8509 10 22462 20 36872 100 97347 200 140549
QT 8906 13786 19583 36944 46257
LT,0.90 4621 10200 14838 30113 39144
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Figure 6.5.5
Results for Example 6.5.2.
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CHAPTER 7
Extreme Value Distributions
7.1
Generalized Extreme Value (GEV) Distribution
The probability density function of the GEV distribution is of the form x–u 1 f ( x ) = --- 1 – k ----------- α α
x–u 1/k – 1 – 1 – k ----------- α
e
1⁄k
(7.1.1)
The range of the variable x depends on the sign of the parameter k. When k is negative (Type II extreme value distribution EVA2(3), CS > 1.1396) the variable x can take values in the range u + α/k < x < ∞ which makes it suitable for flood frequency analysis. However, when k is positive, (Type III extreme value distribution EV3(3), CS < 1.1396) the variable x becomes upper bounded and takes values in the range –∞ < x < u + α/k which may not be acceptable for analyzing floods unless there is sufficient evidence that such an upper bound does exist. When k = 0 (CS = 1.1396) the GEV distribution reduces to the type I extreme value distribution (EVA1(2)) discussed in Section 7.2. The GEV distribution function is of the form (Jenkinson, 1955) in Eq. 7.1.2 x–u F ( x ) = exp – 1 – k ----------- α
1/k
(7.1.2)
Parameter estimation in the generalized extreme value distribution has recently received some attention. Phien and Fang (1989) derived equations to estimate the parameters of the GEV distribution by the ML method. In-Na and Nguyen (1989) introduced a new unbiased plotting position formula for the GEV distribution. The PWM was used by them to estimate the exact plotting positions and a simple but reliable formula was proposed for practical use. This formula can explicitly take into account the skewness coefficient of the data.
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The concept of partial probability weighted moments (PPWM) which can be used to estimate a distribution from censored samples was introduced by Wang (1990a). He derived unbiased estimators of PPWM and applied the method to estimate the parameters of GEV distribution. Wang (1990b) derived expressions for unbiased estimators of PWM and PPWM for use with recorded and historic flood information. He also considered parameter and quantile estimation of GEV distribution. The asymptotic variance of probability weighted moments and normalized quantile estimators for the GEV distribution were derived by Lu and Stedinger (1992a). The normalized estimators are at-site quantile estimators divided by the sample mean. Lu and Stedinger (1992a) derived a regional homogeneity test for GEV/PWM analysis. Simple formulas are derived for sampling variances of quantile estimators of GEV distributions by Lu and Stedinger (1992b). These are based on PWM or L-moments.
7.1.1 Parameter Estimation Method of Moments: The first moment of the GEV distribution is calculated (assume k < 0) by using Eq. 7.1.3. ∞
µ′1 =
∫
f ( x )x dx
(7.1.3)
u + α/k
Substituting f(x) from Eq. 7.1.1 and using the transformation in Eq. 7.1.4 the integral reduces to Eq. 7.1.5. x–u y = 1 – k ----------- α ∞
µ′ 1 =
∫
1/k
α k –y α --- – --- y + u e dy k k
(7.1.4)
(7.1.5)
0
Performing the integration in Eq. 7.1.5 results in the value of Eq. 7.1.6.
µ′1
in
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α µ′ 1 = u + --- [ 1 – Γ ( 1 – k ) ] k
(7.1.6)
The second central moment is calculated by using the substitution in Eq. 7.1.4, ∞
µ2 =
∫
2
α --- { Γ ( 1 + k ) – y k } e – y dy k
(7.1.7)
0
α µ 2 = -----2 k
2
∞
∫ [ Γ ( 1 + k ) – 2y Γ ( 1 + k ) + y 2
k
2k
]e
–y
(7.1.8)
dy
0
α 2 µ 2 = -----2 [ Γ ( 1 + 2k ) – Γ ( 1 + k ) ] k 2
(7.1.9)
Similarly, the third central moment is given by Eq. 7.1.10: α 3 µ 3 = -----3 [ – Γ ( 1 + 3k ) + 3Γ ( 1 + k )Γ ( 1 + 2k ) – 2Γ ( 1 + k ) ] (7.1.10) k 3
The skewness coefficient CS is calculated by Eq. 7.1.11. 3⁄2
C S = µ 3 /µ 2
k ( – Γ ( 1 + 3k ) + 3Γ ( 1 + k )Γ ( 1 + 2k ) – 2Γ ( 1 + k ) ) - (7.1.11) = ----- -----------------------------------------------------------------------------------------------------------------------3⁄2 2 k [ Γ ( 1 + 2k ) – Γ ( 1 + k ) ] 3
Eq. 7.1.11 is solved numerically to get the value of kˆ . The relationship between Cs and k is shown in Figure 7.1.1. Approximate relationships between the value of k and the skewness coefficient obtained through regression analysis are given below. a. EV2(2):
k < 0 (1.14 < CS < 10), R2 = 1 2
3
k = 0.2858221 – 0.357983 C S + 0.116659 C S – 0.022725C S 4 5 6 + 0.002604 C S – 0.000161 C S + 0.000004 C S
(7.1.12)
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Figure 7.1.1. tribution.
Relationship between the skewness coefficient and k for the GEV dis-
b. EV3(3):
k > 0 (–2 < CS < 1.14), R2 = 1 2
3
k = 0.277648 – 0.322016 C S + 0.060278 C S + 0.016759 C S 4 5 6 – 0.005873 C S – 0.00244 C S – 0.000050 C S
(7.1.13)
For negative values of skew, there exists two possible values of k; one is given by Eq. 7.1.13 (EV3(3)) and the other one is given by Eq. 7.1.14.
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c. EV2(3):
k < 0 (–10 < CS < 0), R2 = 0.999978 2
3
k = – 0.50405 – 0.00861 C S + 0.015497 C S + 0.005613 C S 4 5 + 0.00087 C S + 0.000065 C S
(7.1.14)
For negative skew values one can choose the value of k which gives the best fit to the data. To solve Eq. 7.1.11 numerically the NewtonRaphson method is used. An initial value of kˆ is obtained by using the approximations in Eqs. 7.1.12 to 7.1.14. The initial value ko is updated by Eq. 7.1.15. k n + 1 = k n – F ( k n )/F′ ( k n )
(7.1.15)
gr = Γ(1 + rk)
(7.1.16)
dr = Γ′(1 + rk) = Γ(1 + rk) ψ(1 + rk)
(7.1.17)
If we define
and
where ψ(⋅) can be calculated using Eq. 6.2.53, then 3
k – g 3 + 3g 1 g 2 – 2g 1 F ( k ) = – C S + ----- -----------------------------------------2 3⁄2 k ( g2 – g1 )
(7.1.18)
2
k – 3d 3 + 3d 1 g 2 + 6g 1 d 2 – 6g 1 d 1 ----- ----------------------------------------------------------------------2 3⁄2 F′ ( k ) = k ( g2 – g1 )
(7.1.19)
3k ( – g 3 + 3g 1 g 2 – 2g 1 ) ( 2d 2 – 2g 1 d 1 ) – --------- ----------------------------------------------------------------------------------2 5⁄2 2k ( g2 – g1 ) 3
The iteration in Eq. 7.1.15 is repeated until F(k) is sufficiently close to zero. Once kˆ is estimated, it is substituted into Eqs. 7.1.6 and 7.1.9 replacing µ′1 and µ 2 by their sample estimates m′1 and m2 to get αˆ and uˆ 2
2 αˆ = [ m 2 kˆ / { Γ ( 1 + 2kˆ ) – Γ ( 1 + kˆ ) } ]
1⁄2
(7.1.20)
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αˆ uˆ = m′ 1 – --- [ 1 – Γ ( 1 + kˆ ) ] kˆ
(7.1.21)
Maximum Likelihood (ML) Method The application of the ML method to the GEV distribution is given by Jenkinson (1969) and Flood Studies Report (1975). The likelihood function is given by 1 x i – u L = ∏ --- 1 – k ----------- α α i = 1 N
x i – u 1/k – 1 – 1 – k ----------- α
1/k
e
(7.1.22)
Instead of using the likelihood function in Eq. 7.1.22, Jenkinson uses the transformation in Eq. 7.1.23. α – ky x = u + --- ( 1 – e ) k
(7.1.23)
The pdf in Eq. 7.1.1 is now given by Eq. 7.1.24. 1 –y –y(1 – k ) f ( y ) = --- exp ( – e ) e α
(7.1.24)
The likelihood function is given by Eq. 7.1.25. N
N
–y 1 L = -----N- exp – ∑ e i ·e α i=1
–
∑y
i(1 – k)
i=1
(7.1.25)
The log likelihood function is in Eq. 7.1.26, N
log L = – N log α – ( 1 – k ) ∑ y i – i=1
N
∑e
– yi
(7.1.26)
i=1
where from Eq. 7.1.23, x i – u 1 y i = – --- log 1 – k ----------- α k
(7.1.27)
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Differentiating Eq. 7.1.26 with respect to u, α, and k and equating the results to zero results in Eqs. 7.1.28 to 7.1.30. ∂ log L Q – --------------- = ---- = 0 ∂u α
(7.1.28)
1 P+Q ∂ log L – --------------- = --- -------------- = 0 α k ∂α
(7.1.29)
1 ∂ log L P+Q – --------------- = --- R – -------------- = 0 k ∂k k
(7.1.30)
In Eqs. 7.1.28 to 7.1.30, N
P = n–
∑ e –y
(7.1.31)
i
i=1
N
Q =
∑e
– YI + ky
N i
– (1 – k) ∑ e
i=1
(7.1.32)
i=1
N
R = N–
k yi
N
∑ yi +
∑ yi e
i=1
i=1
– yi
(7.1.33)
Eqs. 7.1.28 to 7.1.30 are solved numerically to get uˆ , αˆ , and kˆ . Two methods of solution are available to solve these equations (Flood Studies Report, 1975). Since no further simplification is possible to reduce these equations into one equation in one unknown, the three Equations (7.1.28 to 7.1.30) are solved simultaneously using the Newton method for three unknowns. First, initial estimates of kˆ , αˆ , uˆ are required and these can be the MOM estimates. The initial values k o , α o , u o are then updated as in Eq. 7.1.34 un + 1 un δu n = + αn + 1 αn δα n kn + 1 kn δk n
(7.1.34)
The values of δk n , δα n , and δu n are given by Eq. 7.1.35.
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–1
∂ log L ∂ log L ∂ log L – ---------------- – ----------------- – ----------------2 ∂u∂α ∂u∂k ∂u 2
2
δu n 2 ∂ log L δα n = – ----------------∂α∂u δk n 2 ∂ log L – ----------------∂k∂u
2
∂ log L ∂ log L – ---------------- – ----------------2 ∂α∂k ∂α 2
2
∂ log L ∂ log L – ----------------- – ---------------2 ∂k∂α ∂k 2
2
∂ log L --------------∂u ∂ log L --------------∂α ∂ log L --------------∂k
(7.1.35)
The first order partial derivatives in Eq. 7.1.35 are given by Eqs. 7.1.28 to 7.1.30 and the second order partial derivatives are given in Eqs. 7.1.36 to 7.1.44. 2 1 ∂Q ∂ log L – ---------------- = --- ------2 α ∂u ∂u
(7.1.36)
2 1 ∂Q –1 ∂ log L – ----------------- = -----2- Q + --- ------α ∂α ∂u∂α α
(7.1.37)
2 1 ∂Q ∂ log L – ----------------- = --- ------α ∂k ∂u∂k
(7.1.38)
∂ log L 1 ∂P ∂Q – ----------------- = ------ ------ + ------∂α∂u αk ∂u ∂u 2
2 –1 P + Q 1 ∂ log L – ---------------- = -----2- -------------- + -----2 αk k α ∂α
2 –1 1 ∂ log L – ----------------- = --------2 ( P + Q ) + -----αk ∂α∂k αk
(7.1.39)
∂P ∂Q ------- + ------∂α ∂α ∂P ∂Q ------ + ------∂k ∂k
(7.1.40)
(7.1.41)
2 1 ∂R 1 ∂P ∂Q ∂ log L – ----------------- = --- ------ – --- ------ + ------- k ∂u k ∂u ∂u ∂k∂u
(7.1.42)
2 1 ∂R 1 ∂P ∂Q ∂ log L – ----------------- = --- ------- – --- ------- + ------- k ∂α k ∂α ∂α ∂k∂α
(7.1.43)
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P+Q 1 ∂R 1 ∂P ∂Q ∂ log L 1 P+Q – ---------------- = – ----2 R – -------------- + --- ------ – --- ------ + ------- + ------------2 2 k ∂k k ∂k ∂k k k ∂k k 2
(7.1.44)
where ∂P ------ = ∂u
N
∑e i=1
N
∂P ------- = ∂α ∂P ------ = ∂k ∂Q ------- = ∂u ∂Q ------- = ∂α ∂Q ------- = ∂k
N
∑ –( 1 – k )e
i=1
N
∑ –( 1 – k ) e
∑e
N
N
∑e i=1
–( 1 – k ) yi
–( 1 – k ) yi
i=1
–( 1 – k ) y1
∂y -------i + ∂k
N
∑ yi e
– yi
i=1
i=1
∑ –( 1 – k ) e
– yi
–( 1 – k ) y1
i=1
– yi
∂y i ------∂u
(7.1.45)
∂y -------i ∂α
(7.1.46)
∂y -------i ∂k
(7.1.47)
N
k y ∂y ∂y i ------- – ( 1 – k ) ∑ k e i -------i ∂u ∂u i=1
(7.1.48)
N
k y ∂y ∂y -------i – ( 1 – k ) ∑ k e i -------i ∂α ∂α i=1
N
y k∂y – ( 1 – k ) ∑ ke i -------i + ∂k i=1
N
∑e
k yi
i=1
(7.1.49)
N
– ( 1 – k ) ∑ yi e
k yi
i=1
(7.1.50) N
N
N
– y ∂y – y ∂y ∂y ∂R ------ = – ∑ -------i + ∑ e i -------i – ∑ y i e i -------i ∂u i = 1 ∂u ∂u i = 1 ∂u i=1
N
N
N
– y ∂y – y ∂y ∂y ∂R ------- = – ∑ -------i + ∑ e i -------i – ∑ y i e i -------i ∂α ∂α ∂α ∂α i=1 i=1 i=1
N
N
(7.1.51)
(7.1.52)
N
– y ∂y – y ∂y ∂y ∂R ------ = – ∑ -------i + ∑ e i -------i – ∑ y i e i -------i ∂k i = 1 ∂k ∂k ∂k i = 1 i=1
(7.1.53)
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The partial derivatives of yi are obtained from Eq. 7.1.27 as in Eqs. 7.1.54 to 7.1.56. ∂y 1 yk -------i = – --- e i α ∂u
(7.1.54)
yk ∂y 1 -------i = – ------ ( e i – 1 ) ∂α αk
(7.1.55)
yk y 1 ∂y -------i = – ----i + ----2 ( e i – 1 ) ∂k k k
(7.1.56)
The iteration in Eq. 7.1.34 is repeated until the values of the functions in Eqs. 7.1.28 to 7.1.30 are sufficiently close to zero. An alternative method which does not require the matrix inversion in Eq. 7.1.35 uses the inverse of the matrix of the expected values of the second partial derivatives which depend only on α, u, and k. In this case Eq. 7.1.35 becomes 7.1.57.
δu n δα n δk n
∂ log L --------------∂u α b α h αn f 1 2 log L = --- α h α 2 a α g ∂-------------n n n n ∂α α n f α n g c ∂ log L --------------∂k 2 n
2 n
(7.1.57)
The values a, b, c, f, g, and h in Eq. 7.1.57 depend only on k. These are tabulated by Jenkinson (1969) and are given in Table 7.1.1. The 3 × 3 matrix in Eq. 7.1.57, evaluated at the ML estimates uˆ , αˆ , and kˆ , is the covariance matrix for uˆ , αˆ , and kˆ which is used later to calculate the standard error. It should be noted, however, that the ML solution may not exist for all data. PWM Method The PWMs of the GEV distribution (Hosking et. al., 1985) are of the form in Eq. 7.1.58. βr = ( r + 1 )
–1
α –k u + --- { 1 – ( r + 1 ) Γ ( 1 + k ) } k
(7.1.58)
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Table 7.1.1. Values of a, b, c, f, g, and h in the Covariance Matrix of ML Estimates of GEV Parameters
k –0.4 –0.3 –0.2 –0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.8 1.0
a 1.05 0.92 0.81 0.72 0.65 0.61 0.58 0.58 0.60 0.63 0.68 0.82 1.00
b 1.29 1.29 1.28 1.27 1.25 1.22 1.20 1.17 1.14 1.11 1.08 1.02 1.00
c 0.84 0.73 0.64 0.55 0.48 0.39 0.33 0.27 0.21 0.15 0.10 0.03 0.00
f 0.26 0.26 0.26 0.26 0.26 0.24 0.22 0.19 0.16 0.13 0.09 0.03 0.00
g –0.09 –0.03 0.04 0.10 0.15 0.18 0.21 0.23 0.24 0.24 0.22 0.15 0.00
h 0.80 0.69 0.57 0.46 0.34 0.21 0.09 –0.03 –0.16 –0.30 –0.43 –0.71 –1.00
Source: Flood Studies Report, after Jenkinson, 1969.
The value of parameter k is given by Hosking et. al. (1985) as the solution to Eq. 7.1.59, 3+t –k –k -------------3 = ( 1 – 3 )/ ( 1 – 2 ) 2
(7.1.59)
which is approximated by 2 kˆ = 7.8590C + 2.9554 C
(7.1.60)
2b 1 – b 0 log 2 2 log 2 - – ----------- = ------------- – ----------where C = -----------------3 + t 3 log 3 3b 2 – b 0 log 3 Once the value of k is obtained αˆ and uˆ are estimated by Eqs. 7.1.61 and 7.1.62, where b0, b1, b2 are the sample estimates of β0, β1, β2. ( 2b 1 – b 0 )kˆ l 2 kˆ = ------------------------------------------αˆ = ------------------------------------------ˆ –k – k Γ ( 1 + kˆ ) ( 1 – 2 ) Γ ( 1 + kˆ ) ( 1 – 2 )
(7.1.61)
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αˆ αˆ uˆ = b 0 + --- [ Γ ( 1 + kˆ ) – 1 ] = l 1 + --- [ Γ ( 1 + kˆ ) – 1 ] kˆ kˆ
(7.1.62)
EXAMPLE 7.1.1 Estimate the parameters of the GEV distribution for the annual maximum flow data of the White River near Nora, Indiana given in Table 7.1.2. Method of Moments: From Table 2.1.2 for station 74 we get
n = 62, m′ 1 = 14172, C v = 0.5152, C S = 0.7549 For the value of Cs = 0.7549 we find from Eq. 7.1.13,
ko = 0.0736 After using the iteration outlined in Section 7.1.2 to solve Eq. 7.1.15 we get the MOM estimate
kˆ = 0.0736 We now calculate
Γ ( 1 + kˆ ) = Γ ( 1.0736 ) = 0.9625 Γ ( 1 + 2kˆ ) = Γ ( 1.1472 ) = 0.9339 Now from Eqs. 7.1.20 and 7.1.21 we get
( 0.5152 × 14172 ) ( 0.0736 ) 2 7 αˆ = --------------------------------------------------------------------- = 3.8536 × 10 2 [ 0.9339 – ( 0.9625 ) ] 2
2
αˆ = 6209.4 6209.4 uˆ = 14172 – --------------------- [ 1 – 0.9625 ] = 11012 ( 0.0736 ) ML Method: The initial parameter estimates are taken equal to the MOM estimates. Applying the procedure given in Section 7.1.2 we get the ML estimates
αˆ = 5745.6, uˆ = 10849, kˆ = 0.0050 .
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Table 7.1.2. Annual Maximum Flows in White River near Nora, IN
Year
Ann. Max. Flow
1930 1931 1932 1933 1934 1935 1936 1937 1938 1939 1940 1941 1942 1943 1944 1945 1946 1947 1948 1949
23200 2950 10300 23200 4540 9960 10800 26900 23300 20400 8480 3150 9380 32400 20800 11100 7270 9600 14600 14300
Ann. Max. Flow 22500 14700 12700 9740 3050 8830 12000 30400 27000 15200 8040 11700 20300 22700 30400 9180 4870 14700 12800 13700 7960 9830
Year 1950 1951 1952 1953 1954 1955 1956 1957 1958 1959 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971
Year 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 — —
Ann. Max. Flow 12500 10700 13200 14700 14300 4050 14600 14400 19200 7160 12100 8650 10600 24500 14400 6300 9560 15800 14300 28700 — —
To verify that these are the ML estimates we first calculate N
∑ yi i=1
N
= 36.1616,
∑e
– yi
N
= 62,
i=1
∑e
k yi
N
= 62.1839,
i=1
N
∑ yi e
– yi
∑e
–( 1 – k ) yi
= 61.8703
i=1
= – 25.8384
i=1
Substituting the above in Eqs. 7.1.31 to 7.1.33 we get, P = 62 – 62 = 0, Q = 61.8703 – (1 – 0.005) 62.1839 = 0 R = 62 – 36.1616 – 25.8384 = 0
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If we substitute P, Q, and R in Eqs. 7.1.28 to 7.1.30 we get
∂ log L ∂ log L ∂ log L --------------- = 0, --------------- = 0, --------------- = 0 ∂α ∂k ∂u Therefore the equations are satisfied and the computed parameters are the ML estimates. PWM Method: From Table 3.1.1 we have for station 74, N = 62, l1 = 14172, t = 0.2863, t3 = 0.1828. We have, l2 = 0.2863 × 14172 = 4057.6 l3 = 0.1828 × 4057.6 = 741.65 From Eq. 7.1.60 we get,
2 log 2 C = ------------------------- – ----------- = – 0.0025 3 + 0.1828 log 3 2 kˆ = 7.8590 × – 0.0025 + 2.9554 × ( – 0.0025 ) = – 0.0199
We then calculate using Eq. 6.1.6,
Γ ( 1 + kˆ ) = Γ ( 0.801 ) = 1.0119 From Eqs. 7.1.61 and 7.1.62 we get,
4057.6 × – 0.0199 αˆ = ------------------------------------------------= 5745.3 0.0199 1.0119 × ( 1 – 2 ) 5745.3 uˆ = 14172 + ------------------- [ 1.0119 – 1 ] = 10740 – 0.0199 The parameter estimates obtained in this example are summarized in Table 7.1.3. Table 7.1.3. Parameter Estimates for Example 7.1.1
Parameter MOM MLM PWM
u 11012 10849 10740
α 6209.4 5745.6 5745.3
k 0.0736 0.0050 –0.0199
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7.1.2
Quantile Estimates
The distribution function of x given by Eq. 7.1.2 can be written in the inverse form in Eq. 7.1.63. α k x = u + --- [ 1 – ( – log F ) ] k
(7.1.63)
By substituting F = 1 – 1/T where T is the return period, the T-year quantile estimate is obtained as in Eq. 7.1.64. αˆ xˆ T = uˆ + --kˆ
kˆ
1 1 – – log 1 – --- T
(7.1.64)
Comparing Eq. 7.1.64 with the frequency factor equation in Eq. 7.1.65, αˆ xˆ T = uˆ + --- [ 1 – Γ ( 1 + kˆ ) ] + K T kˆ
αˆ ----2 [ Γ ( 1 + 2kˆ ) – Γ 2 ( 1 + kˆ ) ] ˆk
(7.1.65)
the frequency factor is given by Eq. 7.1.66. kˆ kˆ Γ ( 1 + kˆ ) – [ – log ( 1 – 1 ⁄ T ) ] K T = --------------------------------------------------------------------------1⁄2 2 kˆ [ Γ ( 1 + 2kˆ ) – Γ ( 1 + kˆ ) ]
(7.1.66)
EXAMPLE 7.1.2 Estimate the 100-year flood for the White River data used in Example 7.1.1. Use the parameters estimated in Example 7.1.1. Quantile Estimates by the MOM Parameters: From Eq. 7.1.64 we get,
xˆ T = 11012 + 84366.8 [ 1 – { – log ( 1 – 1 ⁄ T ) }
0.0736
]
For T = 100,
xˆ 100 = 11012 + 84366.8 [ 1 – { – log ( 0.99 ) }
0.0736
]
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xˆ 100 = 35241 cfs Quantile Estimates by the ML Parameters: From Eq. 7.1.64 we get
xˆ T = 10849 + 1149120 [ 1 – { – log ( 1 – 1 ⁄ T ) }
0.0050
]
For example if T = 100 then,
xˆ 100 = 10849 + 1149120 [ 1 – { – log ( 0.99 ) }
0.0050
].
xˆ 100 = 36977 cfs Quantile Estimates by the PWM Parameters: From Eq. 7.1.64 we get
xˆ T = 10740 – ( 288708 ) [ 1 – { – log ( 1 – 1T ) }
– 0.0199
]
For example if T = 100 then
xˆ 100 = 10740 – 288708 [ 1 – { – log ( 0.99 ) }
– 0.0199
]
xˆ 100 = 38417 cfs Quantile estimates for other recurrence intervals are given in Table 7.1.6.
7.1.3 Standard Error Method of Moments Eq. 4.4.11 can be used for calculating the standard error for the GEV distribution. However, the expressions for γ2, γ3, γ4 and ∂ΚΤ /∂γ1 are complicated. An analytical form for sT is very complicated, but it can be evaluated by computing each of its components numerically. We have Eq. 7.1.67, where ∂γ1/∂k can be evaluated by differentiating Eq. 7.1.11 as in Eq. 7.1.19.
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∂k T -------∂K T ∂k ∂K T ∂k ---------- = ---------- -------- = -------∂k ∂g 1 ∂g 1 ∂g 1 -------∂k
(7.1.67)
KT and g1 are given by Eqs. 7.1.66 and 7.1.11, respectively. The values of g2, g3, and g4 are given by Eqs. 7.1.68 to 7.1.70. 2
4
g 4 – 4g 1 g 3 + 6g 2 g 1 – 3 g 1 g 2 = ----------------------------------------------------------2 2 [ g2 – g1 ] 2
(7.1.68)
3
5
k -g 5 + 5g 4 g 1 – 10g 3 g 1 + 10g 2 g 1 – 4g 1 g 3 = ----- ----------------------------------------------------------------------------------------------2 5§2 k [ g2 – g1 ] 2
3
4
(7.1.69)
6
g 6 – 6g 5 g 1 + 15g 4 g 1 – 20 g 3 g 1 + 15g 2 g 1 – 5g 1 g 4 = ------------------------------------------------------------------------------------------------------------------------2 3 [ g2 – g1 ]
(7.1.70)
where gr is defined in Eq. 7.1.16. From Eq. 7.1.66 we get the derivative of KT with respect to k, k ∂K T k G¢ ( 1 + k ) – [ – log ( 1 – 1 § T ) ] log [ – log ( 1 – 1 § T ) ] ---------- = ----- ---------------------------------------------------------------------------------------------------------------------------1§2 k ∂k [ T ( 1 + 2k ) – G ( 1 + k ) ]
1 [ G ( 1 + k ) – { – log ( 1 – 1 § T ) } ] [ 2G¢ ( 1 + 2k ) – 2G ( 1 + k )G¢ ( 1 + k ) ] – --- -------------------------------------------------------------------------------------------------------------------------------------------------------------------3§ 2 2 [ G ( 1 + 2k ) – G ( 1 + k ) ] k
(7.1.71) where G¢ (1 + rk) is given by Eq. 7.1.17. However, this procedure can be used only if k > –1/6 in order for g4 to exist. ML Method The variance-covariance (dispersion) matrix for the ML estimates is given by Jenkinson (1969) as in Eq. 7.1.72.
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var u cov ( α, u ) cov ( u, k ) 1 cov ( α, u ) var α cov ( α, k ) = --N cov ( u, k ) cov ( α, k ) var k
α b α h αf 2
2
α h α a αg αf αg c 2
2
(7.1.72)
In Eq. 7.1.72, a, b, c, f, g, and h depend only on k and are tabulated by Jenkinson (Flood Studies Report, 1975) and are reproduced here in Table 7.1.1. Alternatively, variances and covariances of the parameters can be obtained from the inverse of the expected information matrix as in Eq. 7.1.73, –1 2
2
2
( – ∂ log L ) ( – ∂ log L ) ( – ∂ log L ) - E ------------------------- E ------------------------E -----------------------2 ∂u∂α ∂u∂k ∂u
var ( u ) cov ( u, α ) cov ( u, k ) 2 2 2 ( – ∂ log L ) ( – ∂ log L ) ( – ∂ log L ) - E ------------------------cov ( u, α ) var ( α ) cov ( α, k ) = E ------------------------- E -----------------------2 ∂u∂α ∂α∂k ∂α cov ( u, k ) cov ( α, k ) var ( k ) 2 2 2 ( – ∂ log L ) ( – ∂ log L ) ( – ∂ log L ) E ------------------------- E ------------------------- E -----------------------2 ∂u∂k ∂α∂k ∂k
(7.1.73)
The expected values in Eq. 7.1.73 are given by Eqs. 7.1.74 to 7.1.79 (Prescott and Walden, 1980). – ∂ log L N - = ---------p E ------------------2 2 ∂u 2 αk
(7.1.74)
– ∂ log L N - = -------E ------------------[ 1 – 2Γ ( 2 – k ) + p ] 2 ∂α 2 αk
(7.1.75)
2
2
– ∂ log L N - = ----2 E ------------------ ∂u 2 k 2
2
k 1 2 2q p ---- + 1 – γ – --- + ------ + ----2 6 k k k
(7.1.76)
– ∂ log L N [ p – Γ(2 – k )] E -------------------- = -------2 ∂u∂α αk
(7.1.77)
– ∂ log L N p E -------------------- = – ------ q + --- ∂u∂k αk k
(7.1.78)
2
2
– ∂ log L N E -------------------- = --------2 ∂α∂k αk 2
{1 – Γ(2 – k )} p 1 – γ – ------------------------------------ – q – --k k
(7.1.79)
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In Eqs. 7.1.74 to 7.1.79, p = ( 1 – k ) Γ ( 1 – 2k ) 2
(7.1.80)
q = Γ ( 2 – k ) [ ψ ( 1 – k ) – ( 1 – k )/k ]
(7.1.81)
γ = Euler constant = 0.5772157
(7.1.82)
Prescott and Walden (1983) recommended the use of the inverse of the observed information matrix, which is the matrix in Eq. 7.1.73, without taking the expected values, evaluated at the ML estimates uˆ , αˆ , and kˆ (the same as in Eq. 7.1.35 which is used in the iterative solution) to provide an estimate of the covariance matrix of the ML estimates. The standard error can be calculated by using Eq. 4.4.19 in which the partial derivatives are obtained from Eq. 7.1.64 as Eqs. 7.1.83 to 7.1.85. ∂x -------T- = 1 ∂u
(7.1.83)
∂x T 1 kˆ -------- = --- [ 1 – { – log ( 1 – 1 ⁄ T ) } ] ∂α kˆ
(7.1.84)
∂x α kˆ -------T- = – ----2 [ 1 – { – log ( 1 – 1/T ) } ] ∂k kˆ αˆ kˆ – --- [ { – log ( 1 – 1/T ) } log { – log ( 1 – 1/T ) } ] ˆk
(7.1.85)
PWM Method The covariance matrix for the PWM estimates is of the form (Hosking et al., 1985) in Eq. 7.1.86.
var ( u ) cov ( u, α ) cov ( u, k ) 1 cov ( α, u ) var ( α ) cov ( α, k ) = --N cov ( u, k ) cov ( α, k ) var ( k )
α w 11 α w 12 αw 13 2
2
α w 12 α w 22 αw 23 αw 13 αw 23 w 33 2
2
(7.1.86)
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In Eq. 7.1.86, wij are functions of k and have a complicated algebraic form. wij are evaluated numerically and are tabulated by Hosking et al. (1985). These wij functions are reproduced in Table 7.1.4. Table 7.1.4. Elements of the Asymptotic Covariance Matrix of the PWM Estimators of the Parameters of the GEV Distribution (Hosking et al. 1985)
k –0.4 –0.3 –0.2 –0.1 0.0 0.1 0.2 0.3 0.4
w11 1.6637 1.4153 1.3322 1.2915 1.2686 1.2551 1.2474 1.2438 1.2433
w12 1.3355 0.8912 0.6727 0.5104 0.3704 0.2411 0.1177 –0.0023 –0.1205
w13 1.1405 0.5640 0.3926 0.3245 0.2992 0.2966 0.3081 0.3297 0.3592
w22 1.8461 1.2574 1.0013 0.8440 0.7390 0.6708 0.6330 0.6223 0.6368
w23 1.1628 0.4442 0.2697 0.2240 0.2247 0.2447 0.2728 0.3033 0.3329
w33 2.9092 1.4090 0.9139 0.6815 0.5633 0.5103 0.5021 0.5294 0.5880
The standard error is calculated by Eq. 4.4.22 where the partial derivatives are given in Eqs. 7.1.83 to 7.1.85.
EXAMPLE 7.1.3 Estimate the standard error for the 100-year flood for the White River data in Example 7.1.1. Use the parameter estimates computed in Example 7.1.1. Standard Error by Using the MOM Estimates: We first calculate the following by using Eq. 6.1.6
Γ ( 1 + 3kˆ ) = 0.9129, Γ ( 1 + 4kˆ ) = 0.8983 Γ ( 1 + 5kˆ ) = 0.8895, Γ ( 1 + 6kˆ ) = 0.8858 Γ′ ( 1 + kˆ ) = – 0.4449, Γ′ ( 1 + 2kˆ ) = – 0.3343, Γ′ ( 1 + 3kˆ ) = – 0.2400 Substituting these into Eqs. 7.1.68 to 7.1.70,
γ 2 = 3.8965, γ 3 = 8.6879, γ 4 = 34.9653 . Substituting the above in Eq. 7.1.19,
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∂γ --------1 = – 4.6146 ∂k When T = 100, from Eqs. 7.1.66 and 7.1.71,
K T = 2.8858,
∂K T ---------- = – 3.4033 ∂k
From Eq. 7.1.67,
∂K ---------T- = ( – 3.4033 )/ ( – 4.6146 ) = 0.7375 ∂γ 1 Now substituting the above derivatives in Eq. 4.4.11, sT = 3572 cfs Standard Error by Using the ML Estimates: For the value of kˆ = 0.0050 from Table 7.1.1, using interpolation, we get a = 0.648, b = 1.248, c = 0.475 f = 0.26, g = 0.148, h = 0.333 Given in Table 7.1.5 are the variances and covariances of the parameters evaluated by: (a) Eq. 7.1.72, (b) the expected information matrix in Eq. 7.1.73, and (c) the observed information matrix in Eq. 7.1.35. Table 7.1.5. Variances and Covariances of Parameter Estimates in Example 7.1.3.
Method (a)
Method (b)
Method (c)
var (u)
776108
664669
698307
var (α)
402979
var (k)
7.66129 × 10
346400 –3
7.655 × 10
376157 –3
11.527 × 10–3
cov (u,α)
207086
176180
205595
cov (u,k)
26.0394
23.977
35.4370
cov (α,k)
14.8224
13.8574
23.215
For the case with T = 100 from Eqs. 7.1.83 to 7.1.85:
∂x ∂x ∂x T -------- = 1; -------T- = 4.5472; -------T- = – 59861 ∂α ∂k ∂u
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Substituting these derivatives in Eq. 4.4.19 and using the variances and covariances, for example, from method (b) in Table 7.1.5 (the expected information matrix method) we get
s T = ( 1 ) × 664669 + ( 4.5472 ) × 346400 + ( – 59861 ) × 7.655 × 10 2
2
2
2
–3
+ 2 × 1 × 4.5472 × 176180 + 2 × 1 × – 59861 × 23.977 + 2 × 4.5472 × – 59861 × 13.8574 = 26445364 s T = 5142 cfs
Standard Error by Using the PWM Moments: For kˆ = – 0.0199 , we get by interpolation, the following from Table 7.1.4: w11 = 1.2724, w12 = 0.3974, w13 = 0.3024 w22 = 0.7569, w23 = 0.2230, w33 = 0.5789 Substituting wij in Eqs. 7.1.86: var (u) = 677416; var (α) = 402985.31; var (k) = 9.337 × 10–3 cov (u,α) = 211557; cov (u,k) = 28.0218; cov (α,k) = 20.6617 For T = 100 we have from Eqs. 7.1.83 to 7.1.85
∂x ∂x ∂x T -------- = 1; -------T- = 4.8173; -------T- = – 64630 ∂α ∂k ∂u Substituting the above values in Eq. 4.4.19 we get,
s T = 1 × 677416 + ( 4.8173 ) × 402985 + ( – 64630 ) × 9.337 × 10 + 2 × 1 × 4.8173 × 211557 + 2 × 1 × – 64630 × 28.0218 + 2 × 4.8173 × – 64630 × 20.6617 = 34580687 2
2
2
–3
ST = 5880 cfs These standard errors and others corresponding to different recurrence intervals are listed in Table 7.1.6.
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Table 7.1.6. Quantile Estimates and their Standard Errors (in parentheses) for Example 7.1.
T 10
P (%) 10
20
5
50
2
100
1
200
0.5
MOM 23888.67 (1641.19) 27577.2 (2151.32) 32070.29 (2928.33) 35240.48 (3572.54) 38241.78 (4253.33)
ML 23705.42 (1787.65) 27787.21 (2506.11) 33048.73 (3843.30) 36975.32 (5142.58) 40873.85 (6683.16)
PWM 23963.27 (1902.11) 28319.11 (2724.62) 34051.64 (4305.59) 38417.29 (5880.54) 42827.77 (7784.80)
The probability plots for station 74 by using MOM, ML, and PWM methods are shown in Figure 7.1.2.
7.2
The Extreme Value Type I EV1(2) Distribution
The probability density function of the EV1(2) distribution is given by Eq. 7.2.1. x – β
– ----------- α x–β 1 f ( x ) = --- exp – ------------ – e α α
(7.2.1)
The variable x takes values in the range –∞ < x < ∞. The distribution function of x is given by Eq. 7.2.2.
F ( x ) = exp – e
x–β – ------------ α
(7.2.2)
The EV1(2) distribution is a special case of the GEV distribution discussed in Section 7.1 in which the shape parameter k is equal to zero. The extreme value type I or the Gumbel distribution has been discussed extensively in the hydrologic literature (Gumbel (1958), Yevjevich (1972), Chow (1964) and many other textbooks). Conse-
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Figure 7.1.2. Observed and estimated flows and 95% confidence intervals for the White River data used in Examples 7.1.1 to 7.1.3.
quently, only some of the recent work related to this distribution is discussed herein. An analytical and a simulation study was conducted by assuming the population distribution to be EV1(2) distribution by Majumdar and Sawhney (1965). Distributions were fitted to the generated data. The authors found that Foster’s type III distribution gave as good an estimate as the EV(1) distribution. A number of methods of fitting the EV1(2) distribution to sample data were compared by Lowery and Nash (1970) and by Jain and Singh
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(1986). The method of moments was found to be most accurate, next to the maximum likelihood method. The method of moments was also found to be virtually unbiased and the simplest to apply. The use of censored data in the estimation of Gumbel distribution parameters for annual maximum flood flows was investigated by Leese (1971), who presented equations to be used for ML estimates. He also presented equations to estimate large sample standard errors. Estimation of parameters of Gumbel distribution by probability weighted moments was developed by Landwehr et al. (1979b). The PWM estimates were compared to the MOM estimates and the MLM estimates. The PWM estimates were shown to be comparable to other estimates. Phien (1987) compared the moment, ML, ME (maximum entropy), and PW estimators for the EV1(2) distribution. He concluded that the moment estimator was not as good as the three other estimators. The probability weighted moment method was found to be best in having the least bias and the ML method was the best in terms of root mean square error and efficiency. Raynal and Salas (1986) analyzed and compared six methods of parameter estimation for the EV1(2) distribution by using synthetic data. They recommended the best linear combination of order statistics for sample sizes smaller than 20. The probability weighted moments method was preferred by them for larger samples. The method of moments, probability weighted moments, mixed moments, ML, incomplete means, and least squares methods were compared to estimate the parameters of the EV1(2) distribution by Jain and Singh (1986). The ML method was found to be the best. The difference between ML and other methods was not pronounced. Lattenmeir and Burges (1982) recommend that Gumbel’s m be set to infinity, rather than to sample length to estimate the parameters of the distribution. The population moments were replaced by their sample estimates. Chang and Moore (1983) used the EV1(2) distribution and the index flood method to derive a regional flood frequency curve for small watersheds in southern Illinois. Data from 22 gaging stations were used in their study. Fierentino and Gabriele (1984) developed modifications to ML estimators of EV1(2) distribution parameters to reduce the bias of parameter estimators. Smith (1986) developed a family of statistical distributions and estimators for extreme values based on a fixed number of the largest annual events. He illustrated his method by an application to the sea levels in Venice. Phien (1987) derived expressions for variances and covariances of estimators of the EV(1) distribution. The usefulness of historical and paleological floods in
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quantile estimation was investigated by Guo and Cunnane (1991). They found that type I censored-data ML estimator was a robust model for Gumbel distribution. EV1(2) and EV2(3) distributions were used by Ochoa et al. (1980) to compare the tail behavior of flood data. Quite often EV2(3) distributions were found to provide a better fit to flood data, which raises the issue of which distribution should be used. One of the conclusions of Ochoa et al. was that flood data have paretian or heavy tailed distributions. In a related paper, Shen et al. (1980) discussed the difference in assumptions about the tail behavior of EV1(2) and LP(3) distributions. These assumptions are compared by using the flood predictions based on type I and type II extreme value distributions.
7.2.1 Parameter Estimation Method of Moments The first moment of the EV1(2) distribution is given by Eq. 7.2.3 (Kite, 1977) µ′1 = β + 0.5772157 α
(7.2.3)
The second central moment is given by Eq. 7.2.4. π 2 µ 2 = ----- α 6 2
(7.2.4)
The parameter estimates are obtained by replacing µ′ 1 and µ 2 by their corresponding sample estimates m′ 1 and m 2 to get 6 αˆ = ------π
m 2 = 0.7797 m 2
βˆ = m′1 – 0.45005
m2
(7.2.5)
(7.2.6)
Maximum Likelihood (ML) Method The likelihood function for a sample of size N from an EV1(2) distribution is given by Eq. 7.2.7.
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xi – β
– ------------- x i – β α 1 - – ∑e L = -----N- exp – ∑ ----------- α α i=1 i=1 N
N
(7.2.7)
Taking the natural logarithm of Eq. 7.2.7 results in Eq. 7.2.8. 1 log L = – N log α – --α
N
N
∑ ( xi – β ) –
∑e
i=1
i=1
xi – β – ------------- α
(7.2.8)
Differentiating Eq. 7.2.8 with respect to α and β then equating to zero gives Eqs. 7.2.9 and 7.2.10. ∂L N 1 ------- = – ---- + -----2 ∂α α α
N
1 ∑ ( x i – β ) – ----2 α i=1 N
∂L N 1 ------ = ---- – --∂β α α
∑e
N
∑ ( x i – β )e
xi – β – ------------- α
= 0
(7.2.9)
i=1
xi – β – ------------- α
(7.2.10)
= 0
i=1
Equation 7.2.11 in α is obtained by substituting from Eq. 7.2.10 into Eq. 7.2.9. N
F (α) =
∑ xi e
– xi ⁄ α
i=1
1 – ---N
N
∑ x i – α
i=1
N
∑e
– x i /α
(7.2.11)
= 0
i=1
Eq. 7.2.11 in α cannot be solved analytically. It is solved iteratively by Newton’s method. An initial value of α is required and can be taken as the MOM estimate of α. The value of α is updated as in Eq. 7.2.12, α n + 1 = α n – F ( α n )/F′ ( α n )
(7.2.12)
where F′ ( α ) the derivative of Eq. 7.2.11 and is given by Eq. 7.2.13. 1 dF ( α ) F′ ( α ) = ---------------- = ----α2 dα
N
∑ xi e 2
i=1
– x i /α
N
+
∑e i=1
– x i /α
1 + --α
N
∑ xi e
– x i /α
(7.2.13)
i=1
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The iteration in Eq. 7.2.12 is repeated until F(α) is sufficiently close to zero. After an estimate αˆ is obtained, βˆ is calculated from Eq. 7.2.14.
N βˆ = αˆ log -----------------N – x i /αˆ ∑e
(7.2.14)
i=1
PWM Method The PWMs of the EV(1) distribution are of the form (Greenwood et. al., 1979; Hosking, 1986a) in Eq. 7.2.15, m α { log ( 1 + r ) + ε } β r = ----------- + -------------------------------------------1+r 1+r
(7.2.15)
where ε is the Euler’s number, ε = 0.5772157. Parameter estimates are obtained by substituting β0 and β1 by their corresponding sample estimates b0, b1, to give Eqs. 7.2.16 and 7.2.17. αˆ = ( 2b 1 – b 0 )/ log ( 2 ) = l 2 / log ( 2 )
(7.2.16)
βˆ = b 0 – 0.5772157 αˆ = l 1 – 0.5772157αˆ
(7.2.17)
EXAMPLE 7.2.1 Estimate the parameters of the EV1(2) distribution by using the annual maximum flow data for Sugar Creek at Crawfordsville, Indiana given in Table 7.2.1. Method of Moments: From the data in Table 7.2.1:
N = 53 , m′ 1 = 10638 , C V = 0.5355 We have, 1⁄2
m2
= C v m′ 1 = 5696.3
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Table 7.2.1 Annual Maximum Flows in Sugar Creek at Crawfordsville, IN.
Year
Max. Ann. Flow
1939 1940 1941 1942 1943 1944 1945 1946 1947 1948 1949 1950 1951 1952 1953 1954
17600 3660 903 5050 24000 11400 9470 8970 7710 14800 13900 20800 9470 7860 7860 2730
Year 1955 1956 1957 1958 1959 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973
Max. Ann. Flow 6480 18200 26300 15100 14600 7300 8580 15100 15100 21800 6200 2130 11100 14300 11200 6670 5440 9370 6900
Year 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 —
Max. Ann. Flow 9680 6810 7730 5290 12200 9750 7390 13100 7190 8850 6290 18800 9740 2990 6950 9390 12400 21200 —
From Eqs. 7.2.5 and 7.2.6 we get
αˆ =
6 ⁄ π × 5696.3 = 4441.4
βˆ = 10638 – 0.45005 × 5696.3 = 8074.4 ML Method: Applying the Newton method outlined in Section 7.2.1 we get the following estimate of α αˆ = 4478.6 To verify Eq. 7.2.11 we calculate
1 ---N
N
N
∑ x i = 10638;
∑e
i=1
i=1
– x i /α
= 8.7841 ;
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N
∑ xi e
– x i /α
= 54103
i=1
Thus Eq. 7.2.11 becomes, F ( α ) = 54103 – ( 10638 – 4478.6 ) × 8.7841≈ 0 Therefore Eq. 7.2.11 is satisfied. From Eq. 7.2.14, βˆ = 4478.6 log [ 53/8.7841 ] = 8049.6 . PWM Method: From Table 3.1.1 we get for this data, N = 53, l1 = 10638, t = 0.2966 l2 = 0.2966 x 10638 = 3155.5. From Eqs. 7.2.16 and 7.2.17, αˆ = 3155.5/ log ( 2 ) = 4552.4 βˆ = 10638 – 0.5772157 × 4552.4 = 8010.1 The parameter estimates obtained in this example are summarized in Table 7.2.2. Table 7.2.2. Parameter Estimates for Example 7.2.1.
Parameter MOM MLM PWM
α 4441.4 4478.6 4552.4
β 8074.4 8049.6 8010.1
7.2.2 Quantile Estimates The distribution function of EV1(2), Eq. 7.2.2 can be obtained in the inverse form in Eq. 7.2.18.
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x = b – a log ( – log F )
(7.2.18)
The T-year quantile is calculated by substituting F = 1 – (1/T), where T is the return period to get Eq. 7.2.19. xˆ T = bˆ – aˆ log [ – log ( 1 – 1/T ) ]
(7.2.19)
From Eq. 7.2.19 the frequency factor is obtained as in Eq. 7.2.20. 6 K T = ------- [ – 0.5772157 – log { – log ( 1 – 1/T ) } ] p = – 0.45 – 0.7797 log { – log ( 1 – 1/T ) }
(7.2.20)
EXAMPLE 7.2.2 Compute the 100-year flood for the Sugar Creek data given in Table 7.2.1. Use the three-parameter estimates computed in Example 7.2.1. Quantile Estimate Based on MOM Parameters: From Eq. 7.2.19 we get:
xˆ T = 8074.4 – 4441.4 log [ – log ( 1 – 1/T ) ] For T = 100,
xˆ 100 = 8074.4 – 4441.4 log [ – log ( 0.99 ) ] = 28505 cfs Quantile Estimate Based on ML Parameters: From Eq. 7.2.19:
xˆ T = 8049.6 – 4478.6 log [ – log ( 1 – 1/T ) ] For T = 100,
xˆ 100 = 8049.6 – 4478.6 log [ – log ( 0.99 ) ] = 28652 cfs
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Quantile Estimate Based on PWM Parameters:
From Eq. 7.2.19: xˆ T = 8010.1 – 4552.4 log [ – log ( 1 – 1/T ) ] For T = 100, xˆ 100 = 8010.1 – 4552.4 log [ – log ( 0.99 ) ] = 28952 cfs Other quantile estimates are given in Table 7.2.3.
7.2.3 Standard Error Method of Moments For a two-parameter distribution, the standard error is given by Eq. 4.4.16 where δ is given by Eq. 4.4.17. For the EV1(2) distribution we have Eqs. 7.2.21 and 7.2.22. γ 1 = C s = 1.1396
(7.2.21)
γ 2 = C k = 5.4002
(7.2.22)
The standard error is computed from Eq. 7.2.23. µ 2 2 s T = ----- [ 1 + 1.1396 K T + 1.1 K T ] N 2
(7.2.23)
2 2 If µ 2 = π /6 αˆ and K T from 7.2.20 are substituted into Eq. 7.2.23, it can be written as Eq. 7.2.24,
α 2 2 s T = ----- [ 1.15894 + 0.19187Y + 1.1Y ] N 2
(7.2.24)
where Y is given by Eq. 7.2.26.
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ML Method The standard error is calculated from Eq. 7.2.25 ∂x 2 ∂x 2 2 s T = ------- var ( α ) + ------ var (β ) + 2 ∂α ∂β
∂x ----- ∂α
∂x ----- cov ( α, β ) ∂β
(7.2.25)
From Eq. 7.2.18, we have ∂x ------- = – log [ – log ( 1 – 1/T ) ] = Y ∂α
(7.2.26)
∂x ------ = 1 ∂β
(7.2.27)
The second derivatives of log L are given by Eqs. 7.2.28 to 7.2.30. 1.8237 N ∂ log L ---------------- = – ---------------------2 2 ∂α α
(7.2.28)
∂ log L N ---------------- = – -----2 2 ∂β α
(7.2.29)
0.4228 N ∂ log L ----------------- = ---------------------2 ∂α∂β α
(7.2.30)
2
2
2
The variances and covariances of parameters are obtained from the information matrix as in Eqs. 7.2.31 to 7.2.33 (Hosking 1986; Flood Studies Report, 1975). α var ( αˆ ) = 0.6079 ----N
(7.2.31)
α var ( βˆ ) = 1.1087 ----N
(7.2.32)
2
2
α cov ( αˆ , βˆ ) = 0.2570 ----N 2
(7.2.33)
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The standard error is obtained by substituting Eqs. 7.2.31 to 7.2.33 and 7.2.26, 7.2.27 into Eq. 7.2.25 to get Eq. 7.2.34 (Kite, 1977), α 2 2 s T = ----- [ 1.1087 + 0.5140 Y + 0.6079 Y ] N 2
(7.2.34)
where Y is given by Eq. 7.2.26. PWM Method The variances and covariances of the PWM estimators αˆ and βˆ are given by Hosking (1986a) as in Eqs. 7.2.35 to 7.2.37. α var ( αˆ ) = 0.8046 ----N
(7.2.35)
α var ( βˆ ) = 1.1128 ----N
(7.2.36)
2
2
α cov ( αˆ , βˆ ) = 0.2287 ----N 2
(7.2.37)
The standard error can thus be calculated using Eq. 4.4.22 to get Eq. 7.2.38 where Y = ∂x/∂α is given by Eq. 7.2.26. α 2 2 S T = ----- ( 1.1128 + 0.4574 Y + 0.8046 Y ) N 2
(7.2.38)
EXAMPLE 7.2.3 Compute standard errors for the 100-year flood estimates computed in Example 7.2.2. Standard Error Based on MOM Estimates: From Eq. 7.2.24 we get
( 4441.4 ) 2 2 s T = ----------------------- [ 1.15894 + 0.19187 Y + 1.1 Y ] 53 2
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If T = 100, Y = – log [ – log ( 0.99 ) ] = 4.60015 then 2
s 100 = 9423494.5 , s 100 = 3069.8 cfs Standard Error Based on ML Estimates: From Eq. 7.2.34:
( 4478.6 ) 2 2 s T = ----------------------- [ 1.1087 + 0.5140 Y + 0.6079 Y ] 53 2
where Y = –log [–log (1 – 1/T )]. For T = 100, Y = – log [ – log ( 0.99 ) ] = 4.60015 ; 2
s 100 = 6182808.23 . s 100 = 2486.5 cfs Standard Error Based on PWM Estimates: From Eq. 7.2.38:
( 4552.4 ) 2 2 s T = ----------------------- ( 1.1128 + 0.4574 Y + 0.8046 Y ) . 53 2
For T = 100, Y = – log [ – log ( 0.99 ) ] = 4.60015 ; 2
S 100 = 7915665.81 . s 100 = 2813.5 cfs Standard errors for other quantile estimates, computed by the computer program discussed in Chapter 10 are shown in Table 7.2.3. The probability plots for these data computed by using MOM, MLM, and PWM are shown in Figure 7.2.1.
7.3
Weibull Distribution
The probability density function of the Weibull distribution is given by Eq. 7.3.1, x – m
a b x – m b – 1 – -----------f ( x ) = --- ------------- e a a
b
(7.3.1)
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Figure 7.2.1. Observed and estimated flows and 95% confidence intervals for the Sugar Creek data used in Examples 7.2.1 to 7.2.3.
where a > 0 and b > 0. The variable x takes values in the range x ≥ m. The distribution function of the Weibull distribution is given by Eq. 7.3.2. F( x) = 1 – e
x–m – ------------- a
b
(7.3.2)
The Weibull distribution is a reverse GEV distribution with parameters k = 1 ⁄ b, α = a ⁄ b , and u = m – a (Hosking, 1986a).
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Table 7.2.3. Quantile Estimates and Their Standard Errors (in parentheses) for Example 7.2.2
T
P (%)
MOM
MLM
PWM
10
10
18068.91
18128.23
18254.85
(1633.26)
(1422.12)
(1559.16)
21265.95
21352.07
21531.85
(2063.28)
(1739.83)
(1934.45)
25404.19
25525.01
25773.59
(2635.28)
(2163.65)
(2433.79)
28505.22
28652.04
28952.18
(3069.94)
(2486.54)
(2813.53)
31594.93
31767.66
32119.17
(3506.05)
(2811.06)
(3194.74)
20 50 100 200
5 2 1 0.5
7.3.1 Parameter Estimation Method of Moments The first moment of the Weibull distribution is calculated by Eq. 7.3.3. ∞
x – m
b
a b x – m b – 1 – -----------µ′1 = x --- ------------- e dx a a
∫
(7.3.3)
m
x–m b Substituting y = ------------- in Eq. 7.3.3 we get Eq. 7.3.4. a ∞
∫
µ'1 = ( m + ay ) e dy 1/b
–y
(7.3.4)
0
1 µ'1 = m + a Γ --- + 1 b
(7.3.5)
The second central moment is calculated by Eq. 7.3.6.
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∞
µ2 =
∫
a
2
y
1/b
2 1 –y – Γ --- + 1 e dy b
(7.3.6)
0
µ2 = a
2
2 2 1 Γ --- + 1 – Γ --- + 1 b b
(7.3.7)
Similarly, the third central moment is given by Eq. 7.3.8. µ 3 = a [ Γ ( 3/b + 1 ) – 3Γ ( 1/b + 1 ) Γ ( 2/b + 1 ) + 2Γ ( 1/b + 1 ) ] 3
3
(7.3.8)
The coefficient of skew CS is given by Eq. 7.3.9. 3 µ3 Γ ( 3/b + 1 ) – 3Γ ( 1/b + 1 ) Γ ( 2/b + 1 ) + 2Γ ( 1/b + 1 ) (7.3.9) C S = -------- = ----------------------------------------------------------------------------------------------------------------------------------3 ⁄ 2 3⁄2 2 µ2 [ Γ ( 2/b + 1 ) – Γ ( 1 ⁄ b + 1 ) ]
Eq. 7.3.9 is solved numerically to get the value of bˆ and then Eqs. ˆ . The expressions of Cs in 7.3.7 and 7.3.5 are solved to obtain â and m Eqs. 7.3.9 and 7.1.11 are similar to each other. By replacing k in Eq. 7.1.11 by 1/b and changing the signs we get Eq. 7.3.9. Since b is allowed to take only positive values, by using Eq. 7.1.12, we find for –1.1396 < CS < 2 (1/b > 0). 2
3
1/b = 0.277561 + 0.3219 C S + 0.061566 C S – 0.017376 C S 4 5 6 – 0.00771 C S + 0.00398 C S – 0.00051 C S
(7.3.10)
Newton’s method is used to solve Eq. 7.3.9 numerically. An initial value of 1/b is obtained from Eq. 7.3.10. The value of 1/b is then updated as in Eq. 7.3.11. ( 1/b ) n + 1 = ( 1/b ) n – F ( 1/b )/F′ ( 1/b )
(7.3.11)
g r = Γ ( 1 + r/b )
(7.3.12)
and d r = Γ′ ( 1 + r/b ) = ψ ( 1 + r/b ) Γ ( 1 + r/b )
(7.3.13)
If we define,
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where ψ(·) can be calculated using Eq. 6.2.53. Then, 3
g 3 – 3g 2 g 1 + 2g 1 F ( 1/b ) = – C S + -------------------------------------2 3⁄2 ( g2 – g1 )
(7.3.14)
3
3d 3 – 3g 2 d 1 – 6g 1 d 2 + 6g 1 d 1 3 ( g 3 – 3g 2 g 1 + 2g 1 ) ( 2d 2 – 2g 1 d 1 ) F′ ( 1/b ) = --------------------------------------------------------------------- – --- -------------------------------------------------------------------------------2 5⁄2 2 3⁄2 2 ( g2 – g1 ) ( g2 – g1 ) 2
(7.3.15)
The iteration in Eq. 7.3.11 is repeated until F(1/b) is sufficiently close to zero. Other parameters are calculated by Eqs. 7.3.16 and 7.3.17. 1⁄2 2 1⁄2 aˆ = m 2 / [ Γ ( 1 + 2/bˆ ) – Γ ( 1 + 1/bˆ ) ]
(7.3.16)
ˆ = m′ 1 – aˆ Γ ( 1 + 1/bˆ ) m
(7.3.17)
It can also be shown that Cs = 0 corresponds approximately to b = 3.60235 (Cohen and Whitten, 1988). Kite (1977) gives an approximation for b as a fourth-degree polynomial. As mentioned before in the GEV distribution, as Cs approaches the value of –1.1396, the value of b becomes infinite and the distribution cannot be used. In such a case EV1(2) is used. Maximum Likelihood (ML) Method The likelihood function of a sample of size n from an EV3 distribution is given by N
b L = --- a
N
N
xi – m
- ∏ ------------a
b–1 –
e
- ∑ ------------a xi – m
b
i=1
(7.3.18)
i=1
Taking natural logarithms of Eq. 7.3.18 we get N
log L = N log b – N log a + ( b – 1 )
xi – m
N
xi – m
-------------- - – ∑ log ------------a ∑ a
i=1
b
(7.3.19)
i=1
Differentiating Eq. 7.3.19 with respect to a, b and m and equating the results to zero we get Eqs. 7.3.20 to 7.3.22.
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N
N
x i – m b x i – m x i – m ∂ log L N - – ∑ ------------- · log ------------- = 0 --------------- = ---- + ∑ log ------------ a a a ∂b b i=1 i=1 ∂ log L Nb b --------------- = – ------- + --∂a a a ∂ log L --------------- = – ( b – 1 ) ∂m
N
N
xi – m
- ∑ ------------a
(7.3.20)
b
= 0
(7.3.21)
i=1
b –1 ∑ ( x i – m ) + --ai=1
N
x i – m b – 1 ------------∑ a - = 0
(7.3.22)
i=1
These three equations (7.3.20 to 7.3.22) are solved simultaneously by ˆ . The same using iterative methods to obtain the values of aˆ , bˆ and m procedure used for the GEV method discussed in Section 7.1.2 is also used here. The second partial derivatives are given in Section 7.3.4 as Eqs. 7.3.43 to 7.3.48. The ML solution may not exist for all data. PWM Method The PWMs for the Weibull distribution are given by (Greenwood et al., 1979) Eq. 7.3.23. m a Γ ( 1 + 1/b ) α S = ----------- + ------------------------------1 + s ( 1 + s ) 1 + 1/b
(7.3.23)
The parameter estimates are obtained by replacing α0, α1, α3 by their corresponding sample estimates a0, a1, a3 to get Eqs. 7.3.24 to 7.3.26. ˆ = 4 ( a 3 a 0 – a 1 )/ [ 4a 3 + a 0 – 4a 1 ] m
(7.3.24)
a 0 – 2a 1 - / log ( 2 ) ˆ )/Γ log -----------------aˆ = ( a 0 – m a 1 – 2a 3
(7.3.25)
2
bˆ = log ( 2 ) /log
a 0 – 2a 1 -------------------------2 ( a 1 – 2a 3 )
(7.3.26)
Alternatively, similar to the case of GEV and following the steps of Hosking (1986a), it is found that 1/b is the solution of Eq. 7.3.27, which is similar to Eq. 7.1.59 with t3 replaced by –t3
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3–t – 1/b – 1/b ------------3 = ( 1 – 3 )/ ( 1 – 2 ) 2
(7.3.27)
An approximate solution to Eq. 7.3.27 is obtained as in Eq. 7.3.28, 2 bˆ = 1 ⁄ ( 7.8590 C + 2.9554 C )
(7.3.28)
log 2 2 where C = ------------ – ----------- . 3 – t 3 log 3
(7.3.29)
Other parameters are calculated as: – 1/bˆ aˆ = l 2 / [ Γ ( 1 + 1/bˆ ) ( 1 – 2 ) ]
(7.3.30)
ˆ = l 1 – aˆ Γ ( 1 + 1/bˆ ) m
(7.3.31)
The two methods give different results since in the first method a0, a1, and a3 are used. In the second method a0, a1, and a2 are used. The second method, however, is the natural method while the first gives explicit solutions for the parameters.
EXAMPLE 7.3.1 Estimate the parameters of a Weibull distribution by using the annual maximum flow data from Tippecanoe River at Delphi, Indiana. given in Table 5.1.1. Use all the three methods discussed in Section 7.3.1. Method of Moments: From Table 2.2.1, we have the following for Station 43:
N = 48, m' 1 = 12665, C v = 0.3719, C s = 0.1194 From Eq. 7.3.10 with Cs = 0.1194 we get
1/bˆ 0 = 0.3168 ; bˆ 0 = 3.1566 Using the iterative solution in Eqs. 7.3.11 to 7.3.15 we get bˆ = 3.1561 . We now calculate:
Γ ( 1 + 1/bˆ ) = 0.8951, Γ ( 1 + 2/bˆ ) = 0.8978
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From Eqs. 7.3.16 and 7.3.17: 2 1⁄2
aˆ = ( 0.3719 × 12665 )/ [ 0.8978 – ( 0.8951 ) ] ˆ = – 896.54 m
; aˆ = 15152 ;
ML Method: Using the MOM estimates as initial values for the solution we use the iterative procedure used in Example 7.1.1.
ˆ = 275.31 bˆ = 2.9114; aˆ = 13900.8 ; m To verify that these are the ML solution we calculate: N
xi – m
- ∑ log ------------a
N
= – 9.6421 ;
i=1
xi – m
- ∑ ------------a
b
i=1
N
N
∑ ( xi – m )
–1
= 0.0047;
i=1
x i – m - = 6.8450 log ------------ a
xi – m
- ∑ ------------a
b
= 48
i=1
N
xi – m
- ∑ ------------a
b–1
= 43.2793
i=1
Substituting the above values in Eqs. 7.3.20 to 7.3.22:
∂ log L 48 --------------- = ---------------- – 9.6421 – 6.8450≈ 0 ∂b 2.9114 ∂ log L – 48 × 2.9114 2.9114 --------------- = -------------------------------- + ------------------- × 48 = 0 13900.8 ∂a 13900.8 2.9114 ∂ log L --------------- = – ( 2.9114 – 1 ) × 0.0047 + ------------------- × 43.2793≈ 0 13900.8 ∂m The equations are satisfied and the obtained values are the ML estimates. PWM Method: From Table 3.1.1 we get for station 43: N = 48, l1 = 12665, t = 0.2150, t3 = 0.0348. From Eqs. 7.3.28 and 7.3.29:
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2 log 2 C = ------------------------- – ----------- = 0.0436 3 – 0.0348 log 3 2 bˆ 0 = 1/ [ 7.8590 × 0.0436 + 2.9554 ( 0.0436 ) ] = 2.8713
Numerical solution of Eq. 7.3.27 gives,
bˆ = 2.8792 We calculate:
Γ ( 1 + 1/bˆ ) = Γ ( 1 + 1/2.8792 ) = Γ ( 1.3473 ) = 0.8914 From Eqs. 7.3.30 and 7.3.31:
aˆ = ( 0.2150 × 12665 )/ 0.8914 1 – 2
1 – ---------------2.8792
;
aˆ = 14277.
ˆ = 12665 – 14277 × 0.8914 = – 61.52 m The parameter estimates obtained in this example are summarized in Table 7.3.1. Table 7.3.1 Parameter Estimates for Example 7.3.1
Parameter MOM MLM PWM
b 3.1561 2.9114 2.8792
a 15152 13900.8 14277
m –896.54 275.31 –61.52
7.3.2 Quantile Estimates The distribution function of x, Eq. 7.3.2, can be written in the inverse form x = m + a [ – log ( 1 – F ) ]
1/b
(7.3.32)
The T-year quantile estimate can be obtained by substituting F = 1 – 1/T where T is the return period to get Eq. 7.3.33. The frequency factor is obtained as in Eq. 7.3.34.
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ˆ + aˆ [ log ( T ) ] xˆ T = m
1/bˆ
(7.3.33)
[ log ( T ) ] – Γ ( 1/b + 1 ) K T = ------------------------------------------------------------------------1⁄2 2 [ Γ ( 2/b + 1 ) – Γ ( 1/b + 1 ) ] 1/b
(7.3.34)
EXAMPLE 7.3.2 Estimate the 100-year flood for the Tippecanoe River data and the parameters estimated in Example 7.3.1. Quantile Estimates Based on MOM Parameters: From Eq. 7.3.33 we have:
xˆ T = – 896.54 + 15152 [ log ( T ) ]
˙ ( 1 ⁄ 3.1561 )
For T = 100,
xˆ T = – 896.54 + 15152 [ log ( 100 ) ]
( 1 ⁄ 3.1561 )
xˆ 100 = 23685 cfs
;
Quantile Estimates Based on ML Parameters:
From Eq. 7.3.33 we have, xˆ t = 275.31 + 13900.8 [ log ( T ) ]
1 ⁄ 2 . 9114
For T = 100, xˆ 100 = 275.31 + 13900.8 [ log ( T 100 ) ]
1 ⁄ 2 . 9114
= 23763 cfs
Quantile Estimates Based on PWM Parameters: From Eq. 7.3.33,
xˆ T = – 61.52 + 14277 [ log ( T ) ]
1 ⁄ 2 . 8792
For T = 100,
xˆ 100 = – 61.52 + 14277 [ log ( 100 ) ]
1 ⁄ 2.8972
;
xˆ 100 = 24202 cfs
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7.3.3 Standard Errors Method of Moments The standard error for Weibull distribution is calculated by following the same steps as for the GEV distribution. Eq. 4.4.11 is used to numerically calculate the standard error. The values of γ2, γ3, γ4 are given by Eqs. 7.3.35 to 7.3.37, 2
4
g 4 – 4g 3 g 1 + 6g 2 g 1 – 3g 1 γ 2 = ---------------------------------------------------------2 2 ( g2 – g1 ) 2
(7.3.35)
3
5
g 5 – 5g 4 g 1 + 10g 3 g 1 – 10g 2 g 1 + 4g 1 γ 3 = -----------------------------------------------------------------------------------2 5⁄2 ( g2 – g1 ) 2
3
4
(7.3.36)
6
g 6 – 6g 5 g 1 + 15g 4 g 1 – 20g 3 g 1 + 15g 2 g 1 – 5g 1 γ 4 = ----------------------------------------------------------------------------------------------------------2 3 ( g2 – g1 )
(7.3.37)
where g r = Γ ( 1 + rk ) . Also, ∂K T /∂γ is given by Eq. 7.3.38. ∂γ 1 ∂K T ∂K T ∂ ( 1/b ) ∂K T - · ---------------- = --------------- ⁄ ------------------------ = --------------∂ ( 1/b ) ∂γ 1 ∂ ( 1/b ) ∂ ( 1/b ) ∂γ 1
(7.3.38)
∂γ1/∂(1/b) is evaluated by differentiating Eq. 7.3.9 to get Eq. 7.3.15. ∂ΚΤ /∂(1/b) is obtained by differentiating Eq. 7.3.34 to get Eq. 7.3.39, 1/b ∂K T [ log ( T ) ] log ( log T ) – Γ′ ( 1/b + 1 ) --------------- = -------------------------------------------------------------------------------------1⁄2 2 ∂ ( 1/b ) [ Γ ( 2/b + 1 ) – Γ ( 1/b + 1 ) ]
(7.3.39)
1 { log ( T ) } – Γ [ ( 1/b + 1 ) ] [ 2Γ′ ( 2/b + 1 ) – 2Γ ( 1/b + 1 )Γ′ ( 1/b + 1 ) ] – --- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------3⁄2 2 2 [ Γ ( 2/b + 1 ) – Γ ( 1/b + f 1 ) ] 1/b
where Γ′ ( 1 + r/b ) is given by Eq. 7.3.13. ML Method The standard error is obtained from Eq. 4.4.19 by substituting: ∂x a 1/b ------ = – -----2 [ log ( T ) ] log [ log ( T ) ] ∂b b
(7.3.40)
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∂x 1/b ------ = [ log ( T ) ] ∂a
(7.3.41)
∂x ------- = 1 ∂m
(7.3.42)
From Eq. 7.3.19 N
2 x i – m b x i – m N ∂ log L ---------------------------= – ---– log ------------∑ 2 2 a a ∂b b i=1
∂ log L Nb b ---------------- = ------2- – -----2 2 ∂a a a 2
∂ log L ---------------- = –( b – 1 ) 2 ∂m 2
N
∑ ( xi – m )
2
–2
i=1
N
N
b
N
xi – m
- ∑ ------------a
b
(7.3.44)
i=1
N
b–2
(7.3.45)
i=1
xi – m
i=1
b + --a
xi – m
- ∑ ------------a
- ∑ ------------a
b–1
i=1
xi – m
(7.3.43)
- ∑ ------------a
b(b – 1) – ------------------2 a
x i – m b - + --∑ ------------ a a i=1
∂ log L 1 –1 ----------------- = – ∑ ( x i – m ) + --∂b∂m a i=1
N
2
b
x i – m b - – ----∑ ------------a a2 i=1
N
2
∂ log L N 1 ----------------- = – ---- + --∂b∂a a a
N
2
N
b
x i – m - (7.3.46) log ------------ a
xi – m
- ∑ ------------a
i=1
b–1
x i – m log ------------ a (7.3.47)
2
2 b ∂ log L ----------------- = – -----2 ∂a∂m a
N
xi – m
- ∑ ------------a
b–1
(7.3.48)
i=1
There is no direct method for evaluating these partial derivatives at the parameter estimates, and the expected values must be calculated to get the variance-covariance matrix. For b > 2, Cohen and Whitten (1988) gives expressions for these expected values which result in the following Eqs. 7.3.49 to 7.3.54. 2
a ˆ ) = ----- φ 11 var ( m N
(7.3.49)
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2
b var ( bˆ ) = ----- φ 22 N
(7.3.50)
2
a var ( aˆ ) = ----- φ 33 N
(7.3.51)
a ˆ , bˆ ) = ---- φ 12 cov ( m N
(7.3.52)
2
a ˆ , aˆ ) = ----- φ 13 cov ( m N
(7.3.53)
a cov ( bˆ , aˆ ) = ---- φ 23 . N
(7.3.54)
ψ' ( 1 ) φ 11 = -----------2 b M
(7.3.55)
C – Γ ( 2 – 1/b ) φ 22 = -------------------------------------M
(7.3.56)
where,
2
2
KC – J φ 33 = -----------------2 b M
(7.3.57)
J + ψ ( 2 )· Γ ( 2 – 1/b ) φ 12 = – --------------------------------------------------- . M
(7.3.58)
Jψ ( 2 ) + KΓ ( 2 – 1/b ) φ 13 = – --------------------------------------------------2 b M
(7.3.59)
Cψ ( 2 ) + JΓ ( 2 – 1/b ) φ 23 = ---------------------------------------------------M
(7.3.60)
1 1 2 2 2 M = KC – 2Jψ ( 2 )Γ 2 – --- – Cψ ( 2 ) – K Γ 2 – --- – J b b
(7.3.61)
and
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1 A = 1 + ψ 2 – --- b
(7.3.62)
b – 1 2 2 C = Γ 1 – --- + bΓ 2 – --- ---------- b b b 2
(7.3.63)
1 1 J = Γ 1 – --- – AΓ 2 – --- b b
(7.3.64)
K = ψ′ ( 1 ) + ψ ( 2 ) .
(7.3.65)
2
In the above equations, ψ(.) is the digamma function and ψ′(.) is the trigamma function given by Eqs. 6.2.53 and 6.2.54. If the lower bound m is known, the variances and covariances of the remaining parameters a and b are (Cohen and Whitten, 1988): 2
b var ( bˆ ) = 0.607927 ----- N
(7.3.66)
2
a var ( aˆ ) = 1.108665 ---------2 Nb
(7.3.67)
a cov ( aˆ , bˆ ) = 0.257022 ---- N
(7.3.68)
The standard error in this case is given by Eq. 7.3.69. ∂x 2 ∂x 2 ∂x ∂x 2 s T = ------ var a + ------ var b + 2 ------ ------ cov ( a, b ) ∂a ∂b ∂a ∂b
(7.3.69)
EXAMPLE 7.3.3 Compute the standard errors for the 100-year flood estimated in Example 7.3.2.
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Standard Error Based on MOM Parameters: The following functions are computed first.
Γ ( 1 + 3/bˆ ) = 0.9801, Γ ( 1 + 4/bˆ ) = 1.1444 Γ ( 1 + 5/bˆ ) = 1.4129, Γ ( 1 + 6/bˆ ) = 1.8291 Γ′ ( 1 + 1/bˆ ) = – 0.1345 , Γ′ ( 1 + 2/bˆ ) = 0.1388 Γ′ ( 1 + 3/bˆ ) = 0.3826 Substituting these into Eqs. 7.3.35 to 7.3.37 γ2, γ3, and γ4 are calculated. γ2 = 2.7161, γ3 = 0.1715, γ4 = 11.5019 From Eq. 7.3.15:
∂γ 1 --------------- = 2.9795 ∂ ( 1/b ) For T = 100, from Eq. 7.3.39,
∂K T --------------- = 2.1279 ∂ ( 1/b ) And thus from Eq. 7.3.38,
∂K T ---------- = 2.1279/2.9795 = 0.7142 ∂γ 1 Also from Eq. 7.3.34, KT = 2.3398. Substituting the above in Eq. 4.4.19 we get, s100 = 1635 cfs. Standard Error Based on ML Estimate: As bˆ = 2.9114 is greater than 2, the procedure is applicable. The following functions are computed by using equations given in Section 6.2. Γ ( 1 – 1/b ) = 1.3726 , Γ ( 1 – 2/b ) = 2.8609, Γ ( 2 – 1/b ) = 0.9011 2 Γ ( 2 – 2/b ) = 0.8956, ψ′ ( 1 ) = π /6 = 1.644934, ψ ( 2 ) = 0.42278433, ψ ( 2 – 1/b ) = 0.1735.
Using Eqs. 7.3.61 to 7.3.65 the following parameters are computed. A = 1.1735, C = 1.2331, J = 0.3151, K = 1.8237, M = 0.2081
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Substituting the values in Eqs. 7.3.55 to 7.3.60 and then in Eqs. 7.3.49 to 7.3.54 the variances and covariances of the estimates are computed.
var ( bˆ ) = 0.357321 var ( aˆ ) = 4905741 ˆ ) = 3754200 var ( m cov ( bˆ , aˆ ) = 1120.6975 ˆ ) = – 968.7312 cov ( bˆ , m ˆ ) = – 4054670 cov ( aˆ , m From Eqs. 7.3.40 to 7.3.42 for T=100, the following derivatives are evaluated.
∂x ∂x ∂x ------- = 1, ------ = – 4232, ------ = 1.6897 ∂b ∂a ∂m Substituting the above values in Eq. 4.4.19 the standard error for the 100-year flood is computed.
s 100 = 1 × 3754200 + ( – 4232 ) × 0.357321 + ( 1.6897 ) × 4905741 2
2
2
+ 2 × 1 × – 4232 × – 968.7312 + 2 × 1 × 1.6897 × – 4054670 + 2 × – 4232 × 1.6897 × 1120.6975 = 2629267.51 ∴s 100 = 1621 cfs Standard errors for other return periods are given in Table 7.3.2. Table 7.3.2 Quantile Estimates and Their Standard Errors (in parentheses) for Example 7.3.3.
T 10
P (%) 10
20
5
50
2
100
1
200
0.5
MOM 18838.16 (949.83) 20554.19 (1136.25) 22446.86 (1415.88) 23685.1 (1635.55) 24801.76 (1855.54)
MLM 18787.29 (965.76) 20538.52 (1152.24) 22483.71 (1417.93) 23763.58 (1621.51) 24922.44 (1823.01)
PWM 19011.37 20836.61 22866.07 24202.47 25413.24
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The probability plots for the data from station 43 computed by using MOM, MLM, and PWM are shown in Figure 7.3.1.
Figure 7.3.1. Observed and estimated flows and 95% confidence intervals for the Tippecanoe River data used in Examples 7.3.1 to 7.3.3.
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CHAPTER 8
The Wakeby Distribution
8.1 8.1.1
The Five-Parameter Wakeby Distribution (WAK(5)) Introduction
A random variable x is distributed as a Wakeby distribution if it is distributed as in Eq. 8.1.1 x = m + a[1 – (1 – F)b] – c[1 – (1 – F )–d]
(8.1.1)
where F = F(x) = P(X ≤ x). The Wakeby distribution is analytically defined only in the inverse form in Eq. 8.1.1. Therefore explicit expressions cannot be obtained for either the probability density function or the distribution function. Although moments of x can be obtained as functions of the parameters (Greenwood et al., 1979), the inverse relationship cannot be readily derived. Consequently, moment estimates of the parameters are not feasible. Moreover, maximum likelihood estimates of the parameters are not easily obtained. Only the PWM method is presently considered for this distribution. The Wakeby distribution is potentially useful in flood frequency analysis for several reasons discussed by Greenwood et al. (1979). One of these is the large number of parameters in the Wakeby distribution which permits better fitting of data than by distributions characterized by fewer parameters. Another reason is that it can accommodate a variety of flows ranging from low flows to floods. The Wakeby distribution was proposed for flood frequency analysis by Houghton (1978a). Wakeby distribution was shown to demonstrate the separation effect which other distributions do not exhibit. Hence it was considered to be a superior distribution for flood frequency analysis. This argument for using the Wakeby distribution is not as strong now, in view of the recent work of Ashkar et al. (1992). A procedure to estimate the parameters of a distribution, called the incomplete means procedure was also introduced by Houghton (1978b). Houghton tested this method by using the Wakeby distribution.
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The probability weighted moments method was used to estimate the parameters of the Wakeby distribution by Landwehr and Matalas (1979a). The choice of the estimating algorithm, which minimizes the root mean square error of the quantiles is shown to be unimportant when the upper flood quantiles are of interest and when the lower bound is unknown. A method to estimate the parameters of the Wakeby distribution with a known lower bound was developed by Landwehr et al. (1979b). This algorithm is based on probability weighted moments. These estimates were shown to be neither highly biased nor variable even for very small sample sizes.
8.1.2 Parameter Estimation The method that is discussed here is the PWM given by Greenwood et al. (1979). The probability weighted moments of the Wakeby distribution are given by Eq. 8.1.2 (Greenwood et al., 1979). m a–c a c M 1, 0, k = α k = ------------ + ------------ – --------------------- + --------------------1+k 1+k 1+k+b 1+k–d
(8.1.2)
Parameter estimation requires the solution of five simultaneous equations obtained by substituting k = 0, 1,…, 4 in Eq. 8.1.2 and replacing αk by their sample estimates ak. First the following are defined: {k}= (k + 1)(k + 1 + b) (k + 1 – d) ak, k = 0,1,2,3,4
(8.1.3)
N4–j = (4)j a3 – (3)1+j a2 + 3(2)j a1 – a0, j = 1,2,3
(8.1.4)
C4–j = (5)j a4 – 3(4)j a3 + (3)1+j a2 – (2)j a1, j = 1,2,3
(8.1.5)
ˆ 1, 0, i where ai is M Estimates of the parameters in Eq. 8.1.1 are then obtained as: ˆ = [{3} – {2} – {1} + {0}]/4 m 2
(8.1.6) 1⁄2
( N 3 C 1 – N 1 C 3 )± [ ( N 1 C 3 – N 3 C 1 ) – 4 ( N 1 C 2 – N 2 C 1 ) ( N 2 C 3 – N 3 C 2 ) ] bˆ = ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------2( N 2C3 – N 3C2)
(8.1.7)
dˆ = ( N 1 + bˆ N 2 )/ ( N 2 + bˆ N 3 )
(8.1.8)
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( bˆ + 1 ) ( bˆ + 2 ) aˆ = ---------------------------------bˆ ( bˆ + d )
{1} 0 ˆ ------------ – ------------ – m ˆ 2 + b 1 + bˆ
(8.1.9)
( 1 – dˆ ) ( 2 – dˆ ) cˆ = ---------------------------------dˆ ( bˆ + dˆ )
–{ 1 } 0 ˆ ------------ – ------------ – m 2 + dˆ 1 + dˆ
(8.1.10)
Eq. 8.1.7 gives two values of bˆ , but according to Greenwood et al. (1979) the two values correspond to the same solution but with parameters exchanged, and thus the larger value of bˆ is chosen without loss of generality. The Wakeby distribution parameter estimation, however, may not succeed if any of the following conditions hold (Landwehr et al., 1979b): 1. 2. 3. 4. 5.
Imaginary value of bˆ or bˆ < 0.3 or bˆ > 50 dˆ ≥ 1 ; the mean does not exist Invalid probability density function: f ( m ) = 1/ ( aˆ bˆ + cˆ dˆ ) < 0 Improperly defined distribution function: F(x1) > F(x2) for x1 < x2 for a combination of signs of parameters. The same as 4 but for a combination of parameter values
Landwehr et al. (1979) gave the following procedure for estimating the parameters of the WAK(5) distribution: 1.
Assume m ≠ 0 and determine parameters from appropriate equations. Test for error conditions; if none hold, accept the parameters; if any error condition holds, go to step 2. 2. Assume m = 0, fit a WAK(4) distribution (Section 8.2) using appropriate equations. If successful, accept the parameters; otherwise go to step 3a. 3a. Assume m ≠ 0. Fix a number of allowed iterations. Set b equal to bmax = 50 and choose ∆b. Go to step 3b. 3b. If the allowed number of iterations is exceeded, go to step 4a. If not, set bˆ to ( bˆ – ∆b ) . If bˆ falls below bmin = 0.3 go to step ˆ from Eqs. 8.1.6, 8.1.8, 8.1.9, and 4a. Determine dˆ , aˆ , cˆ and m 8.1.10. Test for error conditions 1 to 3; if none hold, go to step 3c. If either conditions 1 or 2 holds, go to beginning of step 3b to start next iteration. If only error condition 3 holds, decrease the value of ∆b by half, and start new iteration at beginning of step 3b.
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3c. Test for error conditions 4 and 5. If neither holds, accept the parameters; otherwise go to step 4a. 4a. Assume m = 0, proceed as in step 3a but branch to step 4b instead of 3b. 4b. m = 0. Proceed as in step 3b, except that the appropriate equations (m = 0, Section 8.2) are used to compute aˆ , cˆ and dˆ . If number of allowed iterations is exceeded or if bˆ falls below bmin = 0.3 then the algorithm fails; otherwise go to step 4c. 4c. m = 0. If a solution is obtained for which error conditions 1 to 3 do not hold, test for error conditions 4 and 5. If either condition 4 or 5 holds, the algorithm fails; otherwise the algorithm succeeds. Hosking (1986) has given a reparameterized form for the Wakeby distribution as in Eq. 8.1.11. β
–δ
x = ε + ( α/β ) [ 1 – ( 1 – F ) ] – ( γ /δ ) [ 1 – ( 1 – F ) ]
(8.1.11)
The probability weighted moments in this case are given in Eq. 8.1.12 rα r – 1 = ε + α/ ( r + β ) + γ / ( r – δ ) ,
δ < 1
(8.1.12)
The L-moments are of the form in Eq. 8.1.13. αΓ ( 1 + β ) Γ ( r – 1 – β ) γΓ ( 1 – δ ) Γ ( r – 1 + δ ) λ r = --------------------------------------------------------- + ------------------------------------------------------Γ(1 – β) Γ(r + 1 + β) Γ(1 + δ) Γ(r + 1 – δ)
(8.1.13)
In this case, Eqs. 8.1.4, 8.1.5, and 8.1.7 are still used but values of N and C are of opposite sign. Eq. 8.1.7 gives two values of bˆ , and the values of βˆ and δˆ are assigned as in Eqs. 8.1.14 and 8.1.15. βˆ = max ( bˆ 1, bˆ 2 )
(8.1.14)
δˆ = – min ( bˆ 1, bˆ 2 )
(8.1.15)
The values of εˆ , αˆ , γˆ are given by ( 4N 2 – N 1 ) + ( 4N 3 – N 2 ) βˆ γˆ = ( 1 – δˆ ) ( 2 – δˆ ) ( 3 – δˆ ) ----------------------------------------------------------------6 ( βˆ + δˆ )
(8.1.16)
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( 4N 2 – N 1 ) – ( 4N 3 – N 2 ) δˆ αˆ = – ( 1 + βˆ ) ( 2 + βˆ ) ( 3 + βˆ ) ---------------------------------------------------------------6 ( βˆ + δˆ )
(8.1.17)
αˆ γˆ εˆ = a 0 – ----------------- – ---------------ˆ ( 1 + β ) ( 1 – δˆ )
(8.1.18)
The following conditions must be satisfied for these parameters to be valid (Hosking, 1986b). 1. 2. 3. 4. 5.
β + δ > 0 or β = γ = δ = 0 if α = 0 then β = 0 if γ = 0 then δ = 0 γ≥0 α+γ≥0
EXAMPLE 8.1.1 Estimate the parameters of a WAK(5) distribution for the data of Wabash River at Lafayette, Indiana (station 49). The data are given in Table 1.8.1. Parameter Estimates: We will consider the solution in both the original and reparameterized forms. Both methods should give values of bˆ = βˆ ,
ˆ and m ˆ = εˆ . The probability weightdˆ = δˆ , aˆ = αˆ /βˆ , cˆ = γˆ /δ ed moments for station 49 (n = 85) are given by: ao = 52621.18, a1 = 20075.82, a2 = 11730.64 a3 = 8028.83, a4 = 5979.14 a. Original Form: From Eq. 8.1.4 we calculate N1, N2, and N3. N1 = 64a3 – 81a2 + 24a1 – ao = – 7138.6 N2 = 16a3 – 27a2 + 12a1 – ao = 22.541 N3 = 4a3 – 9a2 + 6a1 – ao = – 5626.7 From Eq. 8.1.5 we calculate C1, C2, and C3. C1 = 125a4 – 192a3 + 81a2 – 8a1 = – 4567.2 C2 = 25a4 – 48a3 + 27a2 – 4a1 = 518.782 C3 = 5a4 – 12a3 + 9a2 – 2a1 = – 1026.1 From Eqs. 8.1.7 and 8.1.8 bˆ and dˆ are computed.
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bˆ = 6.5349 , which is the largest of the two roots of Eq. 8.1.7. dˆ = 0.19025 , which is also the smallest of the two roots of Eq. 8.1.7. From Eq. 8.1.3 we calculate: {0} = 321042.27, {1} = 620166.51 {2} = 942798.88, {3} = 1288942.22
ˆ. From Eq. 8.1.6 we get m 1 ˆ = --- [ { 3 } – { 2 } – { 1 } + { 0 } ] = 11755 m 4 From Eq. 8.1.9 and 8.1.10 we get aˆ and cˆ
aˆ = 26777 cˆ = 75090 b. Reparameterized Form: The first part is the same as in (a. original form) above except that Ni and Ci are: N1 = 7138.6, C1 = 4567.2 N2 = – 22.541, C2 = – 518.782 N3 = 5626.7, C3 = 1026.1 From Eq. 8.1.14 and 8.1.15 using Eq. 8.1.7 βˆ and δˆ are computed.
βˆ = 6.5349 δˆ = 0.19025 Using Eqs. 8.1.16 to 8.1.18, γˆ , αˆ , and εˆ are estimated. γˆ = 14286 αˆ = 174990 174990 14286 εˆ = 52621.18 – ------------------------------ – ------------------------------ = 11755 ( 1 + 6.5349 ) ( 1 – 0.1903 ) ˆ, It can be seen by comparing the results in a and b that εˆ = m βˆ = bˆ , δ = dˆ and αˆ 174990 --- = ------------------ = 26777 = aˆ 6.5349 βˆ
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γˆ 14286 -- = ------------------- = 75090 = cˆ ˆδ 0.19025 To check the validity of the parameters, it is noticed that all the parameters are positive and dˆ < 1 . The same is true for the reparameterized form and thus all the conditions are satisfied and the parameters are valid.
8.1.3
Quantile Estimation
The Wakeby distribution in the inverse form is given in Eq. 8.1.1. By substituting F = 1 – (1/T) where T is the return period, the T-year quantile is calculated as in Eq. 8.1.19. – bˆ
dˆ
ˆ + aˆ [ 1 – T ] – cˆ [ 1 – T ] xˆ T = m
(8.1.19)
or in the reparameterized form as in Eq. 8.1.20. – βˆ δˆ xˆ T = εˆ + ( αˆ /βˆ ) [ 1 – T ] – ( γˆ /δˆ ) [ 1 – T ]
8.1.4
(8.1.20)
Standard Error
The mathematical basis for calculating the asymptotic variances and covariances of the parameters does exist, but the calculations are complicated (Hosking, 1986b). However, approximate standard error as well as variances and covariances of parameters may be obtained through numerical simulation (Landwehr et al., 1979b).
EXAMPLE 8.1.2 Estimate the 100-year flood based on Wakeby distribution for the data used in Example 8.1.1. From Eq. 8.1.19 xˆ T is given by:
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xˆ T = 11755 + 26777 [ 1 – T
– 6.5349
] – 75090 [ 1 – T
0.19025
]
For example if T = 100 then
xˆ 100 = 11755 + 26777 [ 1 – ( 100 )
– 6.5349
] – 75090 [ 1 – ( 100 )
0.19025
]
xˆ 100 = 143778 cfs The quantile estimates for different return periods T are given in Table 8.1.1. The results are obtained by using the computer program discussed in Chapter 10. It can be seen from Table 8.1.1 that F ( x 1 ) >F ( x 2 ) for x 1 > x 2 and thus conditions 4 and 5 of Landwehr et al. (1979b) are satisfied. The quantile plot for station 49 by using the PWM method and the WAK(5) distribution is given in Figure 8.1.1.
Table 8.1.1
T 10 20 50 100 200
Quantile Estimates for Example 8.1.2
P (%) 10 5 2 1.00 0.50
Q (cfs) 79810 96214 121499 143779 169200
Figure 8.1.1. Observed and estimated flows for the Wabash River data used in Examples 8.1.1 and 8.1.2.
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8.2
The Four-Parameter Wakeby Distribution (WAK(4))
8.2.1 Introduction The WAK(4) distribution is a special case of the WAK(5) distribution in which the parameter m (or ε in the reparameterized form) is equal to zero. This condition occurs when the lower bound for the observations is known to be zero. If the lower bound is known, but is not equal to zero, then m (or ε) can be made equal to zero by subtracting this known lower bound from all the observations. For the WAK(4) distribution we have x = a[1 – (1 – F )b] – c[1 – (1 – F )–d]
(8.2.1)
and in the reparameterized form β
–δ
x = ( α/β ) [ 1 – ( 1 – F ) ] – ( γ /δ ) [ 1 – ( 1 – F ) ]
(8.2.2)
The probability weighted moments and L-moments can be obtained from Eqs. 8.1.2, 8.1.12, and 8.1.13 by replacing m = ε = 0. 8.2.2 Parameter Estimation The PWM method is used for the estimation of the WAK(4) distribution parameters also. A procedure similar to that used in WAK(5) distribution is used. However, as the number of parameters to be estimated is four rather than five, fewer PWMs will be used. N4–j and C4–j are redefined as: N4–j = –(3)j a2 + (2)1+j a1 – a0, j = 1,2,3
(8.2.3)
C4–j = –(4)j a3 + 2(3)j a2 – (2)j a1, j = 1,2,3
(8.2.4)
ˆ Eqs. 8.1.7 through 8.1.10 are then used to find bˆ , dˆ , aˆ , cˆ replacing m by zero. A similar procedure is used for the reparameterized form (Hosking, 1986). The definition of N4–j and C4–j is of opposite sign to that in Eqs. 8.2.3 and 8.2.4. Eq. 8.1.7 is used to get bˆ 1 and bˆ 2 ; then Eqs. 8.1.14 and 8.1.15 are used to obtain βˆ and δˆ . The other parameters are then calculated by using Eqs. 8.2.5 and 8.2.6.
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( 3N 2 – N 1 ) + ( 3N 3 – N 2 )βˆ γˆ = ( 1 – δˆ ) ( 2 – δˆ ) --------------------------------------------------------------2 ( βˆ + δˆ )
(8.2.5)
( 3N 2 – N 1 ) – ( 3N 3 – N 2 ) δˆ αˆ = – ( 1 + βˆ ) ( 2 + βˆ ) -----------------------------------------------------------2 ( βˆ + δˆ )
(8.2.6)
The same conditions for parameter validity must be satisfied as discussed in Section 8.1.2.
EXAMPLE 8.2.1 Estimate the parameters of the WAK(4) distribution using the data from the Wabash River at Lafayette, Indiana. We will consider the solution in both the original and reparameterized ˆ /βˆ form. The two methods should give values of bˆ = βˆ , dˆ = δˆ , aˆ = α and cˆ = γˆ /δˆ . The probability weighted moments for station 49 (n = 85) are given by a0 = 52621.18, a1 = 20075.82 a2 = 11730.64, a3 = 8028.83 a. Original Form: N1, N2, and N3 are calculated from Eq. 8.2.3 N1 = –27a2 + 16a1 – a0 = –48135 N2 = –9a2 + 8a1 – a0 = 2409.6 N3 = –3a2 + 4a1 – a0 = –7509.8 C1, C2 and C3 are calculated from Eq. 8.2.4 C1 = –64a3 + 54a2 – 8a1 = –40997 C2 = –16a3 + 18a2 – 4a1 = 2387 C3 = –4a3 + 6a2 – 2a1 = –1883.1 From Eqs. 8.1.7 and 8.1.8 bˆ and dˆ are computed.
bˆ = 16.2990 dˆ = 0.07385 From Eq. 8.1.3, {0} = 84307 {1} = 1415200 Substituting the above into Eqs. 8.1.9 and 8.1.10 and noting that
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ˆ = 0, aˆ and cˆ are computed. m aˆ = 33929 cˆ = 259008 b. Reparameterized Form: The first part is the same as in a except Ni and Ci are as follows. N1 = 48135, C1 = 40997 N2 = –2409.6, C2 = –2387 N3 = 7509.8, C3 = 1883.1
ˆ are calcuFrom Eqs. 8.1.14 and 8.1.15 using Eq. 8.1.7, βˆ and α lated βˆ = 16.2990 δˆ = 553008.8 ˆ are calculated By using Eqs. 8.2.5 and 8.2.6 γˆ and α γˆ = 19127.7 αˆ = 553008.8 It can be seen by comparing the results in a and b that
βˆ = bˆ , δˆ = dˆ and αˆ 553008.8 --- = ---------------------- = 33929 = aˆ ˆβ 16.2990 19127.7 γˆ -- = ------------------- = 259008 = cˆ ˆδ 0.07385 We notice that all the parameters are positive and dˆ < 1 . The same is true for the reparameterized form and thus all the conditions are satisfied and the parameters are valid.
8.2.3 Quantile Estimation As discussed in the WAK(5) distribution, by substituting F = 1 – (1/T) where T is the return period, the T-year quantile is calculated by,
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– bˆ
dˆ
xˆ T = aˆ [ 1 – T ] – cˆ [ 1 – T ]
(8.2.7)
or in the reparameterized form – βˆ δˆ xˆ T = ( αˆ /βˆ ) [ 1 – T ] – ( γˆ /δˆ ) [ 1 – T ]
(8.2.8)
EXAMPLE 8.2.2 Estimate the 100-year flood for the data of Example 8.2.1. From Eq. 8.2.7 xˆ T is given by:
xˆ T = 33929 [1 – T–16.299] –259008 [1 – T 0.07385] For T = 100,
xˆ 100 = 33929 [1 – (100)–16.299] –259008 [1 – (100)0.07385] xˆ 100 = 138847 cfs The quantile estimates for different return periods T are given in Table 8.2.1. The results are obtained by using the computer program discussed in Chapter 10. It can be seen from Table 8.2.1 that F ( x 1 ) > F ( x 2 ) for x1 > x2 and thus conditions 4 and 5 of Landwehr et al. (1979c) are satisfied. The quantile plot for station 49 obtained by using the PWM method and the WAK(4) distribution is shown in Figure 8.2.1. Table 8.2.1. Quantile Estimates for Example 8.2.1
T 10 20 50 100 200
P (%) 10 5 2 1 0.5
Q (cfs) 81939 98065 120688 138849 157963
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Figure 8.2.1. Observed and estimated flows for the Wabash River data used in Examples 8.2.1 and 8.2.2.
8.2.4
Standard Error
As in the case of the WAK(5) distribution, the computation of variances and covariances, and consequently the standard error of the parameters is rather complicated. Approximate standard errors as well as parameter variances and covariances may be estimated by simulation.
8.3
The Generalized Pareto Distribution
8.3.1 Introduction The generalized Pareto distribution is a special case of the Wakeby distribution. If γ = 0 and β = k in the reparameterized form of WAK(5) distribution in Eq. 8.1.11, the Pareto distribution results as in Eq. 8.3.1. α k x = ε + --- [ 1 – ( 1 – F ) ] k
(8.3.1)
The distribution function F = F(x) is explicitly written as in Eq. 8.3.2. k F ( x ) = 1 – 1 – --- ( x – ε ) α
1/k
(8.3.2)
The probability density function is written as in Eq. 8.3.3.
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1 k f ( x ) = --- 1 – --- ( x – ε ) α α
1/k – 1
(8.3.3)
The variable x takes values in the range ε ≤ x < ∞ for k ≤ 0 and ε ≤ x ≤ ε + α/k for k > 0. The special case of k being 0 yields the exponential distribution, whereas the special case of k = 1 yields the uniform distribution on [ε, ε + α]. Pareto distributions are obtained when k < 0. The Pareto distribution is the logical choice for modeling flood magnitudes that exceed a fixed threshold (Hosking and Wallis, 1987) when it is reasonable to assume that successive floods follow a Poisson process and have independent magnitudes.
8.3.2 Parameter Estimation Method of Moments The first moment of the generalized Pareto distribution can be calculated by (assume k < 0) ∞
µ'1 =
∫ ε
x --α
k µ′1 = – x 1 – --- ( x – ε ) α
k 1 – --- ( x – ε ) α
1/k
∞
∞
| ∫ +
ε
ε
1/k – 1
(8.3.4)
dx
k 1 – --- ( x – ε ) α
1/k
dx
α µ′1 = ε + -----------1+k
(8.3.5)
(8.3.6)
Similarly other moments are given by Eqs. 8.3.7 to 8.3.9. α µ 2 = -------------------------------------2 ( 1 + k ) ( 1 + 2k ) 2
(8.3.7)
1⁄2
2 ( 1 – k ) ( 1 + 2k ) C s = ---------------------------------------------( 1 + 3k )
(8.3.8)
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3 ( 1 + 2k ) ( 3 – k + 2k ) C k = --------------------------------------------------------( 1 + 3k ) ( 1 + 4k ) 2
(8.3.9)
Given the coefficient of skewness of the data, Eq. 8.3.8 can be solved numerically to obtain kˆ . To numerically solve Eq. 8.3.8 we use a Newton iterative procedure. The initial value of k may be taken as zero for positive skew and –1/2 for negative skew. The value of k is updated as k n + 1 = k n – F ( k n )/F′ ( k n )
(8.3.10)
where F ( k ) = 2 ( 1 – k ) ( 1 + 2k )
1⁄2
1⁄2
/ ( 1 + 3k ) – C S –1 ⁄ 2
(8.3.11) 1⁄2
6 ( 1 – k ) ( 1 + 2k ) – 2 ( 1 + 2k ) 2 ( 1 – k ) ( 1 + 2k ) F′ ( k ) = --------------------------------- + -------------------------------------------------- – -----------------------------------------------2 ( 1 + 3k ) ( 1 + 3k ) ( 1 + 3k ) (8.3.12) The iteration in Eq. 8.3.10 is repeated until F(k) is sufficiently close to zero. The other parameters can then be calculated from Eqs. 8.3.6 and 8.3.7 as 1⁄2 2 αˆ = [ m 2 ( 1 + kˆ ) ( 1 + 2kˆ ) ]
(8.3.13)
εˆ = m′ 1 – αˆ / ( 1 + kˆ )
(8.3.14)
We can also obtain modified moment estimates by considering the smallest observation x1 which can be proved to have a probability density function of the form (Hogg and Tanis, 1988) given in Eq. 8.3.15. f ( x1 ) = N [ 1 – F ( x ) ]
N k f ( x 1 ) = ---- 1 – --- ( x – ε ) α α
(N – 1) -----------------k
N–1
f ( x)
k 1 – --- ( x – ε ) α
(8.3.15) ( 1/k – 1 )
(8.3.16)
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1 (k ⁄ N ) f ( x 1 ) = ------------ 1 – ----------------- ( x – ε ) α⁄N (α ⁄ N )
1 ----------- –1 R ⁄ N
(8.3.17)
The density function in Eq. 8.3.17 is that of a generalized pareto distribution with parameters k* = k/N and α* = α/N. E(x1) can be derived from Eq. 8.3.6. α/N α E ( x 1 ) = ε + ------------------ = ε + ------------1 + k/N N+k
(8.3.18)
By considering Eq. 8.3.18 along with Eqs. 8.3.6 and 8.3.7 and replacing µ′ 1, µ 2 and E(x1) by their sample estimates m′ 1, m 2 and x1 (the only available observation), the value of ε is given as in Eq. 8.3.19. 2 εˆ = – b + b – C
(8.3.19)
m2 b = ( n – 1 ) ----------------------– m′ 1 ( m′ 1 – x 1 )
(8.3.20)
( m 1 – nx 1 ) 2 C = ( m′ 1 ) – m 2 + 2m 2 -----------------------( m1 – x1 )
(8.3.21)
In Eq. 8.3.19,
Other parameters of the distribution are then calculated as 1 2 kˆ = --- [ ( m′1 – ε ) /m 2 – 1 ] 2
(8.3.22)
1 2 αˆ = --- ( m′1 – ε ) [ ( m′1 – ε ) /m 2 + 1 ] = ( m′1 – ε ) ( 1 + kˆ ) 2
(8.3.23)
When studying flood flow data that exceed a certain threshold εo, we can subtract αˆ from the observations and the lower bound becomes zero. The value of ε may also be known or assumed to be zero. In these cases, the estimates of α and k are obtained from Eqs. 8.3.22 and 8.3.23 by replacing ε by zero or its known value.
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Maximum Likelihood (ML) Method The likelihood function for a sample of size n from a generalized Pareto distribution is given by Eq. 8.3.24. 1 N L = --- α
N
∏
i=1
k 1 – --- ( x i – ε ) α
1/k – 1
(8.3.24)
Taking natural logarithms of Eq. 8.3.24 results in Eq. 8.3.25. N
1 k log L = – N log α + --- – 1 ∑ log 1 – --- ( x i – ε ) k α
(8.3.25)
i=1
For convenience, Eq. 8.3.26 is substituted into Eq. 8.3.25 to get Eq. 8.3.27 k y i = 1 – --- ( x i – ε ) α
(8.3.26) N
1 log L = – N log ( α ) + --- – 1 ∑ log ( y i ) k i=1
(8.3.27)
Similar to the likelihood function of the exponential distribution (Section 6.1), the log-likelihood function is an increasing function of ε, and thus the ML estimate of ε is the largest value ε can attain which is naturally estimated by x1, the smallest observation. For the other parameters Eq. 8.3.27 is differentiated with respect to α and k to obtain Eqs. 8.3.30 and 8.3.31. Eqs. 8.3.28 and 8.3.29 are used to derive Eqs. 8.3.30 and 8.3.31. ∂y -------i = ( 1 – y i )/α ∂α
(8.3.28)
∂y -------i = ( y i – 1 )/k ∂k
(8.3.29)
N
∂ log L 1 1 N –1 --------------- = --- --- – 1 ∑ y i – ------ = 0 ∂α α k αk i=1
(8.3.30)
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N
N
1 1 ∂ log L N 1 1 –1 --------------- = ---- --- – 1 – ----2 ∑ log ( y i ) – --- --- – 1 ∑ y i = 0 k kk ∂k k k i=1 i=1
(8.3.31)
Eqs. 8.3.30 and 8.3.31 are solved by a Newton iterative method to obtain the values of αˆ and kˆ , following the same steps for the GEV distribution. αn δα n αn + 1 = + kn + 1 kn δk n
(8.3.32)
The values of δαn and δkn are obtained as in Eq. 8.3.33 –1
– ∂ log L – ∂ log L ------------------- -------------------2 ∂α∂k ∂α = 2 – ∂ log L – ∂ log L -------------------- ------------------2 ∂k∂α ∂k 2
δα n δk n
2
∂ log L --------------∂α ∂ log L --------------∂k
(8.3.33)
The second partial derivatives in Eq. 8.3.33 are given by Eqs. 8.3.34 to 8.3.36. N
1 1 ∂ LogL N –2 ------------------ = -------– -----2 --- – 1 ∑ y i 2 2 ∂α α k α k i=1 2
∂ log L N ---------------- = – ----2 2 ∂k k 2
N
2 2 2 3--- – 1 + ---log ( y i ) + ----2 --- – 1 k k3 ∑ k k i=1
(8.3.34)
N
N
∑ yi
–1
i=1
1 1 –2 – ----2 --- – 1 ∑ y i k k i=1 (8.3.35)
2 2 1 1 ∂ log L ∂ log L ----------------- = ----------------- = ------ --- – 1 αk k ∂α∂k ∂k∂α
N
N
∑ yi
–2
i–1
1 2 N –1 – ------ --- – 1 ∑ y i + --------2 ∂k k ∂k i=1
(8.3.36)
The iterations are repeated until Eqs. 8.3.30 and 8.3.31 are sufficiently close to zero. Initial values of the parameters can be assumed to be the MOM estimates. This procedure can also be used when ε is known or is assumed to be equal to zero.
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PWM Method The L-moments of the generalized Pareto distribution are given by (Hosking, 1990) α λ 1 = ε + -----------1+k
(8.3.37)
α λ 2 = ---------------------------------(1 + k )(2 + k )
(8.3.38)
(1 – k) τ 3 = ----------------(3 + k)
(8.3.39)
Parameter estimates are obtained by replacing λ1, λ2, l3 by their corresponding sample estimates l1, l2, and t3 to get kˆ , αˆ , and εˆ . ( 1 – 3t 3 ) kˆ = -------------------(1 + t3)
(8.3.40)
αˆ = l 2 ( 1 + kˆ ) ( 2 + kˆ )
(8.3.41)
εˆ = l 1 – l 2 ( 2 + kˆ )
(8.3.42)
Hosking (1986b) suggests an alternative method for estimating the parameters to insure better asymptotic efficiency of the estimators. The method is based on λ1, λ2, and Nαn–1, where N α n – 1 = ε + α/ ( n + k ) is estimated by x1:N, the smallest observation. In this case we get the estimates given in Eqs. 8.3.43 to 8.3.45. N ( l 1 – x 1:N ) – 2 ( N – 1 )l 2 kˆ = ----------------------------------------------------------( N – 1 )l 2 – ( l 1 – x 1:N )
(8.3.43)
αˆ = ( 1 + kˆ ) ( 2 + kˆ )l 2
(8.3.44)
αˆ εˆ = x 1:N – -----------------( N + kˆ )
(8.3.45)
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If ε is known or can be assumed to be zero, kˆ and αˆ are given by Eqs. 8.3.46 and 8.3.47. kˆ = l 1 /l 2 – 2
(8.3.46)
αˆ = ( 1 + kˆ ) l 1
(8.3.47)
EXAMPLE 8.3.1 Estimate the parameters of the generalized Pareto distribution by using the data from the Wabash River at Mt. Carmel, Illinois exceeding a threshold of 50,000 cfs which are given in Table 8.3.1. The data cover a period of 65 years (1928 to 1992) with 281 peaks that exceed 50,000 cfs with an average annual number of peaks of λ = 281/65 = 4.32308 . Method of Moments:
From the data in Table 8.3.1 the following statistics are computed. N = 281, m' 1 = 100774.377, C v = 0.4601, C s = 1.5342 a. Direct Moment Estimates: Following the procedure given in Section 8.3.2 to solve Eq. 8.3.3 kˆ is obtained: kˆ = 0.0956
ˆ s is computed and compared to Cs computed directly As a check, C from the data 1⁄2
2 ( 1 – kˆ ) ( 1 + 2kˆ ) Cˆ s = ---------------------------------------------- = 1.5342 = C s ( 1 + 3kˆ ) ˆ and εˆ are computed. From Eqs. 8.3.13 and 8.3.14 α 2 2 αˆ = [ ( 0.4601 × 100774.377 ) ( 1 + 0.0956 ) ( 1 + 2 × 0.0956 ) ]
1⁄2
αˆ = 55443.00 εˆ = 100774.377 – 55443.00/ ( 1 + 0.0956 ) = 50169.23 b. Modified Method of Moments: From Table 8.3.1 x1 is 50400.
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From Eqs. 8.3.20 and 8.3.21, b and C are computed.
( 0.4601 × 100774.377 ) b = ( 281 – 1 ) ---------------------------------------------------------- – 100774.377 = 11848816.97 ( 100774.377 – 50400 ) 2
2
2
C = ( 100774.377 ) – ( 0.4601 × 100774.377 ) + 2 ( 0.4601 × 100774.337 ) × ( 100774.377 – 281 × 50400 )/ ( 100774.377 – 50400 )
2
C = –1.1922135 × 1012 From Eq. 8.3.19 εˆ is computed. 2 12 εˆ = – 11848816.97 + 1184881697 + 1.1922135 × 10
εˆ = 50203.04 ˆ are computed. From Eqs. 8.3.22 and 8.3.23 kˆ and α 1 2 2 kˆ = --- [ ( 100774.377 – 50203.04 ) / ( 0.4601 × 100774.377 ) – 1 ] = 0.0948 2 αˆ = ( 100774.377 – 50203.04 ) ( 1 + 0.0948 ) = 55365.72 ML Method:
ˆ , and kˆ are: Following the procedure given in Section 8.3.2 εˆ , α εˆ = 50400 αˆ = 55142.29 kˆ = 0.0945 To verify that these are the solutions for Eqs. 8.3.30 and 8.3.31 we calculate n
∑ yi
–1
= 310.3325966
i=1
n
∑ log ( y )
= – 26.56062703
i=1
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Substituting in Eqs. 8.3.30 and 8.3.31 we get 1 ∂ log L 1 281 –6 --------------- = ---------------------- ---------------- – 1 ( 310.3325966 ) – --------------------------------------------- = 1.2 × 10 ≈ 0 55142.29 0.0945 ∂α 0.0945 × 55142.29
1 281 1 1 ∂ log L 1 --------------- = ---------------- ---------------- – 1 – -----------------------2 ( – 26.56062703 ) – ---------------- ---------------- – 1 0.0945 0.0945 0.0945 0.0945 ( 0.0945 ) ∂k × ( 310.3325966 ) = 0
PWM Method: From Table 8.3.1 the following is calculated. N = 281, l1 = 100774.38, l2 = 24295.96, t3 = 0.2703, and x1 = 50400 a. Direct Method: From Eqs. 8.3.40 to 8.3.42 we get
1 – 3 × 0.2703 kˆ = ---------------------------------- = 0.1489 1 + 0.2703 αˆ = 24295.96 ( 1 + 0.1489 ) ( 2 + 0.1489 ) = 59983.60 εˆ = 100774.38 – 24295.96 ( 2 + 0.1489 ) = 48564.79 b. Modified Method: From Eqs. 8.3.43 to 8.3.45 we get
281 ( 100774.38 – 50400 ) – 2 × 280 × 24295.96 kˆ = ----------------------------------------------------------------------------------------------------------------- = 0.0814 280 × 24295.96 – ( 100774.38 – 50400 ) αˆ = ( 1 + 0.0814 ) ( 2 + 0.0814 ) 24195.96 = 54685.98 54685.98 εˆ = 50400 – ------------------------------- = 50205.44 281 + 0.0814 The parameter estimates calculated in this example using different methods are given in Table 8.3.2.
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Table 8.3.2. Parameter Estimates for Example 8.3.1
ε 50169.23 50203.04 50400.00 48564.79 50205
MOM MOM (modified) MLM PWM PWM (modified)
α 55443.00 55365.72 55142.00 59983.60 54685.98
k 0.0956 0.0948 0.0945 0.1489 0.0814
8.3.3 Quantile Estimation Using the inverse form x = x(F) in Eq. 8.3.1, for a given return period T, substituting F = 1 – (1/T) the T-year quantile is computed by Eq. 8.3.48 αˆ – kˆ xˆ T = εˆ + --- [ 1 – T ] ˆk
(8.3.48)
The frequency factor is derived as 1⁄2
( 1 + 2k ) –k K T = -------------------------- [ ( 1 + k ) ( 1 – T ) – k ] k
(8.3.49)
EXAMPLE 8.3.2 Estimate the 100-year flood for the Wabash River data used in Example 8.3.1. Use the parameters estimated in Example 8.3.1. The probability of exceedence in this case is defined as P(xT > x > xo) = 1 – 1/λT, where λ is the average annual number of floods exceeding 50,000 cfs which is equal to 4.32308 as given in Example 8.3.1. This will affect the computation of quantiles. Quantile Estimates by the MOM Parameters: Using the Modified MOM estimators, from Eq. 8.3.48 we get
xˆ T = 50203.04 + 584026.58 [ 1 – T
– 0.0948
]
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Table 8.3.1. The Wabash River Flows at Mt. Carmel (Threshold = 50,000 cfs) 126000 148000 66000 156000 136000 122000 183000 162000 85200 56800 56600 138000 81000 51800 90700 13900 160000 118000 50600 137000 151000 172000 52600 248000 152000 64500 61500 143000 108000 53000 134000 115000 84100 105000 85400 76900 99100 73700 122000 62500 54300 58000 144000 55800 127000 55800 107000
56400 128000 106000 110000 232000 60400 60400 50800 100000 55900 167000 53700 56700 126000 100000 59500 164000 81800 56400 124000 64600 77300 65900 72500 65100 80800 69800 53000 195000 128000 114000 110000 149000 74100 75900 99300 168000 70800 104000 125000 77300 97300 140000 54900 66000 199000 99800
105000 93700 277000 85700 77300 122000 106000 93300 79000 130000 126000 57800 64700 162000 71900 63500 81500 51000 84400 108000 185000 55800 94600 82800 146000 66500 57700 78700 85100 129000 75700 104000 139000 50600 53500 178000 110000 50800 76000 130000 67300 149000 78400 96600 83300 68400 84300
56400 112000 76400 116000 51400 59800 63900 81900 88200 62300 162000 67200 85500 51000 286000 73800 61300 60800 91300 134000 106000 70800 106000 122000 149000 53700 85300 144000 54800 116000 67500 56500 86700 91500 105000 134000 97300 84000 141000 52600 124000 196000 84200 54500 74500 104000 57200
61000 155000 96500 89100 77900 70500 73400 180000 83700 302000 133000 92100 105000 235000 213000 96100 77100 73900 55400 55200 87800 52600 106000 93000 147000 61800 101000 154000 52000 121000 86700 57300 97500 112000 88500 76200 140000 87400 154000 95100 131000 131000 54900 78800 101000 224000 54800
50900 63500 63500 152000 51000 285000 114000 197000 106000 132000 83700 67200 110000 202000 127000 90600 12600 73900 86500 181000 141000 79700 97800 57300 77200 133000 82900 550000 62000 51700 54500 51600 103000 134000 71700 57000 63900 60700 81900 171000 111000 50400 50500 69700 88900 76600 —
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This is applicable only for an annual maximum series, however, and it must be noticed here that since the data in this example came from a POT series, the value of T should be replaced by λT where λ = 4.32308 is the average number of peaks per year and the quantiles become
xˆ T = 50203.04 + 584026.58 [ 1 – ( 4.32308T )
– 0.0948
]
For example, if λT = 100 (T = 23.13 years) then
xˆ 23.13 = 50203.04 + 584026.58 [ 1 – ( 100 )
– 0.0948
]
xˆ 23.13 = 256803 cfs Quantile Estimate by the ML Parameters: From Eq. 8.3.48 the quantile estimate is given by:
xˆ T = 50400 + 583516.30 [ 1 – T
– 0.0945
]
This is true only for annual maximum data, replacing T by λT since we are using POT data we get
xˆ T = 50400 + 583516.30 [ 1 – ( 4.32308 T )
– 0.0945
]
For example if λT = 100 (T = 23.13 years) we get
xˆ 23.13 = 50400 + 583516.30 [ 1 – ( 100 )
– 0.0945
]
xˆ 23.13 = 256298 cfs Quantile Estimate by the PWM Parameters: Consider the case of the modified estimates. We have from Eq. 8.3.48
xˆ T = 50205.44 + 671817.94 ( 1 – T
– 0.0814
)
This is true only for annual maximum data. For POT data with λ = 4.32308 we replace T by λT to get
xˆ T = 50205.44 + 671817.94 [ 1 – ( λ T )
– 0.0814
]
For example if λT = 100 (T = 23.13 years) we get
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xˆ 23.13 = 50205.44 + 671817.94 [ 1 – ( 100 )
– 0.0814
] = 260226 cfs
Quantile estimates for other recurrence intervals are given in Table 8.3.3.
8.3.4 Standard Error Method of Moments Hosking and Wallis (1987) give the variances and covariances of the generalized Pareto distribution for the case when ε = 0 in Eqs. 8.3.50 to 8.3.52. ( 1 + k ) ( 1 + 6k + 12k ) 2α var ( αˆ ) = --------- --------------------------------------------------------------N ( 1 + 2k ) ( 1 + 3k ) ( 1 + 4k )
(8.3.50)
1 ( 1 + k ) ( 1 + 2k ) ( 1 + k + 6k ) var ( kˆ ) = ---- ------------------------------------------------------------------------N ( 1 + 2k ) ( 1 + 3k ) ( 1 + 4k )
(8.3.51)
2
2
2
2
2
2
α ( 1 + k ) ( 1 + 2k ) ( 1 + 4k + 12k ) cov ( αˆ , kˆ ) = ---- -----------------------------------------------------------------------------N ( 1 + 2k ) ( 1 + 3k ) ( 1 + 4k ) 2
2
(8.3.52)
When k ≤ –1/4 then the variance of σ2 is infinite and the variances of αˆ and kˆ are not asymptotic order n–1. The above equations are valid only if k > –1/4. The standard error can be obtained by substituting Eqs. 8.3.50 to 8.3.52 in Eq. 4.4.3, where from Eq. 8.3.48 we have 1 ∂x –k ------- = --- [ 1 – T ] k ∂α
(8.3.53)
α –k –α ∂x –k ------ = ------2- [ 1 – T ] + ---T log ( T ) k ∂k k
(8.3.54)
For the modified moment estimates, the variance of x1 is obtained from Eq. 8.3.7 using α* = α/N, k* = k/N as in Eq. 8.3.55
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2
nα var ( x 1 ) = ------------------------------------------2 ( N + k ) ( N + 2k )
(8.3.55)
This should ensure that var εˆ is of order N–2 and thus can be neglected relative to the variances of αˆ and kˆ (order N–1) to obtain approximate confidence intervals. In this case the variances and covariances of αˆ and kˆ are the same as in Eqs. 8.3.50 to 8.3.52. ML Method For the ML method, when ε = 0 we have Eqs. 8.3.56 to 8.3.58 (Hosking, 1986; Hosking and Wallis, 1987). 2
2α var ( αˆ ) = --------- ( 1 – k ) N
(8.3.56)
1 2 var ( kˆ ) = ---- ( 1 – k ) N
(8.3.57)
α cov ( αˆ , kˆ ) = ---- ( 1 – k ) N
(8.3.58)
These results are correct only for k < 1/2. The results for k > 1/2 are much more complicated. The standard error can be obtained by substituting Eqs. 8.3.56 to 8.3.58 in Eq. 4.4.19 also using the partial derivatives in Eqs. 8.3.53 and 8.3.54. For the three-parameter case, the asymptotic variance of εˆ = x 1 is of order N–2 and may be neglected with respect to the variances of αˆ and kˆ (order N–1) for large N to obtain approximate standard errors. PWM Method Hosking (1986a) and Hosking and Wallis (1987) give the variances and covariances of parameter estimates for the case when ε = 0 as in Eqs. 8.3.59 to 8.3.61. α ( 7 + 18k + 11k + 2k ) var ( αˆ ) = ----- ------------------------------------------------------N ( 1 + 2k ) ( 3 + 2k ) 2
2
3
(8.3.59)
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1 ( 1 + k ) ( 2 + k ) ( 1 + k + 2k ) var ( kˆ ) = ---- --------------------------------------------------------------------------( 1 + 2k ) ( 3 + 2k ) N
(8.3.60)
α ( 2 + k ) ( 2 + 6k + 7k + 2k ) cov ( αˆ , kˆ ) = ---- -------------------------------------------------------------------N ( 1 + 2k ) ( 3 + 2k )
(8.3.61)
2
2
2
3
Once again, these results are correct only for k > –1/2. The expressions for k ≤ –1/2 are much more complicated. The standard error is calculated by substituting Eqs. 8.3.59 to 8.3.61 and 8.3.53 and 8.3.54 into Eq. 4.4.22. For the three-parameter case, if the modified estimates in Eqs. 8.3.43 to 8.3.45 are used (Hosking, 1986a), the variances and covariances of αˆ and kˆ are the same as in Eqs. 8.3.59 to 8.3.61 and the asymptotic variance of εˆ is of order N–2, which may be neglected with respect to the variances of αˆ and kˆ (order N–1) for large N to obtain approximate standard errors.
EXAMPLE 8.3.3 Estimate the standard error of the 100-year flow by using the parameters estimated in Example 8.3.1. Standard Error by MOM Parameters: For the case of Modified MOM we can calculate the following from Eqs. 8.3.50 to 8.3.52
var ( αˆ ) = 20805954.28 –3 var ( kˆ ) = 3.2904227 × 10
cov ( αˆ , kˆ ) = 198.246895 Now consider the case where λT = 100 (T = 23.13 years) From Eqs. 8.3.53 and 8.3.54 we get
1 ∂x –k ------- = --- [ 1 – ( λ T ) ] = 3.7315488 k ∂α α –α ∂x –k –k ------ = ------2- [ 1 – ( λ T ) ] + --- ( λ T ) log ( λ T ) = – 441209.53 k ∂k k From Eq. 4.4.3 we get
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2
s T = (3.7315488)2 (20805954.28) + (–441209.53)2 (3.2904227 × 10–3) + 2(3.7315488) (–441209.53) (198.246895) = 277459183.2 sT = 16657 cfs Standard Error by MLM Parameters: From Eqs. 8.3.56 to 8.3.58 we get
2 ( 55142.29 ) var ( αˆ ) = -------------------------------- ( 1 – 0.0945 ) = 19596645.04 281 2
1 2 –3 var ( kˆ ) = --------- ( 1 – 0.0945 ) = 2.9179012 × 10 281 55142.29 cov ( αˆ , kˆ ) = ---------------------- ( 1 – 0.0945 ) = 177.691614 281 Now consider the case with λT = 100 (T = 23.13 years). From Eqs. 8.3.53 and 8.3.54 and replacing T with λT we get
1 ∂x – 0.0945 ] = 3.7339405 ------- = ---------------- [ 1 – 100 0.0945 ∂α – 55142.29 55142.29 ∂x – 0.0945 – 0.0945 ------ = ------------------------2- [ 1 – 100 ] + ---------------------- ( 100 ) log ( 100 ) = – 439818.811 0.0945 ∂k ( 0.0945 )
From Eq. 4.4.19 we get 2
s T = (3.7339405)2 (19596645.04) + (–439818.811)2 (2.9179012 × 10–3) + 2(3.7339405) (–439818.811) (177.691614) = 254032361.9 sT = 15938 cfs Standard Error by PWM Parameters: For the modified case we have from Eqs. 8.3.59 to 8.3.61
var ( αˆ ) = 24710657.06
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var ( kˆ ) = 4.9623848 × 10
–3
cov ( αˆ , kˆ ) = 279.30186 For the case where λT = 100 we get from Eqs. 8.3.53 and 8.3.54
1 ∂x –k ------- = --- [ 1 – ( λT ) ] = 3.84048 k ∂α –α α ∂x –k –k ------ = ------2- [ 1 – ( λT ) ] + --- ( λT ) log ( λT ) = – 453447.26 k ∂k k Substituting in Eq. 4.4.22 we get 2
s T = (3.84048)2 (24710657.06) + (–453447.26)2 (4.9623848 × 10–3) + 2(3.84048) (–453447.26) (279.30186) 2
s T = 412019108.8 sT = 20298 cfs The quantile estimates and their standard errors for different values of λT where λ = 4.32308, the average number of peaks per year are given in Table 8.3.3. Shown are the results for the Modified MOM, MLM, and modified PWM. These results are obtained by using the computer program discussed in Chapter 10. Quantile plots generated by using the MOM, ML, and PWM methods are shown in Figure 8.3.1.
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Table 8.3.3. Quantile Estimates and their Standard Errors (in parentheses) for Example 8.3.2
λΤ P(%) MOM (Modified) 20 5.00 194591 (8255) 40 2.50 222550 (11271) 60 1.67 238073 (13470) 80 1.25 248731 (15211) 100 1.00 256800 (16657) 120 0.83 263267 (17897) 140 0.71 268648 (18985) 160 0.63 273247 (19956) 180 0.56 277255 (20832) 200 0.50 280802 (21632)
MLM PWM (Modified) 194263 195585 (8125) (8999) 222136 224467 (10936) (13042) 237614 240622 (12978) (16007) 248242 251765 (14594) (18352) 256288 260230 (15937) (20299) 262738 267034 (17089) (21969) 268105 272708 (18101) (23433) 272692 277566 (19003) (24739) 276689 281807 (19819) (25919) 280228 285567 (20563) (26996)
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Figure 8.3.1. Observed and estimated flows and 95% confidence intervals for the Wabash River data used in Example 8.3.1 to 8.3.3.
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CHAPTER 9
The Logistic Distribution
9.1
Logistic Distribution
9.1.1 Introduction The logistic distribution function is given in Eq. 9.1.1.
F( x) = 1 + e
x – m ----------- a
–1
(9.1.1)
The variable x takes values in the range –∞ < x < ∞. The probability density function of x is given by Eq. 9.1.2. x – m
x – m
–2
------------ a a 1 -----------f ( x ) = --- e 1+e a
(9.1.2)
9.1.2 Parameter Estimation Method of Moments The moments of x are expressed in terms of parameters m and a (Greenwood et al., 1979) in Eq. 9.1.3. r
d θm µ′r = --------r [ e Γ ( 1 – aθ ) Γ ( 1 + aθ ) ] dθ
θ=0
(9.1.3)
From Eq. 9.1.3 we may write,
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µ′1 = m,
(9.1.4)
π 2 2 µ′2 = m + ----- a 3 2
(9.1.5)
Thus, π 2 2 µ 2 = µ′2 – ( µ′ 1 ) = ----- a 3 2
(9.1.6)
Parameter estimates can be obtained by replacing µ′1 and µ 2 by their corresponding sample estimates m′1, m2 to get: 3 1⁄2 aˆ = ------- m 2 π
(9.1.7)
ˆ = m′1 m
(9.1.8)
Maximum Likelihood (ML) Method The likelihood function for a sample of size N from a logistic distribution is given by Eq. 9.1.9.
1 1 L = -----N exp – --a a
N
N
∑ ( xi – m ) ∏
i=1
1+e
xi – m – -------------- a
–2
(9.1.9)
i=1
The log-likelihood is obtained from Eq. 9.1.9 as in Eq. 9.1.10, N
N
x i – m
– ------------- a 1 log L = – n log ( a ) – --- ∑ ( x i – m ) – 2 ∑ log 1 + e ai = 1 i=1
(9.1.10)
By defining y as in Eq. 9.1.11,
yi = 1 + e
x i – m ------------ a
(9.1.11)
Eq. 9.1.10 is rewritten as Eq. 9.1.12.
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N
N
1 log L = – N log ( a ) – --- ∑ ( x i – m ) – 2 ∑ log ( y i ) ai = 1 i=1
(9.1.12)
Differentiating 9.1.12 and noting that, ∂y -------i = ( y i – 1 )/a ∂m
(9.1.13)
∂y i 2 ------- = ( y i – 1 ) ( x i – m )/a ∂a
(9.1.14)
the derivatives of the log likelihood function are given in Eqs. 9.1.15 and 9.1.16. N
∂ log L N 2 --------------- = – ---- + --∂m a a ∂ log L N 1 --------------- = – ---- – -----2 ∂a a a
∑ yi
–1
(9.1.15)
= 0
i=1
N
N
2
-2 ∑ ( x i – m )y i ∑ ( x i – m ) + ---a
–1
= 0
(9.1.16)
i=1
i=1
Equations 9.1.15 and 9.1.16 are solved numerically to obtain the values ˆ and aˆ . The Newton’s method, which was also used in the GEV of m distribution (Section 7.1) is used here. Initial values of m and a are taken as the MOM estimates. The values are updated according to Eq. 9.1.17 mn δm n mn + 1 = + an + 1 an δa n
(9.1.17)
where the values of δmn and δan are obtained from Eq. 9.1.18. –1
∂ log L ∂ log L – ---------------- – ----------------2 ∂m∂a ∂m = 2 2 ∂ log L ∂ log L – ----------------- – ---------------2 ∂a∂m ∂a 2
δm n δa n
2
∂ log L --------------∂m ∂ log L --------------∂a
(9.1.18)
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The second partial derivatives are given by Eqs. 9.1.19 to 9.1.21. N
N
2 2 2 ∂ log L –1 –2 ---------------- = – -----2 + ∑ y i + -----2 ∑ y i 2 a i=1 ∂m a i=1 2 N 2 ∂ log L ---------------- = -----2 + -----3 2 a a ∂a
2 – -----4 a
N
∑ i–1
N
(9.1.19)
N
4
∑ ( x i – m ) – a-----3 ∑ ( x i – m )yi
–1
i=1
i=1
N
2 2 –1 2 –2 ( x i – m ) y i + -----4 ∑ ( x i – m ) y i a i=1 N
(9.1.20)
N
2 2 ∂ log L N 2 –1 –1 ----------------- = -----2 – -----2 ∑ y i – -----3 ∑ ( x i – m ) y i ∂a∂m a i=1 a a i=1
2 + -----3 a
N
∑ ( xi – m ) yi
–2
(9.1.21)
i=1
The iteration in Eq. 9.1.17 is repeated until Eqs. 9.1.15 and 9.1.16 are sufficiently close to zero. PWM Method The distribution function in Eq. 9.1.1 can be written in the inverse form as Eq. 9.1.22. x = m + a [log (F ) – log (1 – F )]
(9.1.22)
PWMs are obtained by Eq. 9.1.23 1
αs =
∫ [m + a
log ( F ) – a log ( 1 – F ) ] ( 1 – F )5dF˙
(9.1.23)
0
The PWMs of the logistic distribution are of the form (Greenwood et al, 1979), a m α k = ------------ – -----------1+k 1+k
k
1
∑ --s-
(Summation = 0 for k = 0 )
(9.1.24)
s=1
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The PWMs can thus be obtained by Eqs. 9.1.25 and 9.1.26. α0 = m
(9.1.25)
m a α 1 = ---- – --2 2
(9.1.26)
ˆ and aˆ are computed by substituting a0 and Parameter estimates m a1 to their corresponding population estimates α0 and α1. aˆ = a 0 – 2a 1 = l 2
(9.1.27)
ˆ = a0 = l1 m
(9.1.28)
EXAMPLE 9.1.1 Estimate the parameters of a logistic distribution for the annual maximum flows of the Tippecanoe River near Delphi, Indiana (station 43) given in Table 5.1.1. Method of Moments: From Table 2.1.2 for the given data (no. 43) we have:
N = 48, m′1 = 12665, C V = 0.3719 and C S = 0.1194 From Eqs. 9.1.7 and 9.1.8 we get
aˆ =
3 --- ( 0.3719 × 12665 ) = 2596.62 π ˆ = m′1 = 12665 m
ML Method: Applying the procedure given in Section 9.1.2 the following estimates are obtained:
aˆ = 2708.64 ˆ = 12628.59 m To verify Eqs. 9.1.15 and 9.1.16 we calculate
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n
∑ ( xi – m )
= 1757.4
i=1 n
∑ ( x i – m )yi
–1
= 65886
i=1 n
∑ yi
–1
= 24
i=1
Substituting the above values in Eqs. 9.1.15 and 9.1.16 we get
48 2708.64
2 2708.64
Eq. 9.1.15: – ------------------- + ------------------- ( 24 ) = 0
1 ( 2708.64 )
2 ( 2708.64 )
48 –8 - – --------------------------2 ( 1757.4 ) + --------------------------2 ( 65886 ) = – 2 × 10 ≈ 0 Eq. 9.1.16: – -----------------2708.64
The equations are satisfied and the calculated parameters are the ML estimates. PWM Method: From Table 3.1.1 for station 43 we have: N = 48, l1 = 12665, t = 0.2150
ˆ are estimated from Eqs. 9.1.27 and 9.1.28. aˆ and m aˆ = l 2 = tl 1 = 0.2150 × 12665 = 2722.60 ˆ = l 1 = 12665 m The parameter estimates for this example are summarized in Table 9.1.1.
Table 9.1.1 Parameter Estimates for Example 9.1.1
Parameter MOM MLM PWM
m 12665 12628 12665
α 2596.62 2708.64 2722.60
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9.1.3 Quantile Estimation The inverse form x = x(F ) is given by Eq. 9.1.29. x = m + a log [ F/ ( 1 – F ) ]
(9.1.29)
By substituting F = 1 – (1/T ) where T is the return period, the T–year quantile estimates are calculated by Eq. 9.1.30. ˆ + aˆ log ( T – 1 ) xˆ T = m
(9.1.30)
The frequency factor can be obtained by comparing 9.1.30 with Eq. 4.3.2 to get Eq. 9.1.31. 3 K T = ------- log ( T – 1 ) π
(9.1.31)
EXAMPLE 9.1.2 Compute the 100-year flood by using the data and parameter estimates of Example 9.1.1. Quantile Estimates by MOM Parameters: From Eq. 9.1.30 we have:
xˆ T = 12665 + 2596.62 log ( T – 1 ) For T = 100,
xˆ 100 = 12665 + 2596.62 log ( 100 – 1 ) xˆ 100 = 24597 cfs Quantile Estimates by MLM Parameters: From Eq. 9.1.30 we have,
xˆ T = 12628 + 2708.64 log ( T – 1 ) For T = 100,
xˆ T = 12628 + 2708.64 log ( 100 – 1 )
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xˆ 100 = 25075 cfs Quantile Estimates by PWM Parameters: From Eq. 9.1.30 we have
xˆ T = 12665 + 2722.60 log ( T – 1 ) For T = 100,
xˆ 100 = 12665 + 2722.60 log ( 100 – 1 ) xˆ 100 = 25176 cfs Quantile estimates for other recurrence intervals are given in Table 9.1.2.
9.1.4 Standard Error Method of Moments The standard error can be obtained from Eq. 4.4.3. For the logistic distribution we have Eqs. 9.1.32 and 9.1.33 (Balakrishnan, 1992). γ1 = 0
(9.1.32)
γ 2 = 4.2
(9.1.33)
The value of δ is given by Eq. 9.1.34. 2 1⁄2
δ = [ 1 + 0.8 K T ]
(9.1.34)
The standard error is calculated by Eq. 9.1.35, a π a 2 2 2 s T = ----- ----- [ 1 + 0.8 K T ] = ----- [ 3.2899 + 2.6319 K T ] N 3 N 2
2
2
(9.1.35)
where KT is given by Eq. (9.1.31).
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Variances and covariances of the estimated parameters are: 2
4 a var ( aˆ ) = --- ----- 5 N
(9.1.36)
π a ˆ ) = ----- ----- var ( m 3 N
(9.1.37)
ˆ , aˆ ) = 0 cov ( m
(9.1.38)
2
2
ML Method The variances and covariances for the parameters in the ML case are as follows (Hosking, 1986; Antle et al. 1970): 2
a 9 - ----- var ( aˆ ) = -----------------2 (3 + π ) N
(9.1.39)
2
a ˆ ) = 3 ----- var ( m N
(9.1.40)
ˆ) = 0 cov ( aˆ , m
(9.1.41)
From Eqs. 9.1.30 and 9.1.31 we have: π ∂x ------ = ------- K T ∂a 3
(9.1.42)
∂x ------- = 1 ∂m
(9.1.43)
The standard error is computed by using Eq. 4.4.19 to give Eq. 9.1.44. In Eq. 9.1.37. KT is given by Eq. (9.1.31). 2
a 2 2 s T = ----- [ 3 + 2.3007 K T ] N
(9.1.44)
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PWM Method The variances and covariances of the estimated parameters estimated by PWM are given by Hosking (1986) in Eqs. 9.1.45 to 9.1.47.
2
a var ( aˆ ) = 0.7101 ----N
(9.1.45)
2
a ˆ ) = 3 ----var ( m N
(9.1.46)
ˆ) = 0 cov ( aˆ , m
(9.1.47)
By substituting Eqs. 9.1.45 to 9.1.47 into Eq. 4.4.22 and using Eq. 9.1.42 and Eq. 9.1.43 we get Eq. 9.1.48. 2
a 2 2 s T = ----- [ 3 + 2.3361 K T ] N
(9.1.48)
EXAMPLE 9.1.3 Estimate the standard errors for the 100-year flood values computed in Example 9.1.2. Standard Error by MOM Parameters: From Eq. 9.1.35 we get
( 2596.62 ) 2 2 s T = -------------------------- [ 3.2899 + 2.6319 K T ] 48 2
For T = 100, from Eq. 9.1.31,
3 K T = ------- log ( 100 – 1 ) = 2.5334224 π
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2
s T = 2834921.36; s T = 1684 cfs Standard Error by the ML Parameters: From Eq. 9.1.44,
( 2708.64 ) 2 2 s T = -------------------------- [ 3 + 2.3007 K T ] 48 2
For T = 100, from Eq. 9.1.31:
3 K 100 = ------- log ( 100 – 1 ) = 2.5334224 π 2
s T = 2715571.57; s T = 1648 cfs Standard Error by the PWM Parameters: From Eq. 9.1.48,
( 2722.60 ) 2 2 s T = -------------------------- [ 3 + 2.3361 K T ] 48 2
For T = 100, from Eq. 9.1.31,
3 K T = ------- log ( 100 – 1 ) = 2.533422 π 2
s T = 2778721.31; s T = 1667 cfs The quantile estimates for different return periods as well as their standard errors are given in Table 9.1.2. These results are obtained by the computer program discussed in Chapter 10. Quantile plots for station 43 computed by using the logistic distribution and the MOM, ML, and PWM methods are shown in Figure 9.1.1.
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Table 9.1.2. Quantile Estimates and their Standard Errors (in parentheses) for Example 9.1.3
T 10
9.2
P (%) 10
20
5
50
2
100
1
200
0.5
MOM 18370.56 (1002.32) 20310.79 (1198.49) 22770.77 (1471.11) 24596.97 (1683.72) 26409.89 (1900.19)
ML 18580.09 (987.22) 20604.03 (1176.97) 23170.14 (1441.37) 25075.13 (1647.90) 26966.26 (1858.37)
PWM 18647.39 (996.34) 20681.76 (1189.12) 23261.11 (1457.46) 25175.92 (1666.95) 27076.8 (1880.38)
Generalized Logistic Distribution
9.2.1 Introduction The generalized logistic distribution function is given by Eq. 9.2.1 (Hosking, 1986). 1/k – 1
x–ε F ( x ) = 1 + 1 – k ----------- α
(9.2.1)
The variable x takes values in the range α α ε + --- ≤ x < ∞ for k < 0 and – ∞ < x ≤ ε + --- for k > 0 k k The probability density function of x is given by Eq. 9.2.2.
1 f ( x ) = --α
x–ε 1 – k ----------- α
1--- – 1 k
1/k – 2
x–ε 1 + 1 – k ----------- α
(9.2.2)
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Figure 9.1.1. Observed and estimated flows and 95% confidence intervals for the Tippecanoe River data used in Examples 9.1.1 to 9.1.3.
The logistic distribution (Section 9.1) is a special case of this distribution with k = 0, m = ε and a = α. The generalized logistic distribution is equivalent to the log-logistic distribution (Ahmad et al., 1988). The log-logistic distribution function is given by Eq. 9.2.3. 1 –1 – --c
x – a F ( x ) = 1 + ----------- b
(9.2.3)
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The relation between the parameters of Eqs. 9.2.1 and 9.2.3 is, c = –k, b = α/k and a = ε + α/k. 9.2.2 Parameter Estimation Method of Moments The first moment of x is given by α µ′ 1 = ε + --- [ 1 – Γ ( 1 + k ) Γ ( 1 – k ) ] k
(9.2.4)
The central moments are given by α 2 2 µ 2 = -----2 [ Γ ( 1 + 2k )Γ ( 1 – 2k ) – Γ ( 1 + k )Γ ( 1 – k ) ] k 2
(9.2.5)
α µ 3 = -----3 [ – Γ ( 1 + 3k )Γ ( 1 – 3k ) + 3Γ ( 1 + k )Γ ( 1 – k )Γ ( 1 + 2k ) k 3
(9.2.6)
Γ ( 1 – 2k ) – 2Γ ( 1 + k )Γ ( 1 – k ) ] 3
3
The skewness coefficient is thus given by Eq. 9.2.7. µ3 k C s = --------= ----- [ – Γ ( 1 + 3k ) Γ ( 1 – 3k ) + 3Γ ( 1 + k )Γ ( 1 – k )Γ ( 1 + 2k )Γ ( 1 – 2k ) 3⁄2 k µ2 3
3
2
2
– 2Γ ( 1 + k )Γ ( 1 – k ) ] ⁄ [ Γ ( 1 + 2k )Γ ( 1 – 2k ) – Γ ( 1 + k )Γ ( 1 – k ) ]
(9.2.7) 3⁄2
Eq. 9.2.6 can be solved numerically to obtain the value of kˆ given the sample Cs. Substituting kˆ in Eqs. 9.2.4 and 9.2.3 αˆ and εˆ are computed. An approximate solution of Eq. 9.2.7 is given by Ahmad et al., (1988) in Eqs. 9.2.8 and 9.2.9. For CS < 2.5, k = – exp ( – 2.246 + 0.848z – 0.1271 z + 0.4008z ) 2
3
(9.2.8)
and for Cs > 2.5,
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2
–C S k = ----------------------------------------------------------------------2 4.007 + 3.411 C S + 2.985 C S
(9.2.9)
where z = log (Cs ). However, this approximation is good only for positive skew values and for k greater than –1/3. The relationship between the parameter k and the coefficient of skew Cs is shown in Figure 9.2.1. Three regions can be identified in Figure 9.2.1. Region I is for all positive and negative values of Cs with k ranging from –1/3 to 1/3. Region II is for positive Cs and values of k ranging from 1/3 to 1/2. Region III is for
Figure 9.2.1.
Cs vs. K relationship for the generalized logistic distribution.
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negative CS and k values ranging from –1/2 to –1/3. The following results are obtained by fitting curves to these relationships. Region I
–10 < CS < 10; –1/3 < k < 1/3
2 –1 k = ------ tan ( – 0.59484 C S ), ( max. error 0.009 ) 3π
(9.2.10)
or approximately, –C S 2 –1 - . k = ------ tan --------------- 2 – 1/π 3π Region II
( max. error 0.009 )
(9.2.11)
0 < CS < 10; 1/3 < k < 1/2
1 1 –1 k = ------ tan ( 0.03688 – 0.29824 C S ) + --- , ( max. error 0.009 ) (9.2.12) 3π 2 or approximately CS 1 –1 - + 1 ⁄ 2. k = ------ tan – --------------- 3π 4 – 2/π
Region III
( max. error 0.012 )
(9.2.13)
–10 < CS < 0; –1/2 < k < –1/3
1 –1 k = ------ tan ( – 0.036884 – 0.29824 C S ) – 1 ⁄ 2 , (max error 0.009) (9.2.14) 3π or approximately CS 1 –1 - – 1 ⁄ 2 . (max error 0.012) k = ------ tan --------------- 4 – 2/π 3π
(9.2.15)
It is clear from Figure 9.2.1 that for each value of Cs two different values of k exist with opposite signs. The appropriate value of k is chosen by checking whether the observations are bounded from above (k > 0) or from below (k < 0).
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The numerical solution of Eq. 9.2.7 is obtained by using Newton’s method. The initial value of the parameter k is calculated using the above mentioned approximate relationships between k and Cs. The value of k is then updated according to Eq. 9.2.16. k n + 1 = k n – F ( k n )/F′ ( k n )
(9.2.16)
The values of F(k) and F′ ( k ) are given by Eqs. 9.2.17 and 9.2.18, 3
k – g 3 + 3g 2 g 1 – 2g 1 - – CS F ( k ) = ----- ------------------------------------------2 3⁄2 k ( g2 – g1 ) 3
k – d 3 + 3g 2 d 1 + 3d 2 g 1 – 6g 1 d 1 3 k ( – g 3 + 3g 2 g 1 – 2g 1 ) ( d 2 – 2g 1 d 1 ) F′ ( k ) = ----- ---------------------------------------------------------------------- – --- ----- ------------------------------------------------------------------------------2 5⁄2 2 3⁄2 k 2k (g – g ) (g – g ) 2
2
2
1
(9.2.17)
(9.2.18)
1
where, g r = Γ ( 1 + rk ) Γ ( 1 – rk )
(9.2.19)
dg d r = --------r = rg r [ ψ ( 1 + rk ) – ψ ( 1 – rk ) ] dk
(9.2.20)
and Γ(·), ψ(⋅) can be calculated by using Eqs. 6.1.6 and 6.2.53. The iteration in Eq. 9.2.16 is repeated until F(k) is sufficiently close to zero. Once the value of kˆ is obtained, other parameters are calculated by using Eqs. 9.2.21 and 9.2.22, 2 2 1⁄2 αˆ = [ m 2 kˆ / ( g 2 – g 1 ) ]
(9.2.21)
αˆ εˆ = m′1 – --- [ 1 – g 1 ] kˆ
(9.2.22)
where gr is given by Eq. 9.2.19. The Maximum Likelihood (ML) Method The likelihood function for a sample of size N from a generalized logistic distribution is given in Eq. 9.2.23.
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1 L = -----Nα
N
∏
i=1
x–ε 1 – k ----------- α
1/k-1
N
∏
i=1
1/k – 2
x–ε 1 + 1 – k ----------- α
(9.2.23)
The log-likelihood function is given by Eq. 9.2.24. 1 log L = – N log ( α ) + --- – 1 k
N
∑ log
i=1
x i – ε - –2 1 – k ---------- α
1/k
N
∑ log
i=1
x i – ε 1 + 1 – k ---------- α
(9.2.24) By taking partial derivatives of log L with respect to α, ε, and k and equating the results to zero we get a set of three simultaneous equations, the solution of which gives the values of αˆ , εˆ , and kˆ . Ahmad et al. (1988) suggest numerical maximization of the likelihood function using the OPTIMISE facility in GENSTAT, which also gives approximate standard errors for the estimated parameters. Alternatively, we follow the same procedure used for the GEV distribution (Section 7.1.2). We first define yi as in Eq. 9.2.25 x i – ε 1 y i = – --- log 1 – k ---------- α k
(9.2.25)
Substituting Eq. 9.2.25 in Eq. 9.2.24 we get Eq. 9.2.26. N
log L = – N log ( α ) – ( 1 – k )
∑ yi
N
–2
i=1
∑ log ( 1 + e
– yi
)
(9.2.26)
i=1
The partial derivatives of yi are given by Eqs. 9.2.27 to 9.2.29. ∂y 1 yk -------i = – --- e i α ∂ε yk ∂y i 1 ------- = – ------ ( e i – 1 ) ∂α αk yk y 1 ∂y -------i = – ----i + ----2 ( e i – 1 ) ∂k k k
(9.2.27)
(9.2.28)
(9.2.29)
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Differentiating Eq. 9.2.26 with respect to ε, α, and k and equating to zero we get Eqs. 9.2.30 to 9.2.32. ∂ log L Q – --------------- = ---- = 0 ∂ε α
(9.2.30)
∂ log L P+Q – --------------- = -------------- = 0 ∂α αk
(9.2.31)
∂ log L 1 – --------------- = --∂k k
P+Q R – -------------- = 0 k
(9.2.32)
In Eqs. 9.2.30 to 9.2.32, N
P = –N + 2
∑ (1 + e
– yi –1
)
(9.2.33)
i=1
N
∑e
Q = (1 + k)
yi k
N
–2
i=1
N
yi k
i=1
y –1
(1 + e i)
(9.2.34)
i=1
N
∑ yi
R=N+
∑e
– y –1
– 2 ∑ yi ( 1 + e i )
(9.2.35)
i=1
The partial derivatives are given by: N
– y – 2 – y ∂y ∂P ------ = 2 ∑ ( 1 + e i ) e i -------i ∂ε ∂ε i=1
(9.2.36)
N
– y – 2 – y ∂y ∂P ------- = 2 ∑ ( 1 + e i ) e i -------i ∂α ∂α i=1
(9.2.37)
N
– y – 2 – y ∂y ∂P ------ = 2 ∑ ( 1 + e i ) e i -------i ∂k ∂k i=1
∂Q ------- = k ( 1 + k ) ∂ε
N
∑e
i=1
yi k
∂y -------i – 2k ∂ε
N
∑e
i=1
yi k
(9.2.38)
N
– y – 1 ∂y yk – y –2 – y ∂ y ( 1 + e i ) ------ – 2 ∑ e i ( 1 + e i ) e i -------i ∂ε ∂ε
(9.2.39)
i=1
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∂Q ------- = k ( 1 + k ) ∂α
∂Q ------- = k ( 1 + k ) ∂k N
+
∑e
N
∑e
i=1
N
∑e
i=1
yi k
i=1
∂R ------- = ∂ε
∂R ------- = ∂α ∂R ------ = ∂k
yi k
N
yi k
∂ yi ------- – 2k ∂α
∂y -------i – 2k ∂k
N
– 2 ∑ yi e
N
∑e
yi k
i=1
N
∑e
N
– y –1 ∂ y yk – y –2 – y ∂ y ( 1 + e i ) -------i – 2 ∑ e i ( 1 + e i ) e i -------i ∂α ∂α
yi k
i=1
N
– y –1 ∂ y yk – y –2 – y ∂ y ( 1 + e i ) -------i – 2 ∑ e i ( 1 + e i ) e i -------i ∂k ∂k
(1 + e i) + (1 + k)
i=1
N
N
∂ yi
yi k
N
- –2 ∑ ( 1 + e ∑ -----∂α
i=1
i=1
N
N
N
– y –2 – y ∂ y ∂y ) -------i – 2 ∑ y i ( 1 + e i ) e i -------i ∂ε ∂ε
– yi –1
(9.2.42)
i=1
i=1
∂y i
∑ yi e
i=1
- –2 ∑ ( 1 + e ∑ -----∂ε
i=1
(9.2.41)
i=1
N
– y –1
yi k
(9.2.40)
i=1
– yi –1
)
N
– y – 2 – y ∂y ∂y -------i – 2 ∑ y i ( 1 + e i ) e i -------i (9.2.43) ∂α ∂α i=1
N
– y – 1 ∂y – y – 2 – y ∂y ∂y i - – 2 ∑ ( 1 + e i ) -------i – 2 ∑ y i ( 1 + e i ) e i -------i ∑ -----∂k ∂k ∂k i=1 i=1 i=1
(9.2.44)
The ML estimates of ε, α, and k are obtained by solving Eqs. 9.2.30 to 9.2.32 using the iteration in Eq. 9.2.45, εn + 1 dε n εn α n + 1 = α n + dα n kn + 1 kn dk n
(9.2.45)
where the increments dεn, dαn, and dkn are given by Eq. 9.2.46. –1
∂ log L ∂ log L ∂ log L – ---------------- – ----------------- – ----------------2 ∂ε∂α ∂ε∂k ∂ε 2
dε n 2 ∂ log L dα n = – ----------------∂α∂ε dk n 2 ∂ log L – ----------------∂k∂ε
2
2
∂ log L ∂ log L – ---------------- – ----------------2 ∂α∂k ∂α 2
2
∂ log L ∂ log L – ----------------- – ---------------2 ∂k∂α ∂k 2
2
∂ log L --------------∂ε ∂ log L --------------∂α ∂ log L --------------∂k
(9.2.46)
The second partial derivatives are identical in form to those for the GEV (Eqs. 7.1.36 to 7.1.44) with the values of P, Q, R, and their
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derivatives given by Eqs. 9.2.33 to 9.2.44 above. The iteration in Eq. 9.2.45 is repeated until Eqs. 9.2.30 to 9.2.32 are sufficiently close to zero. Similar to the GEV and other cases, the ML estimates may not always exist. PWM Method The L-moments of the generalized logistic distribution are given by (Hosking, 1986) in Eqs. 9.2.47 to 9.2.49. α λ 1 = ε + --- [ 1 – Γ ( 1 + k ) Γ ( 1 – k ) ] k
(9.2.47)
λ2 = α Γ ( 1 + k ) Γ ( 1 – k )
(9.2.48)
τ3 = –k
(9.2.49)
Parameter estimates are obtained by replacing λ1, λ2, τ3 by their corresponding sample estimates, l1, l2, t3 as in Eqs. 9.2.50 to 9.2.52. kˆ = – t 3
(9.2.50)
αˆ = l 2 / [ Γ ( 1 + kˆ ) Γ ( 1 – kˆ ) ]
(9.2.51)
εˆ = l 1 + ( l 2 – αˆ )/kˆ
(9.2.52)
EXAMPLE 9.2.1 Estimate the parameters of a generalized logistic distribution for the annual maximum flow data of East Fork White River at Seymour, Indiana which is given in Table 9.2.1. Method of Moments: From Table 2.1.2 for station 109 we have:
N = 68, m′1 = 32714, C v = 0.506214 and C S = 0.49051 Since Cs = 0.49051 we use Eq. 9.2.10 to get k0 = –0.0602
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Table 9.2.1. Annual Maximum Flow in East Fork White River at Seymour, IN
Year
Ann. Max. Flow
1924 1925 1926 1927 1928 1929 1930 1931 1932 1933 1934 1935 1936 1937 1938 1939 1940 1941 1942 1943 1944 1945
24000 7920 21900 47100 30400 36100 67100 7030 28200 40100 10300 11100 17000 65600 32600 36200 46400 3650 16800 44800 37100 42900
Year 1946 1947 1948 1949 1950 1951 1952 1953 1954 1955 1956 1957 1958 1959 1960 1961 1962 1963 1964 1965 1966 1967 1968
Ann. Max. Flow 15000 33000 28000 78500 54000 28600 44000 13300 6120 11100 42100 33400 30100 62100 28100 59400 23800 52000 54900 25600 10900 33700 60200
Year 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991
Ann. Max. Flow 39200 26300 27900 27000 22700 17500 46400 19300 12700 36000 39900 25400 30200 47000 39800 23800 29600 33400 15400 28400 26700 46500 61200
After applying the iterative procedure in Section 9.2.2, the final estimate of kˆ is:
kˆ = – 0.05515 By using Eqs. 6.1.6 and 6.1.4:
Γ ( 1 + kˆ )Γ ( 1 – kˆ ) = 1.0050201 Γ ( 1 + 2kˆ )Γ ( 1 – 2kˆ ) = 1.02029405 Other parameters are computed from Eqs. 9.2.21 and 9.2.22. 2 αˆ = ( 0.506214 × 32714 ) ( 0.05515 )/ [ 1.02029405 – ( 1.00502012 ) ]
1⁄2
= 9030
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9030 εˆ = 32714 – --------------------------- [ 1 – 1.0050201 ] = 31892 ( – 0.05515 ) ML Method: Applying the procedure outlined in Section 9.2.2 we get the following estimates
εˆ = 30911.830; αˆ = 9305.0205; kˆ = – 0.144152 To verify that these are the ML estimates the following sums are computed: N
∑ (1 + e
– yi –1
)
= 34
i=1
N
∑e
yi k
= 70.359254
i=1
N
∑e
yi k
– y –1
(1 + e i)
= 30.108420
i=1
N
∑ yi
= – 1.4585536
i=1
N
∑ yi
– y –1
(1 + e i)
= 33.270723
i=1
Substituting the above sums into Eqs. 9.2.33 to 9.2.35, P, Q, and R are computed. P = –68 + 2 × 34 = 0 Q = (1 – 0.144152) × 70.359254 – 2 × 30.108420 = 1.3 × 10–4 ≈ 0 R = 68 + (–1.4585536) – 2(33.270723) = 4 × 10–7 ≈ 0 If these values are substituted into Eqs. 9.2.30 to 9.2.32, the equations are satisfied. The calculated estimates are the ML estimates. PWM Method: From Table 3.1.1 the following moments are obtained.
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N = 68, l1 = 32714, t = 0.287028 and t3 = 0.105225 From Eq. 9.2.50,
kˆ = – t 3 = – 0.105225 By using Eqs. 6.1.6 and 6.1.4, the following gamma function values are computed.
Γ ( 1 + kˆ ) = 1.0728803 Γ ( 1 – kˆ ) = 0.94926529 From Eqs. 9.2.51 and 9.2.52,
αˆ = ( 0.287028 × 32714 )/ [ 1.0728803 × 0.94926529 ] = 9219.7 εˆ = 32714 + ( 0.287028 × 32714 – 9219.7 )/ ( – 0.105225 ) = 31097.14 The parameter estimates are summarized in Table 9.2.2. Table 9.2.2. Parameter Estimates for Example 9.2.1
Parameter MOM MLM PWM
ε 31892 30912 31097
α 9030 9305 9220
k –0.05515 –0.14415 –0.10523
9.2.3 Quantile Estimation The distribution function of x in Eq. 9.2.1 can be written in the inverse form x = x(F ) as in Eq. 9.2.53. α k x = ε + --- [ 1 – { ( 1 – F )/F } ] k
(9.2.53)
By substituting F = 1 – (1/T ) where T is the return period, the T-year quantile estimate is given by Eq. 9.2.54. αˆ –k xˆ T = εˆ + --- [ 1 – ( T – 1 ) ] kˆ
(9.2.54)
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Other quantile estimates are given in Table 9.2.3.
The frequency factor KT in Eq. 9.2.55 is obtained by comparing Eq. 9.2.54 with Eq. 4.3.2. k Γ ( 1 + k )Γ ( 1 – k ) – ( T – 1 ) K T = ----- -----------------------------------------------------------------------------------------------------------k [ Γ ( 1 + 2 )Γ ( 1 – 2k ) – Γ 2 ( 1 + k )Γ 2 ( 1 – k ) ] 1 ⁄ 2 –k
(9.2.55)
EXAMPLE 9.2.2 Compute the 100-year flood by using the parameters estimated in Example 9.2.1. Quantile Estimate by the MOM Parameters: From Eq. 9.2.54 we get
xˆ T = 31892 – 163735.268 [ 1 – ( T – 1 )
0.05515
]
For T = 100,
xˆ 100 = 31892 – 163735.268 [ 1 – ( 100 – 1 )
0.05515
] = 79117 cfs
Quantile Estimate by the MLM Parameters: From Eq. 9.2.54,
xˆ T = 30912 – 64550.815 [ 1 – ( T – 1 )
0.14415
]
For T = 100,
xˆ 100 = 30912 – 64550.815 [ 1 – ( 100 – 1 )
0.14415
] = 91552 cfs
Quantile Estimate by the PWM Parameters: From Eq. 9.2.54,
xˆ T = 31097 – 87617.6 [ 1 – ( T – 1 )
0.10523
]
For T = 100, then
xˆ 100 = 31097 – 87617.6 [ 1 – ( 100 – 1 )
0.10523
] = 85578 cfs
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9.2.4 Standard Error Method of Moments Eq. 4.4.11 can be used for calculating standard error from the Generalized Logistic distribution. Just as in GEV and Weibull distributions, expressions for γ2, γ3, γ4 and ∂k T /∂γ 1 are complicated and are evaluated numerically. We have, ∂K ∂γ ∂K ∂k ∂K T ---------- = ---------T- -------- = ---------T- / --------1 ∂k ∂γ 1 ∂k ∂k ∂γ 1
(9.2.56)
where KT , the frequency factor is given by Eq. 9.2.55 as a function of ∂K k, and γ1 = Cs is given by Eq. 9.2.7. The value of ---------T- can be obtained ∂k by differentiating Eq. 9.2.55 to get Eq. 9.2.57. –k
–k
∂K T k [ g 1 – ( T – 1 ) ] [ d 2 – 2g 1 d 1 ] k d 1 + ( T – 1 ) log ( T – 1 ) - – --------- ------------------------------------------------------------------------------ = ----- ---------------------------------------------------------2 1⁄2 2 3⁄2 k 2 k ∂k [g – g ] [g – g ] 2
1
2
(9.2.57)
1
In Eq. 9.2.57, gr and dr are given by Eqs. 9.2.19 and 9.2.20. The value ∂γ of --------1 is given by Eq. 9.2.18. ∂k Also we have:
2
4
g 4 – 4g 3 g 1 + 6g 2 g 1 – 3g 1 γ 2 = ---------------------------------------------------------2 2 [ g2 – g1 ]
2
(9.2.58)
3
5
k – g 5 + 5g 4 g 1 – 10g 3 g 1 + 10g 2 g 1 – 4g 1 γ 3 = ----- ----------------------------------------------------------------------------------------2 5⁄2 k [ g2 – g1 ]
2
3
4
(9.2.59)
6
g 6 – 6g 5 g 1 + 15g 4 g 1 - 20g 3 g 1 + 15g 2 g 1 – 5g 1 γ 4 = --------------------------------------------------------------------------------------------------------------------2 3 [ g2 – g1 ]
(9.2.60)
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where, g r = Γ ( 1 – rk ) Γ ( 1 + rk )
(9.2.61)
The standard error can therefore be evaluated by numerically evaluating its components and substituting them in Eq. 4.4.11. However, 1 this procedure can be used only when k < --- in order for γ4 to exist. 6 ML Method A simple expression cannot be obtained for the standard error in the case of ML or PWM estimates. However, numerical solution of the ML maximization as discussed before (Ahmad et al., 1988) can provide approximate standard errors for the estimated parameters. Efron and Hinkley (1978) suggested the use of the inverse of the observed information matrix (Eq. 4.4.21 without taking the expectation) which is the inverse matrix in Eq. 9.2.46 to obtain approximate variances and covariances of the parameters as in Eq. 9.2.62. –1
∂ log L ∂ log L ∂ log L - – ----------------- – ----------------– ---------------2 ∂ε∂α ∂ε∂k ∂ε 2
var ( ε ) cov ( ε, α ) cov ( ε, k ) var ( α ) cov ( α, k ) = var ( k )
2
2
∂ log L ∂ log L - – ----------------– ---------------2 ∂α∂k ∂α 2
2
∂ log L – ---------------2 ∂k 2
(9.2.62) This approximation has been used and suggested for the locationscale families (Hinkley, 1978), for the Weibull distribution (Cohen, 1965; Lemon, 1975) and the GEV distribution (Prescott and Walden, 1983). The use of this approximation for the generalized logistic distribution has not yet been reported, but may be used here to obtain only approximate values. Given the values of the variances and covariances of the parameters by Eq. 9.2.62, the approximate standard error is calculated by substituting them in Eq. 4.4.19. Also we have from Eq. 9.2.54,
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∂xˆ -------T- = 1 ∂εˆ
(9.2.63)
∂xˆ T 1 – kˆ -------- = --- [ 1 – ( T – 1 ) ] ˆk ∂αˆ
(9.2.64)
∂xˆ αˆ αˆ –k – kˆ -------T- = – ----2 [ 1 – ( T – 1 ) ] + --- ( T – 1 ) log ( T – 1 ) ˆ ˆ ˆ k ∂k k
(9.2.65)
EXAMPLE 9.2.3 Estimate the standard error for the 100-year floods computed in Example 9.2.2. Standard Error by MOM Parameters: We first calculate the following using equations in Chapter 6. g1 = 1.00502, g2 = 1.02029, g3 = 1.04648 g4 = 1.08476, g5 = 1.13695, g6 = 1.20570 For T = 100, from Eq. 9.2.55 we have, KT = 2.80202 From Eqs. 9.2.57 and 9.2.18 we get
∂K T ---------- = – 4.79486 ∂k ∂γ ------ = – 9.2815 ∂k From Eq. 9.2.56,
∂K T ---------- = – 4.79486/ – 9.2815 = 0.516604 ∂γ 1 From Eqs. 9.2.58 to 9.2.60,
γ 2 = 4.80372 γ 3 = 10.53297
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γ 4 = 74.11431 We also have γ1 = Cs = 0.49051 Substituting the above values in Eq. 4.4.11, sT = 10982 cfs Standard Error by MLM Parameters: The elements of the observed information matrix can be calculated by using Eqs. 7.1.36 to 7.1.44 in conjunction with the P, Q, R values and their derivatives in Eqs. 9.2.33 to 9.2.44 at the ML estimates of ε, α, k to get
2 ∂ log L –7 – ------------------- = 2.87664 × 10 2 ∂ε 2
2 ∂ log L 2 ∂ log L –7 – -------------------- = – -------------------- = 1.99300 × 10 ∂ε∂α ∂α∂ε 2
2
2 ∂ log L 2 ∂ log L –3 – -------------------- = – -------------------- = – 2.32747 × 10 ∂ε∂k ∂k∂ε 2
2
2 ∂ log L –6 – ------------------- = 1.24696 × 10 2 ∂α 2
∂ log L 2 ∂ log L –3 – ----------------- = – -------------------- = 4.47741 × 10 ∂α∂k ∂k∂α 2
2
2 ∂ log L – ------------------- = 145.4549 2 ∂k 2
Inverting this matrix as in Eq. 9.2.62 we get,
var ( εˆ ) = 4242159.83 var ( αˆ ) = 957803.02 var ( kˆ ) = 8.38773 × 10
–3
cov ( εˆ , αˆ ) = 488252.74 cov ( εˆ , kˆ ) = 52.850727 cov ( αˆ , kˆ ) = – 21.670565
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From Eqs. 9.2.63 to 9.2.65,
∂x ∂x -------T- = 1, -------T- = 6.51695 ∂α ∂ε ∂x T -------- = – 154595.08 ∂ε Substituting the above values in Eq. 4.4.19, 2
2
2
s T = ( 1 ) ( 4242159.83 ) + ( 6.51695 ) ( 957803.02 ) 2
–3
+ ( – 154595.08 ) ( 8.38773 × 10 ) + 2 ( 1 ) ( 6.51695 ) ( 488252.74 ) + 2 ( 1 ) ( – 154595.08 ) ( 52.850727 ) + 2 ( 6.51695 ) ( – 154595.08 ) ( – 21.670565 ) = 279072980.2 s T = 16705 cfs
Standard errors of floods of other return periods are given in Table 9.2.3. These results are obtained by using the computer program discussed in Chapter 10. Table 9.2.3. Quantile Estimates and their Standard Errors (in parenthesis) for Example 9.2.3
T 10
P (%) 10
20
5
50
2
100
1
200
0.5
MOM 52983 (3510) 60759 (5110) 71089 (8083) 79115 (10982) 87396 (14467)
MLM 54965 (4453) 65042 (6917) 79482 (11721) 91552 (16705) 104807 (23057)
PWM 53887 62919 75439 85576 96408
Quantile plots for station 109 computed by using the GLOG distribution and the MOM, MLM, and PWM are shown in Figure 9.2.2.
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Figure 9.2.2. Observed and estimated flows and 95% confidence intervals for the East Fork White River data used in Examples 9.2.1 to 9.2.3.
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CHAPTER 10
Computer Program
10.1 Introduction A computer program has been developed for the computation and visualization of results of flood frequency analysis methods discussed in this book. The code is written in MATLAB® language, which combines the power of computation with visualization of various results in one program. A standalone program is also currently being developed. Although most of the computations involved in flood frequency analysis are simple to perform, it should be noted that double precision computations are necessary in many cases to reach the desired accuracy. This is very important especially for the case of the log-type distributions (log-normal and log-Pearson). Also, flood data seem to produce badly scaled matrices for the computation of the standard errors, which makes it necessary to use double precision in order to obtain accurate estimates. The computer program performs both at-site and regional analysis of data. Regional analysis is performed by using the index-flood method. Regional parameter estimates are based on the average of atsite moments for the case of MOM and PWM. For the ML method, regional parameter estimates are obtained by pooling the observations together after dividing by the mean at each site (station-year method). Included also are some tests for stationarity and randomness mentioned in Chapter 1. The program produces parameter estimates, quantile and standard error estimates in addition to probability plots and moment ratio diagrams.
10.2 Description of Program The computer program consists of more than 120 MATLAB M-files. These include input and output routines, graph routines, parameter
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estimation routines, quantile estimation routines, and standard error routines in addition to testing routines. The main program is freq.m, which calls three different programs according to the user’s choice. These are freqpwm.m, freqmlk.m, freqmom.m, pwmom.m, avrgpwm.m, lmratios.m, poolobs.m are then used for computing at-site and regional moments. Parameter estimation routines are named by using three letters denoting the distribution followed by ‘fit’ and then either ‘m’ for MOM, ‘l’ for ML, or ‘p’ for PWM. For example LP(3) estimation by ML method uses the routine lp3fit1.m, and weibull estimation by MOM uses the routine, weifitm.m, and so on. Distributions are denoted as follows 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.
evl exp gam gev glg ln2 ln3 log lp3 nrm par pe3 wei wk4 wk5
: : : : : : : : : : : : : : :
Extreme value type I distribution. Exponential distribution. Gamma distribution. Generalized extreme value distribution. Generalized logistic distribution. Two-parameter log-normal distribution. Three-parameter log-normal distribution. Logistic distribution. Log-Pearson III distribution. Normal distribution. Generalized Pareto distribution. Pearson-III distribution. Weibull distribution. Four-parameter Wakeby distribution. Five-parameter Wakeby distribution.
Quantile estimation routines are named by using the three letters of the distribution followed by ‘q’. For example logq.m calculates the logistic quantiles. Standard error routines, like parameter estimation routines consist of the three letters of the distribution followed by ‘std’ and then either ‘m’ for MOM, or ‘l’ for ML, or ‘p’ for PWM. Plotting routines are plotmrd.m, plotlmrd.m, probplot.m. In addition, there are a number of routines for calculating different functions needed for parameter and quantile estimates, these are kpvalue.m, hknew.m, getkgev.m, getkglog.m, getkpar.m, getlp3.m, psi.m, and tablej.m. Programs for testing homogeneity and stationarity are gbtest.m for the Grubbs-Beck test, mwtest.m and mw2test.m for the Mann-Whitney test, and wwtest.m for the Wald-Wolfwitz test. The Wiltshire test for regional homogeneity is performed by wiltshir.m.
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Input to the program is through an input data file which is supplied by the user. The structure of the file is as follows: Line Line Line Line Line
1 2 3 4 5
: : : : :
Region name. Number of sites in the region. Station number 1 ID# Number of observations in station 1 Observations for station 1
For other stations lines 3 to 5 are repeated. An example file is: line line line line line
1 2 3 4 5
Wabash River basin 93 219 54 15000 16300 12000
[name of region] [93 stations] [first station ID# is 219] [54 observations in #219] [the actual 54 observations]
After the user inputs the file name which contains the input data, the user is guided through the program by a series of menus for the choice of analysis methods (MOM, ML, PWM) and statistical distributions. The standard output gives both at-site and regional parameter estimates as well as quantile estimates at 10, 20, 50, 100, and 200 years in addition to standard errors where applicable. The user also can choose different return periods to obtain quantiles as well as standard error estimates. Graphical output consists of ordinary moment ratio diagrams, L-moment ratio diagrams, and probability (quantile) plots. Included with the program is a data file containing data from 93 stations in the Wabash River basin in Indiana. The examples in this book are chosen from these stations.
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