Introduction to Nonparametric Detection with Applications (Mathematics in Science & Engineering,119)

Introduction to Nonparmetric Detection with Applications

This is Volume 1 19 in MATHEMATICS IN SCIENCE AND ENGINEERING A Series of Monographs and Textbooks Edited by RICHARD BELLMAN, University of Southerit California The complete listing of books in this series is available from the Publisher upon request.

Introduction to Nonparametric Detection with Applications Jerry D. Gibson

Department of Electrical Engineering University of Nebraska Lincoln, Nebraska

James L. Melsa

Department of Electrical Engineering University of Notre Dame Notre Dame, Indiana

ACADEMIC PRESS

New York San Francisco London

A Subsidiary of Harcourt Brace Jovanovich, Publishers

1975

COPYRIGHT (B 1975. BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR A N Y INFORMATION STORAGE AND RETRIEVAL SYSTEM. WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER.

ACADEMIC PRESS, INC.

111 Fifth Avenw, New York, New' York 10003

United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. 24/28 Oval Road. London NWl

Library of Congress Cataloging in Publication Data Gibson, Jerry D Introduction to nonparametficdetection with applications. (Mathematics in science and engineering ; ) Bibliography: p. Includes index. 1. Signal processing. 2. Demodulation (Electronics) 3. Statistical hypothesis testing. I. Melsa, James L., joint author. 11. Title. 111. Series. 621.38'0436 15-3519 TK5 102AG48 ISBN 0-12-282150-5

PRINTED IN THE UNITED STATES OF AMERICA

To Elaine

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Contents

Preface

xi

Chapter 1 Introduction to Nonparametric Detection Theory 1.1 1.2 1.3 1.4

Introduction Nonparametric versus Parametric Detection Historical and Mathematical Background Outline of the Book

Chapter 2 Basic Detection Theory 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8

Introduction Basic Concepts Bayes Decision Criterion Neyman-Pearson Lemma Receiver Operating Characteristics Composite Hypotheses Detector Comparison Techniques Summary

Chapter 3 One-Input Detectors

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8

Introduction Parametric Detectors Sign Detector Wilcoxon Detector Fisher-Yates, Normal Scores, or c, Test Van der Waerden’s Test Spearman Rho Detector Kendall Tau Detector

3.9 summary Problems

9 10 12 17 21 24 31 36

38 39 47 51 57 61 65 68 71 12

viii

Contents

Chapter 4 One-Input Detector Performance

4.1 4.2 4.3 4.4 4.5 4.6

Introduction Sign Detector Wilcoxon Detector One-Input Detector ARES Small Sample Performance Summary Problems

Chapter 5 Two-Input Detectors

5.1 5.2 5.3 5.4

Introduction One-Input Detectors with Reference Noise Samples Two Simultaneous Input Detectors Summary Problems

Chapter 6 Two-Input Detector Performance

6.i 6.2 6.3 6.4 6.5 6.6 6.7

Introduction Asymptotic Results for Parametric Detectors ARE of the PCC Detector ARE of the Mann-Whitney Detector ARE of Other Two-Input Detectors Small Sample Results Summary Problems

Chapter 7 Tied Observations 7.1 7.2 7.3 7.4

Introduction General Procedures Specific Studies Summary Problems

Chapter 8 Dependent Sample Performance 8.1 8.2 8.3 8.4

Introduction Dependence and the Constant False Alarm Rate Property Nonparametric and Parametric Detector Performance for Correlated Inputs Summary Problems

Chapter 9 Engineering Applications 9.1 9.2 9.3 9.4

Introduction Radar System Applications Other Engineering Applications Summary

74 75 81 87 90 92 93

95 97 122 152 152

154 155 158 162 164 167 170 170

172 173 176 178 178

181 182 185 188

189

191 192 204 209

Contents

ix

Appendix A Probability Density Functions

210

Appendix B Mathematical Tables

214

Answers to Selected Problems

226

References

232

Index

231

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Preface

Statisticians have long been interested in the development and application of nonparametric statistical tests. Although engineers have unknowingly implemented nonparametric detectors in the past, only recently have engineers become aware of the advantages of nonparametric tests, and hence have only recently become interested in the theoretical analysis and application of these detectors. Because of these two facts, almost the entire development and analysis of nonparametric detectors has taken place in the statistical literature. Engineers are presently highly interested in the application and analysis of nonparametric detectors; however, engineering understanding is proceeding slowly in this area. This is because the basic results of nonparametric detection theory are developed in papers scattered throughout statistical, psychological, and engineering journals and consist almost wholly of mathematically rigorous proofs and statements of theorems or specific applications to nonengineering problems. Consequently, the nonmathematically oriented engineer and the engineer who is unfamiliar with these journals find it extremely difficult, if not impossible, to become educated in this area. It is the purpose of this book to alleviate this problem by presenting a tutorial development of nonparametric detection theory written from the engineer’s viewpoint using engineering terminology and stressing engineering applications. Only those nonparametric hypothesis testing procedures that have already been applied to or seem well suited to the type of statistical detection problems that arise in communications, radar, sonar, acoustics, and geophysics are discussed in this book. The Neyman-Pearson approach to hypothesis testing is employed almost exclusively. Xi

xii

Preface

The book is written at the senior-first-year graduate level and requires only a basic understanding of probability and random variables. Some previous exposure to decision theory would be useful but is not required. The book should find its principal application as a reference monograph for students and practicing engineers. However, the unusually thorough consideration of Neyman-Pearson detection theory in both the parametric and nonparametric cases make it attractive for use as a secondary text for a standard engineering detection theory course or as a primary text for a more in-depth study of detection theory. The book draws heavily on the statistical and engineering literature on nonparametric detection theory, and an attempt has been made to assign complete and accurate credit for previous work and ideas in all instances. Three excellent survey papers on nonparametric detectors by Carlyle and Thomas [ 19641, Carlyle [ 19681, and Thomas [ 19701 have been published, and these papers probably served as the genesis for this book. The authors gratefully acknowledge the contribution of these and other researchers in the field. The authors would like to express their appreciation to their colleagues and students who have assisted in the preparation of this book in various ways. The authors also wish to thank Kathryn Stefl, Linda Schulte, Sharon Caswell, Marge Alles, and Elaine Gibson who patiently typed, retyped, and corrected the manuscript.

CHAPTER

1

Introduction to Nonparametric Detection Theory

1.1 Introduction

The received waveform at the input of a communication or a radar system normally consists of a randomly fluctuating component called noise plus any desired signal components that may be present. The system designer desires to extract information (the signal) from these received data. Since the noise may almost completely obscure the signal, the received waveform must be processed to enhance the extraction of information from the input. When the problem requires the determination of the presence or absence of a signal, an area of statistical inference called hypothesis testing, or decision theory, is useful. In engineering terminology, decision theory is often referred to as detection theory, because of the early applications to radar systems. The latter terminology is employed throughout this book. The bulk of the engineering literature on detection theory is concerned with the determination of optimum detectors utilizing the NeymanPearson' fundamental lemma of hypothesis testing or the Bayesian' approach to hypothesis testing. Both methods require knowledge of a statistics€ description of the interfering noise process. This statistical description may be obtained by actual measurement or it may be assumed. Once a statistical description of the noise has been determined, it can be utilized with one of the previously mentioned procedures to yield a detector which is optimum in the chosen sense. Unfortunately the resulting 'see Chapter 2.

I

2

I

Introduction to Nonparametric Detection Theory

optimum detectors are often difficult to implement. Another class of detectors, called adaptive or learning detectors, operate in a near optimum fashion without a complete statistical description of the background noise. This is achieved by allowing the system parameters or structure to change as a function of the input. Due to their continually varying structure, adaptive detectors are difficult to analyze mathematically and must be simulated on a computer. Additionally, since the detector structure is a function of the input, it is also dependent on the operating environment and as a result, adaptive detectors are generally complex to implement. When a statistical description of the input noise process is not available or when the optimum or adaptive detector is too complex to implement, one often seeks a nonoptimum detector whose performance is satisfactory. A class of nonoptimum detectors called nonparametric or distribution-free detectors exists which is simple to implement and requires little knowledge of the underlying noise distribution. Throughout this book and the literature, nonparametric and distribution-jree are generally used as synonyms. Strictly speaking, the two terms are not completely equivalent. Nonparametric refers to a class of detectors whose input distribution has a specified shape or form but still cannot be classified by a finite number of real parameters; while distribution-free refers to the class of detectors which makes no assumptions at all concerning the form of the input distribution [Bradley, 19681. The purpose of the introductory material in this chapter is to give the reader an intuitive feeling for nonparametric detection theory and to outline the approach used in this book for the systematic study of this subject. 1.2 Nonparametric versus Parametric Detection Before beginning the development of nonparametric detectors, it is instructive to discuss further the differences between parametric and nonparametric detection and to give some specific examples of the types of problems to which nonparametric detectors can be effectively applied. The detection problem considered throughout this book is of the type where a signal is observed, and one of two different decisions must be made. Problems of this type are not unique to the engineering field and can be found in agriculture, business, and psychology as well as many other fields. In agriculture a farmer must analyze the yield of a plot of land and decide whether or not to plant wheat on the land; a businessman must study the sales and growth possibilities of a company and decide whether or not the company would be a good investment; a psychologist must observe the effect of a certain environment on individual behavior and decide whether

1.2

Nonparametric versus Parametric Detection

3

or not the environment is detrimental to human development. In each of these cases, the decision is either yes or no, accept or reject. There are therefore two different possibilities. In this book one of the possibilities will be called the null hypothesis, or simply hypothesis, and the second possibility will be called the alternative hypothesis, or alternative. The hypothesis will be denoted by H and the alternative by K. Consider now the general detection problem where a random signal x is observed, and a decision must be made as to which one of the two possibilities, the hypothesis H or the alternative K, is true. If H and K could be observed directly, there would be no uncertainty in the detection problem, and a correct decision could be made every time. In truth, however, the observations of H and K are usually corrupted by noise. A detector must be devised which attempts to compensate for the distortion introduced by the noise and chooses either H or K as the correct source of the observed data. A generalized detector is shown in Fig. 1.2-1. The input to the detector is a random signal x and the output is a random variable, D ( x ) , which depends on the input x , and takes on the values 0 or 1. If the output of the detector is 0, the hypothesis H is accepted, and if the output of the detector is 1, the alternative K is accepted. The input x is assumed to be a stochastic process. At any instant of time t , , x is a random variable, and x will have a different density function depending on whether the hypothesis H is true or the alternative K is true. The probability density function of x is written asf(x1H) if H is true and asf(x1K) if K is true. DECISION D(X) DETECTOR

Fig. 1.2-1.

)

Generalized detector.

Parametric detectors assume that f ( x l H ) and f ( x l K ) are known and utilize the known form of these probability density functions to arrive at a form for the detector D. If the actual probability density functions of the input x are the same as those assumed in determining D, the performance of the detector in terms of probability of detection and false alarm probability is good. If, however, the densities of the input x are considerably different from the probability density functions assumed, the performance of the parametric detector may be poor. On the other hand, nonparametric detectors do not assume that the input probability density functions are completely known, but only make general assumptions about the input such as symmetry of the probability density function and continuity of the cumulative distribution function. Since there are a large number of density functions which satisfy these

4

1

Introduction to Nonparametric Detection Theory

assumptions, the density functions of the input x may vary over a wide range without altering the performance of the nonparametric detector. Since parametric detectors are based on specified forms of the input probability density functions, their performance may vary widely depending on the actual density of the input x . Nonparametric detectors, however, maintain a fairly constant level of performance because they are based on general assumptions on the input probability density. How do the performance of parametric and nonparametric detectors compare when applied to the same detection problem? The answer to this question pivots about how well the assumptions upon which the two detectors are based are satisfied. For instance, assume that the parametric detector is based on the assumption that the input density is Gaussian, and the nonparametric detector is based on the assumption that the input density is symmetric and the cumulative distribution function is continuous. If the input probability densities are actually Gaussian, the performance of the parametric detector will generally be significantly better than the performance of the nonparametric detector. If, however, the input densities are not Gaussian but are still symmetric, the nonparametric detector performance may be much better than the performance of the parametric detector. Another property of nonparametric detectors is their relative ease of implementation when compared to parametric detectors. The following example will illustrate this property. Consider the .problem of detecting a known constant signal of positive amplitude in additive white noise on the basis of a set of observations x = { x , , x2, . . . , x,}. If the noise has a zero mean Gaussian density with a known variance u2, the optimum parametric detector has the form (see Section 3.2)

(1.2-1)

Here Z:, ,xi is known as the test statistic and To is the threshoZd of the test. The optimum parametric detector for this case sums the input observations, compares the resulting sum to a threshold To, and decides H (D ( x ) = 0) if the sum is less than To or decides K (D ( x ) = 1) if the sum is greater than T,. The simplest nonparametric detector is called the sign test and bases its decision only on the signs of the input observations. For this problem, the

1.2

Nonparametric versus Parametric Detection

5

sign detector has the form (see Section 3.3)

i n r

D(x) =

O

if

x x n

u(xi) < T ,

i- 1

1

if

i= 1

#(xi) > TI

( 1.2-2)

where #(xi) is the unit step function, and T I is again referred to as the threshold of the test. The nonparametric detector of Eq. (1.2-2) must therefore simply determine the polarity of each input observation, count the number of positive observations, compare the total to the threshold T , , and decide or accept H if the total is less than T , or decide K if the total is greater than T I . Since the parametric detector requires that the input observations be summed whereas the nonparametric detector requires only the total of the number of positive observations to be found, the nonparametric detector is considerably simpler to implement. A shorthand notation will often be used to represent the detector rule. For example, Eq. (1.2-2) will be written in the form

( 1.2-3) By this notation, we imply that we decide K if %T-lu(xi)> T I and decide H if Z:,,u(xi) < T I .The probability of Z:,,u(xi) = T , is zero when the test statistic is continuous, however in this discrete case, exact equality has finite probability. Although not indicated notationaly, we will always assign this event to the alternative. As another illustration, suppose that the noise has a Cauchy density, then the optimum parametric detector has the form

n (A,x; n

i- 1

+ A,xi)

K

S T2

H

( 1.2-4)

where A , and A, are known constants and T, is the threshold. The parametric detector given by Eq. (1.2-4) requires each observation to be squared, multiplied by a constant, and then added to the unsquared observation multiplied by another constant. This calculation has to be 'For this book, the unit step function is defined by

6

I

Introduction to Nonparametric Detection Theoty

performed for each observation, the product of the individual results taken, and the product compared to the threshold T,. If the product is less than T,, H is accepted and if the product is greater than T,, K is accepted. The basic form for the nonparametric detector for this case is still given by Eq. (1.2-2). The nonparametric detector is therefore much simpler to implement than the parametric detector when the input noise has a Cauchy density. Many other examples could be given which illustrate the simple implementation schemes required by nonparametric detectors. The most important property of nonparametric detectors is, however, the maintenance of a constant level of performance euen for wide variations in the input noise density. This property, coupled with the usually simple form of the test statistics, make nonparametric detection techniques quite attractive to the engineer. 1 3 Historical and Mathematical Background Since the simplest nonparametric detectors are quite intuitive in nature and are based on fairly elementary mathematical concepts, it is not surprising that papers concerned with nonparametric tests of hypotheses appeared as early as the eighteenth century. In particular, in 1710, John Arbuthnott applied the sign test to a hypothesis concerning Divine Providence [Bradley, 1968; Arbuthnott, 17101. In spite of these few early papers, however, the true beginning of nonparametric detection theory is fixed by Savage [1953] as 1936, when Hotelling and Pabst [1936] published a paper on rank correlation. The development and application of nonparametric detectors was considerably delayed by the overwhelming acceptance by mathematicians and scientists of the normal distribution theory. Since it could be shown that the sum of a large number of independent and identically distributed random variables approached the normal distribution, practitioners in almost all fields viewed and used the normal distribution as the tool for whatever purpose they had in mind. Such was the reputation of the normal distribution that many times, sets of experimental data were forced to fit this mathematical and physical phenomenon called the normal distribution. When sets of data appeared which could not even be forced to fit the normal distribution, it was assumed that all that was needed to obtain a normal distribution was to take more samples. As larger data sets became available, however, it became evident that not all situations could be described by normal distribution theory. When nonparametric tests began to appear, people continued to consider normal distribution theory as the

1.4 Outline of the Book

7

most desirable approach, and nonparametric tests were used only as a last resort. Since the beginning in 1936, however, the number of papers in the statistical literature concerning nonparametric techniques has swelled, until today, the literature on nonparametric statistical inference is quite large. Although nonparametric or distribution-free methods have been studied and applied by statisticians for at least 25 years, only recently have engineers recognized their importance and begun to investigate their applications. Only elementary probability theory is necessary to apply and analyze nonparametric detectors. Therefore unlike many engineering techniques, nonparametric detection was not slowed in development due to the lack of mathematical methods. Most of the definitions and concepts of detection theory required to understand the mathematics contained in this book, excluding basic probability theory, are covered in Chapter 2. 1.4

Outline of the Book

This book consists of essentially three sections: basic theory, detector description, and detector performance. Chapter 2, entitled Basic Detection Theory, comprises the entire basic theory section wherein simple and composite hypotheses and alternatives, type 1 and type 2 errors, false alarm and miss probabilities, and receiver operating characteristics are discussed. In addition, the Neyman-Pearson fundamental lemma is treated in detail, and the important concepts of consistency, robustness, efficacy, and asymptotic relative efficiency are defined. The detector description section is composed of two chapters, Chapter 3, which covers one-input detectors, and Chapter 5 which treats two-input detectors. Some clarification of terminology is necessary here. The terms one- and two-input detectors indicate the number of sets of observations available at the detector input, not the total number of available input samples. For example, a one-input detector has only one set of observations (x,, xZr. . . , x,,) at its input to use in making a decision, whereas a two-input detector has two sets of samples ( y , , y z , . . . , y k ) and (zlrz2,. . . , zm). In these two chapters, the basic rank and nonrank nonparametric detectors are described in detail and examples given of the test statistic calculations. In order to compare a nonparametric detector with another nonparametric or a parametric detector, some kind of evaluation technique must be established. In the performance section of the book, which consists of the remaining chapters, the calculation of certain performance parameters is illustrated, the effects of experimental methods are demonstrated, and

8

I

Introduction to Nonparametric Detection Theory

the performance of the nonparametric detectors for various applications is given. In Chapters 4 and 6, the calculation of the asymptotic relative efficiency for both rank and nonrank detectors is illustrated, and the small sample performance of several of the nonparametric detectors is given. Chapter 4 treats the performance of one-input detectors, while Chapter 6 is concerned with two-input detector performance. When working with signed-rank detectors, a method for treating zero value observations and for assigning ranks to observations with the same absolute value must be adopted. Both zero value observations and equal magnitude observations are generally lumped under the common title of tied observations. Chapter 7 discusses some of the more common techniques for treating tied observations and demonstrates the effects of the various techniques on nonparametric detector performance. All of the nonparame.tric detectors in this book are based on the assumption that the input observations are independent and identically distributed. Although the assumption of independent input observations is quite restrictive, it should not be surprising that the nonparametric detectors are based on this assumption, since the nonparametric tests were developed primarily by statisticians who generally have more control over experimental techniques such as sampling frequency, than do engineers. Since there are many physical situations in engineering which may give rise to dependent input observations, the performance of nonparametric detectors when the input samples are correlated is of prime importance. A treatment of the dependent sample performance of. both one- and twoinput detectors is contained in Chapter 8. In order to apply nonparametric detectors to radar and communication systems, slight modifications of the basic detector test statistics are sometimes required. Chapter 9 presents examples of the engineering applications of the nonparametric detectors covered in this book and discusses the implementation and performance of these detectors for the applications considered. Two appendices complete the book. Appendix A gives a brief description of several standard probability densities that are used in the book, while Appendix B contains mathematical tables which are needed for the work in the book. Answers to selected problems are given after the appendices.

CHAPTER

2

Basic Detection Theory

2.1

Introduction

This chapter is concerned with defining, developing, and discussing the basic detection theory concepts required throughout the remainder of this book. The reader is assumed to have a basic knowledge of probability theory and random variables. While an attempt has been made to make the book self-contained, prior introduction to the basic concepts of detection theory will be helpful since only a limited class of detection problems is treated here. The reader interested in a more comprehensive discussion of detection t h e o j is referred to the extensive available literature [Van Trees, 1968; Melsa and Cohn, 19761. The first section defines the basic concepts of simple and composite hypotheses and alternatives, type 1 and type 2 errors, and false alarm and detection probabilities. Sections 2.3 and 2.4 discuss three of the most common approaches to the hypothesis testing problem, the Bayes, ideal observer, and Neyman-Pearson methods. The Neyman-Pearson technique is developed in detail since it is the test criterion that will be utilized in the rest of this book. Receiver operating characteristics are discussed in Section 2.5. Section 2.6 presents methods for handling composite hypotheses including such concepts as uniformly most powerful, optimum unbiased, and locally most powerful detectors as well as the detector properties of consistency and robustness. The last section of the chapter presents several techniques for comparing the performance of parametric and nonparametric detectors. The principal technique presented is the concept of asymptotic relative 9

10

2

Basic Detection Theory

efficiency (ARE). The relationship of the ARE to the finite sample relative efficiency and the efficacy is given and the advantages and disadvantages of each technique are discussed. 2.2

Basic Concepts

Let the input samples be denoted by x = { x i ; i = 1, . . . , n } and let F be the probability distribution of the input data. The form of the probability distribution F depends on the values of some number of parameters which will be denoted by 8 = { O,, O,, . . . , Om}, m not necessarily finite. If a hypothesis or an alternative uniquely specifies each of the Oi,i = 1, 2, . . . , m , then the hypothesis or alternative will be called simple. The hypothesis or alternative will be called composite if one or more of the Oi are unspecified. To illustrate this concept, consider a Gaussian density which has mean p and variance u2. For this particular density function, there are two parameters, p and u2, which play the part of the Oi,i = 1, 2, and specify the form of the probability density. Three different statements or hypotheses concerning p and u2 are given below: H , : p = 5, H,: p = 2,

u2 = 1 u2

>2

u2 = 1 H,: p > 0, In hypothesis H I both of the parameters are uniquely specified, and therefore H I is called a simple hypothesis. The mean is explicitly given as 2 in hypothesis H,, but the variance is only given as being greater than 2. Since the variance can take.on a range of values greater than 2, hypothesis H, is called composite. In hypothesis H, the variance is fixed at 1, but tlie mean is only stated as being positive. Since p can equal any positive number, hypothesis H, is called composite. Many physical problems give rise to composite hypotheses or alternatives. One such example is a pulsed radar system. The received waveform at the input of a pulsed radar system can be written as (2.2-1) 0< t < T y ( t ) = A cos(wt +) n ( t ) ,

+ +

where A cos(wt + +) is the radar return, or signal, and n ( t ) is the noise. The amplitude A of the target return is not known exactly at the radar receiver due to variations in radar cross section and signal fading. A decision is required as to whether “noise only” ( A = 0) is present or “target and noise” ( A > 0) is present. Since a target is declared for any

2.2

Basic Concepts

11

> 0, the decision problem consists of testing a simple hypothesis ( A = 0) versus a composite alternative ( A > 0). Composite hypothesis testing problems may also arise in radar systems due to other parameters such as phase angle, frequency, and time-ofarrival being unknown. In radar systems the time-of-arrival of a radar return is rarely accurately known, and therefore, radar systems must be capable of detecting targets over a range of possible arrival times. Accurate phase angle information is usually not available due to phase shifts contributed by the target and propagation effects, and if the target is moving with respect to the radar, a substantial frequency shift may occur due to the Doppler effect. In communication systems, composite hypothesis testing problems may occur as a result of signal fading (range of signal amplitudes), multipath propagation (range of arrival times), and phase angle changes due to oscillator drift. Returning now to the general detection problem, it is clear that when testing a hypothesis H versus an alternative K, four possible situations can occur. The detector can A

accept the hypothesis H when the hypothesis H is true; (2) accept the alternative K when the hypothesis H is true; (3) accept the alternative K when the alternative K is true; (4) accept the hypothesis H when the alternative K is irue. (1)

Situations (1) and (3) are correct decisions, and situations (2) and (4) are incorrect decisions. The incorrect decision stated in (2) is called a type I error, and the incorrect decision given by (4) is called a type 2 error. Suppose that r is the set of all possible observations, then let us divide the observation space r into two disjoint subsets rHand rKwhich cover r such that rHu r , = r and rHn r,=O

If x E rH,we accept the hypothesis H, and if x E r,, we accept the alternative K. Since rK and rH are disjoint and cover r, the detector is completely defined once either rK or rH is selected. Because rKis the region for which we decide for the alternative K, it is sometimes referred to as the critical region of the test. If F E H,' the conditional probability density of x may be written as f ( x l H) and similarly, the conditional probability of x is f (XI K) for F E K. The probability of a type 1 error, also called a false alarm, is denoted by a 'F E H denotes that the probability distribution of the input data x is a subset of the probability distributions described in the hypothesis H , or more simply, that the hypothesis H is true. Similarly, F E K denotes that the alternative K is true.

12

2

Basic Detection Theory

and may be expressed as a = P { x E IIKIH} =

J

(2.2-2)

f ( x l H ) dx

TK

The false alarm probability is the probability of deciding K when H is true. By making use of the fact that rK and rH are disjoint and cover r we can also write a as a = i K f ( x l H )d x = = 1

-I

s,

f(xlH) dx-

f ( x l H ) d x = 1 - P { x E IIHIH}

(2.2-3)

H

Equation (2.2-3) simply expresses the fact that a is one minus the probability of correctly identifying H. The probability of a type 2 error, also called a miss, is denoted by p and may be written as (2.2-4)

/3 = P { x E r H I K } = J f ( x l K ) d x rH

The miss probability is the probability of deciding H when K is true. It is also possible to express P in terms of rKas

=

1

-s,

K

f ( x l K ) dx= 1 - P { x E rKIK}

(2.2-5)

K

Since P is the probability of a miss, 1 - p is the probability of detection. The probability of detection is also referred to as the power of the rest and can be written as 1 - p = 1 -L , f ( x l K ) dx=

s,

f ( x l K ) dx= P

{X

E r K ( K } (2.2-6)

K

In other words, 1 -

P is the probability of accepting K when K is true.

2.3 Bayes Decision Criterion In the theory of hypothesis testing, there are several different approaches to choosing the form of a test or detector. Two of the more well-known methods are the Bayes criterion and the Neyman-Pearson criterion. Each of these criteria is a method for choosing the two subsets, r, and rK,of the observation space r to achieve a specific goal. In this section, we consider the Bayes decision criterion which makes use of a systematic

2.3 Bayes Decision Criterion

13

procedure of assigning costs to each decision and then minimizing the total average cost. Let the cost corresponding to each of the four situations given in Section 2.2 be denoted by C,, = cost of accepting H when H is true; CKH= cost of accepting K when H is true; CKK= cost of accepting K when K is true; C,,

= cost of accepting H when K is true.

It is assumed that these costs are uniquely known. The Bayes criterion is a method for selecting the decision regions rH and rK such that the expected value of cost 4 = E {cost} is minimized. The expected cost can be written as 4 = E{cost} = E{costlH}P{H} + E{costlK}P{K) (2.3-1) Here P { H} is the probability, sometimes called the aprioriprobability that

H is true, while P { K} = 1 - P { H} is the probability that K is true. The

conditional expected cost E {costlH} can be expressed as + C,,P { x E = E {costlH = C H H P { x E r,lH

aH

=,,c

(1 - a) +

ac,,

(2.3-2)

In a similar manner, the expected cost given K is = E { c o s t l q = C,,P { x E r,jq + C,,P

aK

= cKK(l

r,p

{ x E r,lK

- P ) + PcHK

(2.3-3)

The total expected cost is, therefore, 3 = 4 H P { H } + 4 t , P { K } = P{H}[CHH

+ a(CKH

-

CHH)]

(2.3-4) + P ( ‘ H K - ‘KK 11 Now let us substitute the integral expressions of Eqs. (2.2-2) and (2.2-5) for a and to write 4 as +P{K)[CKK

4

= cHHP{H} -P{K}(CHK

+

+

cHKP{K}

-

‘KK)J

r K

f(xlK)

Combining the two integrals yields

CR =

cHHP{H)

+

cHKP{K)

- {K}(CHK -

P{H>(cKH

dx

+J

[P{H}(c,H

rK

‘KK

- cHH)J

) f ( x l K ) l dx

TK

f(x~~)dx

(2.3-5)

- c H H ) f ( x l ~ )

(2.3-6)

14

2

Basic Detection Theory

Under the Bayes criterion, the subsets rHand rKare now to be selected such that 9,,given by Eq. (2.3-6) and called the average risk, is a minimum. The minimum value of the average risk is called the Bayes risk, and choosing rK to minimize 9,results in what is called a Bayes solution to the statistical hypothesis testing problem. N o other strategy can yield a smaller average risk than the Bayes criterion. The problem is to select rK such that 3 is minimized. The first two terms in Eq. (2.3-6) are not a function of rK; to minimize the integral we assign to rKall of the values of x for which (2.3-7) Therefore the Bayes detector can be written as

(2.3-9) The quantity on the left-hand side of Eq. (2.3-9) is referred to as the likelihood ratio defined as (2.3-10) The right-hand side is the threshold of the test (2.3-1 1) Hence Eq. (2.3-9) can be written as K

L ( x ) 2 TB H

(2.3- 12)

Tests which have the form of Eq. (2.3-12) are known as likelihood ratio tests. The likelihood ratio is also referred to as a sufficient statistic. In general, a test statistic is any function (normally scalar) of the observation x. A test statistic is sufficient if it can be used to replace the complete observation as far as a detector is concerned. 2This assumption is reasonable since we are simply assuming that the cost for an error is greater than the cost for a correct decision. If this assumption is not valid, the decision rule may still be written in the form of Eq. (2.3-9) but the inequalities must be reversed.

15

2.3 Bqyes Decision Criterion

Example 23-1 In order to illustrate the use of the Bayes criterion, let us consider a simple problem. Suppose that we have a single observation whose density function under H and K is given by H : f ( x ( H ) = - exp(- 71 x 2 )

G

K : f ( x l K ) = -exp(- 81 x 2 ) 2 G

If H is true, x has a variance of one, while if K is true x has a variance of

4. We assume that H and K are equally likely so that P { H} = P { K } =

0.5 and let the costs be

CHK= CKH= 1; Then the likelihood ratio is given by ~ ( x =) $ exp( - i x 2

CHH=

+)’.4

c,,

=0

)’2.

= $ exp(

while the threshold is T, =

+(1 - 0)

$(1 - 0)

= 1

Therefore the Bayes optimal detector is given by K

4 exp( a x ’ ) s 1 H

Let us take logarithms on both sides to obtain K

3x221n2 H

or

1x1

g d E = 1.36 H

Example 23-2 As another example, suppose that the observation densities are given by

If H is true, the observations xi,i = 1, 2, . . . , n are uncorrelated, zero mean Gaussian variables while if K is true, the observations are uncorrelated Gaussian variables with mean p. This problem corresponds to detecting the presence of a known constant signal in additive white noise. We assume that P { H} = 0.75 and P { K} = 1 - P { H} = 0.25 and the costs

16

2 Basic Deiection Theory

are given by

0; CHK= 1 and The likelihood ratio for this problem is given by CHH = CKK =

= exp{ -

[

=2

i= I

7 1 - 2 p i xi 20

CKH

+ np2

i- 1

while the threshold is 0.75(2 - 0) =6 TB = 0.25(1 - 0) Therefore the Bayes optimal detector for this problem is i- I

Once again let us take logarithms on both sides to obtain p

K

p2n

H

20’

- X x i S -+ln6 U2 i-1

If we assume that p

> 0, then we have

Hence the detector sums the observations and then compares the sum to the threshold T,,. If in the expression for the average risk, Eq. (2.3-4), the costs are such that C,, = CKK= 0, and CKH= CHK= 1, the following expression results (2.3-13) 3 = P,= P { H } a P { K } P

+

which is called the total average probability of error. Substituting integral expressions for a and P in Eq. (2.3-13) yields

P,

=

P ( H ) I f ( x l H ) dx+ P(K)/ f ( x l K ) dx rK

rH

(2.3-14)

The ideal observer criterion consists of choosing rHand r K such that Eq. (2.3-14), the total average probability of error, is minimized. The ideal observer criterion can therefore be used when the cost of correct decisions is assumed or known to be zero, and the cost of incorrect decisions is taken

I7

2.4 Neyman-Pearson Lemma

to be one. The ideal observer criterion still requires the a priori probability of the hypothesis and the alternative to be known.

2.4 Neyman-Pearson Lemma If neither the cost function nor the probabilities of H and K are known, the Neyman-Pearson criterion can be used. However the main advantage of the Neyman-Pearson over the Bayes and ideal observer criteria is that it yields a detector which keeps the false alarm probability less than or equal to Some prechosen value and maximizes the probability of detection for this value of a. The Neyman-Pearson criterion is based on the NeymanPearson fundamental lemma of hypothesis testing, which can be stated as follows.

Neyman-Pearson Lemma Necessary and sufficient conditions for a test to be most powerful of level a. for testing H: f ( x l H ) against the alternative K: f ( x l K ) are that the test satisfy P { xE rKIH}

= J f ( x l H ) dx=

a.

(2.4-1)

rK

and (2.4-2) for some A. The form of the test given by Eq. (2.4-2) can be derived by using the method of Lagrange’s undetermined multipliers to maximize the probability of detection or power of the test subject to the constraint given by Eq. (2.4-1). Since maximizing 1 - p is equivalent to minimizing p, the following equation must be minimized by choosing rH and rK, A = p + A(a - ao) where A is an undetermined Lagrange multiplier. Using Eqs. (2.2-2) and (2.2-5) for a and p, we obtain the following expression

(2.4-3) For A

> 0, the expression for A in Eq. (2.4-3) will be minimized if all of the

18

2

Basic Detection Theory

x’s for which f(xlK) < Xf(xlH) are placed in rH,and all of the x’s for which f(xlK) > Xf(xlH) are placed in rK.Since in the continuous case the situation where f(xlK) = Xf(x1H) has probability zero, this situation can be included in either rH or rK.The allocation of the points wheref(x1K) = Xf(xlH), however, can affect the false alarm probability. Maximization of the power of a test subject to the constraint of a fixed false alarm probability therefore results in a test of the form (2.4-4) and once again we have obtained a likelihood ratio test. The threshold X is obtained by satisfying the constraint on the false alarm probability (2.4-5) It is sometimes convenient to think of the likelihood ratio L(x) as simply a random variable which is a function of the observation x. In this case, one can think of L(x) as possessing a density function which will depend on whether H or K is true. We will write these densities asf[L(x)lH] and f[L(x)lK]. Since L(x) is a ratio of probability densities, L(x) > 0 so that f [ L ( H ] = f[LIK] = 0 for L < 0. The false alarm and miss probabilities can now be written as a = P { L(x)

> AIH}

= / m f [ L ( ~ ) l H ]dL

(2.4-6)

/3 = P{L(x)

< XIK}

=/f[L(x)lK] dL

(2.4-7)

h

h

0

We can use Eq. (2.4-6) to obtain X by setting a = a. and solving for A. We note that h must be nonnegative. The sufficiency of Eqs. (2.4-1) and (2.4-2) can be shown in the following manner. Let r denote the observation space and let r , ( h ) be that critical region where f(xlK) > Af(x1H) gives a test of maximum power among all tests which have a false alarm probability of a,,.In other words, r , ( h )is the Neyman-Pearson critical region as defined by Eqs. (2.4-1) and (2.4-2). Now let robe any region in r which is the critical region of another test that also has a false alarm probability of a,,.If Eqs. (2.4-1) and (2.4-2) are sufficient conditions for a most powerful test, then the power of the test which has critical region rl will be greater than or equal to the power -of

2.4

19

Neyman-Pearson Lemma

the test which has critical region ro.The regions rl and romay have a common part which will be denoted by rol.A diagram of the relationship among rl, ro,and rolis shown in Fig. 2.4-1.

Fig. 2.4-1. Critical regions of two tests.

Let rl - rolrepresent the set of samples in r, not contained in To,. Then, for every observation in r , , f ( x l K ) > Af(xlH), so that we have

1

(2.4-8)

AP { x E (rl - To1)lH Adding P { x E rolIK} to both sides of Eq. (2.4-8) yields P { X E (rl - rol)lK} + p { x E rollK}

> AP { x E (rl - rol)lH 1

+ P { x E rollK1

The left-hand side of Eq. (2.4-9) is simply P { x test, and therefore, P{X E

rlpq > A P { X

E

E

(2.4-9)

r l l K } , the power of the

(r,- rol)lH)+ P { X

E

rolpq

(2.4-10)

Working with the right-hand side of Eq. (2.4-lo), we have A P { x E (l-1 - rol)lH} + P { X E rollK} = AP { x E r , p 1 - AP { x E r o l p 1 + P { x E rolpq (2.4-11)

Since both tests have a false alarm probability of a0, we know that

P { x E rllH} = a0 = P {x E I',lH} so that Eq. (2.4-1 1) becomes AP{X E (rl - rol)lH} + P { X E rolIK1 = AP { x E rolH1 - AP { x E rollH1. + P { x E rollK1 = AP { x E

(r, - rol)lH1 + P { x E rollK1

Substituting this result into Eq. (2.4- 10) yields P { X E rlpq > AP { x E (r, - rol)lH1 + P { x The points in

(2.4-12) E

rolpq

(2.4-13)

ro- rolare not contained in the region TI, and therefore, as

20

2 Basic Detection Theory

shown in Fig. 2.4-1, f ( x l K ) < X f ( x l H ) for all observations in Using this result in Eq. (2.4-13) gives

P{xEr,lK}

ro - rol.

> P{xE(ro - ro,)lK} + P{xEro,lK} = P{xEr,lK} (2.4- 14)

Since P {x E I',lK} represents the power of the test corresponding to rl while P {x E r 0 l K } is the power for the test corresponding to ro,Eq. (2.4-14) shows that the Neyman-Pearson test has the greatest power of any test of level ao. Hence, Eqs. (2.4-1) and (2.4-2) are sufficient conditions for the test to be most powerful.

Example 2.4-1 To illustrate the use of the Neyman-Pearson lemma, let us consider the following hypothesis and alternative: H: f ( x l ~= ) ( l / V G ) exp(-x2/2) K : ~ ( X I K=)( l / V G ) exp[ - ( x - 112/2]

We assume that a. = 0.2. The likelihood ratio is given by

f( 4 K ) L ( x ) = -= exp[(2x - 1)/2] f( 4 H ) and the Neyman-Pearson test is therefore, K

exp[(2x - 1)/2] S A H

By taking logarithms on both sides, we reduce the test to K

x S lnA+t H

Now we must select X to satisfy the constraint on the false alarm probability. The false alarm probability is given by

where @[.I is the cumulative distribution function for a zero mean, unit variance Gaussian random variable (see Appendix B). Hence, we wish to find X such that 1 - @[ln X

+ $1 = a. = 0.2

From Appendix B, we find that In A Neyman-Pearson test becomes K

+4

x S 0.84 H

= 0.84, so that

X

= 1.4 and the

2.5 Receiver Operating Characteristics

The corresponding power for the test is 1-

p

= 1

-J

m

1

0.84

- exp[ -

( x - 1)2

6

]

dx=

21

a[-0.161

= 0.4364

2.5 Receiver Operating Characteristics A plot of the probability of detection 1 - /3 versus the false alarm probability a is sometimes useful in describing the performance of a detector. Plots of this type are called receiuer operating characteristics (ROC). An expression for the receiver operating characteristic may be obtained by treating the output of the detector as a random variable D ( x ) that depends on x and takes on the values 0 or 1; if D ( x ) = 0, we accept H and if D ( x ) = 1, we accept K. Consider the expected value of the random variable D (x). If F E H, the expected value of D ( x ) equals Since D ( x ) = 0 for x E r,, Eq. (2.5-1) can be written as E{D(x)lH}

=I f ( x l H ) d x = a

(2.5-2)

rK

which is another expression for the false alarm probability given by Eq. (2.2-2). If F € K, the expected value of D ( x ) becomes

1 - i H f ( x l K )dx= 1 -

p

(2.5-3)

which is the probability of detection. The expected value of the random variable D ( x ) therefore defines the receiver operating characteristic curves and will be denoted by Q, (F),

Q D ( F )= E { D ( x ) }

(2.5-4)

It is clear that the operating characteristic of the detector Q,(F) depends both on the probability distribution of the input data and the form of the detector. Since the random variable D ( x ) takes on only the values 0 and 1, the operating characteristic is bounded by O < Q D ( F )4 1

(2.5-5)

In terms of the operating characteristic of the detector, we can express the

22

2 Basic Detection Theory

false alarm and detection probabilities as (2.5-6)

Q,(F E H ) = a

(2.5-7) Q, ( F E K ) = 1 - p A typical receiver operating characteristic is illustrated in Fig. 2.5- 1 for the detection problem of Example 2.3-2, where

H: K:

F is a zero-mean, unit-variance Gaussian distribution F is a mean p > 0, unit-variance Gaussian distribution

From Fig. 2.5-1 it can be seen that as p increases, the probability of detection for a specified false alarm probability increases, and as the threshold T increases, the probability of accepting K becomes small. Although the receiver operating characteristic presented in Fig. 2.5-1 is for a specific problem, it illustrates that operating characteristics depend only on the detection and false alarm probabilities and are not a function of a priori probabilities or costs. The receiver operating characteristic is therefore quite general and can be very useful in analyzing detector performance.

5m 4

m 0

a a 0

0.5 PROBABILITY OF MLSE ALARM

a

1.0

Fig. 2Sl. Typical receiver operating Characteristic.

Another interesting property of the operating characteristic is that for any likelihood ratio test, the slope of the operating characteristic at a particular point is equal to the value of the threshold T required to achieve the false alarm and detection probabilities which define that point. This fact can be'proven by first considering a likelihood ratio detector for testing H versus K which has the form K

L(x) 3 T H

(2.5-8)

2.5 Receiver Operating Characteristics

23

The false alarm and detection probabilities of this test can be written by the use of Eqs. (2.4-6) and (2.4-7) as a = PF =

and 1-

LW

f ( L ( x ) l H ) dL

0 = P D = STm f ( L ( ~ ) ( dL K)

(2.5-9)

(2.5- 10)

Differentiating both Eq. (2.5-9) and Eq. (2.5-10) with respect to T, yields3 dpF/dT

= -fLIH

(2.5-1 1)

dpD/dT

= -fLlK

(2.5- 12)

and dividing Eq. (2.5-12) by Eq. (2.5-1 l), gives (2.5-13) It must be shown that

Working with the probability densities of the observations, the probability of detection can be written as = p [ L ( x )>

where rK = {x: L(x)

TIK]

f ~ l K ( ~ dx l ~ )

(2.5-14)

rK

> T}. Rewriting Eq. (2.5-14)

(2.5-15) which becomes upon making the change of variablesy = L(x),

Differentiating Eq. (2.5-16) with respect to T yields dpD/dT

= - TfLIH

(2.5- 17)

'We will employ subscripts on density functions whenever there is a possibility of confusion.

2 Basic Detection Theory

24

which can be equated to Eq. (2.5-12) to get

-

TfL(H

(TIH)

= -fLIK

(TIK)

or (2.5-18) Using Eq. (2.5-18) with Eq. (2.5-13) gives the result dPD/dPF = T

(2.5-19)

which shows that the slope of any one of the operating characteristic curves at a particular point equals the value of the threshold required to obtain the PD and PF values which define that point. The conditions of the Neyman-Pearson lemma may be stated in terms of the receiver operating characteristic as follows: If (? is the class of all detectors satisfying QD(FEH) a (2.5-20) then the optimum detector fi in this class for hypothesis H, alternative K, and false alarm probability a satisfies the conditions Q i ( F E K) 2 QD(F E K) forall D E (? (2.5-21) These statements simply say that the detector b is the most powerful test with level less than or equal to a which belongs to the class ‘2. 2.6

Composite Hypotheses

In developing the Neyman-Pearson criterion, no limit was put on the number of parameters which are necessary to define the probability densities under H and K. Each of these parameters, however, must be exactly specified in H and K ; that is, the hypothesis and alternative must be simple. Since composite hypotheses and alternatives occur often in detection problems, it is important that a procedure be established to treat these cases. Consider the problem of testing the composite hypothesis H versus the simple alternative K H : f ( x l H ) = f(x, 8 ) with 8 E Q K : ~ ( X I K ) = f(x, 8,)

(2.6-1)

where Q is a set of possible values for 8 and 8, B Q is a specific value of 8. The general approach to the problem is to reduce the composite hypothesis to a simple hypothesis, and then test the simple hypothesis versus the

2.6 Composite Hypotheses

25

simple alternative. That is, instead of determining the most powerful test which satisfies D ( x ) f ( x I H )d x = J w D ( x ) f ( x , 8 ) d x < a

(2.6-2)

-W

for every 8 E 52, find the most powerful test which satisfies

for a specific 8 = 8,. The detection problem now becomes that of testing a simple hypothesis versus a simple alternative as given by H': ~ ( X I H = ) f ( x , 8,) K : f ( x l K ) = f ( x , 01)

(2.6-4)

The most powerful detector for testing H' versus K as specified by Eq. (2.6-4) can be found by utilizing the Neyman-Pearson fundamental lemma. If the detector for testing H' versus K , which is a member of the larger class of tests which satisfy Eq. (2.6-3), also belongs to the smaller class of tests which satisfy Eq. (2.6-2), then the detector for testing H' versus K is also most powerful for testing H versus K as given by Eq. (2.6-1). In many detection problems, the composite hypothesis may be reduced to a simple one by taking a weighted average of the density under the hypothesis f ( x l H ) = f ( x , 8) utilizing an assumed probability density h ( 8 ) for 8. That is, a test is found among the larger class of tests which satisfy

1

D ( x ) f ( x , 8 ) dx h ( 8 ) d o < a

(2.6-5)

instead of among the smaller class which satisfies Eq. (2.6-2). This results in a new hypothesis of the form (2.6-6) If the density for 8 is properly chosen, the detector for testing the new simple hypothesis and alternative will be most powerful for the original problem of testing a composite hypothesis versus a simple alternative. The question arises as to what criterion to use to select the proper value of 8, or correct density h ( 8 ) . In the case of a finite number of 8 parameters, the following procedure is quite useful. Inspect each 8 E Q and choose for 8, that 8 which most resembles the alternative parameter 8,; that is, select for 8, that 8 E Q which would seem to be most difficult to test against 8,. The resulting most powerful detector, must then be checked to see if it satisfies Eq. (2.6-2) for every 8 E 52. If it does, the value for 8, 8 = 8,, is called the least fuuorable choice of all 8 E 52.

26

2 Basic Detection Theory

For the case where 8 takes on a range of values, the density h ( 8 ) should be chosen such that it gives as little information about 8 as possible, and such that the weighted average over all 8 E 52 yields a simple hypothesis statement which is most difficult to test against the alternative. A most powerful test is found by the fundamental lemma and it is checked to see if it satisfies Eq. (2.6-2) for every 8 E 52. The distribution of 8, h ( 8 ) , which yields a test that satisfies Eq. (2.6-5) for every 8 E 52 is called the least favorable distribution of 8. The choice of a least favorable distribution of 8 usually requires the system designer to have some intuitive feeling of the 8 variation, such as symmetry or a specific order. When such intuitive guidelines are not available, the selection of a least favorable distribution is extremely difficult. A common example of the assignment of a least favorable distribution occurs in the problem of detecting narrowband, noncoherent pulses. The signal to be detected for this case has the form S ( t ) = A ( t ) cos[wt + 91 where 9, the phase angle, is unknown. Since no information about 9 is available, 9 is usually assumed to have the density function

(y

0 Q 9 2a otherwise which demonstrates maximum uncertainty in the value of 6, and hence gives little information concerning the unknown parameter. Helstrom [ 19681 proves that this distribution of 9 is truly least favorable. In certain cases, one detector may maximize the power of the test for all F E K even when K is composite. Detectors with this property are called uniformly most powerful (UMP). The concept of a uniformly most powerful test is best illustrated by means of two simple examples. h(+) =

Example 2.6-1 Consider the problem of testing the simple hypothesis H versus the composite alternative K where H and K are given by H: x = v K: x = m + v and x is the observation, v is zero mean, Gaussian noise with variance u2, and m is a fixed but unknown parameter. The probability densities under H and K can be written as

2.6

27

Composite Hypotheses

and

Using these two densities and forming the likelihood ratio yields [ ( x - m)’ - x’] .XP{ -

2a’

K

} HS T

which can be further simplified to yield mx 2 -

m’ = TI‘ T’ + U’ H 2a’ If m is assumed to be greater than zero, this result can be expressed by

(2.6-7) The threshold T”’ must be chosen to fix the false alarm probability a at the desired level ao. Writing the expression for a and equating it to the desired false alarm probability ao, gives

= S m lexp( - 2a2 ) dx= X’

T”’

a.

(2.6-8)

(I

Since the parameter m does not appear in the above expression for ao, the threshold T”’ is independent of m, and consequently, the test is independent of m. The test given by Eq. (2.6-7) with T“’ determined from Eq. (2.6-8) is therefore the most powerful test of level a. independent of the value of m, and therefore, Eq. (2.6-7) defines a UMP test for testing H versus K. A second example will illustrate a simple detection problem where a UMP test does not exist.

Example 2.6-2 Consider testing the composite hypothesis H versus the composite alternative K shown below H: x = - m + u K: x = m + u where x is the observation, u is zero mean Gaussian noise with variance a’, and m is a fixed but unknown parameter. The probability densities under

28

2 Basic Detection Theory

H and K can be written as

and

Forming the likelihood ratio and simplifying yields 2mx Z T' U2

Assuming m

H

> 0, the test can be written in the form (2.6-9)

Writing the expression for the false alarm probability and equating this expression to the desired level ao,gives a = JTOPf(xIH) dx=

Jm T" %a

erp[ -

(x

+ ml2 2a2

]

dx= a.

The above expression for a. clearly depends on the parameter m, and hence if T" is chosen to fix a. utilizing this expression, T" will also be a function of m. Since the threshold T" and therefore the test depends on m, the test defined by Eq. (2.6-9) is not the most powerful test of level a. for all values of m, and thus is not a UMP test for testing H versus K. When a UMP test does not exist for a composite hypothesis or alternative, a detector which is most powerful in some suitably restricted sense may prove useful. Unbiased statistical tests are of great importance because there is a large class of problems for which a UMP unbiased test may be defined, even though a UMP test does not exist. Unbiasedness is a simple and reasonable condition to impose on a detector. Definition: Unbiased Detector A detector D (x) is unbiased at level a if QD(F E H ) a (2.6-10a) and (2.6-lob) In other words, a detector is unbiased if the probability of detection for all

2.6

Composite Hypotheses

29

F E K is never less than the level of the test. The reasonableness of requiring a test to be unbiased can be seen by noting that if the conditions given in Eq. (2.6-10) are not satisfied, the probability of accepting H will be greater under some distributions contained in the alternative K than in some cases when the hypothesis H is true.

Definition: Optimum Unbiased -Detector If A is the class of unbiased detectors of level a, a detector D is optimum unbiased, or UMP unbiased, for hypothesis H and alternative K at level of significance a if Q,(F E K ) 2 Q,(F E K ) forall F E K and D € A (2.6-11) Although Eq. (2.6-10) states the concept of unbiasedness quite clearly, it is difficult to employ this definition to find most powerful unbiased tests because of the presence of the inequalities. For this reason when it is desired to restrict the class of tests being considered to only unbiased tests, the concept of similar regions or similar tests is often useful. Definition: Similar Tests A detector or test D ( x ) is said to be similar of size a for testing the hypothesis H if QD(F)

=Iw D ( x ) f ( x I H )d x = i x / ( x l H ) d x = -03

(2.6-12)

for all possible F E H [Kendall and Stuart, 1967; Fraser, 1957a; Lehmann, 19591. From Eq. (2.6-12) it can be seen that for a test to be similar, the test must make incorrect decisions at the maximum allowable rate a for all densities that belong to the composite hypothesis. This does not seem to be an intuitively desirable property for a detector to have, and indeed, it is not. The importance of similar tests is that they provide a route by which an optimal unbiased test can be found, since unbiasedness is a much more reasonable property for a detector to possess. If an unrestricted UMP test for a composite hypothesis cannot be found and it is desired to find an optimal unbiased test for this#hypothesis,the procedure is as follows. Find a detector that is most powerful similar for testing a boundary hypothesis A (i.e., a hypothesis on the boundary between H and K) versus the original alternative. Since all unbiased tests of the original hypothesis are similar tests of the boundary hypothesis, if the test of the boundary hypothesis is most powerful similar, and it can be shown that this test is unbiased, then the test is optimal unbiased for the original composite hypothesis. Although this procedure may seem circuitous and unlikely to be very

30

2

Basic Detection Theory

useful, the fact is that similarity and unbiasedness are closely related, and unbiased tests can frequently be found from analyzing the similar regions of a particular hypothesis testing problem. In this book, a rigorous proof that certain tests are UMP similar will not be given, since the demonstration of such a property generally requires mathematics beyond the intended scope of the book. The concept of similarity will be used frequently, however, and in these cases, the required proofs will be referenced for the more mathematically advanced reader. A general example of the type of problem that can be solved by the above technique is that of testing the mean of a Gaussian distribution with known variance: H: p

< 0,

a’isunknown

versus K : p > 0, a’is unknown The hypothesis and alternative being tested are composite since it is desired to detect any positive mean value as opposed to any negative mean value, and not specific values of mean. In addition the variance is unknown. Since the hypothesis and alternative are composite and no fixed form can be written for the probability densities under H and K , the Neyman-Pearson lemma cannot be immediately employed to solve the problem. Suppose, however, that it is desired to limit the detectors considered to only unbiased tests, and a UMP detector within this limited class is required. The first step in solving this problem is to replace the unknown a’ with some estimate 6’ and find a detector for testing the boundary hypothesis H : p = 0,

f Gaussian with a’

=

6’

K : p = h, f Gaussian with a’ = 6’ by using the Neyman-Pearson lemma. If the resulting test statistic is independent of a’, then the detector is UMP similar for testing the original H and K. If the detector can then be shown to be unbiased, the required UMP unbiased detector for testing H versus K has been found. Locally most powerful (LMP) tests are useful when certain weak signal assumptions are valid. Locally optimum or LMP detectors maximize the power of the test for a subset of the probability distributions specified by the alternative K which are close in some sense to the hypothesis H. Since detectors are most useful under weak signal conditions, LMP detectors are very important. Two other properties of interest when considering nonparametric detectors are consistency and robustness.

2.7 Detector Comparison Techniques

31

Definition: Consistericy Considering the input data set x and a fixed alternative K , a detector D is said to be consistent if the probability of detection approaches one for any F E K as the number of samples becomes large, (2.6- 13)

The limiting process must be taken in such a manner that the detector has the same false alarm rate a under the hypothesis H for each n. Some detectors, although not nonparametric in the sense already defined, are robust or asymptotically nonparametric. Definition: Asymptotically Nonparametric A detector D ( x , , . . . , x,,) is robust or asymptotically nonparametric if lim QD(x,,

n+w

,, ,

,xa)( F

)=a

for all F E H

(2.6- 14)

2.7 Detector Comparison Techniques In general, there are many possible solutions to a given detection problem, each of which may produce different results. In order to evaluate the performance of these detectors, some meaningful criterion must be established. Two types of comparisons are usually made: (1) A comparison of a parametric and a nonparametric detector under some subclass of the hypothesis and alternative, and (2) a comparison of two nonparametric detectors under the same hypothesis and alternative. One method for comparing two detectors under the same hypothesis and alternative is to compute the relative efficiency of one detector with respect to the other. Many different definitions of relative efficiency exist in the statistical literature, but the definition most commonly used is due to Pitman [1948] and may be stated as follows: Definition: Relative Efficiency If two tests have the same hypothesis and the same significance level and if for the same power with respect to the same alternative one test requires a sample size N , and the other a sample of size N,, then the relative efficiency of the first test with respect to the second test is given by the ratio e I a 2= N 2 / N l . It is clear from the above definition that the relative efficiency is a function of the significance level, alternative K, and sample size of the two detectors and is therefore highly dependent on the experimental proce-

32

2

Basic Detection Theoty

dures used when taking the data. In order to completely compare two detectors using the definition given above, the relative efficiency must be computed for all possible values of a,K, N,, and N,. Besides the fact that this computation may be quite difficult, there does not usually exist for small N, a sample size N , such that the powers of the two detectors are exactly equal. To circumvent this problem, some kind of interpolation of the power of the second test as a function of N must be performed. Hodges and Lehmann [ 19561 have cited a specific example in which they compared two detectors utilizing linear interpolation to achieve the desired power and then compared their results with those of Dixon [1953, 19541 who utilized polynomial interpolation to obtain equal powers. The values obtained for the relative efficiency utilizing the two interpolation methods were markedly different, and the two methods did not even yield the same trend (increasing or decreasing in value) for the relative efficiency as the alternative was varied. The finite sample relative efficiency is difficult to compute, highly dependent on experimental methods, and even peculiar to the mathematical techniques employed for its computation. For these reasons, a simpler expression somewhat equivalent to the finite sample relative efficiency is usually employed. The simpler expression for the relative efficiency is obtained by holding the significance level and power constant, while the two sample sizes approach infinity and the alternative approaches the hypothesis. This technique yields the concept of asymptotic relative efficiency (ARE). Definition: Asymptotic Relative Efficiency The ARE of a detector D,, with respect to detector D, can be written as n2

El,, = lim n1-m n,

(2.7- 1)

n2+m

K+H

where n , and n2 are the smallest number of samples necessary for the two detectors to achieve a power of 1 - j3, for the same hypothesis, alternative, and significance level. The alternative K must be allowed to approach the hypothesis, because for the case of consistent detectors against fixed alternatives for large sample sizes, the power of the detectors approaches 1. If K is allowed to approach H as n , and n2 + 00, the power of the detectors with respect to the changing alternative converges to a value between the significance level a and 1. In many cases the ARE defined by Eq. (2.7-1) is just a single, easy-to-use number, independent of a. The ARE of two detectors can also be written in terms of the ratio of the efficacies of the two detectors if certain regularity conditions are satisfied

2.7

Detector Comparison Techniques

33

[Capon, 1959al. Given a test statistic S,, for deciding between the hypothesis H: 9 = 0 (signal absent) and the alternative K : 9 > 0 (signal present), the regularity conditions can be stated as: (1) S,, is asymptotically Gaussian when H is true with mean and variance E { S , J H } and var{ S , J H } , respectively; (2) S,, is asymptotically Gaussian when K is true with mean and variance E { S,,IK} and var{ S , J K } ,respectively; (3)

lim

K-+H

var{ Sn I K } = 1; var{S,IH}

constant independent of n and dependent only on the probability densities under H and K; (7)

lim var{S,,lH} = 0.

n-+m

The regularity conditions are essentially equivalent to requiring that the detector test statistic be consistent as defined by Eq. (2.6-13) [Capon, 1959bl. The expression in regularity condition ( 5 ) is called the efficacy of the detector. It can be shown that if the test statistics S,, and S,* for two detectors satisfy the regularity conditions, that the ARE of the S,, detector with respect to the S,* detector is equal to the ratio of the efficacies of S,, and S,*, that is [Pitman, 1948; Capon, 19601 (2.7-2) The definitions of the asymptotic relative efficiency in Eqs. (2.7-1) and (2.7-2) and the regularity conditions listed above are valid only for detectors which have one input. Since both one- and two-input detectors are treated in the sequel, it is necessary to state the analogous definitions for two-input detectors.

Definition: A RE for Two-Input Detectors The asymptotic relative efficiency of detector D, with respect to detector D4 is given by (2.7-3)

2

34

Basic Detection Theory

where m,,n, and m,, n, are the sample sizes of the two sets of observations for detectors D, and D,,respectively. Of course, the ARE in Eq. (2.7-3) can also be written in terms of the efficacies of the two detectors if the following regularity conditions are satisfied. Given a test statistic S,,,,, for testing the hypothesis H : 8 = 0 versus the alternative K: 8 > 0, 8 a signal-to-noise ratio parameter, the regularity conditions can be stated as: (1') S,, is asymptotically Gaussian when H is true with mean and variance E { S,,IH} and var{ Sm,IH},respectively; (2') S,, is asymptotically Gaussian when K is true with mean and variance E { S,,IK} and var{ S,l K}, respectively; (3')

lim

K+H

var{ S m n I K } = 1; var{ S,,IH }

is independent of m and n, and depends only on the probability densities under H and K ;

(7')

lim

m, n-oo

var{ S,,IH} = 0.

When two test statistics S,, and S i n satisfy the regularity conditions, the ARE of the detector based on S,, with respect to the S& detector is equal to the ratio of the efficacies of S,,,,, and S i n , which is ESm,S& = 6 , /GS& (2.7-4) Perhaps not surprisingly, the efficacy can be related to the distinguishability of the test statistic probability densities under the hypothesis and alternative [Hancock and Lainiotis, 19651. In order to demonstrate this fact, consider a one-input detector with test statistic S, and define as a measure of separability the quantity (2.7-5)

which is designated the output signal-to-noise ratio (SNR,,h for reasons that will become evident momentarily. That Eq. (2.7-5) is a valid measure of the distinguishability of the test statistic probability densities when H and K are true can be seen by noting that as the difference in the locations

2.7 Detector Comparison Techniques

35

or mean values becomes larger or the variance of the test statistic under K becomes smaller, SNR,,, in Eq. (2.7-5) increases. Define further the input signal-to-noise ratio as SN\, = e (2.7-6) where 8 = 0 when H is true, and 8 > 0 when K is true. The interpretation of Eqs. (2.7-5) and (2.7-6) as signal-to-noise ratios is now clearer, since 8 is obviously a measure of the SNR at the detector input and Eq. (2.7-5) is the SNR after the input has been operated on by the detector. Returning to the goal of relating SNR,,, to the efficacy, consider the regularity condition (4) which becomes for weak signals (8' << l),

Substituting Eq. (2.7-7) into Eq. (2.7-5), and forming the ratio of SNR,,, to SNR, yields (2.7-8)

- dE { Sn I K 1/ d o SN% {var{SnlH f

SNRout --

lim

n--tm

>>

SNR,,

(2.7-9) e-o

(2.7- 10)

The efficacy of a detector test statistic is thus a measure of the improvement in SNR afforded by the detector for 8 small and n large. The efficacy is extremely useful for calculating the ARE of various nonparametric detectors, and its use is demonstrated in later chapters. The ARE is in general independent of a, and hence the ARE is not dependent on experimental methods as the finite sample relative efficiency is. The ARE has two obvious deficiencies, however. Since no detector is able to base its decision on an infinite number of observations, the ARE does not give the engineer or statistician the relative efficiency of the two detectors for the finite number of samples used in the detection procedure. The requirement that K+H is also a severe limitation since few practical problems require the testing of a hypothesis and alternative that differ only infinitesimally from each other. How useful then is the ARE as a criterion for comparing detectors when the number of observations is not large or

36

2

Basic Detection Theoiy

when the alternative is not close to the hypothesis? Calculations of small sample relative efficiencies using a common test statistic and similar test conditions indicate that the ARE and the small sample relative efficiency, in general, rank the tests in the same order; however, the actual values of the ARE should be used only as an approximation of the small sample relative efficiency of two detectors. When the ARE is used to compare two detectors, the detector used as the basis for the comparison is usually chosen to be the optimum, or most powerful, detector, assuming one exists, for the hypothesis and alternative being tested. When this is the case, the ARE of the nonoptimum detector with respect to the optimum detector is some number less than one. If however, deviations from the assumptions on which the optimum detector is based occur, the ARE of the nonoptimum detector with respect to the optimum detector may then become greater than one. This is because the optimum detector may be highly susceptible to changes in design conditions, whereas the suboptimum detector may be based on more general assumptions and thus be less sensitive to environmental changes. It should also be noted when comparing a suboptimum nonparametric detector to an optimum parametric detector that the ARE is, in general, a lower bound for the relative efficiency. The relative efficiency tends to be largest when the sample size is small and then decreases to the lower bound given by the ARE as the number of samples becomes large. This statement is only true, however, for vanishingly small signals; that is, K close to H. 2.8

Summary

In this chapter the basic detection theory concepts have been introduced, several approaches to the detection problem have been discussed, and various detector evaluation and comparison techniques have been described. The initial discussion explained the difference between simple and composite hypotheses and alternatives and gave examples of physical situations where simple and composite hypotheses arise. The false alarm and detection probabilities were then defined and their use in the receiver operating characteristic was explained. The importance and generality of the receiver operating characteristic for evaluating the performance of detectors was described in detail and several properties of the ROC were pointed out. The Bayes criterion, the ideal observer criterion, and the NeymanPearson criterion were introduced and developed. The applicability of the Bayes criterion, which minimizes the average risk, and of the ideal observer

2.8 Summary

37

criterion, which minimizes the total average probability of error, was discussed. The use of the Neyman-Pearson criterion when costs and a priori probabilities are unknown or when the false alarm probability must be maintained at or below some predetermined level was described. The Neyman-Pearson fundamental lemma was derived and the sufficiency of the Neyman-Pearson lemma for defining a most powerful test was proven. A modification to the Neyman-Pearson lemma for determining a most powerful test when the hypothesis or alternative is composite was presented and discussed, thus introducing the concept of a least favorable distribution. The important concept of a UMP test was introduced and two examples were given to further clarify the subject. Since UMP tests cannot always be found, optimum unbiased and locally most powerful tests were defined and their applicability discussed. The two asymptotic properties of consistency and robustness were also defined. The important problem of comparing two detectors was then considered with the concepts of finite sample relative efficiency, asymptotic relative efficiency (ARE), and efficacy being introduced. The difficulty of computing the finite sample relative efficiency due to its dependence on the significance level, alternative, and sample size was pointed out. Two expressions for the ARE were given, one in terms of efficacies, and the usefulness of the ARE described. The regularity conditions necessary for the relation between the ARE and efficacies to hold were given.

CHAPTER

3

One-Input Detectors

3.1 Introduction

This chapter is concerned with introducing and developing several one-input parametric and nonparametric detectors that can be utilized in communication and radar systems. The term one-input as used here is intended to delineate the number of sets of input data samples, not the number of data samples themselves. In this chapter parametric detectors for testing for a positive signal embedded in Gaussian noise for various simple and composite hypotheses and alternatives are derived utilizing the Neyman-Pearson fundamental lemma. The parametric detectors are developed primarily to be used as a baseline for the evaluation of the performance of the nonparametric detectors in Chapter 4 and to illustrate the application of the NeymanPearson lemma. The Student’s t test, the uniformly most powerful unbiased detector for testing for a positive shift of the mean when the noise is Gaussian with an unknown variance, is derived and shown to be asymptotically nonparametric and consistent. Six nonparametric detectors, the sign detector, the Wilcoxon detector, the normal scores test, Van der Waerden’s test, the Spearman rho detector, and the Kendall tau detector are presented and discussed. The basic form of each of the detector test statistics is given and the computation of each test statistic is illustrated. The important properties and the advantages and disadvantages of each detector with respect to implementation and performance are also discussed.

38

3.2 Parametric Detectors

39

3.2 Parametric Detectors One of the classic problems in detection theory and one that has received much attention in the literature is the detection of a deterministic signal in additive, white Gaussian noise. The noise, which will be denoted by u, is usually assumed to be zero mean with known variance u;. The input process to the detector is sampled at intervals such that the observations obtained are independent. The input to the detector is denoted by x = { x,, x2, . . . , x n } , assuming n samples of the input to have been taken, and x will consist of either noise alone, xi = ui, or signal plus noise, xi = si + ui. The noise alone possibility is called the null hypothesis or simply, hypothesis, and is denoted by H,. The signal plus noise possibility is called the alternative hypothesis, or alternative, and is denoted by K,. If F ( x ) is the cumulative distribution function, p is the mean, and u2 is the variance of the input, the detection problem may be written as testing H,:

F ( x i ) is Gaussian with pi = 0, u,? = u;

versus KO: F(xi) is Gaussian with pi = si, u,? = u; for i = 1, 2, . . . , n. The problem is therefore simply one of testing the mean of a normal distribution. Since each of the xi are independent and identically distributed, the joint density of the xi under the hypothesis H,, which will be denoted by f(xlHo), equals the product of the n identical densities, thus (3.2-1) Similarly, the density under the alternative KO,denoted by f(xl KO)can be written

Utilizing the Neyman-Pearson fundamental lemma, the likelihood ratio can now be formed

40

3

One-Input Detectors

which is then compared to a threshold T ,

]fr

(3.2-4)

HO

To simplify Eq. (3.2-4), take the natural logarithm of both sides (2skxk

'k')

-

20:

k- 1

2 In T = T'

and rearrange the terms

or 2

x n

k-1

KO

skxk 2 2ugT'+ Ho

n

2 si

k- 1

Since each s, is a known signal, Xi-lsi is known and is the energy in the transmitted signal. The above expression can now be rewritten as

The optimum detector under the Neyman-Pearson lemma is therefore

i

k - 1' k X k (

< To + > To

Do(x) = 0, Do(x) = 1,

accept H , accept KO

(3.2-5)

where the threshold Tois determined by setting the false alarm probability a. A somewhat simpler detection problem results when all of the sk are assumed to be equal to some unknown positive constant C. The detection problem then becomes that of testing H I : F ( x J is Gaussian with pi = 0, u,? = ug

versus K , : F ( x j ) is Gaussian with p j = C

> 0, u,? = ui

where i = 1, 2, . . . , n. The detector given by Eq. (3.2-5) becomes

41

3.2 Parametric Detectors

or

5

k-,

.k{

< TI * > T, =$

D , ( x ) = 0, D,(x) = 1,

acceptH, accept K,

(3.2-6)

which is sometimes called the linear detector. As before, the threshold T I may be determined under the NeymanPearson criterion by fixing the false alarm probability a. To determine T I , the density of the test statistic under the hypothesis H I must be obtained. Since each of the xi have a Gaussian distribution with zero mean and variance ui, the statistic S = Et_Ixk has a Gaussian distribution with zero mean and variance nu;

The expression for the false alarm probability may then be written (3.2-7) By setting a = a0,the threshold T I may be determined from Eq. (3.2-7),

Making a change of variables with y = ( S / f i a. =

m

1

uo)

exp( 7 -Y2 d) y = 1 - a[

-1

TI

fi(I0

(3.2-8)

where a[-1 is the cumulative distribution function of a zero mean Gaussian random variable with unit variance. Solving Eq. (3.2-8) for T I yields TI = fiu0@-'[ 1 -

(yo]

(3.2-9)

A slightly more general detection problem is obtained if the noise power ui is assumed to be unknown. The hypothesis and alternative to be tested are then

H,: F(x)is Gaussian with p = 0 and unknown variance u2 K,: F ( x ) is Gaussian with p > 0 and unknown variance u2 Lehmann and Stein [ 19481 have shown that for a < f an optimum detector for testing H , versus K, does not exist in the unrestricted sense. In the same paper they show that for a

> 4,

a detector which implements the

42

3 One-Input Detectors

Student’s t test is uniformly most powerful for testing H , versus K,. Since false alarm levels greater than one-half are of little practical interest, a detector which is optimum in a restricted sense will now be considered. If attention is limited to the class of unbiased rules, the optimum detector for testing H , versus K, may be derived as follows. The input to the detector is assumed to be n independent and identically distributed random variables xi, i = 1, 2, . . . , n. The density function of the xi under the hypothesis H, is therefore given by (3.2- 10) and similarly the joint density of the xi under K, may be written as j ( x l K , ) = ( 2 r ~ ~ ) - exp{ ” / ~-

i

(Xj -

j-1

20,

]

d2

(3.2-11)

Since the densities of p and u2 are unknown, these parameters will be estimated by maximizing the two density functions in Eqs. (3.2-10) and (3.2-1 1) with respect to the unknown parameters. Under the hypothesis H,, u2 can be estimated by utilizing

(a/au2)

l n f ( x ( ~ , )= o

(3.2- 12)

Taking the natural logarithm of Eq. (3.2-10) and substituting into Eq. (3.2- 12),

Equating this to zero and solving for u2 yields (3.2- 13) as the estimate of the variance under H,. The estimates of p and u2 under K2 are found by simultaneously solving the equations

a lnf(xlK,) acl

=0

and

a lnf(xlK2) = 0 au2

Let us first consider the partial derivative with respect to p :

3.2 Parametric Detectors

43

Equating this result to zero and solving for p gives

P,

=

1 " J'

2 xj

(3.2-14)

1

which is independent of the variance. Similarly for the estimate of u2 under K29

Equating this result to zero and rearranging yields .

n

(3.2- 15)

where jiKis given by Eq. (3.2-14). Forming the likelihood ratio

and substituting the estimates of p and u2 into this expression,

n

2I xi'

J-

J-

1

(3.2- 16)

3 One-Input Detectors

44

Equation (3.2-16) can be rewritten by using

I:xi'= x n

n

i- 1

i- I

(Xi

-

&),+ nil;

Using this expression, the likelihood ratio becomes n/2

(3.2- 17)

The quantity in the brackets is 1 + [ t Z / ( n - I)] where t has the t distribution with n - 1 degrees of freedom when H , is true. Since L ( x ) in Eq. (3.2-17) is a monotonic function of t 2 , the test based on t Z is equivalent to the likelihood ratio test defined by Eq. (3.2-17). Since t 2 = 0 when L(x) = 1, and t 2 becomes infinite when L(x) approaches infinity, a critical region of the form L(x) > A > 1 is equivalent to a critical region t 2 > B. The critical region is therefore defined by the extreme values for t, both positive and negative. Since the alternative K, is concerned only with positive signals and since t Z is a monotonic function of t, the test statistic for the hypothesis H , versus the alternative K , becomes n

Since under H , the test statistic 6 has a limiting Gaussian distribution with zero mean and unit variance, the threshold T may be approximately determined from (3.2- 19) @ ( T )= 1 - (Yo for a large number of observations. The probability of detection or power of the test can then be found from the expression 1 - p = 1 - O [ T - p] since the test statistic has a Gaussian distribution with mean p and unit variance under K,. For small n, T can be exactly determined by utilizing the equation (Yo = P ( 6 > TIH,} where a0 is the desired false alarm rate, by using tables of the t distribu-

45

3.2 Parametric Detectors

tion. Knowing T, the power of the test can be found from 1 - /3= P ( 6 > TIK,} = 1 - P ( 6 < T J K , } using tables of the noncentral t distribution (see Hogg and Craig, 1970 for a discussion of the noncentral t distribution). Since the threshold T is chosen according to the Neyman-Pearson criterion utilizing the distribution of the Student’s t statistic given by the left-hand side of Eq. (3.2-18), the false alarm probability clearly does not depend on the parameters p or 6’. The Student’s t test given by Eq. (3.2-18) is therefore invariant for any parameter values under H , and K,. Even though the Student’s t test has been derived utilizing a generalized form of the Neyman-Pearson lemma, the use of maximum likelihood estimates in the probability densities prevents the Student’s t test from being UMP for testing H , versus K,. Indeed, all that the calculation which led to Eq. (3.2-18) has done is to illustrate how the Student’s t test might be derived; no optimum properties can be inferred from the derivation, even for a restricted class of tests. Fraser [1957a] has shown that the Student’s t test given by Eq. (3.2-18) is uniformly most powerful similar for testing H , versus K,. By the discussion of similarity and unbiasedness in Section 2.6, the Student’s t test will be the UMP unbiased test of H , versus K,,if it can be shown that the Student’s t test statistic in Eq. (3.2-18) is unbiased. As stated in Section 2.6, a detector is said to be unbiased at level a if Q,(F) < a forall F E H and if Q,(F)

>a

forall F

E

K

To check to see if the Student’s t test is unbiased, let t, be chosen such that (3.2-20)

when p = 0 and where S is defined by

Consider a value of p

> 0; then

46

3

Therefore,

One-Input Detectors

Q, (F) > a

for all F E K2

(3.2-2 1)

and from Eq. (3.2-20), E { D ( x ) l H , } = Q, ( F ) = a

for all F E H,.

(3.2-22)

The Student’s t test is therefore the UMP unbiased test of level a for testing the hypothesis H, versus the alternative K,. To evaluate the performance of the Student’s t test for a nonparametric problem, let the form of the underlying noise distribution be unknown so that the hypothesis and alternative become H,:

p = 0, u2 finite, F ( x ) otherwise unknown

K,:

p

> 0, u2 finite,

F ( x ) otherwise unknown

Since the detector defined by Eq. (3.2-18) is the optimum unbiased detector under the assumption of Gaussian noise, the false alarm rate will not in general remain constant for arbitrary input noise distributions. As the number of input observations becomes large, however, the Student’s t test exhibits an interesting property. The bracketed quantity in the denominator of Eq. (3.2-18) can be shown to converge in probability to u2 as the number of observations becomes infinite. By the central limit theorem, which states that the distribution of the sum of n independent and identically distributed random variables is asymptotically Gaussian [Hogg and Craig, 19701, E x i / fi has a limiting Gaussian distribution with zero mean and variance a2 under H,. The test statistic on the left-hand side of Eq. (3.2-18) thus has a limiting Gaussian distribution with zero mean and unit variance under H,. Since the Student’s t test statistic is asymptotically Gaussian, if the threshold T is chosen according to Eq. (3.2-19), the false alarm probability will remain constant for arbitrary input noise distributions if the number of samples is large. The Student’s t test is therefore asymptotically nonparametric. Since Zxi/ds has a limiting Gaussian distribution with mean fi p and variance u2 under the alternative K,, the mean of the test statistic in Eq. (3.2-18) approaches infinity as n + 00, and the probability of detection goes to one. The Student’s t test is thus consistent for the alternative K, by Eq. (2.6-13). The Student’s t test is therefore the optimum unbiased detector for the hypothesis H , versus the alternative K2 and is an asymptotically nonparametric detector for testing the hypothesis H , versus the alternative K3. Since the detector is only asymptotically nonparametric for H , versus K,, the false alarm rate will not remain constant, and may be significantly greater than the desired level, for small n and non-Gaussian input noise.

3.3 Sign Detector

47

Bradley [ 19631 has noted and studied an interesting phenomenon that occurs when the Student’s t statistic is being used to test for a location shift of a probability density function. When testing for a location shift, it seems intuitive that the hypothesis should be accepted or rejected according to whether the sample mean, K = 2:,Ixi/n, essentially the numerator of Eq. (3.2-18), is small or large, respectively. In H2and K,, however, the variance is unknown and thus must be estimated from the set of observations. The test statistic in Eq. (3.2-18) therefore depends not only on the sample mean but also on the sample variance. The statistic may thus exceed the threshold not due to a positive location shift, but due to a small value of the sample variance in the denominator. Large values of the test statistic so obtained are called irreleuant, since the hypothesis is rejected not due to a change in the characteristic being tested but due to a chance value of a parameter which has no relationship to the hypothesis being tested. Such a parameter is sometimes called a nuisance parameter. Bradley’s results show that irrelevant values of the test statistic adversely affect the power of the test for small sample sizes with the effects of irrelevance virtually negligible for large sample sizes. This result is intuitive since one would expect that the sample variance would approach the true variance at large sample sizes.

3 3 Sign Detector For a detector to be classified as nonparametric or distribution-free, its false alarm rate must remain constant for a broad class of input noise distributions. A detector whose test statistic uses the actual amplitudes of input observations is not likely to have a constant false alarm rate for a small number of observations. This is clear from intuition and from the previous discussion on parametric detectors and the Student’s t test. The use of the input observation amplitudes may be avoided by discarding all amplitude information and using only polarity information, or by ranking the input data and using the relative ranks and polarities in the detection process. The simplest nonparametric detector would logically seem to be one that utilizes only the polarities of the input data observations. A detector which bases its decision on the signs of the input data may be derived as follows. Let the input data consist of n independent and identically distributed samples of the input process denoted by x = {xl, x2, . . . , x,,}. The detection problem consists of deciding whether noise alone is present or noise plus a positive signal is present. If the probabilityp is defined by p = P{x,

> O}

= 1

- e(0)

(3.3-1)

48

3 One-Input Detectors

where F,(x,) is the cumulative probability distribution of the xi, the hypothesis and alternative to be tested may be stated as H,: p = 1/2, F otherwise unknown (3.3-2) K,: p > 1/2, Fotherwise unknown Since the hypothesis and alternative described above are composite, an optimum test cannot be found by a straightforward application of the Neyman-Pearson lemma. If, however, an optimum test of level a is found for testing the simple hypothesis F(xlH,) versus the simple alternative F(xlK,) and it can be shown that this test maintains a level < a for all densities under H,, then the test is also most powerful for testing H4 versus K.,. This is simply an application of the modification to the NeymanPearson lemma described in Section 2.6, where the least favorable distribution of the unknown parameter is found. To find the most powerful test of H4 versus K,, letf' be the conditional density of xi given that the input is positive andf- the conditional density given that the input is negative, f' = f ( x i l x i > 0); f - = f ( x i l x i < 0) (3.3-3) The f' and f - densities are illustrated in Figs. 3.3-la and 3.3-lb7 respectively.

b

a

Fig. 331. Sketches off' and f - .

A typical density under the alternative f(xilK4) can be written as f(XiIK4)

= Pf+ +(I - P)f-

(3.3-4)

To choose the density f(xilH,) which yields a most powerful test for all densities in H,, the procedure described in Section 2.6 is sometimes useful. Since H4 provides little information concerning f(xil H,), the density f ( x i l H 4 ) should provide as little help as possible for testing against f(xilK4);f ( x i l H 4 ) should therefore be as close tof(xilK4) as possible. Let (3.3-5) f(XiIH4) = 2 (f' + f - ) The density f ( x i l H 4 ) has zero median, and hence belongs to H,; and f(x,lH,) closely resembles f ( x i l K 4 ) since f ( x i l H 4 ) has the same shape as If(x,IK4) on the positive and negative axes.

49

3.3 Sign Detector

Since the xi are independent and identically distributed, the likelihood ratio may be written as (3.3-6) The ratio of the signal present density to the signal absent density in Eq. (3.3-6) has different values when the input is positive and when it is negative, for xi > 0

f (xiIK4) f (xilH4)

(3.3-7) for xi

<0

(3.3-8)

If n, = Z~,,u(x,)is used to denote the number of positive observations, the likelihood ratio may be written using Eqs. (3.3-7) and (3.3-8) (3.3-9)

L ( x ) = (2p)"' [2(1 - p)]"-"+

Eq. (3.3-9) may be simplified to yield L ( x ) = 2pn+(1 - p)"-"+

(3.3-10)

which may be compared to a threshold To to yield K4

(3.3-11)

2 p n +(1 - p)"-"+ 3 To H4

This may be written as

which simplifies to

[ &I"+:

< [2(1 -p)]-"TO=

T,

H4

Taking logarithms of both sides to the base p/( 1 - p) yields n

n+ =

2 .(xi) i- 1

4

>(

logp,(,-,,TI=

T

(3.3-12)

H4

Since it can be shown that the detector in Eq. (3.3-12) maintains a level 1957a; Lehmann, 19471, the most powerful detector for testing the nonparametric hypothesis H4 versus the

< a for all densities under H4 [Fraser,

3

50

One-Input Detectors

nonparametric alternative K4 is

T

i= 1

D ( x ) = 0, D ( x ) = 1,

accept H, accept K4

(3.3-13)

and is usually called the sign detector. The threshold T is determined by setting the false alarm rate a, which requires a knowledge of the density of the detector test statistic under the hypothesis H,. The function u ( x i ) can take on the value 0 with probability 1 - p or the value 1 with probabilityp. Since n , is the total number of the positive observations out of the n samples, it is clear that n , has a binomial distribution with parameters n a n d p [Hogg and Craig, 19701. The probability density of n + under the alternative K , is thus

(

k = n , = 0, 1,. . . , n

b ( n + ; n,p) = i ) p k ( l -

(3.3-14)

Since under the hypothesis H4, p = 1 - p = 3, the density of n + under H, is given by b ( n + ; n,

i)

=

(:)(

i ) ' , k = n + = 0, 1 , . . . , n

(3.3-15)

The expression for the false alarm rate a may be written utilizing Eq. (3.3-15) a

i (;)( f)'

k= T

(3.3- 16)

The threshold T is found by setting a equal to the desired false alarm rate a,, and finding the smallest integral value for T which satisfies Eq. (3.3-16). The inequality is necessary since n, has a discrete density and only certain

values of a may be exactly obtained. After the value for the threshold is determined, the probability of detection may be written utilizing Eq. (3.3-14) 1-

p

= k- T

(i)Pk(l -

(3.3- 17)

By inspection of Eq. (3.3-16), it can be seen that the false alarm rate a does not depend on any parameters which are a function of the background noise density; and a, as well as T, is therefore constant for all possible distributions under the hypothesis H,. Since by the Neyman-Pearson lemma the sign detector is the most powerful detector of level a for testing H , versus K,, and since a and T are constant for all distributions under H,, the sign detector given by Eq. (3.3-13) is the UMP test of level a for testing H , versus K,. Furthermore, the sign detector is also nonparametric for testing H , versus K,. This result is quite significant since it is one of the

3.4

Wilcoxon Detector

52

few problems in which an unrestricted optimum detector exists for a nonparametric hypothesis and alternative. Due to the general nature of the hypothesis H4 and the alternative K4, the sign detector exhibits several invariance properties under H4and K4.In particular, since the sign detector bases its test statistic only on the polarities of the input data, the false alarm and detection probabilities of the sign detector are invariant to a nonlinearity of the form x,: =

#(Xi)

where I)passes through the origin and is wholly contained in the first and third quadrants. This is because the nonlinearity I) does not change the signs of the input data xi, i = 1, 2, . . . , n, and therefore P {x,! > 0} = P { x i > O}. This invariance property also holds even if the nonlinearity varies with time. Although the sign detector is not optimum for all hypotheses which are a subclass of H4,it is nonparametric for all subclasses of H4and quite easy to implement. The ARE of the sign detector with respect to the optimum parametric detector for Gaussian noise is 6476, when the noise is actually Gaussian; however, if the background noise is non-Gaussian, the sign detector may prove to be more efficient than the parametric detector. These statements are proven in Chapter 4. 3.4

Wilcoxon Detector

In the previous section, the sign detector was shown to be uniformly most powerful, as well as nonparametric, for testing the hypothesis H4 versus the alternative K.,. The noise density under the hypothesis H4 was very general, the only assumption being that the noise density had zero median. If, however, the noise density is assumed to be symmetric or even, f ( x ) = f (- x ) , the sign detector does not directly use this symmetry property, and a more efficient nonparametric detector may exist for testing the new hypothesis. In particular, if the noise density is symmetric and a positive signal is received, not only will there be more positive samples than negative samples, on which the sign detector bases its decision, but the positive samples will probably have larger absolute values than the negative samples. The amplitude data that are discarded by the sign detector are utilized by detectors that process both polarity and rank information. Rank detectors are based on a portion of elementary statistics called order statistics or rank statistics. The individual observations of the input process xi are stored until all n samples have been taken, and the observa-

52

3 One-Input Detectors

tions are then ranked in order of increasing absolute value,

h,l < I X k J <

*

*

*

< IxkJ

(3.4-1)

Detectors which use this rank information have test statistics of the general form n

S

=

2 v(Ri)u(xi)

i- 1

where R, is the rank of the ith observation, v(.) is some function of the ranks, and u ( x i ) is the unit step function previously defined. The function q(.) can be chosen in many different ways, each producing a different test. Perhaps the simplest and most obvious way to choose q(*) is to let v ( R i ) = Ri, the rank of the ith observation. To simplify notation, let Ri = j so that the resulting detector may be written (3.4-2)

where

d,=

0

for x4

j

for x$

<0 >0

(3.4-3)

The detector defined by Eqs. (3.4-2) and (3.4-3) is usually called the Wilcoxon detector [Wilcoxon, 19451. The question quickly arises as to how to rank two or more observations that have the same absolute value. Since in most cases the cumulative distribution function of the noise is assumed to be continuous, the probability of two observations having the same absolute value is theoretically zero. In practical applications, however, this possibility must @;I dealt with. There are many possible solutions, such as randomization and ra
K,: A CDF G ( x i ) exists such that G ( x i ) E H , and G(xi)

< F ( x i ) for all xi, and G ( x i ) < F ( x i ) for some xi.

The hypothesis is simply a statement that the noise density is symmetric about a zero mean or median, and K, is just a statement of a general

3.4

53

Wilcoxon Detector

positive shift of the median. Since under the hypothesis H , the noise density is symmetric, all possible partitions of the sample ranks are equiprobable. Assuming n samples of the input have been taken, each sample may be positive, in which case d, equals its rank i, or negative, in which case 4 equals zero; thus there are 2" possible sets of values for Zl,,4. The threshold T for the Wilcoxon detector in Eq. (3.4-2) is determined by setting the false alarm probability a,which can only take on certain discrete values equal to an integer from 2" to 1 times 2-". To illustrate the method for determining the threshold, let m be an integer such that 1 < m < 2". For a false alarm probability of m/2", the threshold T is set such that only m of the 2" possible values of Zl-ld,. exceed T. As the number of observations n becomes large, the statistic X7-14 is approximately Gaussian with mean p,, = n(n 1)/4 and variance u,' = n(n + 1)(2n + 1)/24 [Hajek, 19691.' For this case the threshold T can be determined from

+

@ [ ( T-

&)/.,I

= 1

-a

(3.4-4)

where @[ -1 is the CDF of a zero mean Gaussian random variable with unit variance. By setting a = ao,the desired false alarm rate, tables may be used to find the value of T that satisfies Eq. (3.4-4). has mean p,, = n(n + 1)/4 and variance That the test statistic uf = n(n + 1)(2n + 1)/24 for large n can be shown as follows. The expected value will be computed first. Let S = Zl-14 and then straightforwardly,

=

i-1

{i j-1

iP{Rj = i}E{u(x4)}

(3.4-5)

where Rj is the rank of thejth sample. Since the noise density under the hypothesis H , is symmetric with respect to the orign,

Using this result, Eq. (3.4-5) can now be written as E { S }=$

i{ i

i=l

iP{Rj= i}

(3.4-7)

j-1

'Hajek gives a proof of the normality of the Wilcoxon test statistic for large R. The proof is not repeated here due to its prohibitive length.

54

3

One-Input Detectors

In order to determine P { Rj = i } , note that the total possible number of permutations of the ranks of the observations is n!; similarly, the total possible number of permutations of the ranks given that the rank of thejth observation is i equals (n - l)!. The desired probability can now be found as ( n - l)! 1 P{Rj = i } = =(3.4-8) n! n Substituting this probability into Eq. (3.4-7), 3.4-9) Since it is well known that Z ; - l i = n(n the expected value becomes

+ 1)/2 [Selby and Girling, 19651,

(3.4- 10) E . { S } = $ [ a ( . + l)] = CL, which is the mean of the Wilcoxon test statistic. To calculate the variance of the test statistic, write dias iu(x,) and use the usual expression for the variance

(3.4-1 1) The quantity inside the brackets in Eq. (3.4-11) can be expanded remembering that there is only one set of ranks, which means that 2

(

~ i= 1

i =) i i 2 i- I

(3.4- 12)

and therefore the variance of the test statistic becomes i- 1

= E ( + Ei = i 1. )

(3.4-13) Using Eq. (3.4-8) to evaluate this expectation, Eq. (3.4-13) may be written as (3.4- 14)

55

Wilcoxon Detector

3.4

Since Z ; - l i 2 = n(n + 1)(2n + 1)/6 [Selby and Girling, 19651, the variance of the Wilcoxon detector test statistic is var{S) = &n(n

+ 1)(2n + 1)

(3.4-15)

as was to be proven. Although the Wilcoxon test statistic is easily implemented utilizing a digital computer, 2 i - l d k can be put into a form which is more analogous to an engineering situation. Consider the n(n + 1)/2 pairwise sums xi xi with i 4 j [Lehmann, 19591. The Wilcoxon rank sum equals the number of these pairwise sums that are positive; in other words

+

n

n

S =

n

2 dk= 2 2 .(xi + xi)

k-1

(3.4- 16)

i=l j-i

Example 3.4-1 Let x1 = -3, x2 = 2, and x3 = - 1. For this problem n = 3 and the ranks of xl, x2, and x j are 3, 2, and 1, respectively. The Wilcoxon test statistic equals the sum of the ranks of the positive observations, 2:,,dk = 2. To simplify its evaluation, the pairwise sum may be tabulated as follows: i = 1, i = 1, i = 1, i-2, i = 2, i = 3,

j = 1: j = 2:

j = 3: j=2: j = 3: j = 3:

u(xI u(xI u(xI u(x2 u(x2 u(x3

+ x,) = 0 + x2) = 0 + x3) = 0

+ x2) = 1

+ x3) = 1

+ x 3 ) =0 2=

3

3

2 2 #(Xi + Xi)

;==I j - i

Therefore, 3

2

k- 1

dk =

3

3

2 2 .(xi + x,)

=2

i - 1 j-i

The function on the right-hand side of Eq. (3.4-16) can be implemented as a digital system that processes the n input pulses x = { xI,x2, . . . , x,} and products a stair-step dc voltage level at the output. Like the sign detector, the Wilcoxon detector possesses some very important invariance properties. The false alarm rate a of the Wilcoxon detector remains constant even when a nonlinearity of the form xi = $ ( x i ) ,

i = 1, 2,

. . . , rl

(3.4- 17)

56

3 One-Input Detectors

perturbs the input data sequence, where \c, is continuous, odd, and strictly increasing. The transformation preserves the sign of each observation since \c, is odd. Since 1\c,1 is a continuous, strictly increasing function of 1x1, it leaves the order of the absolute values unchanged, and hence the ranks remain the same. It is clear that the false alarm rate is not affected by symmetrical nonlinearities which transform the input under the hypothesis H, when the observations are identically distributed; but what happens when the samples are not identically distributed due to the nonlinearity varying with time or some equivalent phenomenon? The answer is that the false alarm rate remains constant for the Wilcoxon detector, and in fact, for every symmetric detector based on the ranks of observations. The above fact may be stated in the form of a lemma.

Lemma 3.4-1 Let D ( x , , . . . , x,,) be symmetric in its n variables and such that Q D ( F ) = EF{D(xlr...,x,,)} = a (3.4- 18) when the xi's are a sample from any continuous distribution F which is symmetric with respect to the origin. Then, (3.4- 19) E { D ( x , , .. .,X")} = a if the joint distribution of the xi's is unchanged under the 2" transformations x i = + x l , . . . , x; = + x n .

Proof: D ( x , , . . . , x,,) can take on values 0 or 1 and is the expression for a detector which has as its input the n samples xi, i = 1, . . . , n. Equation (3.4-18) is given, and it is desired to prove that Eq. (3.4-19) holds if the joint density of the observations is invariant under the transformations x i = + x l , . . . , x; = +x,. The discrete definition of expected value is given by E { g(x)) = 2 g(xm)P {x = m

Xm}

(3.4-20)

The probability of each point in the joint density of the xi's after the transformation is 1/(2" n!), considering each xi and both plus and minus signs. Using this probability density and Eq. (3.4-20) in Eq. (3.4-18),

-

where the outer summation extends over all n! permutations ( j , , . . . ,in), and the inner summation over all 2" possible allocations of the plus and minus signs. If the detector is symmetric (invariant under all n! permutations of the input observations), the outer summation will cancel with the

3.5 Fisher-Yates, Normal Scores, or c1 Test

57

n! in the denominator, and Eq. (3.4-21) becomes D ( k x , , . . . , & X") = a (3.4-22) 2" Suppose now that the joint density of the xi's is invariant under the 2" transformations stated in the lemma. The conditional probability of any sign combination of the input samples xl, . . . , x,,, given the magnitudes IxII,. . . , Ix,,~, is simply 1/2". Equation (3.4-22) is now seen to be equivalent to the equation (3.4-23) E { D ( x , , . * . X")IIXII, * * I."l} =a Since F is any continuous distribution which is symmetric with respect to the origin, Eq. (3.4-23) may be written as (3.4- 19) E { D ( x , , . . . , X")} = a

x

9

a

,

which proves the lemma [Lehmann, 19591. The Wilcoxon detector is therefore nonparametric even if each observation taken has a different density function as long as the joint density of the input observations is symmetric with respect to the origin. If each of the input samples is assumed to be independent, the Wilcoxon detector, as well as any other symmetric rank detector, is nonparametric for the hypothesis H6: J ; ( x i )= J ; ( - x i ) , i = 1, 2 , . . . , n

where none of thex's are assumed to be equal. Although the invariance of the Wilcoxon detector to certain nonlinearities is of great practical importance, the Wilcoxon test is also a very efficient detector. When compared to the Student's t test or the optimum parametric detector, the Wilcoxon detector has an ARE of 95% for a Gaussian noise density, and furthermore, has an ARE of at least 86.4% for any choice of background noise. These facts are discussed in more detail in Chapter 4.

3 5 Fisher-Yates, Normal Scores, or c1 Test There are many modifications and generalizations of the one-sample Wilcoxon test. One modification has the following form. Assume n independent and identically distributed observations x,, x2, . . . ,xn have been obtained. The observed data are ranked by ordering the absolute values of the xi such that

58

3

One-Input Detectors

Let these observations be replaced by the constant expected values of the observations having corresponding size ranks in a sample of n observations from a standardized Gaussian distribution with zero mean and unit variance. The ordered sample from the Gaussian distribution will be denoted by (3.5-1) The detector for this test takes the form given by Eq. (3.4-2) namely,

i d i<>{TT

* *

D(x) = O D(x) = 1

(3.4-2)

In this case, however, the 4. have the form,

4. =

0, if x,

<0

E { z , } , ifx,

>0

(3.5-2)

where the zi are the ordered values of lykil. Since they’s are a sample from a Gaussian distribution with zero mean and unit variance, the zi’s are an ordered sample of size n from a distribution with density (3.5-3) The detector defined by Eqs. (3.4-2) and (3.5-2) is the normal scores test, sometimes called the Fisher-Yates test or the c, test [Bradley, 1968; Lehmann, 19591. The normal scores test is locally most powerful among tests based on the signed ranks for testing the hypothesis H,: f ( x ) is Gaussian with zero mean and variance u2 versus the alternative K,: f ( x ) is Gaussian with mean p

(3.5-4)

> 0 and variance u2

To prove this, let A = {rl, r2, . . . , r,,} be the ranks of the input data set x = { xl, x2, . . . , x,,}. Considering the ranks only, there are n! possible permutations of the ri, i = 1, 2, . . . , n. Sincef(x) is symmetric under H,, each of the permutations of the ri are equally probable, and therefore the probability density of the ri, denoted byf(RIH,), equals l/n!. To obtain the most powerful test for H, versus K , utilizing the Neyman-Pearson fundamental lemma, the deasity function, f ( R I K , ) of the ranks under the alternative must be determined. Using x to designate the specific input data set and using a dummy variable w in the density, f(R IK,) may be expressed as f(RIK7) =Jf(WIK7) dw

(3.5-5)

3.5

59

Fisher-Yates, Normal Scores, or c , Test

Transforming the density under the integral sign into the density of the order statistics and properly adjusting the limits, Eq. (3.5-5) becomes f(RIK7) =

5

JJ

0

*

< lXkll <

.

J

*

f ( w ~ l K 7~)

W R

< 1x41

' ' '

(3.5-6)

--

where f(wRIK7)is the density of the order statistics lxkll < Ixk21 < * < ) x h ) under the alternative K,. Multiplying and dividing under the integral sign in Eq. (3.5-6) by f ( w R I H J ,yields

0

< I X k J < . < 1x41 '

(3.5-7)

'

Regressing for a moment to consider the densities under H , and K,, it is clear that 1

-exp[ G

(x -

d2

2u2

U

]

1

-exp[ -

=

G

1

exp(-f(x-l)2) U

=-

1

mi

exp( -

+

+ p2)

2u2

U

differs from

v/2n

(2- 2pX

(x2

-21-1 x U

+ 2)) U2

only by constant multiplying factors in the exponent equal to 1/u and l/a2, and by a scale factor l/u. Instead of using the exact densities under H, and K,, it is equivalent then to utilize a Gaussian density with zero mean and unit variance for the hypothesis and a Gaussian density with mean 6 = p / u and unit variance for the alternative. Substituting these densities into Eq. (3.5-7) and simplifying,

60

3

One-Input Detectors

Equation (3.5-8) may be written as

which is the probability density function of the ranks under the alternative K,, and where the Iuli are the order statistics of a sample of absolute values from a Gaussian distribution with zero mean and unit variance. To find the most powerful test under the fundamental lemma, the likelihood ratio must be formed,

n!

(3.5- 10)

L ( R ) in Eq. (3.5-10) is therefore the test statistic for the most powerful test of H , versus K,. For weak signals, S is small, and Eq. (3.5-10) may be approximated by L(R)= E( 1

+ SXl0li) = 1 + SXE{IWli}

(3.5-1 1)

An equivalent statistic to that given by Eq. (3.5-1 1) is Z:,,E { Iuli} which is the test statistic of the normal scores test in Eqs. (3.4-2) and (3.5-2). The normal scores test is thus locally most powerful among all signed rank tests for detecting weak signals in Gaussian noise [Fraser, 1957b; Hoeffding, 19511. The calculation of the normal scores test statistic is illustrated by the following example.

Example 3.5-1 Given the 12 observations listed below, it is desired to calculate the value of the normal scores test statistic given by Zl- ,di where di is specified by Eq. (3.5-2). The observations and their ranks are: Observation Rank(i)

-3.6 0.8 -2.1 3.7 6.4 -2.5 5 6 2 4 7 12

5.1 4.2 11 9

-1.1 3

-0.5 3.9 4.8 1 8 10

The E { z i } for the ranks i = 1, 2, . . . , 12 can be found from Table B-5 to be i

1

2

3

E{z~) 1.63 - 1.12 -0.79

4 5 -0.54 -0.31

6 7 8 9 10 11 12 -0.10 0.10 0.31 0.54 0.79 1.12 1.63

We associate with each observation according to its rank a value of

3.6

61

Van der Waerden’s Test

E { z i } ; that is, the observation 5.1 with rank 11 corresponds to E { z I 1 } = 1.12 after the expected normal scores transformation. To find the value of the normal scores test statistic, we simply sum the E {zi} for each positive observation to obtain SNs= 1.12 + 0.54 - 1.12 + 0.10 + 1.63 + 0.31 + 0.79 = 3.37 The normal scores test has an ARE of 1 compared to the Student’s t test under the null hypothesis that satisfies all of the latter’s assumptions. The efficiency of the normal scores test is offset by the fact that the test is quite cumbersome for practical use since not only must the observations be ranked, but the expected normal scores transformation must be performed, which requires the use of a table of stored values. The normal scores test is quite important theoretically, however, since it combines the generality of nonparametric tests with the efficiency of parametric testing procedures.

3.6 Van der Waerden’s Test Van der Waerden proposed a test that is closely related to the normal scores test [Van der Waerden, 1952, 19531. His test utilizes the inverse normal transformation and is called an inverse normal scores test. A discussion of the inverse normal transformation and how it is used in the test statistic is given below. The input to the detector is assumed to be a set of n independent and identically distributed observations x = { x , , x2, . . . , x,,} drawn from a continuous but otherwise unknown distribution. The xi are stored until all n samples have been taken, and then the observations are ranked according to their absolute values,

As can be seen, xk, is the ith smallest of the input observations, and the expected proportion of the x population that lies below xk, is i / ( n 1).

+

This fact is not obvious but it is a result of a basic property of order statistics:

The distribution of the area under the density function between any two ordered observations is independent of the form of the density function [Mood and Graybill, 19631. To prove this, the joint density of the area between ordered samples will be derived and shown to be independent of the density of the observations

f (XI.

62

One-Input Detectors

3

Let x be a continuous random variable with density f ( x ) for - 00 < x A random sample of size n is taken from the density f ( x ) and arranged by order of magnitude

< 00.

bk,I

< bk,l <

*

* *

< Ix/J

To simplify notation, let yi = Ixkl. The joint density of the yi is given by g ( ~ 1 3 ~.2* ,

9Yn)

= n ! f ( ~ l ) f ( ~* 2* )* f ( Y n )

for 0 < y ,

<

*

*

-

< 00


(3.6-1)

where f ( y i ) is f ( x ) [Mood and Graybill, 19631. Define ui as the area under f ( x ) which is less than the ith ordered observation yi, ui = F ( y i )

= J-”m f ( x ) dx,

i = 1,2,. . .,n

(3.6-2)

where F ( y i ) is the cumulative distribution function of f(y). The joint density of the ui may be found by using Eq. (3.6-2) to transform the density of the yi in Eq. (3.6-1). The transformation takes the general form

where the term J ( y , , y 2 , . . . , y n ) is the Jacobian. Using Eq. (3.6-2) to compute the Jacobian of the transformation yields

-

f(Yd

0

...

0

f(Y2)

0

0

...

0

... =f

f ( y n)

( ~ l ) f ( ~ *2 *)

*

f(rn) (3.6-4)

3.6

63

Van der Waerden’s Test

The density in Eq. (3.6-5) can now be used to find the expected proportion of the x population that lies below the ith-ordered observation, which is just equal to

which yields i E{ui} = n + l

(3.6-7)

Example 3.6-1 Let n = 3 and i = 2. Equation (3.6-6) becomes E { u,} = / 1 / u 3 ~ u 2 3 ! u du, , du, du, 0

0

= 3 ! / 1 / u 3 i u 2 u 2 du, du, du, 0

0

= 3!/I/”u: 0

=3 ! i

0

1

du, du,

u; 7 du,=

[

1 3! 3 ! 4 ] = i

(3.6-8)

Substituting the values for i and n into Eq. (3.6-7) yields E { u , } = 2/ (3 + 1) = 1/2 which is the same as Eq. (3.6-8). Define p = F ( x i ) = i / ( n + 1) and let y be a Gaussian distributed random variable with zero mean and unit variance. As previously defined, let @(y)denote the CDF of y , and define the value of y whose CDF equals p as $., From the above discussion it is clear that

@ [ I p ]=

i

p =n+l

To obtain an expression for the population quantile inverse of Eq. (3.6-9) which yields 1

n + l

(3.6-9)

&, simply take

the

(3.6-10)

which is the inverse normal transformation. According to Eq. (3.6- lo), therefore, the inverse normal transformation is performed by replacing the ith smallest of the n-ordered observations with the population quantile that the ith-ordered observation would represent if the sampled distribution were Gaussian with zero mean and unit variance.

64

3

One-Input Detectors

The Van der Waerden detector which utilizes the inverse normal transformation has the general form given by Eq. (3.4-2)

2 di( <> TT

i-1

The

*

D ( x )= O

(3.4-2)

D(x)= 1

4 for this test have the values (3.6-1 1)

where xk, is the input observation that has the ith smallest absolute value. The following example illustrates the calculation of the test statistic. Example 3.6-2 We wish to calculate the value of Van der Waerden's test statistic, 27- ,d, where the 4 are given by Eq. (3.6-1 I), for the n = 12 observations in Example 3.5-1. For each possible rank i = 1, 2, . . . , 12, it is necessary to compute p = i/(n 1). The positive population quantiles, lp,are then found from Table B-2 for eachp. The negative quantiles are assigned by symmetry. Following the above procedure and rounding the population quantiles to two decimal places yields

+

i p = i/(n

+ 1)

SP

i

p = i/(n

sb

+ 1)

1 2 3 4 5 6 0.0769 0.1538 0.2308 0.3077 0.3846 0.46 15 - 1.43 - 1.02 - 0.74 - 0.50 - 0.29 - 0.10 7 0.5385 0.10

8 0.6154 0.29

9 0.6923 0.50

10 0.7692 0.74

11 0.8462 1.02

12 0.9231 1.43

A correspondence between the population quantiles and the observations can be established by using the ranks (i) of the observations listed in Example 3.5-1. The value of Van der Waerden's statistic is determined by summing the population quantiles which correspond to positive observations to produce S ,, = 1.02 + 0.5 - 1.02 + 0.10 + 1.43 + 0.29 0.74 = 3.06

+

The inverse normal transformation is closely related to the expected normal scores transformation utilized in the normal scores test discussed in the previous section. Of course, this is not too surprising since both transformations are based on the zero mean Gaussian distribution with unit variance. For a finite number of samples, it is clear from looking at the two test statistics in Eqs. (3.5-2) and (3.6-11) that the two transforma-

3.7 Spearman Rho Detector

65

tions are not equivalent; however, as the number of observations becomes large the two transformations approach each other, and at infinite n they are equivalent. This fact may be illustrated by letting y i denote the ith smallest of n observations which make up a random sample taken from a zero mean Gaussian distribution with unit variance. The expected value of yi will be denoted by E { yi}.As proven earlier, the expected proportion of the y population less than or equal to yi is i / ( n + I), and l, is the exact population quantile of the zero mean, unit variance Gaussian distribution for this proportion. As the number of observations approaches infinity, the value of y j approaches E { y ; } , and simultaneusly yi approaches $., At 1 = @ - ’ [ i / ( n + l)] and the two infinite n, therefore, y j = E {yi}= , transformations are equivalent. Van der Waerden’s test has essentially the same advantages and disadvantages as the normal scores or Fisher-Yates test. Van der Waerden’s test is difficult to implement practically, due to the necessity of the transformation; however, like the normal scores test, Van der Waerden’s test has an ARE of 1 with respect to the Student’s t test under assumptions satisfying the latter’s null hypothesis. Van der Waerden’s test does have one advantage over the Fisher-Yates test described in the previous section. The inverse normal transformation requires cumulative probability tables of the zero mean Gaussian distribution with unit variance which are generally quite extensive and readily available; whereas the expected normal scores transformation requires tables of E { yi} which are usually less common and less complete. The difficulty of implementing the required transformations, however, limits the practical usefulness of both detectors [Bradley, 19681.

3.7 Spearman Rho Detector In his classic book Rank Correlation Methodr, Kendall discusses in detail two rank correlation coefficients, the Kendall tau ( 7 ) coefficient and the Spearman rho (p) coefficient. In Chapter 2 of his book, Kendall introduces a generalized correlation coefficient r and shows that 7 , p, and the common product-moment correlation of two random variables are all particular cases of the general correlation coefficient [Kendall, 19621. The Kendall 7 and the Spearman p coefficients were originally used to test for the independence of two sets of ranked data, and both statistics can be written in many different forms. Although the basic forms of the 7 and p coefficients are generally not used when testing hypotheses, equivalent statistics are used and have proven quite valuable when applied to various detection problems.

66

3

One-Input Detectors

Consider the case where two separate sets of n observations are taken, each set taken from a different population. Let one set of data be denoted by xi, i = 1, 2, . . . , n, and the other set by yi, i = 1, 2, . . . , n. Each set of data is ranked individually with Rxi the ranks of the x-observations, and R,: the ranks of the y-observations. Using this notation, one basic form of the Spearman p coefficient that can be used to test independence of the ranks of the two sets of data is

12

n+l

If the two sets of observations are now paired x 1 withy,, x , withy,, and so on, and the pairs of observations are arranged according to the magnitudes of the first variate in the pair, then p depends only on the new ranks R, of the second variate in the pair. The expression for p may now be rewritten as p =

=

=

-

12 n -n

x ;=I

( j -

-

-n) +( Rl i 2

”)

[ 12[ n3 - n

)1+ n(

iRi - 2n(

+ ( n ~+ l )

( -n) R+ il

12 [ i R i - ( n7+ l) i n3 - n i l l n3 - n

2

1 ] ’

,’]

$liRi - n(

(3.7-1)

The coefficient p reduces to this last form since

5 (F ) i = s l ( F

~

) R i = n( n + l

)

1

i- 1

The last term in Eq. (3.7-1) is constant, and hence an equivalent statistic for the Spearman p coefficient is n

2 iRi

(3.7-2)

i-I

The equivalent statistic given by Eq. (3.7-2)can be used as the test statistic for a one-input detector or a two-input detector. Only the one-input application is considered in this chapter. To apply the Spearman p test, n samples are taken from the input process x = { xI,x,, . . . , x,} and ranked in the order of increasing magnitudes bk,l

< bk,I <

*

*

*

< 1xk.J

3.7 Spearman Rho Detector

67

The one-input detector based on the statistic in Eq. (3.7-2) takes the form i= 1

T

=$ =$

D ( x ) = 0, D ( x ) = 1,

accept H accept K

(3.7-3)

where

0 for xi < 0 (3.7-4) iRi for xi > 0 where Ri is the rank of the ith observation and Eqs. (3.7-3) and (3.7-4) are called the Spearman rho ( p ) detector. The threshold T is determined as usual by fixing the false alarm probability a. Consider the application of the Spearman p detector to the problem of testing the hypothesis of symmetry H, versus the alternative of a general positive shift K,,

4Ri =

H,: A x ) = !(-XI K,: A CDF G ( x ) exists such that G ( x ) E H ,

and G ( x ) < F ( x ) for all x and G ( x ) < F ( x ) for some x Under the hypothesis H,, the test statistic ZiR, can take on 2" n! possible values, each of which are equally likely. The probability density under H, is therefore

-

-

For a false alarm probability a,, = m/2" n ! , where m is any number from 1 to 2" * n!, the threshold T in Eq. (3.7-3) is chosen such that only m of the 2" n! possible values for the test statistic exceed T under the hypothesis. Since the false alarm probability does not depend on any parameters of the noise density and is clearly invariant for all possible distributions under H,, the Spearman p detector is nonparametric for testing H , versus K,. Like the Wilcoxon detector, the false alarm rate of the Spearman p detector remains constant even when the input data are transformed by a nonlinearity + ( x i ) as long as is continuous, odd, and strictly increasing. This follows from Lemma 1 proven in Section 3.4 on the Wilcoxon detector and the fact that the Spearman p detector is a symmetric rank test under H,. The Spearman p detector defined by Eqs. (3.7-3) and (3.7-4) is a linear rank detector and is quite amenable to digital implementation. The relative efficiency of the Spearman p test is high when compared to several other tests for a variety of sampled populations. For these reasons the Spearman p detector is quite important in hypothesis testing and has many practical applications.

-

+

68

3

One-Input Detectors

3.8 Kendall Tau Detector

The Kendall r coefficient is closely related to the Spearman p coefficient just discussed. Although the tests are not mathematically equivalent for finite n, they are asymptotically equivalent as n approaches infinity. The two tests can be compared on several different points; however, the detector based on the Kendall r coefficient has the advantage that its test statistic may be computed recursively and is thus much more easily handled mathematically than the Spearman p test statistic. This fact will be discussed in more detail later. The original rank correlation coefficient proposed by Kendall is based on two sets of data samples, say x and y, each set with n components x = {xl, x2, . . . , x,,} and y = { y , , y 2 , . . . ,y,} [Kendall, 19621. Both sets of data are ranked separately with the x’s having ranks R , and the y’s having ranks I?,, . The samples are arranged in the order in which they were taken { xl, x 2 , . . , x,,} and { y , , y2, . . . ,y,,}, and then each observation is replaced with its rank. This procedure yields two sets of ranks of the form { R , , , RXl,. . . , Rxm}and R,,,, . . . , R,,}. Consider the first pair of x-observation rank members Rx,Rx,. If the two ranks are in increasing order from left to right, a + 1 is scored for the pair, but if they are in decreasing order, a - 1 is scored for the pair. The same procedure is The rank score of the x-rank followed for the first pair of y ranks pair is then multiplied times they-rank pair score which will yield a + 1 or a - 1. If the ranks are in the same order, the resulting score will be + 1, but if the ranks are in different orders, the score will be - 1. The rank R,, is next paired with R,, and Ry, is paired with R,,,. The procedure continues until R,, and R,,, have been compared to every following rank. R,, and Ryl are next compared to every following rank, assigning scores as outlined above. After the ( n / 2 ) ( n - 1) possible pairs of ranks have been scored, the scores are all added together to equal a value denoted by S. If P and Q are the total number of positive and negative scores, respectively, S = P - Q, and the Kendall tau (7) coefficient may be written as

{Is.,,

r =

S

f n ( n - 1)

-

p-4? t n ( n - 1)

(3.8- 1)

which is the basic form given by Kendall. Since the maximum and minimum values of S are 4 n(n - 1) and - 4 n ( n - l), it can be seen from Eq. (3.8-1) that - 1 < 7 < 1, - 1 indicating perfect negative correlation and 1, perfect positive correlation. To better illustrate the calculation of the Kendall r coefficient, a simple example will be considered.

+

3.8 Kendall Tau Detector

69

Example 3.8-1 Suppose two sets of observations are received x = { x I ,x2, x 3 } and y = { y I , y z , y 3 }and , have the following ranks:

Observation: x1 x 2 x 3 YI Y2 Y3 Rank: 2 3 1 3 1 2 Consider the first pair of x-ranks x I x 2 .Their ranks 2, 3 occur in increasing order, and therefore a score of + 1 will be assigned to this pair. Consider now the first pair of y-ranks y I y 2 .Their ranks occur in decreasing order, and therefore a score of - 1 will be assigned to this pair. The score of the first x-y observation pair is thus equal to (+ 1)( - 1) = - 1. The procedure is continued by pairing the first and third x-y observations and then the second and third x-y observations. The table below lists each x-y observation pair and their respective scores. Score

-1

= actual score

The total possible score is (n/2)(n - 1) = $(3 - 1) = 3. The Kendall r coefficient may thus be evaluated as actual score = -- -1 - -0.333 7 = maximum possible score 3 which indicates a significant amount of negative correlation. It is obvious, however, that in order to have a meaningful measure of rank correlation, a much larger number of observations is required. Since the purpose of this example was simply to illustrate the calculation of the Kendall 7 coefficient, only a small number of samples was used. The expression for the Kendall 7 coefficient can also be explicitly expressed in terms of ranks of observations. Let two sets of observations be received, namely, x = {xl, . . . , x,,} and y = { y l , . . . ,y,,},and denote the ranks of the x and y observations as R , = { R x , ,. . . , Rxm}and R , = { %,, . . . , R,,}, respectively. Using these ranks the Kendall 7 coefficient may be expressed as

i#j

3

70

where sgn

(

0

)

One-Input Detectors

is defined by sgn(t)

1, t > O [-I, t < O The Kendall 7 coefficient in Eq. (3.8-2) is a linear function of the statistic =

(3.8-3)

and therefore, T' can be used as the test statistic when testing hypotheses. The Kendall T test statistics considered thus far require two different sets of observations and are therefore two-sample detectors. To obtain a one-input detector that is directly related to Kendall's basic 7 correlation coefficient, let n observations of an input process be obtained and be denoted by x = (xI,x2,. . . , x,,}. Similar to the basic Kendall 7 coefficient, the rank of each observation is compared to the rank of every succeeding observation and the number of times that R., > R, is totaled. The test statistic may be written as follows n-1

K, =

n

X 2

-

4,)

(3.8-4)

i=l j=i+l

where u( .) is the unit step function defined earlier, and R., is the rank of the jth observation. Another form of K, does not utilize ranks explicitly, but compares each observation with every following observation and sums the number of times that xi > xi.This procedure may be written as the statistic

x

n-1

K,, =

n

2

U ( X ~-

xi)

(3.8-5)

i=l j-i+l

This statistic K,. is clearly amenable to a recursive form. For instance, if after n samples have been received and K,. has been calculated, another sample is taken, the statistic based on all n + 1 observations may be calculated using n

(3.8-6) i= 1

where K,. is given by Eq. (3.8-5). In order for the test statistic K, or K,. to be of any utility, a specific rank order of the input samples must be expected; that is, the detector that uses the test statistic K, or K,. decides if the rank order of the input observations is the same, within reasonable limits, as a specific rank order that the system designer wishes to detect. This requirement may seem to limit the practical utility of the K,(K,,) detector, but there are cases where a specific amplitude modulated sequence may be received. A particular

3.9 Summary

71

pulsed radar application is discussed in Chapter 9. The Kendall 7 coefficient as well as the statistics K, and K,. are nonlinear, and therefore special consideration is necessary when calculating their statistical properties. The Spearman p coefficient discussed previously is the projection of the Kendall 7 coefficient into the family of linear rank statistics, and this fact may be used to circumvent some of the computational problems [Hajek and Sidak, 19671. The detector based on the Kendall 7 coefficient is very efficient when compared to the Student’s t test under the assumption of Gaussian noise. This fact, along with the properties that the statistic may be computed recursively and easily implemented digitally, makes the Kendall 7 detector of great practical importance.

3.9 Summary In this chapter the most powerful (parametric) detectors for testing for the presence of a known signal and an unknown positive signal in zero mean, Gaussian noise with known variance were derived. The variance or noise power was then assumed to be unknown, and the UMP unbiased detector for testing for a positive signal in zero mean, Gaussian noise was derived by finding maximum likelihood estimates for the mean and variance. It was then shown that the resulting detector, the Student’s t test, is both asymptotically nonparametric and consistent. The simplest nonparametric detector, the sign detector, was derived and shown to be the UMP detector for testing for the positive shift of the median of a distribution. The invariance of the sign detector for certain nonlinear operations on the input data was explained. The Wilcoxon detector was then presented and its mean and variance derived. An alternative form for the Wilcoxon detector test statistic was given and its equivalence to the basic form of the test statistic illustrated by an example. The invariance of the Wilcoxon detector as well as any symmetric detector to any nonlinear transformation of the input data which is continuous, odd, and strictly increasing was then proven. The normal scores test was next presented and proven to be the locally most powerful detector for testing for the mean of a Gaussian distribution with known variance. A closely related detector, Van der Waerden’s test, was then discussed, and the equivalence of the normal scores and Van der Waerden’s tests for an infinite number of observations was shown. The fact that both detectors may be cumbersome to apply due to the transformations required was pointed out.

72

3 One-Input Detectors

The last two detectors to be discussed were the nonparametric Spearman

p and Kendall7 detectors, which are both related to the common product-

moment correlation coefficient. Several different forms for the test statistic of each detector were given, and an example was utilized to illustrate the calculation of the Kendall 7 coefficient. It was also pointed out that the Spearman p coefficient is the projection of the nonlinear Kendall 7 coefficient into the family of linear rank statistics. Problems 1. Given the 20 observations listed below, it is desired to test the hypothesis H , against the alternative K , using the one-input Wilcoxon detector. The observations are: 1.4 6.2 - 4.4 6.5 7.7 4.1 8.1 2.4 - 0.2 1.1 3.0 7.1 4.9 4.3 - 0.4 0.6 - 0.1 5.4 5.2 3.9 The desired false alarm probability is a = 0.1. Hint: Use the Gaussian approximation to the distribution of the Wilcoxon detector test statistic. 2. Given a set of 10 independent, identically distributed observations of a random variable x with cumulative distribution function F ( x ) , it is desired to test the hypothesis

H: F ( 0 ) = 0.5 against the alternative K:

F ( 0 ) < 0.5

using the sign detector. Write an expression for the power function of the test and find the false alarm probability a if the threshold T = 8. Assume that F(0IK) = P [ x < OIK] = 0.1 and find the probability of detection. 3. Use the sign detector to test the hypothesis of a zero median against the alternative of a positive shift of the median for the three sets of observations given. Let a = 0.1 and do not randomize to obtain the exact a. (i) (ii) (iii) 4.

tions

1.7 0.8 - 0.2 1.3 - 1.8 2.2 9.3 1.4 - 1.2 3.3 6.4 2.1 - 1.2 1.1 0.0 3.1 2.6 - 0.6

1.4 - 0.6 - 1.1 1.9 0.6 0.4 8.8 6.1 - 0.9 - 2.1 2.4 1.4

Use the Wilcoxon detector to test H , versus K , given the 10 observa-

0.4 2.0 1.2 - 0.2 - 0.8 0.9 2.2 0.6 0.1 Let a = 0.1 and use the exact distribution of the test statistic.

1.3

Problems

73

5. Repeat Problem 4 using the Gaussian approximation. 6. For n = 12 observations from a Gaussian distribution with mean p = 32.3 and variance u2 = 1, it is desired to compare the performance of the sign detector and the optimum Neyman-Pearson parametric detector for testing the hypothesis

H: F(31) = 0.5

against the alternative K: F(31)

< 0.5

The desired significance level (Y is 0.08. Find the threshold T to insure the desired false alarm probability, the exact value of a, and the probability of detection 1 - p for both the sign detector and the optimum Neyman-Pearson detector for this problem. 7. For the 10 observations in Problem 4, compute the value of Van der Waerden’s test statistic. 8. Calculate the value of the normal scores test statistic for the 10 observations given in Problem 4. (Use the table of expected values of normal order statistics in Appendix B.) 9. Calculate the value of the Spearman p detector test statistic for the 10 observations in Problem 4. Discuss how the threshold of the detector can be selected to obtain a significance level a = 0.05 when testing H , against

4.

10. Investigate the accuracy of the Gaussian approximation when determining the threshold for (i) the sign test, and (ii) the Wilcoxon detector. Let (Y = 0.05 and consider the cases where n = 8, 10, and 12 observations are available. Assume the hypothesis and alternative to be tested are H , and K,.

CHAPTER

4

One-Input Detector Performance

4.1

Introduction

In this chapter the various one-input detectors discussed in Chapter 3 are compared using the concepts qf ARE and finite sample size relative efficiency and sample calculations of the ARE are given. Since ARE calculations for all detectors and various background noise densities clearly cannot be given here, calculations of the ARE of the sign detector and of the Wilcoxon detector with respect to the optimum parametric detector for Gaussian noise are presented as illustrations of ARE calculations for the two types of nonparametric detectors discussed in this book, those utilizing only polarity information and those utilizing both polarities and ranks. The ARE of the sign detector with respect to the linear detector, given by Eq. (3.2-6), when detecting a positive dc signal for various background noise densities is calculated in Section 4.2 using the basic definition of the ARE given in Section 2.7. In Section 4.3, the ARE of the Wilcoxon detector with respect to the linear detector when detecting a positive dc signal for various background noise densities is calculated utilizing the expression for the ARE in terms of the efficacies of the two detectors. It is also proven in Section 4.3 that the Wilcoxon detector is never less than 86.4% as efficient as the linear detector for large samples and for testing for a location shift regardless of the background noise. The asymptotic relative efficiencies of the six one-input nonparametric detectors developed in Chapter 3 with respect to the linear detector, when testing for a location shift for various background noise densities, are given in Section 4.4. In 74

4.2

Sign Detector

75

addition, the results of a study using a different asymptotic efficiency definition are also included in this section. Finite sample size performance is discussed in Section 4.5 and the results compared with that indicated by the asymptotic performance measures. 4.2

Sign Detector

The general detection problem that will be considered is the detection of a dc signal in background noise which has a symmetric but otherwise unknown probability density. The hypothesis and alternative can therefore be written as H s : p = 1/2 or p = 0, f symmetric but otherwise unknown (4.2-1) Ks: p > 1/2 or y > 0, f symmetric but otherwise unknown

where p is the mean off and p = P { x > 0} = 1 - F(0). To compute the ARE utilizing Eq. (2.7-1), the asymptotic false alarm and detection probabilities of the two detectors to be compared must be calculated, set equal to each other, and solved for the ratio of n1 to n2. Since the optimum detector for the present problem when the noise is Gaussian distributed is the linear detector given by Eq. (3.2-6), this detector will be used as the basis of comparison for calculating the ARE. In order to determine the asymptotic probability of detection for the linear detector, the distribution of the test statistic under the alternative K, must be ascertained. Letting n, be the number of samples of the input process, the linear detector may be written as

5

< TI * D , ( x ) = 0, accept H , (4.2-2) k-1 > Tl * D , ( x ) = 1, accept& where the xi, i = 1, 2, . . . , n, are the input samples. At present, the only xk(

assumption on the background noise density is that it is symmetric. If the input observations xi are assumed to be elements of a random sample, the distribution of the test statistic C",l,,x, as n, becomes large is asymptotically Gaussian with mean n l y and variance n,02 by virtue of the central limit theorem. Utilizing the expression for the probability of detection given by Eq. (2.2-6) and the asymptotic probability density of the detector test statistic, the asymptotic probability of detection may be written as

Making a change of variables with y = ( x - n1y ) / c u, the asymptotic

76

4

One-Input Detector Performance

probability of detection becomes

where the last step follows since the integrand is symmetric with respect to the origin. Equation (4.2-3) may clearly be written as (4.2-4) which is the asymptotic probability of detection for the linear detector. The asymptotic false alarm probability may be determined similarly by using the fact that the noise has zero mean under H , and by noting that if the xi are the elements of a random sample, then the test statistic has a limiting Gaussian distribution with zero mean and variance n1u2.Using Eq. (2.2-2), the asymptotic false alarm probability may be written as

Making the change of variables y = x / G u, Eq. (4.2-5) becomes

which is the asymptotic false alarm probability for the linear detector. An expression for the threshold may be obtained by solving for T , in Eq. (4.2-6),

which yields

T, =

6u a-'[ 1 - a&]

(4.2-7)

Substituting Eq. (4.2-7) for T I in Eq. (4.2-4) yields 1-

where

@-I(-)

pol = (D

is the inverse of

nlp - G u @ - l [ l

@[-I.

- aD,]

1

(4.2-8)

4.2

77

Sign Detector

The asymptotic false alarm and detection probabilities must now be calculated for the sign detector defined by Eq. (3.3-13). Let n2 denote the number of samples taken, and thus for the present detection problem, the sign detector may be written as

< T2 * > T, *

n2

i- 1

D,(x) = 0, accept H, D,(x) = 1 , accept K ,

(4.2-9)

Following the development in Section 3.3, let k equal Zy-,u(xi), the number of positive observations in the sample. Under the alternative K,, u ( x i ) can take on the value 1 with probability p , and therefore k = 2%lu(xi)is binomially distributed with parameters n, andp. The probability density of k under K, is therefore

which has a mean n,p and a variance n,p(l - p ) . As n, becomes large, the binomial density in Eq. (4.2-10) is asymptotically Gaussian with mean n2p and variance n,p (1 - p ) . The asymptotic probability of detection can now be found using the asymptotic probability density and Eq. (2.2-6) to get

which upon making the change of variables y becomes

(nzp

--I-, -

- TZ)/VW(I - p )

1 e-~'/2 (&

G

The asymptotic probability of detection for the sign detector under the alternative K,, is therefore (4.2-1 1)

Under the hypothesis H,, p = 1 - p = 1 / 2 and the binomial density of

78

One-Input Detector Performance

4

k becomes

(4.2-12)

As n2 becomes large, the density in Eq. (4.2-12) is asymptotically Gaussian with mean n2/2 and variance n2/4. Using this result in Eq. (2.2-2), the asymptotic false alarm rate may be written as

(4.2-13) which by making the change of variables z = ( x - n 2 / 2 ) / ( G /2) be-

which is the asymptotic false alarm rate of the sign detector. Solving for the threshold in Eq. (4.2-14) 1-

(YDz

=@

2T, - n2

or @-"l which yields

(YDJ

=

2T, - n,

T2=4(n2+V+q1

G -

(4.2-15)

(YD2])

Substituting Eq. (4.2-15) into the expression for the asymptotic probability of detection, Eq. (4.2-1l), yields

=@

2

which becomes after factoring

6 p -G

+ @-yl -

(YD,

79

Sign Detector

4.2

The asymptotic relative efficiency of the sign detector with respect to the linear detector can now be determined by equating Eqs. (4.2-8) and (4.2-16) and Eqs. (4.2-6) and (4.2-14) and solving for the ratio of n 1 to n2.

- P D , = - PO,,

(YD,= (YD,=

(Y

6(2p - 1) + W ' ( 1 - a) G C L -U

Factoring out

cp-yl - a) =

2

i

S

'

6 on the left-hand side,

and dividing both sides by

Solving for

G ( 2 p - 1) + cp-'(l - a)

1

6,

V G ,

U2

-n1 -- -

p2n2 r

-

"+2 2

i S Y ] i

m 1

-2

cp-yl - a)

+

(2p - l)a2

CLvm-

80

4

One-Input Detector Performance

Letting nl and nz approach infinity and using the definition of ARE given by Eq. (2.7-1), the ARE of the sign detector with respect to the linear detector is (4.2- 18) Ka+ Ha

as the alternative & approaches the hypothesis H,. The expression for the ARE in Eq. (4.2-18) is not very useful; however, by using the basic assumptions of this detection problem, namely that f(x) is symmetric, p > 1/2 (under the alternative &), and p is small (since K, + H8), a more informative expression may be obtained. From the definition of p, p = P{x, > 0} = 1 - F ( 0 ) = 1

-I f(x)dx 0

-m

wheref(x) is a symmetric density function with mean p and variance u2. Making a change of variablesy = x - p ,

(4.2-19) since f(y) is a zero mean symmetric density function with variance u2. Since p is small, p may be approximated by (4.2-20) P = t +Pf(O) Substituting this expression for p into Eq. (4.2-18),

(4.2-21) As K, approaches H,, p + 0 and the ARE of the sign detector with respect to the linear detector is E2,

, = 4u2f2(0)

(4.2-22)

for testing the hypothesis H8 versus the alternative K,. It should be noted that in this case the ARE is independent of the false alarm probability a. To obtain numerical results, a specific form for the noise density f must be assumed. Assuming that the noise is Gaussian with zero mean and

4.3

variance u2,

Wilcoxon Detector

81

f(0) = 1 / m u

Substituting this result into Eq. (4.2-22),

E2, = 4u2(1/ fiu)Z= 4u2(1/27m2) (4.2-23)

The sign detector is therefore asymptotically only 63.7% as efficient as the linear detector when the background noise.is Gaussian with zero mean and variance u2. This result is not surprising since the linear detector was shown to be optimum for detecting a positive dc signal when the noise is Gaussian. Assume now that the background noise has a double exponential or Laplace distribution, (4.2-24)

which when evaluated at the origin becomes

Substitutingf(0) into Eq. (4.2-22), (4.2-25)

which is the ARE of the sign detector with respect to the linear detector when the background noise has a density given by Eq. (4.2-24) [Thomas, 19701. It is clear then that although the sign detector is only 63.7% as efficient as the linear detector when the noise is Gaussian, the sign detector is twice as efficient as the linear detector when the noise has a double exponential density. This illustrates the point made previously that although a detector may be optimum for one set of design conditions, the “optimum” detector may be less efficient than a suboptimum detector when deviations from the design conditions occur. 4 3 Wilcoxon Detector

As an example of the calculation of the ARE for detectors which use both polarity and rank information, the ARE of the Wilcoxon detector

82

4

One-Input Detector Performance

with respect to the linear detector will now be calculated. In order to perform these calculations, a slightly different approach must be taken. As discussed in Section 2.7, the basic definition of ARE is under certain regularity conditions equal to the ratio of the efficacies of the two detectors being compared. The efficacy of a detector is defined as (4.3-1) where S is the detector test statistic and 8 = 8, is the hypothesis being tested [Kendall and Stuart, 19671. For the present detection problem, 8 = p which equals zero under the hypothesis. In order to derive the ARE, the expected value and variance of both test statistics must be calculated and then substituted into Eq. (4.3-1) to find the efficacy of each detector. The test statistic of the linear detector defined by Eq. (3.2-6) will.be denoted by S , , and its mean and variance will be computed first. The mean of S, is E{Sl}=E

K1I x

xk =

"I

&=I

E{xk} =

x "I

p= nip,

&= 1

(4.3-2)

and the variance is given by

x "I

var{Sl} = var{

"I

xk] =

k- 1

2

var{xk}

&- 1

"I

=

2

(4.3-3)

u2= n,u2

k= 1

Substituting these values into Eq. (4.3-1), the efficacy of the linear detector becomes

To compute the mean of the Wilcoxon test statistic, a different form of the test statistic must be used. Define U, as

The Wilcoxon test statistic will be denoted by- S,- and can now be written using U, as "2

s2

=

2

x "2

i = l j=1

Kj

4.3

83

Wilcoxon Detector

Taking the expected value of S , yields (4.3-5) If xi and xi are elements of a random sample, then the expected value in Eq. (4.3-5) becomes nz

E(S2) =

n2

2 2 p=

i = l j-1

n:p

(4.3-6)

where p is the probability that xi is greater than xi. To determine p, note that p is just the probability that xi - xi < 0. The density function of the sum of two independent random variables simply equals the convolution of their respective densities; for instance, if z = w + y ,

f,(z)

=JW

-w

f w ( z-

v ) f , ( v )dv

The cumulative distribution function of z can now be found fromf,(z) as follows

(4.3-7) The density of 3 - xi can be obtained from,Eq. (4.3-7) by letting z = xi xi, - y in Fw( -) equal x - p, and y in f,(.) equal x , to yield F ( x j - xi)

= F(I xj 00

-w

- xi

+x

- p ) f ( x ) dx

(4.3-8)

Since p is the probability that xi - xi < 0, p can be found from Eq. (4.3-8) as p = F(0) =

J-:

F(0

+ x - p ) f ( x ) dx

(4.3-9)

Using this result in Eq. (4.3-6) and taking the partial derivative yields a/aP{E(S,)}

=a/aP[n;p]

(4.3-10)

84

4

One-Input Detector Performance

The variance of the Wilcoxon test statistic is given by

var{S2} = &n2(n2 + 1)(2n2 + 1)

(3.4- 15)

which was calculated previously. The efficacy of the Wilcoxon test statistic can now be found from Eqs. (4.3-1), (4.3-lo), and (3.4-15) to be

(4.3-1 1) The ARE of the Wilcoxon detector with respect to the linear detector can now be found by taking the ratio of Eq. (4.3-1 1) to Eq. (4.3-4), (4.3-12) Equation (4.3-12) is the ARE of the Wilcoxon detector with respect to the linear detector for a background noise density f(x). If the background noise density is Gaussian, then

and

dx= - (4.3-13) 2 a G Substituting Eq. (4.3-13) into the expression for the ARE in Eq. (4.3-12) yields =-

4

1

= 12d[

-1

1 2aG

e-Xz/d

2

12a2 4af

3

= -= T

(4.3- 14)

4.3

85

Wilcoxon Detector

The Wilcoxon detector is therefore 95.5% as efficient as the linear detector for the problem of detecting a positive dc signal in Gaussian noise. This result is quite amazing since the linear detector is the optimum detector for this detection problem. If the noise has a Laplace density

then,

Substituting this result into Eq. (4.3-12) yields E 2 , 1= 12.”

z] l

2= 12u2 = 3 (4.3-15) 802 2 The Wilcoxon detector is thus 1.5 times as efficient as the linear detector when the noise has a Laplace distribution. The ARE of the Wilcoxon detector with respect to the linear detector for the two densities considered thus far is quite high. Because of this fact, the question arises as to whether a nonzero lower bound for the ARE exists for this case, and if so, what is the minimum value? Hodges and Lehmann [ 19561have proven a theorem which states that a lower bound for the ARE of the Wilcoxon detector does exist, and that the minimum value is 0.864 for the detection of a positive shift. Following their development, the derivation of the lower bound requires the minimization of the ARE given by Eq. (4.3-12), or equivalently, the minimization of

for a fixed u2. Since Eq. (4.3-12) is invariant to changes of location or scale, u2 and E { x } may be chosen to have the values u2 = 1 and E { x } = 0. The problem thus becomes the minimization of (4.3- 16) subject to the conditions m

S__f(x)dx=

1

and

J- x 2 f ( x ) dx= m

m

1

(4.3-17)

86

4

One-Input Detector Performance

Using the method of Lagrange undetermined multipliers, the above conditions can be satisfied by minimizing

- 2 v 2 - x ’ ) f ( x ) } dx

J-;{P(x)

(4.3- 18)

Equation (4.3- 18) may be minimized by taking the derivative with respect tof(x) and equating the result to zero,

=JW

--m

{2f(x) - 2h(02 - x ’ ) } dx= 0

(4.3- 19)

Solving Eq. (4.3-19) for f ( x ) , m

00

21-_f(x) d x = 2]-_X(e2

- x 2 ) dx

yields

x(e2- x 2 )

for x 2 G

e2

for x 2 > e2

o

(4.3-20)

The parameters X and 8 may be determined from the conditions in Eq. (4.3-17) as follows 00

/-mf(x) dx= J -e e h ( 0 2 - x’) dx= $he3 = 1

(4.3-21)

and 00

J-_xY(x)

dx=l”

--m

Ax2(02 - x’) d x = $X05 = 1

(4.3-22)

Solving Eq. (4.3-21) for X and substituting into Eq. (4.3-22),

or e2

=5

(4.3-23)

Using the value of 8 in Eq. (4.3-21), and solving for A,

$he3 = g h [ 5 v 3 ] = I or

A=-

3 20v3

(4.3-24)

4.4

87

One-Input Detector AREs

To determine the minimum value of the ARE, the density in Eq. (4.3-20) may be substituted into Eq. (4.3-16) to yield

Using the values for 8 and A given in Eqs. (4.3-23) and (4.3-24) to evaluate Eq. (4.3-25), f ( x ) } 2 d x = %(\15)5(

3

5v3

(4.3-26)

Remembering that u2 = 1, the lower bound on the ARE for the Wilcoxon detector may be found by utilizing Eqs. (4.3-12) and (4.3-26),

(4.3-27) The lower bound on the ARE given by Eq. (4.3-27) means that for large samples and for testing for a location shift, the Wilcoxon detector will never be less than 86.4% as efficient as the linear detector regardless of the background noise density. 4.4

One-Input Detector AREs

Table 4.4-1 gives the asymptotic relative efficiencies of the nonparametric detectors discussed in this book with respect to the Student’s f test or the linear detector for various background noise densities when applied to the problem of detecting a positive location shift. All of the detectors have a surprisingly high value of ARE with respect to the linear detector when the background noise is Gaussian, considering the fact that the linear detector is the optimum (UMP) Neyman-Pearson detector for the Gaussian noise case. The sign detector has the lowest ARE for all cases shown except for the Laplace noise density case. The lower efficiency of the sign detector for most of the cases is not surprising since the sign detector uses only polarity information. The’high ARE value of the sign detector for the Laplace density is an unexpected result, however it can be shown that the sign detector is weak signal optimum for this case [Carlyle, 19681.

88

4

One-Input Detector Performance

Table 4.4-1 Asymptotic relative efficiencies of certain one - input nonparametric detectors with respect to the optimum detector for Gaussian noise Nonparametric detector Sign Wilcoxon Normal scores ( c , ) Van der Waerden Kendall tau ( 7 ) Spearman rho (p)

Bounds for ARE Noise density Gaussian Uniform Laplace Exponential Lower Upper 0.637 0.955 1.000 1 .000 0.912 0.912

0.333 1.000

0 0.864

2.000 1.500

1.o

0.312 0.312

00

1.o

The high ARE values of the Wilcoxon detector have already been discussed quite fully. The very high lower bound of the Wilcoxon detector coupled with the relative ease with which it may be implemented make the Wilcoxon detector one of the most useful detectors for a wide variety of applications. The normal scores and Van der Waerden’s tests are as efficient as the linear detector for Gaussian noise, and it has been shown by Savage and Chernoff [1958] that these two detectors always have an ARE > 1 when compared to the linear detector. This is because both the normal scores and Van der Waerden’s detectors utilize Gaussian transformations to compute their statistics, and the linear detector is based on the assumption of Gaussian background noise. The result is that the normal scores and Van der Waerden’s detectors are as efficient as the linear detector when the noise is Gaussian and more efficient when the noise is non-Gaussian. The Kendall 7 and the Spearman p detectors both have a high ARE relative to the linear detector when the noise is Gaussian, but they are much less efficient when the noise has an exponential distribution. The Kendall 7 and Spearman p detectors have the same ARE since the two tests are equivalent for large sample sizes. The utility of any asymptotic relative efficiency computation or asymptotic performance indicator depends on how well the asymptotic results predict finite sample performance. The ARE definition used thus far in the text, besides requiring the assumption of a large number of observations, is based on the additional assumption that the alternative is arbitrarily close to the hypothesis. More specifically, the ARE is only a weak signal performance indicator. This may sometimes prove to be a fatal restriction if the prediction of detector performance for a finite number of samples and large signals is desired. In order to eliminate the limitation to weak signal detector performance evaluation, Klotz [ 19651 developed asymptotic

4.4

One-Input Detector ARES

89

efficiency curves by examining the exponential rate of convergence to zero of the false alarm probability while maintaining a constant probability of detection. Since a is allowed to vary, it is no longer necessary for the alternative to approach the hypothesis to prevent the detection probability from becoming one when considering consistent tests. The end result is that the asymptotic efficiency values obtained by Klotz have a better agreement with finite sample detector performance than the ARE. Using the above approach, Klotz computed the relative efficiencies of the one-input Wilcoxon, normal scores, and sign detectors with respect to the Student’s t test or linear detector when the input noise density is Gaussian. As the distance p between the hypothesis and alternative probability densities increases from zero, the efficiency of the normal scores detector decreases from an initial value of 1.0, while the efficiency of the Wilcoxon detector increases from an initial value of 0.955, until they cross between p = 1.OOO and p = 1.125. The efficiencies of both detectors then decrease together as p increases. The asymptotic efficiency of the sign test peaks at an efficiency value of approximately 0.76 for p = 1.5 and then approaches the relative efficiency of the normal scores and Wilcoxon detectors as p becomes larger. For p > 3.0 the sign detector has essentially the same relative efficiency as the rank detectors. The three nonparametric detectors are also compared by Klotz for logistic and double exponential noise distributions. For the case of a logistic distribution, the Wilcoxon detector has the highest relative efficiency value for small signals and maintains this superiority as p approaches 3.0. The sign test is the most efficient detector for p = 0 and a double exponential distribution, but the Wilcoxon detector becomes more efficient than the sign test for p > 0.55, and the normal scores detector becomes more efficient than the sign test for p > 1.15. The Wilcoxon detector outperforms the other two detectors for p = 3.0. Of the conclusions that can be drawn from these results, perhaps the most important is that the Wilcoxon detector maintains a reasonably high performance level for the three probability densities and the various location shifts considered. Another important conclusion is that the ARE in Eq. (2.7-1) is generally valid only for the case of vanishingly small signals, and relative detector performance may be opposite to that predicted by the ARE for moderate to large signal inputs (location shifts). It should also be noted that the sign test performs much better for moderate to large location shifts than is indicated by its ARE values. It is clear from the preceding discussion that although nonparametric detectors utilize only polarity and rank information, the loss in efficiency is many times quite small when compared to the optimum linear detector; and if the background noise varies from the Gaussian assumption of the

90

4

One-Input Detector Performance

linear detector, the nonparametric detectors may be infinitely more efficient than the linear detector. It can also be concluded that asymptotic results, in general, and the ARE, in particular, may not accurately predict detector performance for finite sample sizes. For this reason, detector performance studies when only a small number of observations are available for the detection process are extremely important. The following section discusses some of these investigations that have been reported in the literature. 4.5

Small Sample Performance

Small sample detector performance studies have generally been shunned by researchers since such studies are limited in general interpretation by the sample size, false alarm and detection probabilities, and the hypothesis and alternative being tested. However, investigations of this type are important since conditions under which the detector will be applied are more accurately duplicated than when using asymptotic results, and hence the performance evaluation is more accurate. In this section, three interesting studies of one-input nonparametric detector performance are discussed and the results compared to those obtained using asymptotic procedures. The finite sample performance of the one-input Wilcoxon and normal scores detectors for a Gaussian noise density and location shift alternatives has been investigated by Klotz [1963]. Klotz calculates both the power of the Wilcoxon detector and its relative efficiency with respect to the Student’s t test for false alarm probabilities in the range 0.001-0.1 and for sample sizes from 5 to 10. The location shift values considered vary from p = 0.25 to 3.0, with results also obtained for p = 0 and p + co. One general result is that the relative efficiency tends to decrease slightly for all significance levels a and sample sizes n considered as p varies from 0 to 1.5. Furthermore, for all sample sizes and all false alarm probabilities except a = 0.001, the relative efficiency is always greater than or equal to the ARE value of 0.955. The relative efficiency values are larger for smaller sample sizes and tend to be concave downward as a function of increasing a for fixed n, with the maximum occurring for a 2 0.05. The power and relative efficiency of the normal scores test with respect to the Student’s t test was also computed by Klotz [1963]. This part of the investigation considered significance levels in the interval 0.02 < a < 0.1 and n in the same range used for the Wilcoxon detector. The decision regions for the normal scores and Wilcoxon detectors are the same for the combinations of a and n given by (n < 7, a < 0.10), (n = 8, a < 0.054), (n = 9, a < 0.027), and (n = 10, a < 0.0186). The relative efficiency val-

4.5

Small Sample Performance

91

ues when computed for the normal scores detector with respect to the Student’s t test are all less than 1.0 for p < 1.5. This is an important result since the normal scores detector has an ARE with respect to the t test which is always greater than or equal to 1.0. Additionally, the normal scores detector has a higher relative efficiency than the Wilcoxon detector only for p < 1.25, with the relative efficiencies becoming approximately the same for p = 1.25, and the Wilcoxon detector showing an advantage in relative efficiency for some a and n combinations. Notice that this result is in good agreement with the asymptotic calculations performed by Klotz [ 19651 and reported in Section 4.4. Arnold [ 19651 has expanded on the work of Klotz and has computed the power of the Wilcoxon, sign, and Student’s t tests for nonnormal shift alternatives and a small number of input observations. In this investigation, the hypothesis of zero median is tested against various positive shifts of the median for the t distribution with degrees of freedom t , 1, 2, and 4. Sample sizes of n = 5 through 10 observations and false alarm probabilities in the 0.01 to 0.1 range are used in the calculations. The results are that both the Wilcoxon and Student’s t detectors exhibit a significant loss in power for the long-tailed t distributions (degrees of freedom = 4, 1) especially when compared to their detection probabilities for Gaussian alternatives. In fact, Arnold’s results show that the sign test maintains a higher detection probability than either the Wilcoxon or Student’s t test for a Cauchy distribution (degrees of freedom = 1). The Wilcoxon detector does have a higher power than the t test for this case, but in general, its performance is closer to that of the t test than the sign detector. The surprisingly poor performance of the Wilcoxon detector for the Cauchy distribution can be traced to the fact that reverse orderings of positive and negative observations are equally likely for the Cauchy distribution even though the rank sums of these reverse orderings may be radically different [Arnold, 19651. The finite sample relative efficiency of the sign test with respect to the Student’s t test is calculated by Blyth [1958] by fixing the sample sizes and a,and defining loss functions which depend on the detection probabilities of the two tests. The loss functions are then investigated as the ratio of the mean to the standard deviation of the underlying noise distribution, which is Gaussian, is varied. For a = 0.05 and n = 11, the Student’s t test has a 0.1 to 0.2 advantage in power over the sign test for values of the mean to standard deviation ratio from about 0.3 to 1.3. For ratios greater than or equal to 1.5, the powers are essentially the same. Other than the three investigations described in this section, seemingly few additional studies of nonparametric detectors have been conducted. Some engineering applications which consider only a finite number of

92

4

One-Input Detector Performance

input observations are discussed in Chapter 9. However, considering the very interesting, useful, and sometimes surprising results obtained in the studies described in this section, additional work in this area is clearly needed. 4.6

Summary

The purpose of this chapter has been to give examples of the calculation of the ARE of one-input rank and nonrank nonparametric detectors as well as to compare the performance of the six one-input nonparametric detectors discussed in Chapter 3 using the concepts of ARE and finite sample size relative efficiency. In reviewing the results of the comparison, the words of caution on applying the ARE expressed in Section 2.7 should be kept in mind. If the limitations of the ARE as a performance indicator are realized, the ARE can be an extremely useful performance comparison technique, since it allows many detectors to be compared both generally and simply. The results in Section 4.5 indicate, however, that whenever possible, a finite sample study should be performed which duplicates as nearly as possible the detector application environment. The calculation of the ARE of the sign detector with respect to the linear detector utilized the basic definition of the ARE given by Eq. (2.7-1). The asymptotic false alarm and detection probabilities of the two detectors were calculated and the ratio of the two sample sizes determined by equating the false alarm probabilities of the two detectors to each other and the detection probabilities of the two detectors to each other. This approach is, of course, very general and can be used whenever the asymptotic false alarm and detection probabilities can be found. The ARE of the Wilcoxon detector with respect to the linear detector was determined by using the expression for the ARE in terms of the efficacies of the two detectors. This technique is preferred to the approach described in the previous paragraph whenever it is simpler to find the mean and variance of the two detectors rather than the asymptotic false alarm and detection probabilities. Since the mean and variance of the linear detector test statistic are easily calculated and the variance of the Wilcoxon test statistic was determined in Chapter 3, the approach utilizing the efficacies of the two detectors was employed in this case. Following the calculation of the ARE of the Wilcoxon detector with respect to the linear detector in Section 4.3 is the derivation of an extremely powerful result. This result, originally proven by Hodges and Lehmann [1956], shows that the ARE of the Wilcoxon detector with respect to the linear detector will never be less than 86.4% for detecting a

93

Problems

positive location shift regardless of the background noise density. Since many communication and radar systems require the detection of a positive shift of the noise density and since the linear detector is commonly employed in these communication and radar systems, the high lower bound on the ARE of the Wilcoxon detector with respect to the linear detector indicates that the Wilcoxon detector might be gainfully employed in these systems. In Section 4.4, the ARE of the six nonparametric detectors discussed in Chapter 3 with respect to the linear detector for various background noise densities are tabulated and discussed. The ARES shown are surprisingly high, even for the case of Gaussian noise for which the linear detector is the most powerful detector. This section also contains the results of a study by Klotz [1965] using a different definition of asymptotic efficiency. This definition is important since it does not require the alternative to approach the hypothesis, which seems to be the primary weakness of the ARE in predicting finite sample size performance. A few detector performance evaluations based on a small number of input observations are discussed in Section 4.5. Using these results, some idea of the utility of the asymptotic performance measures can be obtained. From the power and relative efficiency values presented in this chapter, it can be concluded that the performance of nonparametric detectors compares very favorably with that of parametric detectors. Further, nonparametric detector performance tends to remain high for a variety of input noise distributions. These last two facts are the primary motivating factors behind the consideration of nonparametric detection procedures for engineering applications.

Problems 1. Compute the ARE of the sign detector with respect to the linear detector for a uniform noise distribution given by

*(.)

=

{ i:

-1/2

<x <

1/2

otherwise

2. Calculate the efficacy of the sign detector when testing for a location shift 0 of a symmetric probability density. 3. Repeat Problem 2 for the one-input Student’s t detector. 4. Confirm the ARE values of the sign detector given in Table 4.4-1 for the Gaussian, uniform, and Laplace distributions using the results of Problems 2 and 3.

94

4

One-Input Detector Performance

5. Use Eq. (4.3-12) to calculate the ARE of the Wilcoxon detector with respect to the linear detector for a uniform probability density over the interval - 1/2 < x < 1/2. 6. Calculate the finite sample relative efficiency of the sign detector with respect to the linear detector when testing the hypothesis

H: F(31) = 0.5, F Gaussian with u2 = 1 against the alternative

F(31) < 0.5, F Gaussian with y = 32.3 and u2 = 1 at a significance level of a = 0.08 Assume that the sign detector uses 12 observations. 7. Show that if a one-input detector has a test statistic that satisfies the regularity conditions in Chapter 2, then K:

where erf x =

-/,xe-Yz 6

dy

Hint: See Capon [1959b, 19601. 8. Use the result of Problem 7 to show that if the test statistics of two detectors, say S,, and S,* satisfy the regularity conditions in Chapter 2, then the ARE of the S,, detector with respect to the S,* detector is the ratio of their efficacies,

[see Capon, 1959b, 19601.

CHAPTER

3

Two-Input Detectors

5.1 Introduction

The discussion thus far has centered on detectors, both parametric and nonparametric, which have only one set of input observations on which to base their decision. Since many systems such as some communication and radar systems have only one input available, the previous treatments are well justified. It would seem intuitive, however, that if a one-input detector performs well, a two-input detector should perform even better because additional information is available for the decision process. Several two-input statistical detection procedures have been described and developed in the literature. These two-input detectors generally can be classified under two headings: (1) detectors with one input and a reference noise source, and (2) detectors with two simultaneous inputs. Any one-input communication system which has available or has a provision for obtaining a set of reference noise samples can be classified under the first heading. The reference noise samples are compared by the detectors to samples of the input to be tested to ascertain the presence or absence of a signal. What are some of the techniques which can be used to obtain these reference noise samples? If the noise is stationary, the noise can be sampled during the system dead time; that is, when it is known that no data will be received. These samples can then be stored and used for the remainder of the detection process. If the noise is quasi-stationary; that is, stationary only for relatively short periods of time, the noise process 95

96

5

Two-Input Detectors

must be sampled and stored periodically to assure that the reference noise samples are from the same process as the samples of the input to be tested. This technique has the disadvantage that the signal detection procedure must be interrupted periodically to allow new noise samples to be taken and stored. If the noise is nonstationary, it is possible that space, angle, frequency, or some other diversity technique can be used to obtain at least quasi-stationary noise samples. Regardless of the technique, it is imperative that the reference noise samples be representative of the noise present in the input used in the signal detection process. The difficulty in obtaining accurate noise observations can severely limit the application of the detectors under the first classification. Detectors with two simultaneous inputs are less common than one-input detectors with or without a reference noise source. These two-input detectors in general perform some kind of correlation between the two sets of input observations. If the two inputs are correlated, a signal is declared present; if the inputs are uncorrelated, noise only is decided to be present. This procedure highlights a fairly stringent requirement; under no-signal conditions, the two inputs must be independent noise processes. This requirement necessitates that the two-input receivers be physically separated by enough distance that the noises received in the two inputs are independent. It is evident then that communication systems with two simultaneous inputs are somewhat more complex, and thus less common, than the one-input systems. This chapter is concerned with defining, developing, and discussing several two-input parametric and nonparametric detectors which can be applied to signal detection problems. The parametric detectors are presented as a baseline for discussion and for use later in Chapter 6 as a basis for nonparametric detector performance evaluation and comparison. Two different types of detection problems are considered in this chapter: (1) the detection of a positive location shift of a probability density function, and (2) the detection of a random process. In the following sections, the discussion of the two types of detection problems start with parametric hypotheses and alternatives and their corresponding detectors, with the hypotheses and alternatives being successively generalized until a nonparametric problem results. This approach gives the reader an idea of exactly how restrictive some of the classical parametric detectors are and at the same time emphasizes the role of assumptions in statistical hypothesis testing, which is an interesting topic in itself. First, detectors which have one input and a set of reference noise samples are discussed. These detectors are most useful for testing for a positive location shift of the probability density function. A two-input

5.2 One-Input Detectors with Reference Noise Samples

97

version of the Student’s t test which is asymptotically nonparametric is derived initially, followed by thorough developments of the MannWhitney detector and the normal scores test. Next, detectors which have two simultaneous inputs and perform some kind of correlation calculation between the two inputs are developed. These detectors find application to problems which require the detection of a random signal with a given probability density function. Detectors of this type that are discussed include the sample correlation coefficient which is asymptotically nonparametric, the Spearman p detector, and the Kendall T test. For each of the detectors, the derivation or background of the test statistic is given, the probability density function of the test statistic under appropriate hypotheses and alternatives derived, and important properties such as consistency, unbiasedness, whether or not a detector is nonparametric, and under what conditions the detector is most powerful (optimum) are established. Performance evaluation of the detectors is deferred until Chapter 6. 5.2 One-Input Detectors with Reference Noise Samples In this section detectors which have one input with a set of reference noise observations are treated. it is assumed throughout this discussion that one of the techniques discussed in Section 5.1 has been applied to obtain a set of independent reference noise observations which are available for use in the detection process. The detection problem to be considered in this section will be that of detecting a positive dc signal in additive noise. The optimum unbiased detector for this detection problem when the noise is Gaussian, the two-input Student’s t test, is derived, and it is shown to be asymptotically nonparametric when the noise is not Gaussian. A two-input version of the Wilcoxon detector sometimes called the Mann-Whitney detector, is discussed next followed by a development of the two-input version of the normal scores detector. 5.2.1

Parametric Detectors

Assume that m independent noise samples have been taken under no signal conditions. These noise samples will be denoted by x = {x,, x2, . . . , x,}, and are assumed to have a cumulative distribution function denoted by F ( x ) and a probability density function denoted by f(x). Next, n independent observations are taken which will be denoted by

5

98

Two-Input Detectors

y = { y 1 , y 2 ., . . , y n } ,and these observations are assumed to have a cumulative distribution function denoted by G ( y ) and a probability density function denoted by g(y). It is desired to detect the presence or absence of a positive dc signal in the y observations by comparing them by some technique to the x observations. The detection problem can therefore be written as testing hypothesis H9 versus the alternative K,, H,: K,:

F(x) = G(y), F ( X )= G ( Y -

-00

e),

<x for

< 00, e >o

-00

< y < 00

where 8 is the amount of the location shift of the distribution function and is the received signal-to-noise ratio (SNR).Of course, a form for the most powerful detector for testing the very general hypothesis H , versus the general alternative K9 cannot be found utilizing the likelihood ratio technique. If however, the distribution functions F and G are assumed to be Gaussian, a detector can be derived which is the most powerful unbiased detector for this particular problem. If the probability densities f ( x i ) and g(y,) are assumed to be Gaussian with means 9, and 9,, respectively, and with equal variance, say u2, the detection problem can be written as testing H,,:

9, = g2,f and g Gaussian with unknown variance a’

versus K,,:

9 , # g2, g2 > 0,f and g Gaussian with unknown variance u’

One method for testing H I , versus K,, would be to find the joint densities of x and y under H , , and K,,, and then form the likelihood ratio of the joint density given K,, is true to the joint density given H,, is true. This approach transforms the problem into the familiar framework used in Section 3.2. Since f and g are Gaussian and independent, the joint density of x and y given H , , is true can be written as w

7

Y P I , ) = f (xlHIo)g(YlHIo)

(5.2-1)

5.2

One-Input Detectors with Reference Noise Samples

99

and the joint density given that K , , is true can be written as h(x, YIK,,) = f(xlHlo)g(YlK,o)

(5.2-2) The problem of testing H I , versus K , , has been transformed into the equivalent problem of testing the density in Eq. (5.2-1) versus the density given by Eq. (5.2-2). Since the parameters 8, and u2 in Eq. (5.2-1) and the parameters B,, B2, and u2 in Eq. (5.2-2) are unknown, these parameters must be eliminated from the expressions before the likelihood ratio is formed. Following the procedure utilized in Section 3.2, these parameters will be estimated by the method of maximum likelihood. The parameters 8, and u2 in Eq. (5.2-1) will be estimated by utilizing the expressions (5.2-3) (a/ae,) In 4% YIH,,) = 0 and (a/au2) In h(x, ~IH,,)

=0

(5.2-4)

Taking the natural logarithm of Eq. (5.2-1) and substituting into Eq. (5.2-3)

yields

x m

i l l

xi

n

+ x y , = ( m + “)el j-I

5

100

Two-Input Detectors

or (5.2-5) Taking the natural logarithm of Eq. (5.2-1), substituting il, for using Eq. (5.2-4),

el, and

Solving for u2

Substituting J1, and @, into Eq. (5.2-1) yields

(5.2-7) To find the estimates of el, 8,, and u2 given that Klo is true, the following expressions must be solved simultaneously (5.2-8) in h ( x , YK,) = 0

(awl)

and

(ape,)

In h ( x , YIK,,) = 0

(a/auz)

In h ( x , YIK,,)

=

o

(5.2-9)

(5.2-10)

Taking the natural logarithm of Eq. (5.2-2) and substituting into Eq.

(5.2-8),

= (& ~ ~ ( x ; - 6 1 ) ( - l ) = o i= 1

5.2 One-Input Detectors with Reference Noise Samples

I01

or

(5.2-11 ) Similarly for

i,,

-

Substituting (5.2-lo),

i1, and i2for 8 ,

-(m+ n)/2

2

-

c

i- I

-

[

(xi

i- I

(5.2-12)

I

and 8, in h ( x , yIK,,) .and utilizing Eq.

(xi -

ln(2no2) -

2” Yj

I’

m

ew[-

-(m2+ ( 2 [ 5 n,

1

=

82

i:

J ~ , -~ Y(vj - b2)2 j- I

2 ( x i - J , , , ) ~+ 2 ( y j m

i-1

-

61, K 12 +

i: j- 1

j-

(Yj -

~2

8,)‘1( 2 I

=0

which when solved for 6’ yields

Substituting Eqs. (5.2-11), (5.2-12), and (5.2-13) for 8 , , O,, and u2, respectively, in Eq. (5.2-2) yields the maximum value of the density

(5.2- 14) Of

The generalized likelihood ratio can now be formed by taking the ratio Eq. (5.2-14) to Eq. (5.2-7)

5

I02

Two-Input Detectors

I).

Working with the first term in the denominator of Eq. (5.2-15)

x m

i

[ x i - ( k ) ( j l x k +

i- 1

j-

=

=

5

i- 1

x m

1

mK +nJ

[xi

-

[xi

- x12+ m E

i- 1

2

+

[

i= 1

[

-

m+n

mK +njj m + n

(5.2- 16)

and similarly with the second term in the denominator,

mK +njj j - 1

i= 1

k- 1

j- 1

j- 1

n

=

2 [ y j -ylz+ j= 1

[

n jj-

Expanding the last term in Eq. (5.2-16), m[P-

mE

+ w ]’=

n+m

(m

.[ 72

m+n

+

l2

(5.2-17)

( mXm ++nnjj 12]

+ nI2Z2 - ( 2 m ~ 2+ 2 n ~ j j ) ( m+ n ) + ( m x + nJI2 (m + n)’

=

m[

-

mn2(X - j j ) 2

(m

- 2 ~ ( + m njj) ~

mX +nJ m+n

+ n)’

1

(5.2- 18)

and similarly for the last term in Eq. (5.2-17),

(5.2- 19) Using the results of Eqs. (5.2-18) and (5.2-19) and substituting Eqs. (5.2-16)

5.2 One-Znput Detectors with Reference Noise Samples

I

and (5.2-17) back into Eq. (5.2-15), yields

L ( x , Y) =

2 (xi - E ) +~ 21

i- 1 m

I

- ( m +n ) / 2

n

m

103

(

j-

~

-j J)’

n

2 (xi- E)’ + 21 ( y , - j j 1 2 + ( m + n ) i- 1

m n ( i -J)’

J-

(m

+

(m+ n)/2

m + n

(5.2-20)

2 (xi - ?l2 + 21 (Yj - Y>’

i- 1

J-

since (X - J ) 2 = ( J - $, ( J - K)2 will be substituted for (K -y)’ in Eq. (5.2-20). Utilizing this substitution and taking the square root of the second term under the brackets and multiplying and dividing the denominator by

d(n

+ m - 2 ) / ( n + m - 2)

yields

+

where t has the Student’s t distribution with n m - 2 degrees of freedom when H I , is true. Equation (5.2-20) can therefore be written as

+

1

(m+n)/2

(5.2-2 1) n+m t2 - 2 L(x, y) in Eq. (5.2-21) is a monotonic function of t 2 and therefore a test based on t 2 will be equivalent to a test utilizing L(x, y) as the test statistic.

L(x, Y) = ( 1

5

I04

Two-Input Detectors

Since t 2 = 0 when L(x, y) = 1 and t 2 + 00 as L(x, y) + 00, an alternative acceptance region of the form L ( x , y ) > A > 1 is equivalent to an alternative acceptance region of the form t 2 > B; that is, the alternative K,,is accepted for positive and negative values of t with magnitudes greater than B. Since the signal to be detected, /I2 is assumed , to be positive and t 2 is a monotonic function of t , the test statistic for testing hypothesis H,, versus the alternative K , , can be written as

i

dzt[ i: J- 1y

1 n+m-2

)[ E i=l

[xi.-

1

j-- l

ix -"1 x

i

1i

1 " 2 " Xkl2+ [ y j - iz ykl2]} k- 1 j- I k c1

KlO

2 T,

Hlo

(5.2-22)

which is the two-input version of the Student's t test. For small m and n, the threshold T can be determined by fixing the false alarm probability at its desired value, say a,, and using tables of the t distribution to find the threshold; that is, let S denote the left-hand side of Eq. (5.2-22), then T can be determined from (5.2-23) a = a, = P { S > TIH,,} using tables of the t distribution (see Appendix B). After finding T using the above equation, the probability of detection can be found by using the expression 1 - p = 1 - P { S < TIK,o} and tables of the noncentral t distribution. [See Hogg and Craig, 1970, for a discussion of the noncentral t distribution.] For m and n large the statistic S under H,, is asymptotically Gaussian with zero mean and unit variance. This can be seen if the definitions (5.2-24a)

x = -1 s;=-2

"

"

i-1

and

(5.2-24b)

E X i i-1

n J'1.

[

(5.2-24~) k-1

k- 1

(5.2-24d)

5.2 One-Input Detectors with Reference Noise Samples

I05

are substituted into the left-hand side of Eq. (5.2-22) to yield

In Eq. (5.2-25) the second quantity in the denominator in brackets is just the estimate of the variance u2, which will be denoted by 6,. Substituting into Eq. (5.2-25),

The quantity j j - X in Eq. (5.2-26) is Gaussian distributed, since x and y are independent and Gaussian distributed, and for m and n large and H,, true, p - X has zero mean and variance 02[(1/m) + (1/.)I. Since for m and

n large, the denominator of Eq. (5.2-26) approaches ud(l/m) + ( l / n ) , S is asymptotically Gaussian with zero mean and unit variance by the central limit theorem. The threshold can thus be found as was done in Chapter 3 by specifying a and utilizing a = 1- @(T)

(3.2-19)

and with this value of T, the power of the test can be determined from 1-

p

=

@[ T - p ]

since S is asymptotically Gaussian with mean p = 8, - el, and unit variance when K , , is true. It was stated earlier that the two-input version of the Student’s t test given by Eq. (5.2-22) is UMP unbiased for testing H I , versus Klw That this statement is true can be seen by employing an argument directly analogous to that used when discussing the one-sample Student’s t test in Section 3.2. Kendall and Stuart (19671 have shown that the two-sample Student’s t test statistic is the UMP similar test for the present hypothesis and alternative. By the discussion of similarity and unbiasedness in Section 2.6, the Student’s t test will be the UMP unbiased test of H,,versus K , , if it can be shown that the test statistic on the left-hand side of Eq. (5.2-22) is unbiased. This result can be demonstrated in a manner analogous to that employed for the one-sample Student’s t test in Section 3.2. Let t, be

I06

5

Two-Input Detectors

chosen such that

(5.2-27) when 8, = 8,, and where SD is defined by

Consider now a case such that 8,

> 8,;

[ ( J -2) - ( 8 2 - 4 ) l SD Therefore,

QD(F) > a

and from Eq. (5.2-27), E { D ( x , y)lH,,} = a

for all F

E

K,,

for all F E HI,

1

> talKlo

=a

(5.2-28) (5.2-29)

The two-sample Student's t test is thus unbiased of level a for testing HI, versus Klw Since the Student's t test is also the UMP similar test for this same problem, it follows that the test defined by Eq. (5.2-22) is the UMP unbiased detector for testing H I , versus Klw The next logical generalization of the present detection problem is to let the density functions under the hypothesis and alternative to be continuous but otherwise unknown. The resulting detection problem can

5.2

I07

One-Input Detectors with Reference Noise Samples

be stated as testing versus

HI,: 8 , = 8,,

u2 finite,f and g otherwise unknown

K,,: 8, < O,,

u2 finite,f and g otherwise unknown

Since no specific form is assumed for either f or g, the problem can be easily classified as being nonparametric or distribution-free. Since the Student’s t test is the UMP unbiased detector for the assumption that f and g are Gaussian, its false alarm rate will not remain constant for arbitrary input probability densities; however, for m and n large in Eq. (5.2-22), the Student’s t test is asymptotically nonparametric. This can be seen by again considering Eqs. (5.2-24a-dH5.2-26). For a large number of x and y observations, both 7 and X are asymptotically Gaussian by the central limit theorem, and their difference is thus also asymptotically Gaussian with zero mean and variance ( u 2 / n ) ( u 2 / m ) under H I , . The discussion of the denominator in Eq. (5.2-26) still holds for

+

HI,, namely that for m and n large, uv(l/m)

+ (l/n)

approaches

ui(l/m) + ( l / n ) . The Student’s t test statistic on the left-hand side of Eq. (5.2-22) is thus asymptotically Gaussian with zero mean and unit variance for HI, true when the number of x and y observations are many. If the threshold is chosen according to Eq. (3.2-19), then the Student’s t test is asymptotically nonparametric when H I , is true. The power of the detector can be found as before. When K , , is true, the test statistic has a limiting Gaussian distribution with mean i m m (0, - 0,) and variance ( u 2 / n ) ( u 2 / m ) ;thus as m and n approach infinity, the mean of the test statistic approaches infinity and the probability of detection goes to one. By Eq. (2.6-13), the Student’s t test is thus consistent for the alternative K, The two-input Student’s t test given by Eq. (5.2-22) is therefore the UMP unbiased detector for testing H I , versus K,,,and is asymptotically nonparametric under H , , and consistent for K,,.The two-input version suffers from the same problem of irrelevance in its rejection region as does the one-input t test discussed in Section 3.2.

+

,.

5.2.2

Wilcoxon and Mum-Whitney Detectors

In this section the problem to be considered is again that of detecting a positive location shift of a probability density function. Two sets of

5

108

Two-Input Detectors

observations are assumed to be available; one is a set of reference noise samples denoted by ,x = { x I ,x,, . . . ,x,,} with distribution function F ( x ) , and the other is the set of observations to be tested denoted by y = { y l , y,, . . . ,y,} with distribution function G (y). The hypothesis and alternative of interest can be written as H,,:

F ( x ) = G(y)

for all x and y

K,,:

F ( x ) 2 G(y)

for all x and y and F ( x )

> G(y) for some x and y

This is simply a statement of the desire to detect a positive shift of the probability density or distribution function of the y observations, and is another way of writing the hypothesis and alternative H I , and K I I , respectively. For the purpose of this discussion, F and G are assumed to be continuous in order to eliminate the possibility of tied observations. The detector originally proposed by Wilcoxon in 1945 consists of algebraically ranking the m .+ n available observations and then summing the ranks of the xi and comparing the statistic to a threshold, (5.2-30) From Eq. (5.2-30), it can be seen that the hypothesis is accepted when the value of the test statistic is greater than a fixed threshold. Since normally in engineering applications the alternative is accepted for larger values of the test statistic, the test will be written in terms of the ranks of the y observations, (5.2-3 1) which is entirely equivalent to the detector in Eq. (5.2-30). A slightly modified version of the detectors in Eqs. (5.2-30) and (5.2-31) which is ideally suited for testing H , , versus K , , was proposed by Mann and Whitney [ 19471. The original test proposed by Mann and Whitney has the form K

r n n

U

=

2 j-1

i-1

U ( X ~-

y j ) 5 T, H

(5.2-32)

where u ( x ) is the unit step function. This test or its equivalent was proposed by a number of authors over the years, including Wilcoxon, and Kruskal [1957] summarizes the proposals of some of the lesser known authors. Strictly speaking, the detector described by Eqs. (5.2-30) and (5.2-31) is a Wilcoxon detector, and that described by Eq. (5.2-32) is a Mann-Whitney detector. However, in spite of a lack of a consensus in the literature, the name Mann-Whitney will be used to denote the detectors

109

5.2 One-Input Detectors with Reference Noise Samples

represented by Eqs. (5.2-30) through (5.2-32), which include the two-input Wilcoxon test. The U detector in Eq. (5.2-32) counts the number of times that an x observation exceeds a y observation and accepts the hypothesis when the value of the test statistic is greater than T2.This detector, like the detector in Eq. (5.2-30), is not in the usual form for engineering applications, but it can be rewritten as

xx m

n

j-1

i-1

U' =

u(yj - xi)

K

(5.2-33)

3 T3

H

which is exactly equivalent to the detector in Eq. (5.2-32). The S, S', U, and U' statistics are all related by straightforward expressions which greatly facilitate the computation of probabilities for the detectors that employ these statistics since only one table is required to supply the necessary probability information. The relationship stated without proof by Mann and Whitney [1947] in their original paper is (5.2-34) U = mn + [ m(m 1)/2] - S'

+

where S' is based on the ranks of the y observations and is given by Eq. (5.2-31). The proof of Eq. (5.2-34) will be approached by first deriving a relation between U and S and a similar relation between U' and S'. To simplify the derivation, let the x and y observations be jointly represented by a single set of observations denoted by w,where w is given by w = { w p w 2 , . . . , wn, W n + l , . * - ,w n + m } = {x,, x 2 , . * X n , Y , , * * * d,} Utilizing this definition, S in Eq. (5.2-30) can be written as 9

S =

n i- 1

x

n

n+m

;=I

j-1

R,= 2

(5.2-35)

u ( w i - wj)

which is simply another method of computing and summing the ranks of the x observations. This can be seen by noting that the statistic on the right-hand side of Eq. (5.2-35) compares each xi with each observation in the total set of observations w, records a 1 each time xi 2 wj, and sums all of the 1's generated in this manner, thus computing the rank of each x i . Continuing with the proof, Eq. (5.2-35) can further be written as n

s = ;2 =I

x

j-1

x n

m+n

U(Wi

- 9 )=

;=I

n+m

xx n

2 U ( W i - w,) + i - 1 j==n+l

n

j-1

U(Wi

-

9)

(5.2-36) Working now with the last term in Eq. (5.2-36), note that this statistic is simply comparing the x observations with themselves. Using this fact, the

5

110

Two-Input Detectors

value of this term can be found as follows. For each value of i there will be a value of j = i such that wi = 9;and thus, u(wi - wj) = u(0). Therefore, using the definition of the unit step function in Chapter 1, there will be a total of n ones generated by the above process for i = j . Since the maximum value of the last term in Eq. (5.2-36) is n2, the maximum possible score excluding the cases for i = j is n2 - n = n(n - 1). Of these n(n - 1) terms, only n(n - 1)/2 will equal one since each observation is being compared to every other observation twice, and in one of these cases xi > xi and in the other, xi < xi. Summing the contributions of the i - j and i # j cases, yields a value of the last term in Eq. (5.2-36) of n + [n(n - 1)/2] = n(n + 1)/2. Using this fact and noting that the first term on the right-hand side of Eq. (5.2-36) is U as given in Eq. (5.2-32), S can now be expressed as [Hajek, 19691 (5.2-37) s = u + [n(n + 1)/2]

u = s - [n(n + 1)/2]

(5.2-38)

In an exactly analogous manner, an expression relating S' and U' can be found, j-1 m

i-1

m+n

m

m

2 2 U ( W j - W i ) + j2 2 -1 = U' + [ m ( m + 1)/2] =

j-1 i=m+l

i-1

or

U' = S'

-[m(m

U(Wj

-

Wi)

(5.2-39)

+ 1)/2]

(5.2-40)

The original result of Mann and Whitney [1947] in Eq. (5.2-34) can now be obtained from Eq. (5.2-40) and the fact that U U' = mn; thus

U' = S' - [ m ( m or rearranging,

U

= mn

+ 1)/2]

+

= mn -

+ [ m ( m + 1)/2]

-

S'

U (5.2-34)

which is the desired result, Eq. (5.2-34). Using Eqs. (5.2-37H5.2-40) and U + U' = mn, any necessary relationship between U,S, U',and S' can be found. As noted earlier, these relationships are particularly useful when one is working with the U' statistic, but only has a table of probabilities for the S statistic available.

5.2 One-Input Detectors with Reference Noise Samples

111

Capon [1959b] has presented and analyzed a normalized version of the

U' statistic given by

(5.2-41) All calculations performed for the U' statistic are applicable to the V statistic with the effects of normalization incorporated, and therefore only the U and U' statistics will be treated in the ensuing discussion. Since the mean and variance of detector test statistics are important when approximations are used to compute false alarm and detection probabilities, the mean and variance of the basic test statistic treated in this section, the S statistic, when the hypothesis H I , is true will be derived. The use of the relationships in Eqs. (5.2-37)-(5.2-40) to obtain the means and variances of the S', U,and U' test statistics will then be illustrated. From Eq. (5.2-30), S = Zl-IRx,and therefore the expected value of S is n

E{&,} =

n+m

2 2 kP{&, i - 1 k-1

=

k } (5.2-42)

where the last step follows from the definition of expectation. To compute the probability that the rank of the ith observation is k, note that when H I , is true, the total possible permutations of the ranks of the first n observations in the set of N = n + m observations is just the number of . permutations of N things taken n at a time, or (n + m ) ! / ( m ) ! Similarly, the number of permutations of the ranks of the first n observations given that the rank of the ith observation is k is (n m - l)!/(m)!. Using these results, the desired probability is found to be

+

Substituting this result into Eq. (5.2-42) yields,

n E { S } = Z /Since Z y , , k = N ( N

+ 1)/2

2N Fk = $

i l l k=l

N

x k k-1

[Selby and Girling, 19651,

+

1) n N(N - n(N + 1) E { S } = -. (5.2-44) 2 2 N which is the mean of the S statistic when H I , is true. The mean of the U statistic can be obtained from Eq. (5.2-32) by noting

5

112

Two-Input Detectors

that

=$

q+l

mn

(5.2-45)

j-1 i = l

This result can also be obtained by taking the expected value of Eq.

(5.2-38),

E{U}

=

E { S } - E{n(n

+ 1)/2}

which upon substituting for E ( S ) from Eq. (5.2-44) becomes

+ 1) - f n ( n + 1) = +n(n + m + 1) - +n(n + 1) = t ( n 2 + mn + n - n2 - n) = tmn

E { U }= f n ( N

which is the same as Eq. (5.2-45). Since U

+ U’= mn,

E { U’}= mn - E { U } = mn - (mn/2) = m n / 2

(5.2-46)

and from Eq. (5.2-39) it follows directly that E { S ’ } = E { U ’ } + + m ( m + l ) = t m n + f m ( m + 1) = fm(n

+ m + 1) = f m ( N + 1)

(5.2-47)

The variance of the S statistic is somewhat more difficult to compute than the mean value. The derivation here parallels that of Hajek [1969], but is less general than his result since only the variance of the S statistic is derived here. The derivation utilizes the fact that the variance of S can be written as n

var{ S } =

2

i- 1

var{ R,

}

xx n

+ j-1

n

k-1

cov{ R,,,,, &, }

(5.2-48)

j#k

The first term on the right-hand side of Eq. (5.2-48) can be expanded as

n

=

N

2 1-1 2 [Z - E { & , ) I 2 P { R , , , , = Z}

(5.2-49)

i-1

When H I , is true, P{R,,,, = I} = 1/N using the same reasoning that was employed in the one-input case in Section 3.4, Eq. (3.4-8). Substituting

5.2 One-Input Detectors with Reference Noise Samples

113

back into Eq. (5.2-49) yields

i- 1

(5.2-50)

N

i-1 I - 1

To compute the contribution of the second term of Eq. (5.2-48), consider only the expression for the covariance, cov{%, N

=

N

kk1 = {[% - E(Rw,)] *[kk - E(kk)I)

22 [ g - E ( %,}][A

g#h-1

- E { k k } ] * P { k= , &kk = A}

(5.2-51)

Using the law of conditional probability, the joint probability of the ranks in Eq. (5.2-51) can be written as

P{R,

=

&kk= h }

= P { k , = g } P { k k=

-

4%

=g}

( N - 2)! 1 ( N - l)! N ( N - 1)

( N - l)! N!

Using this result in Eq. (5.2-51),

and substituting for the double summation yields

-

N

2

[g-

}(.12}

(5.2-52)

8'1

Since E { k,} = 27.]IP { k,= I } = X7,l l / N , the first term on the righthand side of Eq. (5.2-52) becomes

The expression for the covariance thus reduces to

5

I14

Two-Input Detectors

Substituting Eqs. (5.2-50) and (5.2-53) into Eq. (5.2-48) yields

=

[ ( N - l)n - n(n - l ) ] N ( N - 1)

I- 1

(5.2-54)

Since E { R , , , } = Z ; , , k P ( R ,

= k } = Z;,,k/N

From Selby and Girling [1965], N ( N + 1)(2N + 1 ) $p= 6 I= 1

-

4N (N - 1) mn(N + 1 )

=

x N

and

k-1

+ 2 - 3N

-

E,

k =

Eq. (5.2-54) be-

N ( N + 1) 2

3

12

(N - 1) mn(N + 1 ) 12 = 12 ( N - 1) which is the variance of S when H , , is true.

-

(

)

(5.2-55)

5.2 One-Input Detectors with Reference Noise Samples

I15

The variance of S’ can be found in an exactly analogous manner by letting the upper limit on the summations in Eq. (5.2-48) equal m . The variance of S’ thus computed is the same as the variance of S, that is var{S’} = g m n ( N

+ 1)

(5.2-56)

Using Eq. (5.2-38) to find the variance of U yields var{ U >= E

( [ ( s- t n ( n + 1)) - E { S - t n ( n + 1)}12)

but E { S - fn(n

+ l)} = E { S } - $ n ( n + 1)

Therefore,

+ 1 ) s + fn’(n + 1), - 2SE { S }+ n ( n + 1)s + 1 ) s - i n 2 ( . + 11, + E , { s } + n ( n + ~ ) E { s } + a n 2 ( . + I)’} = E { S 2 } - E 2 { S } = var{S} = g m n ( N + 1) (5.2-57)

var{ U } = E { S z - n ( n -n(n

Similarly, var{ U’}= & m n ( N

+ 1)

Since Capon’s [ 1959bl V test is simply V = (1/ m n ) U’,then E { V } = ( l / m n ) E { U‘} = 1/2

(5.2-58) (5.2-59)

and from Eq. (5.2-46),

=-

m2n2

N + l var{U’} = 12mn

(5.2-60)

Capon obtained these same results, Eqs. (5.2-59) and (5.2-60) by utilizing the approach employed by Mann and Whitney [1947] in their original paper. Mann and Whitney [1947] have shown in their original paper on the U statistic that the U statistic is asymptotically normally distributed when HI, is true, if F ( x ) is continuous, and if the limit of the ratio m / n exists as n and m tend to infinity in any manner. In the same paper, Mann and Whitney prove that the U test is consistent under K,,. Lehmann [ 19511 has shown that U is asymptotically normally distributed when K,, is true if F ( x ) and G(y) are continuous and if the limit of m / n exists as m and n approach infinity. Capon [1959b] has proven that the V statistic is asymptotically normally distributed when either HI, or K,, is true in a manner that differs from the proofs of either Mann and Whitney [1947] or Lehmann [ 19511.

116

5

Two-Input Detectors

The two-input version of the Wilcoxon detector or the Mann-Whitney detector is just as efficient as its one-input form. The S , S ’ , U,U‘,or V detectors are all extremely efficient for testing a positive location shift of the probability density function. This is not surprising since the MannWhitney U detector was specifically designed for the problem of testing whether one of two random variables is stochastically larger than the other. Engineering problems in this category are, using Capon’s [ 1959bl nomenclature, the dc detection problem and the noncoherent detection problem. These and other applications of nonparametric detectors are discussed in Chapter 9.

5.2.3 Normal Scores Detector The two-input version of the normal scores detector can be obtained in a manner directly analogous to that used to derive the one-input form of the detector. A slightly abbreviated derivation is given here, but no important. steps are excluded. Assume that two sets of independent, identically distributed observations, x = { x , , . . . , x,,} and y = { y , , . . . ,y,}, which have probability densities f(x) and g(y), respectively, are available at the input. The problem is to find the locally most powerful detector for testing H13: f(x) = g(y),f(x) is Gaussian with mean p , and variance u2 versus K13: f(x) = g ( y - S ) , g(y) is Gaussian with mean p2 and variance u2, and 8 = pz - p ,

To simplify notation, let the combined data set be represented by w = { w , , . . . , w,,, w,,+,, . . . , w,,+,} = { x , , . . . ,x,,, y , , . . . ,y,} with joint probability density h(w). The hypothesis and alternative to be tested can now be stated as H I , : h(w) = IIY. ,f(wi),f(wi) is Gaussian with mean p, and variance u2 K14:

h(w) = I I ~ - , f ( w i ) I I ~ - , , , , g ( w if)(,w i ) as given in HI4 and g(wi) Gaussian with mean p2 and variance u2

Let R = { r , , r2, . . . , rN} be the ranks of w = {w,, w2, . . . , wN}, respectively. In order to find the locally most powerful test of H , , versus K14, the likelihood ratio of the probability densities af the ranks of the observations, wi, i = 1, 2, . . . , n + m = N, underX,, and H , , must be formed. Since the density of the observations is symmetric under Hi4 and there are N ! possible permutations of rank assjgnments, the probability density of the ranks when H , , is true is l/N!. Since when K , , is true, h(w) is no

ZI 7

5.2 One-Input Detectors with Reference Noise Samples

longer symmetric, another approach must be taken to find the probability densities. The desired probability density f R ( RIKl4) can be expressed as fR(RIK14)

=J

f ( R , Z~lK14)d z ~

O
(5.2-61)


where zR is a dummy variable that represents the order statistics 0 < lwk,l < . . . < IwkNland f (R, ZRIK,,) is the joint density of the ranks and order statistics, given K,, is true. Using the law of conditional probability, Eq. (5.2-61) can be written as

which, since f(R ItR,K14) = l/N!, becomes

Multiplying and dividing under the integral sign by f(tRIHl4) yields

Following the argument in Section 3.5, instead of using the exact densities under HI, and KI4, equivalent density functions except for constant multiplying factors will be used. The density under the hypothesis will be Gaussian with zero mean and unit variance, and the density under the alternative will be Gaussian with mean 6 = ( p 2 - p,)/a and unit variance. Substituting for the densities of the order statistics in Eq. (5.2-64) and simplifying, fR

(

I

~ e x p( t(

5

i- 1

z;))

dz,

-

*

dz,

(5.2-65)

118

5

Two-Input Detectors

An examination of Eq. (5.2-65) reveals that the density of the ranks given can be written as

Kl4

where the ( ' I R , are the order statistics of a sample of absolute values from a Gaussian distribution with zero mean and unit variance. The likelihood ratio can now be formed

N

= exp(- $ma2)E

(5.2-67)

In order to find the locally most powerful test from this result, 6 is assumed to be small (weak signal case), and the exponential is approximated by the first two terms of its series expansion. Carrying out these operations on Eq. (5.2-67) produces the approximate result 1

+6

N i-n+l

N

1u14) = 1

+6 2

i=n+l

E{lu14}

(5.2-68)

An equivalent statistic to Eq. (5.2-68) is N

Cl(R) =

2 i-n+l

E{l'lR,}

(5.2-69)

which is the two-input version of the normal scores test statistic. The test is thus performed by ranking the two sets of received observations w = { w , , . . . , w,,, w , , + ~ ., . . , wN} = { x , , . . . , xn, y l , . . . ,y,}, and replacing the rank R = j by the expected value of thejth standard normal order statistic. The values thus obtained for the m y observations are then summed to obtain c , ( R ) which is compared to a threshold. If c , ( R ) exceeds the threshold, the alternative is accepted, otherwise, the hypothesis is accepted. Terry [1952] obtained the result in Eq. (5.2-69) in a slightly different fashion. He also showed that the variance of c , ( R ) , when the hypothesis that both sets of observations come from the same population is true, is

5.2

I19

One-Input Detectors with Reference Noise Samples

given by

mn- 1) var{cl(R)IH13} = N ( N

i-l

[ E { l t ~ l & } ] ~ (5.2-70)

It is emphasized that Eq. (5.2-70) is actually valid for the more general hypothesis than Hi3 that the two densities are equal and continuous, no other assumptions being required. The mean and variance of the normal scores test statistic under the general hypothesis Hi3: f(x) = g(y), f ( x ) and g(y) are continuous will now be derived following Kendall and Stuart [ 19671. Define % as 1, if the j t h rank is due to a y observation 0, otherwise The normal scores test statistic in Eq. (5.2-69) can thus be expressed as $'(

cl(R) =

N

2 E{j}q, J-

(5.2-71)

1

where E { j}denotes the expected value of the observation that has rank j when N observations are drawn randomly and independently from a standard normal distribution. The expected value of c , ( R ) given that Hi3 is true can be expressed as N

N

m ~ { c ~ ( ~ ) l=~ i '~ ~{ }j } ~ { 4 j l ~=i ' ~ }

2

2~

j= 1

J-1

{ j (5.2-72) }

The expected value under the summation sign in Eq. (5.2-72) is just the expected value of the j t h order statistic taken from a standard normal distribution. Since the probability density of the order statistics is simply equal to N! times the original density function, Eq. (5.2-72) can be written as m E{c,(R)IH;,} = -

N

2 N!E{z,}

J-1

(5.2-73)

where zj is an observation that has r a n k j in a sample of N observations taken from a zero mean, unit variance Gaussian density function. The expected value of z, will not, in general, be equal to zero, however, the summation over all j of E { z,} will be equal to zero from the symmetry of the Gaussian distribution. The expected value of c , ( R ) given Hi3 is true thus becomes

E{c,(R)IH;,}

=m

( -~ I)! (0)= o

(5.2-74)

5

I20

Two-Input Detectors

The variance of c , ( R ) when Hi3 is true can be found from the basic expression for the variance

1

N

2 E{j)q/lHi3

var{c1(R)IHi3) = -r(

j- I

(5.2-75)

by realizing that only the q, are random variables, since E { j } is simply a constant for eachj. Using this fact, Eq. (5.2-75) can be expanded as var{cdR)IH;3) = N

=

2

j - 1

N

2~

i- 1

N

2

{ i } E { j ) var{q/) j- 1

( E { j ) ) 2var{q,) +

N

N

2 2 E{

i-1 j - 1 i# j

i } { ~j ) cov{qi, q) (5.2-76)

Working now with the covariance expression alone,

cov{qi, $1 = E { [ q i - ~(qi)][qj-

‘(%)I}

where qk = E(qk). The first term on the right-hand side of Eq. (5.2-77) becomes upon expansion N

N

N

N

n

N

N

The expression for the covariance thus simplifies to

(5.2-78) Writing Eq. (5.2-76) with the double summation expanded yields

5.2

One-Input Detectors with Reference Noise Samples

I21

Remembering that 2 y - I E ( j } = 0 and noting that var(q} = ( 1 / N ) x:-l[qk = var{%}, independent of j , the variance of c , ( R ) becomes N

var(c~(R)IH;3)= var{q}

( ~ [ j ] )+’ -

(5.2-80)

The variance of q can be determined from l

var{ 4 ) = -

N 2 [qk - qkI2

&=I

. ~r

(5.2-81) Substituting this result into Eq. (5.2-80),

(5.2-82)

which is the variance of the two-input normal scores test statistic when Hi3 is true. Terry (19521 has shown that the ratio

has a limiting Gaussian distribution with zero mean and unit variance as N + 00, as long as m / N is bounded away from 0 and 1. Terry [ 19521 also demonstrated that the critical values of c , ( R ) can be computed using an approximation based on the Student’s t distribution. Fisher and Yates

122

5

Two-Input Detectors

[ 19491 tabulated probabilities which can be used in the computation of the cI test, and Terry [ 19521 used these tables to compare the exact distribution

of c , ( R ) for small n, m,and N,N 4 10. The normal scores test thus combines the generality and relative simplicity of a nonparametric detector with the power of a parametric detector when testing for a weak dc signal in Gaussian noise. The normal scores detector is even more efficient than this fact indicates. The efficiency of the c, test is discussed in more detail in Chapter 6. 5.3

Two Simultaneous Input Detectors

In the book thus far, the discussion has been limited to consideration of the detection of a positive dc signal in statistically defined or undefined noise. In this section the problem to be investigated will be that of detecting a stationary random process embedded in noise backgrounds of various statistical descriptions. This detection problem commonly arises in sonar, acoustics, geophysics, and radio astronomy. The detectors developed in this section have two simultaneous inputs. When no signal is present, the two inputs are assumed to be independent noise processes. When a signal is present, however, it is present at both inputs simultaneously along with the independent noise process. Each input is assumed to be sampled at a rate such that each of the resulting observations are independent random variables. An optimum detector in the Neyman-Pearson sense is derived first in this section by assuming that the signal process and the two noise processes are independent Gaussian processes with known means and variances. The detection problem is then slightly generalized to the case where each of the signal and noise processes are still Gaussian, but the means and variances are assumed to be unknown. The UMP unbiased detector for this problem, the sample correlation coefficient, is derived. The restriction of the detection problem to Gaussian processes is then lifted, and the application of the polarity coincidence correlator (PCC) and the rank correlation detector to this most general detection problem is discussed.

5.3.1 Parametric Detectors In this section the signal and noise processes are assumed to have Gaussian distributions with known means and variances, and the UMP

5.3

123

Two Simultaneous Input Detectors

detector for this problem is determined by utilizing the Neyman-Pearson fundamental lemma. Let x = { x I ,xz, . . . , x,,} and y = { y l , y z r. . . ,y,,} denote the n independent, identically distributed observations obtained from the two simultaneous inputs. If no signal is present, x and y will consist of samples of independent noise processes. If a signal is present, x and y will consist of samples of a random signal plus the independent noise process in each channel. Therefore, when a signal is present, the two inputs will be correlated. The detection problem can thus be explicitly stated as testing HIS: X ( x , y) is Gaussian with r = 0 versus K15: X(x, y) is Gaussian with r

>0

where X(x, y) is the joint cumulative distribution function of x and y, and r is their correlation coefficient. Assume now that each x has distribution function F ( x ) with mean px and variance u:, and that each y has distribution function F ( y ) with mean py and variance a.; If the signal has a known mean ps and an unknown variance uf, the correlation coefficient under K,,can be written as r= q

x

- P X ] [ V - Py]IK15}

'

(5.3-1)

axuy

Considering the case where the two noise processes are zero mean with identical variances u;, the received processes at the two inputs can be represented by x = s n 1 andy = s + nz, where s denotes the signal and n, and n2 denote the noise processes. Since n 1 and nz are independent and zero mean, Eq. (5.3-1) can be rewritten as

+

(5.3-2) In amving at Eq. (5.3-2), use has been made of the fact that the variance of the sum of two independent Gaussian processes equals the sum of their respective variances.

5

124

Two-Input Detectors

The modified detection problem can be stated as testing

Hi,:

X(x, y) is Gaussian with r

=0

versus 0 ’,

K;5: X(x, y) is Gaussian with r = -> O (7’,

+ u;

To determine the most powerful detector for this problem, the probability densities when H i , and K;,are true must be obtained. When H i , is true, the joint probability density is simply the product of the individual x and y densities, ”

where it has been assumed that n observations are available. When true, the joint probability density becomes

n n

X(X,

YIK;,) =

i=l

4,is

1

27r(u,2 + u;)2

(xi - P , ) - 2r(xi -

- H)

~ s ) ( ~ i

uo’

+ u,’

+

(yi - ~ 1 , ) ~ u;

+ u,’

(5.3-4)

Since the mean value of the signal p, is assumed to be known, ps can be set equal to zero without loss of generality. Equation (5.3-4) thus becomes n

X(X,

YI&)

=

fl

i=l

27r(u,2

+

1

Equation (5.3-5) can be further simplified by noting that u,” + u; = u,”/ ( 1 - r), and therefore

5.3 Two Simultaneous Input Detectors

125

Using Eqs. (5.3-3) and (5.3-6) to form the likelihood ratio, X(X,

L(x'

YIK;,)

= x ( x , YlHi,)

=

fi

0 0 '

;,(

+ ,-,;)dm

-I [ -x,? - y,? - rx,? - ry? 24(1 + r)

+ x,? - 2rxiyi + $1

Taking the natural logarithm of Eq. (5.3-7) and comparing the result to a threshold yields

(5.3-8) Since the first term to the right of the equals sign in Eq. (5.3-8) and the multiplier of the summation in the second term are constants, both can be included in the threshold setting. This modification results in a test of the form n

2 ( x i + yi)'

i- 1

K

3 T' H

(5.3-9)

which is the most powerful detector in the Neyman-Pearson sense for testing Hi,versus K;, when all processes have zero mean. If much less information is known about the signal and noise processes, a most powerful detector can usually still be derived. For instance, assume in the previous problem that it is still known that the probability densities are Gaussian, but the parameters px, p,,, ,:u u;, and r are all unknown. Following a procedure already used twice in earlier sections, the parameters can be estimated using the method of maximum likelihood, the

5

I26

Two-Input Detectors

resulting estimates substituted for the appropriate parameters in the density functions, and a most powerful detector determined from the likelihood ratio of the two probability densities. T h e detection problem now being considered is that of testing the hypothesis

H I , : X(x, y) is Gaussian with r = 0 and p,, py, u;, u; unknown versus K16: X(x, y) is Gaussian with r > 0 and p,, py, u:, I$ unknown A most powerful detector for testing HI6 versus K16 will now be found using the method outlined above. Writing the probability density under

and taking its natural logarithm yields n 2 2 2 In ~ ( x YIH,,) , = - 7 ln[ ( 2 ~ )OxOy] Yi -

2

Py

(5.3-10)

i-1

In this equation, px, 4, uz, and u;, are all unknown and therefore must be estimated using the method of maximum likelihood. Taking the derivative of Eq. (5.3-10) with respect to each of the parameters p,, py, uz, and and equating all four of the results to zero produces, (5.3-1 1) (5.3-12)

all of which must be solved simultaneously to obtain the desired parameters. Equations (5.3-1 1) and (5.3-12) can be solved immediately for fix and

127

5.3 Two Simultaneous Input Detectors

fi,,,

since they are essentially independent of 1 "

fix = -

2 xi

and

i-1

u,'

and uy, to yield 1 "

fiy = -

2 yi

(5.3- 15)

i-1

Equations (5.3-13) and (5.3-14) can then be solved just as simply to yield 1 " 6; = - 2 (Xi-

fix)2

(5.3- 16)

1 2 " ( y i- fiy)2

(5.3-17)

i-1

and -2 uY

E

n

i l l

where jix and jiy are given by Eqs. (5.3-15). Using these estimates, the probability density under HI, now becomes

(5.3-18) Now writing the probability density when the alternative K , , is true

and taking the natural logarithm of the density

(5.3-20) Taking the derivative of Eq. (5.3-20) with respect to the parameters px, cc,,

5

128 a:, a;,

Two-Input Detectors

and r individually and equating to zero,

a In X(x, yIK,,) aa; -(xi i- 1

-0

(5.3-21)

=o

(5.3-22)

-n 20:

= -2 PX)

2(1

-

+ r(Xi -2

~

1')

x

)( U i

2~;

-~

}=O

y )

(5.3-23)

axay

OX

a In X(x, y(KI6)= -n aa;

1

1

2(1 - r2) (5.3-24)

(5.3-25)

Equations (5.3-21) and (5.3-22) reduce to (5.3-26)

and (5.3-27)

5.3

129

Two Simultaneow Input Detectors

These two equations can be satisfied if n

n

2 xi-

or equivalently,

npx =

2 yi-

npy = O

i- 1

i- 1

(5.3-28) Equations (5.3-23) and (5.3-24) can be factored as

a In W x , yIK16) -- - 1 au; 2031 - r 2 )

+ r 2

(Y; -

n(1 - r 2 ) -

(xi -

Py)(X;

-

Px)

)

~

2

x

)

ox

i-1

0yax

i= 1

2

=O

(5.3-30)

and also Eq. (5.3-25) as

r2) + ((1 l -+ r2$

i

}=O

(5.3-31)

i-1

Eliminating the factors outside the brackets and adding Eqs. (5.3-29) and (5.3-30) produces

(5.3-32)

130

5

Two-Input Detectors

After eliminating the factor outside the brackets and multiplying by (1 - r 2 ) / r , Eq. (5.3-31) is subtracted from Eq. (5.3-32) to yield

or n(1 - r’) =

(1 - r’)

5

(Xi

- Px)(Y;

i-1

- Py)

axay

which can be solved for r to obtain 1

;=

2” ( X i

i-1

- ANY; SXey

fiy)

(5.3-33)

The estimates of u,’ and a? , can be found by substituting Eq. (5.3-33) for r in Eqs. (5.3-29) and (5.3-30),

and

and thus, (5.3-34) (5.3-35) With Eqs. (5.3-28), (5.3-33), (5.3-34), and (5.3-35) specifying the maxi-

5.3 Two Simultaneous Input Detectors

131

mum likelihood estimates for fix, Ci,, i, 6:, and 6;, respectively, the probability density under K]6 becomes

The likelihood ratio can be formed by dividing Eq. (5.3-36) by Eq. (5.3-18). Since Eqs. (5.3-15) are identical to Eqs. (5.3-28), and Eqs. (5.3-16) and (5.3-17) are the same as Eqs. (5.3-34) and (5.3-39, respectively, the likelihood ratio immediately simplifies to

(5.3-37) If L ( x , y) in Eq. (5.3-37) is compared to a threshold,

but this is the same as

or equivalently,

The detector for testing the hypothesis H I , versus the alternative K,, thus

I32

5

Two-Input Detectors

1

"

is 1 " K

2 T' (5.3-38) H

which simply consists of comparing the sample correlation coefficient defined by r^ in Eq. (5.3-33) to a predetermined threshold. Throughout the derivation of the detector described by Eq. (5.3-38), it has only been claimed that a most powerful detector has been sought. This is because since the maximum likelihood estimates of parameters were used in the probability densities, the Neyman-Pearson lemma does not necessarily yield the most powerful detector for testing H I , versus K16, but only a most powerful detector in some unspecified sense. It can be shown that the detector defined by Eq. (5.3-38) is the UMP unbiased detector for testing HI6 versus K16. The proof of this fact is outlined below. It can be shown that the detector given by Eq. (5.3-38) is UMP similar for testing H I , versus Kl6 by following step-by-step Lehmann's proof [Lehmann, 19471 that the circular serial correlation coefficient is UMP similar (Lehmann also demonstrates that the circular serial correlation coefficient is UMP unbiased). Accepting this fact and remembering the discussion in Section 2.6, it only remains to be shown that the test statistic, the sample correlation coefficient, is unbiased. First of all, note that r^ can be expanded and written in the form

(5.3-39)

133

5.3 Two Simultaneow Input Detectors

where

and 1 " Now let the value pa be chosen such that

(5.3-40) when r = 0. Consider now a case when K16 is true, that is r

=

P( w

> 0, then

- > ~s ~ ~1, I K , , }

> p { xy -v - IE(XY) - E ( X ) E ( Y ) l > SfPalK,6) -

=

Therefore,

P{

XY

-v - IE(XY) - E ( X ) E ( Y ) l > palK16

QD(F) > a

st

for all F E K16

I

=a

(5.3-41)

and from Eq. (5.3-40),

E { D ( x , y)lH16} = a

for all F E

HI6

(5.3-42)

The sample correlation coefficient is thus unbiased at level a for testing HI6 versus K16. Since the detector defined by Eq. (5.3-38) is also UMP similar for this problem, Eq. (5.3-38) specifies the UMP unbiased detector for testing HI6 versus K16. It should be noted that this detector is not UMP unbiased for testing the hypothesis r = rl versus r = r2, where rl # 0 and r2 # 0 [Kendall and Stuart, 19671. To determine the required threshold setting to achieve a chosen false alarm rate, it is necessary to know the probability density of the sample correlation coefficient when HI6 is true. Kendall and Stuart [1969] have

5

I34

Two-Input Detectors

noted that the quantity (5.3-43) has a Student’s t distribution with n - 2 degrees of freedom under H I 6 . The threshold, say T,, that achieves the desired false alarm rate a. can thus be found from tables of the Student’s t distribution and then transformed according to T, T; = I (5.3-44) [ n - 2 T:]’

+

to obtain the threshold for the sample correlation coefficient. It is not too difficult to derive the large-sample mean and variance of i, however, these quantities are of limited usefulness and thus only the results are stated here. As the number of samples, n, approaches infinity, .the distribution of i very slowly approaches a Gaussian distribution with mean r and variance (1 - r 2 ) * / n [Kendall and Stuart, 19671. In fact, the distribution of F tends to normality so slowly that such an assumption is not valid for n less than 500; hence, the statement concerning the asymptotic mean and variance made at the beginning of the paragraph. Kendall and Stuart [1969] have derived the large sample variance of F, and they give an expression for the mean of F from which the asymptotic value is easily obtained [Kendall and Stuart, 19671. The slow convergence of the distribution of i to normality should serve to emphasize to the reader that great care must be exercised when using asymptotic distributions of test statistics. The next logical step in the further relaxation of the structure of the hypothesis and alternative is to assume that the joint probability density X(x,y) is completely unknown. The hypothesis and alternative to be tested thus become H17:

versus

X(x,

Y)

= F(x)G(y)

K17: r > 0, X(x, y) unknown The reader may have noticed that in the hypothesis H,,, no specific statement is made concerning the value of the correlation coefficient r, whereas the correlation coefficient is specified to be positive in Kl,. The alternative K,,thus implies that some decision is to be made concerning the value of r, but Hi7 seemingly says nothing at all about r. Actually, Hi7 does make a statement about the value of r. This can be seen by the following discussion.

5.3

Two Simultaneous Input Detectors

I35

Consider the hypothesis H I , , which states X(x, y) = F(x)G(y). This simply says that the joint distribution function of x and y factors into the product of the marginal distributions of x and y, which means that x and y are independent. If x and y were Gaussian distributed, then independence would imply r = 0. One would thus be tempted to conclude that in general, zero correlation implies independence. This is not true, however. In fact, Kendall and Stuart [I9671 have cited the example of a bivariate distribution with uniform probability over a unit circle centered at the means of x and y. For this example, x and y are shown to be nonindependent and also to have zero correlation. In spite of this last discussion and realizing that there is some inconsistency in its specification, we will let r = 0 imply H I , for our purposes. When H I , is true, the sample correlation coefficient i is nonparametric for sample sizes as small as 11. The sample correlation coefficient is also consistent when K,,is true for distributions under K,,for which x and y dependent is equivalent to r > 0, which are essentially those cases being considered here. When dependence between x and y is accompanied by r = 0, i is not consistent under K,,nor does i remain nonparametric for distributions under H,,. As should be clear from the preceding discussion, there are several difficulties with employing i as a detector test statistic, particularly when one desires a detector that is nonparametric or distribution-free. Perhaps the major difficulty is that i, and hence its distribution, depends on the actual observation values x andy. This produces two problems. First, for a small number of observations, say 5 to 10, the distribution of i is fairly difficult to compute. Second, the dependence of the distribution of i on the sample values means that F is only asymptotically nonparametric, although F tends to normality rapidly when H I , is true. Because of these problems, the possibility of finding a detector that is nonparametric and whose distribution is relatively simple to compute, or can be easily tabulated, is attractive. The following sections discuss three detectors that satisfy both of these properties.

5.3.2 Polarity Coincidence Correlator The polarity coincidence correlator (PCC) is a nonparametric detection device that, as its name suggests, uses only the polarities of the two sets of observations to arrive at a decision. The PCC has been applied to problems in geophysics [Melton and Karr, 19571 and acoustics [Faran and Hills, 1952a, b], and has been studied extensively by various authors [Ekre, 1963; Kanefsky and Thomas, 1965; Wolff et al., 19621. Actually, the PCC detector was first investigated by Blomqvist [1950] under the name of

136

5

Two-Input Detectors

medial correlation test. The PCC has been shown to be inferior to the familiar correlation detector given by

x n

K

xiy, 2 T H

i-1

when the noises and signal are stationary random processes, but clearly superior when the noise is nonstationary. The test statistic for the PCC detector is n

Spec =

2

i- 1

(5.3-45)

sgn(xi) sgn(Yi)

where the xi andy,, i = 1, 2, . . . , n are the two sets of observations which consist of either noise alone xi = uli and y, = u2,, or signal plus noise xi = s, + uli andy, = s, + u2,. The s,, u l i , and uZi sequences are assumed to be stationary, independent random variables with the same probability distributions of all orders. The probability distributions are also assumed to have zero median, to be symmetric with respect to the origin, have finite mean square value, and to be absolutely continuous. The hypothesis and alternative to be tested can be stated as H I , : xi = u,,; yi = u2, (5.3-46) K,,: xi = s, + u,,; y, = s, u2, i = 1, 2, . . . , n. The probability distribution of S,, when H I , and K , , are true can be determined as follows. When H I , is true, that is when noise alone is present, the probability of the term under the summation sign being + 1 is

+

p{sgn(xj) sgn(yi) = + l I ~ l 8 } = P {xi andy, > 01H,,} + P {xi andy, = p{xj =

> OIH~~}P{Y~ > OIH,,)

(f ) 2 + (+)2=

+

< OIH,,) + P{xi < OIH,,)P{yi < OIH1,}

and similarly,

t

P {sgn(xj) sgn(yi) = - ~ I H I ~ = } When the alternative K,, is true, the probability of the value P {sgn(xj) sgn(Yj) = + l I ~ 1 8 1 = P{xiandy, W

P{uli W

+ 1 is

> OIK,,} + P{xiandy, < OIKI,)

> -uanduZi >

P { u l i < - u and uZi

-u}f,(u)du

< - u } ~ , ( u )du

(5.3-47)

5.3 Two Simultaneous Input

I37

Detectors

which is simply the probability that both arguments of the sgn( .) terms are greater than or less than - u, integrated over all possible values of u, the signal to be detected. Since the two channels are independent and the noise distributions are symmetric with respect to the origin, the probabilities under the integral signs can be written as P { o , , > -uandozi > - u } = P { o l i > - u } P { o z i > - u } = {1

- F,(-u)}2

and

P{uli < -uanduzi The probability of a

<

- u } = P{uli

<

-u}P{uzi

<

- u } = F:(-u)

+ 1 given K , , in Eq. (5.3-47) thus becomes

P { sgn(xi) sgn(Yi) = + 1IK1*} =J-l[l =

J-:[

- ~ u ( - u ) ] % ( ~ ) d u + J - w00F,2(-u)f,(u)du 1 - 2F,(-u)

+ 2 F , 2 ( - u ) ] f , ( u )du

(5.3-48)

Making a change of variable by letting h = - u , Eq. (5.3-48) can be written as

P { sgn(xi) W(

Yi)

=

+ 1 IKIg}

=1I -2F,(X) :[ + 2F:(X)]f,(X) dX A p +

(5.3-49)

since f,(.) is symmetric, and where p + is defined by Eq. (5.3-49). Further, since F,(A) has zero median, F,(X) can be expressed as (5.3-50) F,(h) = F,(O) + Q @ ) = 1 / 2 + QW

where Q (0) = 0 and Q (A) is positive or negative if X is positive or negative, respectively. Using this definition, the expression for p + becomes

P+

-I1 -: 2 ([ 4 ) - 2 Q @ ) + 2 ( : ) + 2Q(h) + Q'@)]fS(X) =

+ 2 J-mm QZ(h)fS(h)d X > t

(5.3-5 1)

Hence, for any nonzero signal that is present in both channels, p + is greater than 4. The probability distribution of the test statistic S , can now be determined by noting that each of the terms in Eq. (5.3-45) is independent and hence

P { S,

=

k

- "p} = ( j l ) p * q n - *

= b(k;n,p)

5

I38

Two-Input Detectors

where b ( k ; n, p) denotes the binomial distribution with probability of success p depending on HI, and K18. When Hi8 is true, p = 4 while p = p + when K , , is true. Tables of the binomial distribution are readily available [National Bureau of Standards, 1950; Romig, 1953; Appendix B], and therefore this probability is easily calculated. Since the probability of false alarm a can be written as a = P{S,,,

a

TIH,,} =

5 (;)(1/2)71/2)n-k

(5.3-52)

k = T'

Equation (5.3-52) and the fact that T = T' - np can be used to find the threshold T to achieve the desired false alarm rate. With p = p and T as chosen above, the probability of detection 1 - p can be computed for the specified detection problem. The following example illustrates the use of Eqs. (5.3-5OX5.3-52), the selection of the threshold T, and the calculation of 1 - p, the probability, of detection. +

Example 53-1 Let

=-(0, 1,

fu(u)

then,

-$
F,(u)

and

f,(s) =

=su,

fo(X) dX=

u

( 0,"

- f < S < )

otherwise

+f

- 5

By comparing Fu(u) with Eq. (5.3-50), it can be seen that for this example Q ( o ) = u. Hence, using Eq. (5.3-51),

=$+2[&+&]=$ Suppose that it is desired to fix a = 0.25 and n = 10, then a =

10

2

k- T'

(:)(

i)< k

0.25

From Table B-1, T' = 7 for a = 0.172, and the test threshold is T = T' np = 7 - lo(+) = 2. The probability of detection for this value of T is 1 -

p=k

10

- T =7

= 0.57

(;)(

$)k(

f)'"-*

5.3

Two Simultaneous Input Detectors

139

It is frequently necessary to evaluate detector performance when the number of observations available is very large. Since tables of the cumulative binomial distribution are easily accessible, the exact probabilities can be calculated for these cases. For large sample sizes (numbers of observations) the desired probabilities can also be computed using the normal approximation to the binomial distribution. For n large, the statistic

(5.3-53) has a Gaussian distribution with zero mean and unit variance, since S,, has a binomial distribution [Feller, 19601. The mean and variance of S,, under the hypothesis and alternative can be found by noting that

x n

SPCC

=

;=I

wi

is a summation of n independent, identically distributed random variables. The mean and variance are therefore given by E {SPCC} = nE P i ) var{ Spcc}= n var{ wi}

which become

+ ( + l ) p =2p - 1 - p ) + (+1)2p = 1

E{w;} = (-1)(1 - p ) E{W;}

= (-1)2(1

var{ wi} = E { w;} - E’{ wi} = 4p( 1 - p ) The probability p = 4 if H , , is true, and p = p if K18 is true. Equation (5.3-53) can now be used to obtain expressions for the false alarm and detection probabilities as follows. The false alarm probability can be written as +

since E { w i l H l , } = 0 and var{wilH,,} = n. Equation (5.3-54) can be simplified to

/“;I

c ~ = l - @[ T

(5.3-55)

by noting that SkCc has a Gaussian distribution with zero mean and unit variance or by making a change of variable. Similarly, since E { wilK1,) = 2p+ - 1,

var{ wilK1g} = 4p+ (1 - P +

5

140

Two-Input Detectors

the probability of detection is given by 1-

p

=

P{SKC > TIK,,}

= 1

T - n(2p+ - 1)

-@

[ 4 w + (1 - P +

11

1

I

(5.3-56)

Equating LY to the desired false alarm rate and using Eq. (5.3-55), the threshold T of the test can be determined. Using the resulting value of T, the probability of detection can then be computed.

Example 53-2 Reworking Example 5.3- 1 with the normal approximation of Eq. (5.3-53), the false alarm probability is given by = 1

- @[ T / m ] < 0.25

From a table of the cumulative normal probability distribution, it is found that T / m = 0.68 (or T = 2.15) for a = 0.248. Using this value of T , the probability of detection is l-P=l--@ = 1

(

2.15 - 10(1/3) [80/9]+

- @{ -0.396) = 1 - 0.346

= 0.654 Comparing the results of Examples 5.3-1 and 5.3-2, it is seen that a higher detection probability is achieved by using the normal approximation. This is true because a false alarm rate much closer to the acceptance level is attainable by passing to a continuous probability density. Whether the detector based on the normal approximation will actually perform better depends on the accuracy of the approximation in this case [Feller, 19601.

has a binomial distribution for any signal Since the test statistic ,S and noise probability distributions which belong to the class described at the beginning of this section, the PCC detector is nonparametric for the detection problem considered here. That is, the false alarm rate is independent of the probability distribution of the input. That the PCC detector is also consistent can be established by solving Eq. (5.3-55) for T and substituting into Eq. (5.3-56) to obtain l-p=l-@

=1-@

{ {

fi@-yl-

- n(2p+ -1)

ff)

[4nP+ (1 - P + W'(1 -

ff)

> 31

-fi(2p+ -1)

2[P+ (1 - P +

1 >I2

I

I

(5.3-57)

5.3 Two Simultaneous Input Detectors

141

Taking the limit of Eq. (5.3-57) as n + 00, 1 - p + 1, and hence from Eq. (2.6-13), the PCC detector is consistent for the present hypothesis and alternative. The PCC detector is at least asymptotically unbiased as can be seen by a straightforward manipulation of Eq. (5.3-57). That the PCC detector is unbiased for small sample sizes can be seen by noting that the PCC is a bivariate generalization of the sign test in Section 3.3, and as such, it is uniformly most powerful for testing the hypothesis and alternative H19: p = 5 1 ; K19; p >; wherep is the probability of the summand of S,, being + 1 [Carlyle and Thomas, 19641. Since a detector that is UMP is necessarily unbiased, the PCC detector is unbiased for testing H I , versus KI9. Note that given the assumptions concerning the probability distributions of the signal and noises at the beginning of this section, the problem of testing H I , versus K,, is equivalent to testing H I , versus K19. The PCC detector thus possesses the desirable properties of being nonparametric for HI8,consistent for K18, and unbiased for testing HI8 versus KI8. In addition, the PCC detector is simple to implement and its test statistic has a probability distribution that is extensively tabulated. All of these facts combine to make the polarity coincidence correlator a highly attractive detector for testing H I , versus K I 8 .The performance of the PCC detector relative to other detectors for testing other hypotheses and alternatives still must be evaluated, and this evaluation is accomplished in Chapter 6. 5.3.3

Spearman Rho Detector

It is clear from the previous section that a large amount of information available in the input observations is being discarded by the PCC detector. At the same time, however, the development of the sample correlation coefficient r in Section 5.3.1 indicates that the utilization of the actual values of the observations in the test statistic causes the false alarm rate to be dependent on the probability density of the observations.The Spearman rho ( p ) detector developed in this section utilizes only the ranks of the input observations, and therefore employs more of the information available in the observations than the PCC detector while retaining the desirable property of being nonparametric. The test statistic for the Spearman p detector is given by [Spearman, 19041

5 [R(xi)- i ( n

P'

i- 1

fin(.

+ l ) ] [ ~ ( y i )- f ( n + 113

+ l)(n - 1)

(5.3-58)

5

142

Two-Input Detectors

where R ( x i ) and R ( y i ) are the ranks of the xi and y i , i = 1, 2, . . . , n, observations. Specifically, R ( x i ) = 1 if xi is the smallest in magnitude of the xi, i = 1, 2, . . , , n, and R ( x i ) = 2 if xi is the second smallest in magnitude and so on. The observation with the largest magnitude has rank n. For the case when there are no tied observations, the p test statistic in Eq. (5.3-58) is actually just the sample correlation coefficient in Eq. (5.3-33) with the xi andy, replaced by their corresponding ranks. To see this, consider the sample correlation coefficient with the xi and yi replaced by their ranks, which is given by

The sample mean of the ranks is easily computed to be

and (5.3-60b) where use has been made of the fact that X 7 m l i = n(n + 1)/2 [Selby and Girling, 19651. The sample variance can be found similarly

1

i:

i-1

and

-

1 2 " [i [ R ( X i )- R ( x ) I 2 - i-1

F] 2

( n + l)(n 12

- 1)

(5.3-6 1b)

5.3 Two Simultaneous Input Detectors

since n

i2=

n(n

i= I

143

+ 1)(2n + 1) 6

[Selby and Girling, 19651. Substituting the above results into Eq. (5.3-59) yields Eq. (5.3-58) as desired. Another form of the Spearman p test statistic can be obtained by defining

n

substituting the results in Eqs. (5.3-61a, b) into Eq. (5.3-62), pT = g n ( n 2 - 1)

+ g n ( n 2 - 1)

n

and rearranging, to obtain

i= 1

Using Eq. (5.3-63) and Eqs. (5.3-61a, b) in Eq. (5.3-59) yields

Hence, an equivalent form for the Spearman p test statistic is P=

- 6PT n ( n 2 - 1)

(5.3-64)

where pT is given in Eq. (5.3-63) [Gibbons, 19711. A detector that uses the test statistic pT instead of p is called the Hotelling-Pabst test in recognition of their work concerning the nonparametric property of the Spearman p detector [Hotelling and Pabst, 19361. The advantage of using the Hotelling-Pabst test is that it requires

144

5

Two-Input Detectors

fewer arithmetic operations than the Spearman p detector. Note that the Hotelling-Pabst test statistic varies in an inverse fashion relative to the p statistic; that is, pT is large when p is small, and pT is small when p is large [Conover, 19711. The Spearman p detector is useful in engineering applications for testing the hypothesis and alternative stated earlier in Section 5.3.1 X(x, Y) = F ( x ) G ( y ) K,,: r > 0, X(x, y) unknown

H17:

where r is the correlation coefficient. Since H I , specifies that the x and y observations are independent, each of the possible ranks of the xi andy,, i = 1, 2 , . . . , n, are equally probable and the Spearman p detector is nonparametric for testing the hypothesis H17. The mean and variance of the Spearman p detector under the hypothesis of independence, H I , , can be calculated by first expanding Eq. (5.3-58) to obtain

-

i- I

n ( n 2 - 1) n

(5.3-65)

To determine E { p I H I 7 } and var{plH,,} then, it suffices to compute E {El-,R( x i ) R (yi)lH17}and var{Zl, , R (xi) R (yi)IHl7} and use these results along with Eq. (5.3-65) to obtain the desired mean and variance. Since, when H I , is true, the observations xi, i = 1, 2, . . . , n and y,., i = 1, 2, . . . , n are independent, their ranks R ( x i ) and R ( y i ) ,i = 1,2, . . . , n will also be independent, hence

145

5.3 Two Simultaneous Input Detectors

and Var[

5 R (xi)R (Yi)IH17}

i= 1

var{ R (Xi)IHl7} var{

=

(Yi)IHI,}

(5.3-67) +n(n - 1) COV{R(xi)R(xj)lHl7} COV{R(Y~)R(Y~)IHI~} It is thus necessary to compute the mean, variance, and covariance of the ranks. When HI, is true, each of the ranks are equiprobable, hence 1 " (5.3-68) E{ R(xi)lH~,}= E { R ( Y ~ ) I H I ~=} - Z i= 2 +

i=l

and

(5.3-69) The variance is thus var{ R (xi)IHl7} = var{ R (yi)lHI7} = h(n2- 1) The covariance can be determined from the expression

(5.3-70)

(xi)R (Xj)lH~7}

'Ov{

=

{ (xi)R (X,)lH17} n

= n(n - 1)

{ R (xi)lH~7}E{ (Xj)IH~7}

(;I1)

2

n

xi

xxij-i l l j-1 i#j

:2

(5.3-71)

since there are n(n - 1) equiprobable combinations of the ranks. Expanding the first term in Eq. (5.3-71) as

x n

xn i j = ( i i )2 - i i '

i-1 j - 1 i #J

i=

I

i= I

and factoring l/[n2(n - l)] out of all terms, yield cov{ R ( x i ) R (Xj)lH~7}

(5.3-72)

5 Two-Input Detectors

146

Finally, from Eq. (5.3-65) {PIHi7)

=

12nE { (xi)IH~7}E{ (J’i)IH~7} 3(n + 1) n-1 n(n2 - 1)

=o (5.3-73)

1 n-1

(5.3-74)

I -

since for a, b constants var{x

+ u } = var{x}

and

var{bx} = b2 var(x}

The exact probability density of p when Hi7 is true can only be obtained by enumeration and has been tabulated by Kendall [1962]. For applications of p where n > 10, the asymptotic distribution of the random variable Z = p m is very useful, since Z is asymptotically Gaussian distributed with zero mean and unit variance [Gibbons, 19711. The following three examples illustrate the selection of the detector threshold T and the calculation of the detector test statistic: Example 53-3 Suppose it is desired to test HI, versus K,, given two sets of n = 10 observations and a maximum allowable false alarm rate of a = 0.1. From the table of the Spearman p test statistic quantiles in Appendix B, the value of the threshold is seen to be TI = 0.4424. If instead of the exact distribution the normal approximation is used, the value of the threshold (from tables of the standard Gaussian distribution in Appendix B), is T2 = 1.282. The two thresholds are quite different in value and at first glance one suggests that the normal approximation may not be too accurate for n = 10. However, recall that the random variable Z = p m is the statistic that is being compared to T2. In order to compare the thresholds = = 3 to obtain T2 = for p, T2 must be divided by 0.4273, which is much closer to TI.

147

5.3 Two Simultaneous Input Detectors

Example 53-4 Let the two sets of observations for testing H I , versus K,, in Example 5.3-3 be given by the tabulation shown with the ranks indicated directly below each observation. The value of p is 10

lO[(lO)* - 11

P=

[ l o - 11

12[4 + 70 + 81 + 42

E

=

-

+ 10 + 50 + 3 + 64 + 24 + 61 990

4.29 - 3.67 = 0.62 i : 1 5.2 R(x,): 4 yi: 3.2 R(y,): 1

2 11. 10 7. 7

x,:

3 9.1 9 8. 9

4 7.3 7 6.4 6

33 -9

5 3.2 2 6.1 5

6 6.1 5 10. 10

7 1.2

I 4.2 3

8 8.6 8 1.2 8

9 7. 6 5.1 4

10 4.1 3 3.3 2

Since 0.62 > T, and 0.62= 1.86 > T,, both the exact test and the normal approximation cause K , , to be accepted.

Example 53-5 Suppose that for the same hypothesis and alternative in Example 5.3-3, the two sets of observations shown in tabulation are taken with the ranks indicated directly below the corresponding observation in the table. Note that in this case, since negative observation values are present, it is necessary to take the absolute values of the observations before ranking. For example, the rank of y 3 = - 2.1 is R ( y 3 )= 7, and the rank of y,, = 2.3 is R ( y , ) = 8. Only magnitudes are used in assigning values for the ranks. i: 1 x,: 1.5

R(x& yi:

7 2.4

R(Y~): 9

2

3

4

8

3 -3 10

- 0.7

- 2.1

4 0.4

- 2.1

2

7

0.6

5 1.1 6

- 0.6 4

6 3.4 10 1.7 5

1 0.2 1 1.9 6

8 1 5 - 0.2 1

-

9 - 0.4 2 2.3 8

10 -2.3 9 0.5 3

The value of the Spearman p test statistic is thus P =

12[63 + 8

+ 56 + 30 + 24 + 50 + 6 + 5 + 16 + 271 - 3.67

= 3.45 - 3.67 = -0.22

Since -0.22

< T, and

-0.66

990

< T2,the hypothesis Hi7

is accepted.

5 Two-Input Detectors

I48

The Spearman p detector has a test statistic that is reasonably easy to compute, an exact distribution that has been tabulated, an asymptotic distribution that is easy to use, and it is nonparametric for testing the hypothesis of independence between the two sets of observations. The performance of the Spearman p detector relative to other detectors is considered in Chapter 6.

5.3.4 Kendall Tau Detector The Kendall tau ( 7 ) detector is another rank correlation statistic that is closely related to the Spearman p detector. The Kendall 7 detector is generally considered to be less efficient computationally than the Spearman p detector, but the Kendall7 detector possesses the advantage that its test statistic has an asymptotic distribution that tends to normality very rapidly [Conover, 1971; Kendall and Stuart, 19671. Perhaps the best way to introduce the Kendall7 test statistic is to treat a specific example, as done by Kendall [ 19621. Consider the two sets of observations in Example 5.3-4 whose ranks are given by the data shown in the tabulation where the letter column headings are included to allow an easy comparison of ranks. Of the xi ranks, consider the first pair of ranks AB. These ranks are 4 and 10, respectively, and clearly occur in the natural counting order. A score of 1 is therefore assigned to this pair. The pair of xi ranks BC is 10, 9. These ranks occur in inverse order and hence a score of - 1 is assigned to this pair. The pair of yi ranks AB is 1, 7 and the score assigned for this pair is 1. Now the two scores for the pair AB are multiplied together which yields a 1. The products of the two scores for each pair of ranks are then summed to obtain a total score, and the Kendall 7 statistic is calculated by dividing this result by the maximum total score.

+

+

+

A i : l 4

R(x,): R ( y, ) :

1

B

C

D

E

F

G

2 10 7

3 9 9

4 7 6

5 2 5

6 5 10

7 1 3

H 8

I 9

8 8

J 1

6 4

0 3 2

In order to further illustrate this procedure, the following example enumerates the steps in the calculation of the test statistic for the above set of ranks.

5.3 7bo Simultaneour Input Detectors

149

Example 53-6 Each pair of ranks from the preceding tabulation is listed (see table) with the score assigned to that pair of ranks: The total number of positive. scores, usually denoted by P, is 32 and the total number of negative scores, denoted by Q, is 13. Letting S = P - Q, the total score is thus S = 32 - 13 = 19. The maximum value of S is 45 and its minimum value is -45. The Kendall T statistic is therefore given by actual score 7' - P-Q = 19 = +0.42 maximum possible score 45 45 which indicates a significant amount of positive correlation between the x and y rankings. This result, of course, is as anticipated, since the Spearman p detector has already indicated significant positive correlation between the two rankings.

Pair Score AB AC AD AE AF AG AH A1

AJ

BC BD BE

+ 1 +1 +1 - 1

+ I

- 1 + 1 +1 - 1 - 1 + 1 +1

Pair

Score

Pair

Score

Pair

Score

BF BG BH BI BJ CD CE CF CG CH CI CJ

- 1 + 1 - 1 + 1 +1 + I +1 - 1 +1 +1 +1 +1

DE DF DG DH DI DJ EF EG EH EI El FG

+ 1 - 1 + 1

FH FI FJ GH GI GJ HI HJ IJ

- 1 - 1 +1 + 1

+ 1 +1 + 1 + 1 + 1 + 1 - 1 - 1 + 1

+1 - 1 + 1 +1 +1

The expression for the test statistic can be generalized by noting that each pair of the possible rank combinations is equally probable and the total number of possible combinations is given by

Using S to indicate the actual score, the Kendall defined as r =

Using the fact that P

Y

f n ( n - 1)

T

test statistic can be (5.3-75)

+ Q = f n(n - I), Eq. (5.3-75) can also be expressed

150

5 Two-Input Detectors

as 7 =

P-Q in(. - 1)

= I -

$.(.

2Q

- 1)

(5.3-76) (5.3-77) (5.3-78)

There are several simpler ways of calculating 7 than the approach used in Example 5.3-6 [Kendall, 19621. One of the better methods consists of arranging one set of ranks in the natural order with the rank for the corresponding observation listed under it. When this is done, all scores resulting from the natural ordered set of ranks are positive and only the scores due to the second set of ranks need be considered to obtain P. This method is illustrated in the following example. Example 53-7 Consider again the two sets of ranks from Examples 5.3-4 and 5.3-6 which are listed below with the xi ranks written in natural order. Note that the ranks of they, observations are also reordered. G E J A F I D H C B i: R(xj): R(y,):

7 5 1 0 1 1 2 3 4 3 5 2 1

6 9 4 8 3 2 5 6 7 8 9 10 10 4 6 8 9 7

In the above y rankings, the number of ranks to the right of R (y,) = 3 that are greater than 3 are 7 and hence, P = + 7. Similarly for R ( y s ) = 5 , P = + 5 . Continuing this procedure

P= +7+5+6+6+0+4+3+1+0=32 From Eq. (5.3-78), then 2(32) 1 = 1.42 - 1 = +0.42 45 as before. Of course, the number of negative scores Q also could have been counted using the above procedure and Eqs. (5.3-76) or (5.3-77) utilized to calculate 7. 7=--

The mean and variance of the Kendall 7 test statistic are not difficult to calculate, however, readable derivations are available from Kendall [ 19621

5.3

Two Simultaneous Input Detectors

I51

and hence this work is not repeated here. The mean of the test statistic when H , , is true is zero as might be expected, since all possible scores are equally likely and range in magnitude between 1 and - 1. The variance of the T statistic under HI, is given by

+

var{~lH,,}=

2(2n 9n(n

+ 5) -

1)

(5.3-79)

The utility of knowing the mean and variance of T under Hi7 is that the Kendall T test statistic tends to normality very rapidly, and therefore the asymptotic distribution can prove important in hypothesis testing. The approach usually employed to prove asymptotic normality is to show that the moments of S for n large tend to the moments of the Gaussian distribution. The second limit theorem is then invoked to establish asymptotic normality [Kendall, 1962; Kendall and Stuart, 19691. This approach can also be used to demonstrate the asymptotic normality of the Spearman p test statistic. The following example illustrates the application of the Kendall T statistic to the detection problem of Example 5.3-3.

Example 53-8 Again it is desired to test H , , versus K , , at a false alarm rate of a = 0.1 given two sets of n = 10 observations. From the table in Appendix B of quantiles of the Kendall statistic, the value of the threshold is T3 = 0.33. For n = 10, the variance of the statistic is var{T} =

2(20

i:o

+ 5) -- - - - 0.06

90(9)

For this case, the random variable T/has zero mean and unit variance under H,,. From tables of the standard Gaussian distribution in Appendix B, the required quantile is 1.282. The threshold for the Kendall T statistic is therefore T4= 1.282= 0.313. Since from Examples 5.3-6 and 5.3-7 7 = 0.42, which is greater than both T3 and T4,both the exact test and the normal approximation accept K17’

Note that in the previous example and Example 5.3-4, the Kendall T statistic is smaller than the Spearman p statistic for the same problem, but that both produced the same decision when testing HI, versus K17 at a level a = 0.1. These two occurrences remain true in general. Although the Spearman p and Kendall T statistics are computed in seemingly very different ways, the two statistics are highly correlated when H , , is true. The correlation between p and T varies from 1 when n = 2 to a minimum value of 0.98 when n = 5, and then approaches 1 as n + 00. The

5

I52

Two-Input Detectors

Spearman p and Kendall 7 detectors are thus asymptotically equivalent under H,,. Whether 7 and p remain highly correlated when the probability densities of the observations are correlated depends on the form of these distributions. Indeed, the correlation between p and 7 may be as small as zero or as large as 0.984 when the observations have a bivariate Gaussian distribution with correlation less than or equal to 0.8 [Kendall and Stuart, 19671. 5.4

Summary

Two different classes of two-input detectors have been developed in this chapter. Of the detectors in the first class, those that have one input plus a set of reference noise samples, the Student's t test was derived and shown to be asymptotically nonparametric and consistent as well as unbiased. Two nonparametric detectors in this class, the Mann-Whitney and normal scores detectors, were also developed and their use demonstrated. Two simultaneous input detectors, which include the sample correlation coefficient, the polarity coincidence correlator, the Spearman p detector, and the Kendall 7 detector, were developed next. These detectors are mainly useful for detecting a random signal and each perform some form of correlation between the two inputs or transformed versions of the two inputs. It was shown that the sample correlation coefficient is asymptotically nonparametric and is unbiased, while the PCC was demonstrated to be nonparametric, unbiased, and consistent. The nonparametric Spearman p and Kendall T detectors possess the important property of a rapid tendency to normality, and hence for moderate n have a very accessible distribution function. The following chapter develops the relative performance of the detectors described here and therefore allows useful comparisons between the detectors to be made. Applications of some of the detectors discussed are described in Chapter 9. Problems 1. In using the S' form of the Mann-Whitney detector given by Eq. (5.2-31), it is desired to test for a positive location shift in they, observations.There are n = 30 xi (reference noise) samples and m = 20y, samples.

If the value of the Mann-Whitney test statistic is given by 817, at what minimum significance level a will the hypothesis be rejected? Use the Gaussian approximation.

Problems

I53

2. Demonstrate that Eq. (5.2-74) is true by selecting a few cases from the table of expected normal scores. 3. Show that the test statistic in Eq. (5.3-9) has a x2 distribution with n degrees of freedom when H i , is true. 4. Suppose it is desired to test H I , against K , , using the Spearman p detector given the two sets of n = 5 observations i : l xi: 3.1 yi: 2.3

2 2.6 2.0

3 2.4 4.2

4 0.8 0.4

5 1.1 1.9

For a significance level a = 0.1, determine the threshold from both the exact distribution and the normal approximation. Compare the decisions reached using the two separate thresholds. 5. Test H I , versus K,, at a level a = 0.1 using the two sets of observations in Problem 4 and the Kendall7 test. Compute the threshold using the exact distribution only. 6. Given the n = 10 reference noise observations it is desired to test the m = 10 input observations for a positive location shift using the MannWhitney detector. Let a = 0.25 and use the exact distribution. n 10: 0.82 0.11 0.21 m = 10: 1.11 0.65 0.10

- 0.32 - 0.60 - 0.27 - 0.41 1.42 1.08 - 0.08 0.40

0.63 0.54 0.95 0.33

- 0.73 0.55

7. Use the PCC detector to test the two sets of n = 12 observations given below for the presence or absence of a stationary random signal at a significance level a = 0.10. Assume that all probability distributions have zero median. i: 1 2 xi: 0.81 1.63 yi: 0.76 0.89

3

- 0.42 - 0.66

4 0.08 0.53

5

- 0.11 0.21

6

- 0.55

- 0.30

7

- 0.03 - 0.45

8 9 0.90 0.78 1.07 0.95

10 11 - 0.04 2.21

- 0.01

1.77

12 1.43 1.39

8. Repeat Problem 7 using the Spearman p detector with a = 0.1. 9. Repeat Problem 8 for the sample correlation coefficient. 10. Compare the effort required in Problems 7-9 to calculate the values of the test statistics and to arrive at a decision. Which test seems preferable, the PCC detector, the Spearman p detector, or the sample correlation coefficient?

CHAPTER

6

Two-Input Detector Performance

6.1 Introduction

In Chapter 5 several parametric and nonparametric detectors which have one input plus a reference noise source or two simultaneous inputs were developed. These detecters provide the capability for tackling a wide variety of detection problems. However, as soon as one considers a specific application, a decision must be made as to which of the detectors to use. One is thus immediately led to the necessity of evaluating the performance of the two-input detectors presented in the preceding chapter. The tools available for the performance evaluation of two-input nonparametric detectors are the same as those which were utilized in Chapter 4 for the performance evaluation of the one-input detectors, namely the small sample relative efficiency and the ARE. In this chapter asymptotic results for the two-input detectors are derived first, followed by a presentation of some of the small sample performance comparisons given in the literature. Since ARE derivations for all of the detectors developed in Chapter 5 clearly cannot be given here, only examples of ARE computations are presented. In particular, the AREs of the PCC and the Mann-Whitney detectors with respect to optimum parametric detectors are derived in Sections 6.3, and 6.4, respectively, preceded by the development of some of the necessary asymptotic results for parametric detectors in Section 6.2. The AREs of other nonparametric detectors are then presented and discussed in Section 6.5. Section 6.6 details small sample comparisons of 154

155

6.2 Asymptotic Results for Parametric Detectors

the Mann-Whitney and normal scores detectors with respect to the parametric Student’s t test and the implications are discussed. A summary and conclusions concerning the utility and validity of the asymptotic and small sample performance indicators are given in Section 6.7. 6.2 Asymptotic Results for Parametric Detectors

In this section asymptotic results for parametric detectors with two simultaneous inputs are derived for use in evaluating the ARE of the PCC detector in Section 6.3. In particular, expressions for the probabilities of detection and false alarm for the sample correlation coefficient and the optimum Neyman-Pearson detectors derived in Section 5.3.1 are obtained when the number of samples available for the detection process is large. Actually, a nonnormalized form of the sample correlation coefficient called the correlation detector is used for the development here. The presentation in this section and Section 6.3 follows Wolff et al. [ 19621. The correlation detector test statistic is given by nl

Sc =

X xiyi

(6.2- 1)

i- 1

where the x i , y i , i = 1, 2, . . . , n, denote the two sets of observations obtained from the two simultaneous inputs. The test statistic in Eq. (6.2-1) is a sum of n, independent and identically distributed random variables, since the xi and y i sequences are independent, identically distributed random variables. For the case where the number of observations n, becomes large, the central limit theorem [Hogg and Craig, 19701 can be invoked to demonstrate that the statistic [ S, - E { S,}]/[var{ S,}]f has a Gaussian distribution with zero mean and unit variance. Assuming that the signal and noise are both Gaussian processes, the hypothesis and alternative being considered here can be explicitly stated as testing Hzo:

x and y are independent, zero mean Gaussian processes each

with variance ui versus

Kzo: x and y are zero mean Gaussian processes and consist of signal plus noise with the signal variance = u,’ and noise variance = ui The expected value and variance of the correlation detector test statistic

156

6

Two-Input Detector Performance

when Hzo and K,, are true can be found by inspection to be E

0

(6.2-2)

nIu$

(6.2-3)

{ ~ c l ~ = z o ~

var{ SJH,)

=

var{ s , I K , )

= n,(u$

+ 20,203

(6.2-5)

Now, the probability of a false alarm for the correlation detector is given by

where, as before, @ {. } is the unit normal cumulative distribution function. Similarly, the probability of detection can be written as l-P=l-@

= I - @

TI - nIu,’

[ n,(u$ + 2u;U:)l

[ @-‘[I

1

I

- a] - ( u & ? / u ; ) G

1

(6.2-7)

where this last expression results by solving Eq. (6.2-6) for TI and substitut1 ing into the first expression for 1 - fl and then factoring u;np out of the radical in the denominator. The optimum Neyman-Pearson detector for testing Hzo versus Kzo was derived in Section 5.3.1 and has a test statistic given by (6.2-8) i- 1

The mean and variance of Soptwhen H,, is true can be calculated as follows

= 2n,4

(6.2-9)

6.2 Asymptotic Results for Parametric Detectors

157

i=j

+ E { X$?IH~O} + E { x : x ; ~ H ~]~-} 4n$7: = 12n20,4

+ 4n2(n2 - 1)0,"- 4400"

(6.2-10)

which yields (6.2-1 1)

var{ SoptIH20} = 8 n 2 4

The middle term in Eq. (6.2-9) vanished since xi and yi are independent and zero mean when H20 is true. For the same reasons, terms which have xi or yi to the first power drop out of the variance calculation. The final result in Eq. (6.2-1 1) is obtained from Eq. (6.2-10) since E { x,41H20} = E { y41H2,} = 30;

for Gaussian random variables [Papoulis, 1965; Melsa and Sage, 19731 and E { XiZYi2IH20) = E

{ x:lH2O}E { Y:lH20}

=

00"

for xi and yi independent under H2@ Performing similar calculations when K20 is true, yields n1

E{SoptIK20} =

2 ( ( 4+ 0s') + 20:

i- 1

= 2n2(4

+ 20s')

+

( 4 + 4>} (6.2-12)

158

6

Two-Input Detector Performance

+ 20;) 4- n2(n2 - 1)(4(0,24+ 80,2(0,2 + us’) + 44‘) - 4n,2(a,2+ 2u:y = 8n2(~; + 20;)~ 2

= 3n2(40,2

2

( 1 ) ;

(6.2- 13)

Since Sop,is a sum of independent and identically distributed random variables, the central limit theorem can again be employed as n2 becomes large. The false alarm probability is therefore given by

Similarly, the probability of detection is 1-/3=1-0

T2 - 2n2(0,2 + 2 4 ) (0;

= I - @

+ 20,’)[

W l ( 1 - a) 1

8n2]

-a 2),./‘,.(

+ (20,2/0,2)

I

(6.2- 15)

where this last expression is obtained by solving Eq. (6.2-14) for T2 and substituting. Equations (6.2-6), (6.2-7), (6.2- 14), and (6.2- 15) are the desired asymptotic expressions for the parametric detector false alarm and detection probabilities. These equations are used with similar results for the PCC detector in the following section to calculate the ARE of the PCC and the correlation and Neyman-Pearson detectors. 6 3 ARE of the PCC Detector Expressions for the false alarm and detection probabilities of the PCC detector when the sample size is large have been derived in Section 5.3.2

159

6.3 A R E of the PCC Detector

and are given by Eqs. (5.3-55) and (5.3-57), which are repeated here for convenience with n = n3 a = 1-

@[ T

/ G

]

(6.3-1)

and l-P=l-@

W ' ( l - a) -$(2p+ 2[P+ (1 - P +

-1)

)I:

1

(6.3-2)

In order to evaluate Eq. (6.3-2), it is necessary to calculate p + given by Eq. (5.3-49) when K20 is true. It is desirable also to evaluate Eq. (6.3-2) for other non-Gaussian probability distributions, and hence a general expression for p + is required. A general expression for p + can be obtained if interest is limited to the case of vanishingly small signal-to-noise ratios and series expansions of the distribution functions F,(A) and F:(A) in Eq. (5.3-49) are employed. This development is somewhat tedious, however, and would contribute little to the present derivation; hence only the result is given here. The final expression for p + is [Wolff et af., 1962; Kanefsky and Thomas, 19651 (6.3-3) P + = i + 2(F: (A)lA-o)2tJ,2 where Fi(A)lA*o = [(d/dh)Fu(A)]1,,, and Fu(A) is the noise probability distribution function. Since for F,(A) zero mean and Gaussian, F: (A)lA=o = l / G uo,

Eq. (6.3-3) becomes 1 P+ = 2

+ -1 -

0,.

(6.3-4) uo' Substituting Eq. (6.3-4) into the probability of detection expression yields

1-p-

1-@

72

W ' ( 1 - a) - v/.3 (2/a)(u,2/uo')

'[ ( V 4 ) -

(1/722)(of/&]

= 1 - @( @ - y l - a) - (

2 6 /n)(u:/u;))

(6.3-5)

where the second-order term in u,'/uf has been neglected since it is assumed that u,"/u,' << 1. According to Eq. (2.7-1), the ARE is the limiting ratio of the sample sizes required by the two detectors to achieve the same false alarm and

160

6

Two-Input Detector Performance

detection probabilities. By equating the detection probabilities of the two detectors to be compared and solving for the ratio of the sample sizes, the necessary limits can be taken and the ARE calculated. Following this procedure, the ARE of the PCC detector with respect to the correlation detector can be found by equating Eqs. (6.2-7) and (6.3-5) to yield ( P - y 1 - a) -

(u:/U;)c

[ 1 + 2(0,2/0,2)] 4

1

= 1 - (PI W ' ( 1 - a) - ( 2 6 / n ) ( u : / u , l > )

(6.3-6)

or W ' ( 1 - a) -

(U;/u;)G

[ 1 + 2(u:/u,2)] 4

= W ' ( 1 - a) - ( 2 6 /n)(o,2/u,2)

(6.3-7)

-

Since u,'/u,' << 1, the denominator on the left-hand side of Eq. (6.3-7) is 1, and hence Eq. (6.3-7) reduces to n1/n3 =

(2/r)2

The ARE of the PCC detector with respect to the correlation detector is therefore E,,,,

= (2/n)'= 0.405

(6.3-8)

Equating the probabilities of detection of the PCC and NeymanPearson detectors,

(6.3-9)

simplifying and solving for n 2 / n 3 yields ndn3 = (2/4/2 The ARE of the PCC detector with respect to the Neyman-Pearson detector for testing H2,, against K,, is

EXc,,,, = ( 2 / ~ ) , / 2 = 0.202

(6.3- 10)

It is clear then from Eqs. (6.3-8) and (6;3-10) that for Gaussian input noises and a Gaussian signal, the PCC detector does not perform as well as either

6.3 A R E of the PCC Detector

161

of the parametric detectors. This result should have been anticipated, however, since both of the parametric detectors were derived based on the assumption of Gaussian noise. It is enlightening to investigate the relative performance of the PCC detector with respect to the parametric detectors when the Gaussian noise assumption is violated. The expression for the detection probability of the correlation detector remains unchanged, however, p + must be recomputed for the PCC detector. Assuming a double exponential distribution and using Eq. (6.3-3), p + becomes

P+

= (1/2) +

(4%)

(6.3-1 1)

Substituting into Eq. (6.3-2) and simplifying,

= 1

- @(@-'(I

- a) -

2

6 (u:/ui))

(6.3-12)

where again use has been made of the assumption that u,'/u,'<< 1 to evaluate the denominator. Equating Eqs. (6.2-7) and (6.3-12), 1-@{

W ' ( 1 - a) - (u,2/u,')$ [ 1 + 2(uyu,')]

= 1

- @(@-](I

- a) - 2 ~ ( u , ' / u , ' ) )

(6.3- 13)

and simplifying, yields nl/n3= 4

(6.3- 14)

The ARE of the PCC detector with respect to the correlation detector when the noise has a double exponential distribution is EP CC, c

=4

(6.3- 15)

Similarly, it can also be shown that the ARE of the PCC detector with respect to the Neyman-Pearson detector for Gaussian noise when the noise actually has a double exponential distribution is given by [Wolff et af.,19621 (6.3- 16) EPCC, NP = 3-5 The PCC detector thus may perform much better than optimum detectors if the parametric assumptions are unjustified. Further discussion of the

6

162

Two-Input Detector Performance

ARE of the PCC detector is included in Section 6.5. The following section illustrates the caleulation of the ARE for a two-input rank detector.

6.4 ARE of the Mann-Whitney Detector In this section the ARE of the Mann-Whitney detector with respect to the two-input version of the Student’s t detector when the input noise is Gaussian is calculated. Comparing the two detectors for Gaussian noise, of course, gives some advantage to the Student’s t detector, since it is the optimum parametric detector for testing for a location shift in Gaussian noise with unknown means and variances. Rather than follow the approach of the previous section and equate detection probabilities and then solve for the ratio of the sample sizes, the concept of efficacy defined in Section 2.7 is used. The Student’s t detector given by Eq. (5.2-22) is asymptotically equivalent to using the test statistic (6.4-1) when the sample sizes n and m become large, since the denominator of Eq. (5.2-22) approaches the variance of J -3, and hence is only a constant scale parameter that can be ignored. For the present purposes then, Eq. (6.4-1) is taken as equivalent to the Student’s t test statistic. The calculation of the efficacy requires knowledge of both the mean and variance of the test statistic. The mean of Eq. (6.4-1) is (6.4-2)

E { J -X} = 8

where 0 denotes the amount of the location shift when the alternative is true. The variance given that the hypothesis is true, that is, given both x and y have the same distribution with no signal present, is var{g - x I H , , ) =

~ { (-P)~IH,,} p - ~ ’ { -px I H , , )

= E { J 2 I H l 0 }- 2 E { j E l H l o }

+ E {X2IHIo} (6.4-3)

since the hypothesis and alternative being considered are H , , and K , , and the two sets of observations are assumed to be independent and identically distributed. The efficacy of the Student’s t test is therefore lim

m,n+oo

( m + n) mn

{ [ ( a / W E u -X)]le-o}2 var{ j j - XIH,,}

= -1 U2

(6.4-4)

6.4 A R E of the Mann-Whitney Detector

163

In order to compute the mean value of the Mann-Whitney detector, the form of the detector in Eq. (5.2-32), which has a test statistic of the form n

m

u =j X X u(xi -Y,) -1 i-1

(6.4-5)

is used. The expected value of Eq. (6.4-5) can be found as

E{u}=

n

2

;=I

m

~ ~ { u ( x ~ - y= ,m) n} P { x > y } i-I

= mnP{y - x

<0}

(6.4-6)

The probability that y - x < 0 can be found by using Eq. (4.3-7) and identifying the terms in Eq. (4.3-7) as follows z =y -x - y in F , ( . ) + y - 8 in Fy(.) Y i n f , ( . ) - + y in&(*)

Therefore, P { Y - X ’ < O } = SW F, ,(Y-X+Y-@)fu(Y)&

or P{v-x

< O}

=

Fy-,(0)

=JW

-W

F , ( Y - @ ) f , ( Y )ctu

(6.4-7)

since P [ y - x = 01 is zero for continuous distribution functions. Using Eq. (6.4-7), the mean value of the Mann-Whitney test statistic is (6.4-8)

and thus,

(6.4-9)

The variance of the Mann-Whitney test statistic was derived in Section 5.2 and is given by Eq. (5.2-55). The efficacy of the Mann-Whitney detector can therefore be calculated

164

6 Two-Input Detector Performance

using Eqs. (5.2-55) and (6.4-9) as

(6.4- 10)

+

where m n + 1 has been cancelled with m + n since m and n are assumed to be very large. Taking the ratio of Eq. (6.4-10) to Eq. (6.4-4), the ARE of the Mann-Whitney detector with respect to the Student's t test is found to be (6.4- 1'1) For the case where fy ( y ) is Gaussian,

(6.4- 12) 2 a G The ARE of the Mann-Whitney detector with respect to the Student's t test when the input noise is Gaussian is therefore = (&)(&)

=

EMw,,= 3/77 = 0.955

1

(6.4- 13)

This result is particularly encouraging since it says that the Mann-Whitney detector is 95.5% as efficient as the optimum parametric detector when the assumptions of the latter are satisfied. When the noise density is not Gaussian, the ARE in Eq. (6.4-13) may be considerably greater than one. A discussion of asymptotic results for other probability densities is included in the following section.

6.5 ARE of Other Two-Input Detectors Now that two different approaches to the computation of the ARE have been given, the ARES of the nonparametric detectors developed in Chapter 5 relative to optimum parametric detectors for Gaussian noise are listed in this section. Asymptotic relative efficiencies are given for cases when the parametric detector assumption of Gaussian noise is satisfied and for cases when the noise has one of several non-Gaussian probability densities.

4.5 A R E of Other Two-Input Detectors

I65

N o additional derivations of the ARE are included here due to the lack of space and the fact that the remaining derivations are available in the literature. It is hoped that by following the available derivations in the literature and the examples given in Sections 6.3 and 6.4, the reader can construct any required derivations. Table 6.5-1 lists the AREs of the two-input nonparametric detectors with respect to the parametric Gaussian noise theory detectors. The table is divided into two parts: detectors that have one input with a reference noise source and detectors that have two simultaneous inputs. The parametric comparison test under the former division is the Student's t detector. The parametric comparison test under the latter heading is the sample correlation coefficient for all detectors except the PCC detector, which is compared to the correlation detector and the optimum bivariate Gaussian parametric detector. The comparison test being used is indicated in the table by the equation number of the detector as it is listed in the text. Table 65-1 Asymptotic relative efficiencies of certain two-input nonparametric detectors with respect to the optimum detector for Gaussian noise

Noise density Nonparametric detector

Comparison test

One input with a reference noise source: Mann-Whitney Eq. (5.2-22) Normal scores Eq. (5.2-22) Two simultaneous inputs: PCC PCC Kendall tau Spearman rho

Eq. (6.2-1) Eq. (5.3-9) Eq. (5.3-38) Eq. (5.3-38)

Gaussian

Uniform

Double exponential

0.955 1.000

1 .000 -

1.50 -

0.405 0.202 0.9 12 0.912

1 .000 1 .m

4.000

3.500 1.266 1.266

'Many of the results in this table were taken from Bradley [1968].

The most striking thing about the results listed in the table is that many of the ARE values are very close to or in excess of one. For instance, all of the ARE values for non-Gaussian noise probability density functions are > 1. More significant, however, are the values of the ARE for the Gaussian noise density, since the comparison tests being used are all based on the assumption of Gaussian noise, and hence are at least asymptotically optimum for that noise distribution. If nonparametric detectors can achieve AREs very close to one with respect to the parametric detector for the given noise density and achieve AREs greater than one for other noise densities, a substantial case can be made for their application in cases where the noise density may fluctuate.

166

6

Two-Input Detector Performance

From Table 6.5-1 it is evident that, at least asymptotically, the nonparametric tests discussed here compare favorably with the optimum parametric detectors for Gaussian noise. Suppose now that it has been decided that a nonparametric detector is to be employed rather than a parametric detector since fluctuations in the background noise density are known to occur. Which nonparametric detector should be used? Limiting consideration to the detection of a location shift with detectors that have one input plus a set of reference noise samples, the normal scores test would be preferred to the Mann-Whitney detector on the basis of Gaussian noise alone. However, since the noise density is known to fluctuate, Gaussian noise results are not enough upon which to base a decision. Hodges and Lehmann [1961] have presented some results relevant to this specific problem. In particular, they compute the A R E of the Mann-Whitney detector with respect to the normal scores test for several different noise densities. Their results are given in Table 6.5-2. The ARE values in the table seem to further strengthen the selection of the normal scores detector. This is because for most of the cases, the ARE is fairly close to one, except for the uniform and exponential densities for which the ARE is zero. The Mann-Whitney detector only performs substantially better for Cauchy noise densities but may be considerably worse than the normal scores test. The only general conclusion that should be drawn from these results is that the ARE calculations of one nonparametric detector with respect to another nonparametric detector can be performed and can yield valuable information for the selection of a test statistic. Table 6 5 2 Asymptotic relative efficiency of the Mann-Whitney detector with respect to the normal scores detector

Noise distribution

ARE

Uniform Exponential Gaussian Logistic Double exponential Cauchy

0 0

3/97

= 0.955

1; 1.05 3n/8 1; 1.18 1.413

97/3

The high ARE values listed in Table 6.5-1 and the fact that the background noise density is many times at least partially unknown in engineering applications make nonparametric detectors an attractive alternative to their parametric counterparts. However, asymptotic results do not convey the complete picture of detector performance since an

6.6 Small Sample Results

I67

infinite number of observations is never available for the decision process. Of interest then, is the small sample efficiency of the detectors. This important subject is treated in the following section.

6.6 Small Sample Results Asymptotic power efficiency studies have been the primary tool for evaluating the performance of nonparametric detectors. The reason for this of course is that small sample studies require a large number of computations to be performed and the applicability of the results is still limited to the specific alternative and sample sizes considered. In spite of this, several interesting and useful small sample studies have been conducted. The results of some of these investigations are presented here. Klotz [1964] and Terry [1952] studied the performance of the normal scores detector with respect to the Student’s t test for normal shift alternatives and a small number of observations. Both authors considered the case where a = 0.05 and the number of observations is n = m = 4. Klotz also compared the two detectors for n = m = 5. Their results are in good agreement with the normal scores detector being more powerful for a location shift < 0.25 and approaching the power of the Student’s t test for location shifts > 3.0. The Student’s t test possesses its maximum advantage over the normal scores detector for location shifts around 1.5. Terry and Klotz deviate in this region, since Terry indicates a power advantage as great as 0.2 for the Student’s r test while Klotz predicts a power advantage of only 0.02. Terry notes however, that his data should be used only for general comparisons of the shape of the power functions and not for comparing specific values. Dixon [ 19541 investigated the power of the Mann-Whitney detector under normal shift alternatives for small sample sizes and various false alarm probabilities. The specific sample sizes considered were n = m = 3, 4, 5 with false alarm probabilities in the range 0.03-0.10. In general, the Mann-Whitney detector has a very high relative efficiency for small sample sizes with the efficiency decreasing from -0.965 to 0.90 as the amount of the location shift of the alternative varies from 0 to 5.0. For vanishingly small alternatives and finite sample sizes n = m = 5 or less, the Mann-Whitney detector always has a power efficiency with respect to the Student’s t test greater than the ARE value of 0.955. Dixon’s results seem to indicate, at least for the Mann-Whitney detector, that the large sample size assumption of the ARE is not what limits its applicability, but the assumption that the alternative approaches the hypothesis. In fhct, for vanishingly small location shift alternatives, the ARE is a lower bound for

168

6

Two-Input Detector Performance

the relative efficiency of the Mann-Whitney detector with respect to the Student’s t detector for small sample sizes. The power of the Mann-Whitney detector with respect to the Student’s t detector was also computed by Leaverton and Birch [1969]. Their study utilizes sample sizes of n = m = 4, 7, 9, 13, 25, and a false alarm probability of 0.05. The only significant difference between the poyer of the two detectors occurs for the number of observations n = m = 4. The power difference for n = m = 7 and 9 is only slight with essentially no difference in the power of the two detectors when n = m = 13 and 25. A Monte Carlo study by Neave and Granger [1968] also evaluated the power of the Mann-Whitney and normal scores detectors for finite sample sizes and compared them to the Student’s t test. The false alarm probabilities considered were a = 0.01 and 0.05, and sample sizes n = m = 20 and n = m = 40. For normal alternatives, the results were that the normal scores, Mann-Whitney, and Student’s t detectors performed very nearly the same. For nonnormal alternatives, the normal scores test generally maintained its superior performance over the other two detectors, even when the variance of the alternative differed from that of the hypothesis. The Mann-Whitney detector had a higher power than the Student’s t detector when the hypothesis and alternative had equal variances, with the roles being reversed for unequal variances. This is an interesting result since both the Student’s t test and the Mann-Whitney detector are based on the assumption of identical variances. Leaverton and Birch [1969] also address this problem. Neave and Granger further demonstrate that the Student’s t , Mann-Whitney, and normal scores detectors produce highly correlated results for the cases considered. The nonnormal alternatives utilized consisted of a linear combination of nonidentical Gaussian probability densities, and hence, was bimodal and nonsymmetric. Witting [ 19601 studied the efficiency of the Mann-Whitney detector with respect to both the Student’s t test and what Witting called the standard normal test for Gaussian and uniform alternatives. The standard normal test is simply the difference in the sample means of the two sets of observations normalized by the common standard deviation of the two populations. Both equal sample sizes and unequal sample sizes ( m corresponding to the number of reference observations) were used with a in the range 0.02-0.06. The Mann-Whitney detector is 80-83% as efficient as the standard normal test for n = m = 10 and uniform background noise densities. The efficiency increases to approximately 90% when n = rn = 20. Both these results are for the case where the alternative approaches the hypothesis. As the difference between the hypothesis and alternative becomes larger, the efficiency decreases. For the small sample size cases

-

6.6 Small Sample Results

169

m = n = 4, m = n = 5, and m = 6, n = 4, the Mann-Whitney detector is greater than 95% as efficient as the Student’s t test but only 73%-79% as efficient as the standard normal detector when the noise densities are Gaussian. This last result is, of course, not as bad as it sounds since the standard normal test is the most powerful detector for the Gaussian noise case being considered. All of the comparisons presented thus far in this section have evaluated the performance of nonparametric detectors with respect to parametric detectors under parametric alternatives. Naturally it is also of interest to compare the small sample performance of nonparametric detectors under nonparametric alternatives. One such study was conducted by Lehmann [1953] with sample sizes m = n = 4 and 6, and a = 0.1. The tests compared were the median detector, the Mann-Whitney detector, and the most powerful rank detector for testing the hypothesis G = F versus the alternative G = F2. The median detector considered is just a nonzero mean, two-input version of the sign test. The hypothesis and alternative investigated is H : G = F versus K : G = F k where k is an integer. Notice that K is a member of the class of alternatives G < F for which the Mann-Whitney test is designed. The detectors were compared for alternatives with k = 2 and 3. For m = n = 4, the Mann-Whitney and “optimum” rank detector performed almost identically with the median detector producing a significantly lower probability of detection. When the sample sizes were increased to six, the three detectors maintained their relative performance positions with the “optimum” detector increasing its slight advantage over the Mann-Whitney detector. Since F k is the distribution function of k random variables with distribution F, the alternative considered by Lehmann is that the x’s have the distribution F and the y’s have the same distribution as the maximum of k x’s. Gibbons [1964] employs the same alternative but tests it against the hypothesis that the x’s are distributed as the smallest of k random variables with distribution F; that is, the x’s are from the distribution 1 - (1 - F)k. If the distribution F is symmetric, the distribution of the x’s and y’s under the alternative are mirror images. The results are similar to those obtained by Lehmann. The most powerful rank detector, the MannWhitney detector, and the normal scores detector all had essentially the same power. The studies included k = 2, 3, and 4, a = 0.01, 0.05, and 0.10, and m,n = 2, 3,4, 5, and 6. When normal alternatives differing in location were considered, the tests again performed almost identically for the cases investigated. The studies briefly summarized in this section constitute some of the more important finite sample size performance evaluations of non-

170

6

Two-Input Detector Performance

parametric detectors. The results presented considered extremely small to moderate sample sizes and both parametric and nonparametric alternatives. Generally, the Mann-Whitney and normal scores detectors performed as indicated by their ARE values. Deviations from this statement occur when the alternative becomes increasingly separated from the hypothesis. For this case, both nonparametric detectors encountered some loss in relative efficiency. 6.7

Summary

In this chapter the performance of nonparametric detectors has been studied in some detail. Asymptotic power expressions for parametric detectors were derived and derivations of the ARE of detectors with two inputs and one input with a set of reference noise observations were given. Asymptotic relative efficiency values of the nonparametric detectors presented in Chapter 5 with respect to parametric detectors were given as well as ARE comparisons between nonparametric detectors. Several interesting finite sample relative efficiency and power investigations were discussed. The ARE is seen to be a good approximation to the small sample efficiency of nonparametric detectors for the important case of small signals. When the signal becomes large (the hypothesis and alternative differ significantly in location), the efficiency can dip below that predicted by the ARE. All of the investigations in the present chapter have assumed independent observations and that no tied ranks or zero value observations have occurred. In practice, neither of these assumptions may be valid. The effects of these two possibilities are investigated in the following two chapters.

Problems 1. Use Eq. (6.4-11) to compute the ARE of the Mann-Whitney detector with respect to the Student’s t test for a uniform input probability density over the interval - 1/2 < y < 1/2. 2. Show that if the noise has a Laplace or double exponential density the ARE of the Mann-Whitney detector with respect to the Student’s t test is E M w , t= 1.5

3. Discuss the desirability of using the Mann-Whitney detector and the

z 7z

Problems

two-input Student’s t test to test the hypothesis H: K:

F ( x ) and G (y) are zero mean with variances u: and ui, respectively, against the alternative F ( x ) is zero mean and G (y) has mean 8, with population variances u: and ,:a respectively.

4.

From Eq. (5.3-77) the Kendall r=1-

7

test statistic can be written as

4Q n(n - 1)

where Q is the total number of negative scores. Demonstrate that

.x-2 +x 2 n-1

Q=

i=l j==i+l n-I..

=

n

u(yj - xi)

n

i=l j=i+l

[1

-

sgn(y, - yi) sgn(xj - xi>]

5. Using the results of Problem 4, calculate E { r I K } , where K is the alternative that x and y are bivariate Gaussianly distributed with correlation coefficient r > 0.

Hint:

-1, 0, +1,

po

[Kendall and Stuart, 1969, 19671. 6. From Eq. (5.3-79), note that var{rlH} = 4/9n for n large and H the } in hypothesis that r = 0. Using this expression and the E { T ~ Kobtained Problem 5, compute the efficacy of 7 . 7. For n large, E { P I K } = r and var{PlH} = 1/n [Kendall and Stuart, 19671. Use these results to obtain an expression for the efficacy of the sample correlation coefficient i. 8. Calculate the ARE of the Kendall 7 detector with respect to the sample correlation coefficient for bivariate Gaussian alternatives by using the efficacies calculated in Problems 6 and 7.

CHAPTER

7

Tied Observations

7.1 Introduction

The treatment of ties in nonparametric detectors is a subject which normally receives only passing attention since the underlying noise distributions are usually assumed to be continuous, and thus, the probability of equal observations is theoretically zero. In practical applications, however, even if the noise distribution is continuous, ties can occur due to imprecise measurement when taking samples. It is obvious that even the most sophisticated instrument is not infinitely accurate, and an experimenter must therefore be prepared to deal with tied observations when applying detectors which utilize polarity and rank information. Another case in which ties may occur, which are not due to the imprecision of measurement instruments, is when the noise distribution is discrete. The discrete case is of less importance than the continuous case for the applications considered here and is treated briefly following the discussion of the continuous case. Before proceeding any further, it should be explained that the word ties as used in the statistical literature actually refers to two different situations. When an observation that has a value zero occurs, some assignment of a plus or minus sign to this zero or some other adjustment must be made to use this observation in a rank test. Although in this situation a tie between two observations has not occurred, treatment of this case is included under treatment of ties. The second situation is the case where two or more observations have the same absolute value, which of course can be zero, 172

7.2 General Procedures

z 73

and ranks must be assigned to the tied observations. Several different methods for resolving ties and zeros in nonparametric tests are discussed in general terms in Section 7.2. Section 7.3 contains the results of a few of the specific applications of these techniques previously reported by researchers in the literature. 7.2

General Procedures

For the initial discussion, the background noise density is assumed to be continuous but tied observations occur due to the sample being “rounded-off” or “truncated” by the measuring instrument. One of the most common methods for treating ties is the technique of randomization. Under this method when a zero observation is obtained, a plus or minus sign is assigned by performing a purely random experiment. Similarly, when two or more observations have the same absolute value, ranks are assigned to each observation at random. The main advantage of this method is that it does not distort the distribution of the test statistic under the hypothesis if (1) when randomization is used to assign plus or minus signs to zeros, the noise density under the hypothesis is symmetric, and (2) when randomization is used to assign ranks to tied observations, the observations are identically distributed. The main disadvantage of randomization is that the test outcome or detector output is a function of not only the observations but of another random experiment. This random experiment can change the probability distribution when the alternative is true, and hence may affect the probability of detection. A method similar to randomization that can be used only to assign plus or minus signs to zeros is to assign a positive sign to half the zeros and a negative sign to the other half. If there is an odd zero, randomly give it a positive or negative sign. The procedure that is probably used most often to assign ranks to tied observations is the use of midranks. Under this technique each tied observation is assigned the average of the ranks the Observations would have if they were not tied, which is called the midrank. The midrank is defined as (7.2- 1) midrank(x) = t ( N + M + 1) where N is the number of observations < x and M is the number of observations < x [Putter, 19551. The basic idea behind the use of midranks is that by assigning the average rank to tied observations, the test statistic thus produced has approximately the value that would be obtained if the test statistic were averaged over all possible resolutions of ties.

174

7 Tied Observations

Example 7.2-1 Assume that there are five observations with the absolute values lx,l = 1, (x21 = lxjl = Ix41= 3, and lx,l = 4. The rank of x1 is clearly 1. To determine the ranks of the three tied observations, the midrank technique is used. The number of observations less than the tied observations is 1, x , , and therefore N in Eq. (7.2-1) equals 1. The number of observations less than or equal to the absolute value 3 is 4, and thus M = 4. The midrank can now be computed by substituting N and M into Eq. (7.2-1) midrank = t ( N

+ M + 1) = t ( l + 4 + 1) = 3

(7.2-2)

This result can be found alternately by finding the average of the ranks the three observations would have if they were not tied. Since x1 has the rank 1, the tied observations would normally have the ranks 2, 3, and 4, the average of which is (2 + 3 + 4)/3 = 3. The ranks of the five observations as assigned, by using midranks are thus R , = 1, R, = R , = R, = 3, and R, = 5, where Rj is the rank of the j t h sample. Midranks have the disadvantages that they can take on fractional values and that they can have the same value repetitively, both of which may affect the distribution of the test statistic. Very few tables of probabilities exist which utilize midranks for tied observations since the task of such a tabulation (accounting for all possible midrank values for all possible numbers of tied observations) is monumental to say the least. The procedure that the use of midranks emulates is the actual calculation of an average, median, or midrange value for the test statistic. Under this technique, plus and minus signs and ranks are assigned to zero and tied observations in every possible way, and the value of the test statistic is computed for each case. The average, median, or midrange of the test statistic is then calculated and used as the test statistic. Similarly, the false alarm probability can be calculated for each value of the test statistic, and the average, median, or midrange of these values computed to yield a level of significance ii. This value of ii is then used as the test statistic by comparing if to the desired significance level a and rejecting if ti < a. Both of the above methods, an “average” test statistic or significance level, have the disadvantage that the results may be very different from what the actual values should be and may not even equal the test statistic or a value for any possible sign and rank assignment. When this occurs, the distribution of the test statistic is distorted. Although the computation of the midrange is simple, the calculation of the average or median is not nearly as simple or straightforward, and both may be quite difficult to compute. Another procedure that is most often used in cases when zero value

7.2 General Procedures

I 75

observations are obtained but can also be used when tied observations occur is to ignore the zeros and ties completely and reduce the total number of samples accordingly. For the treatment of zeros when the background noise density is symmetric, this technique yields an equivalent test with a smaller sample size. The smaller sample size can cause a reduction in the power of the test when compared to tests based on the original sample size. In any event, some information is clearly being discarded along with the zero or tied observations, and the effect of this information loss will vary depending upon the particular detection problem. The method recommended by Bradley [ 19681 for treating zero value and tied observations is to compute the test statistic once by assigning plus and minus signs to zero Observations and ranks to tied observations in the way that is most likely to cause rejection of the hypothesis, and once in the way which is least likely to cause rejection of the hypothesis. The test statistic and false alarm probability are calculated for both cases previously described. If both values for the false alarm probability are less than the desired significance level, the hypothesis is rejected; if both values are greater, the hypothesis is accepted. If the two values for the false alarm probability fall on opposite sides of the chosen level of significance, more observations must be taken. The detector output is reported and is accompanied by a statement of the upper and lower bounds on the false alarm probability. Of course, the statement of probability bounds has little practical utility in automatic detection problems. One final approach for use in the continuous probability density case is to compute the value of the test statistic for all possible cases and then select for use that value which maximizes the probability of deciding that the hypothesis is true. This procedure results in a detector that is conservative in that it minimizes the false alarm probability, and hence, reduces the power of the test. Turning now to the case where the noise density is discrete, a slightly different situation is obtained. If the measurement instruments are considered to be sufficiently accurate, the sample values are the actual values, and thus when tied observations occur, the assignment of signs or ranks is not one of resolving an ambiguous situation, but one of maintaining the true situation. Since there is actually no change when the samples alone are considered, the same methods used for allocating signs and ranks in the continuous case can be applied to the discrete case. Some caution is necessary in their application, however, since some of the underlying assumptions of the previous procedures may be violated. For instance, all of the previous methods except that proposed by Bradley and the technique of discarding the tied or zero value observations are

176

7 Tied Obseroations

essentially based on the assumption that all possible resolutions of the zeros or tied observations are equally probable. Further, the procedure recommended by Bradley [1968], as he points out, loses some of its importance since the probability bounds can no longer be taken as absolute upper and lower bounds on the test statistic. Actually, simply eliminating zero-valued observations from consideration is much more logical in the discrete case than for a continuous probability distribution. This is because for a discrete population and a precisely accurate measuring instrument, the zeros are truly unsigned observations, while in the continuous case the zero is an inaccurate measurement of a signed variable. Again, of course, some loss of power with respect to a test based on the original sample size may be experienced.

73 Specific Studies Relatively few specific investigations of the effects of the procedures described in Section 7.2 on nonparametric detector performance have been conducted. However, Putter [ 19551 has investigated the effects of applying randomization and midranks to the sign and the Mann-Whitney detectors. Putter computes the power of the sign detector for testing the one-sided alternative of a positive location shift using both a randomized and a nonrandomized test as the number of zeros varies from zero to ten. The randomized test is performed as described in Section 7.2, while the nonrandomized test consists of simply dropping the zeros from consideration. The false alarm probability is 0.05 and the alternative tested is that the median is twice that of the hypothesis. For a total sample size of ten, the powers of the randomized and nonrandomized tests are identical for zero and ten zero-valued observations with the nonrandomized test having a significant advantage when the number of zeros is in the intermediate range of three to six. Putter also demonstrates theoretically that for the same alternative considered above, the nonrandomized test is uniformly more powerful than the randomized test. Additionally, he shows that the ARE of the randomized test with respect to the nonrandomized test is .always less than one when zero-valued observations are present. When considering the Mann-Whitney detector, Putter’s nonrandomized test assigns midranks to each group of tied observations. Although he was unable to obtain results for small samples, Putter proves that the ARE of the randomized test with respect to the nonrandomized test is less than one whenever ties are present. Another very interesting study of the effects of zeros and ties on the Wilcoxon detector was conducted by Pratt [1959]. Pratt introduces his

7.3 Specific Studies

z 77

subject quite ably with an example which illustrates the loss of power that can result by simply dropping zero-valued observations from consideration and performing the test on the remaining observations.

Example 73-1 [Pratt, 19591 Consider the problem of testing the hypothesis that a distribution is symmetric about zero versus the alternative of a positive shift of probability. A single set of 13 observations given by -18 has been taken, and it is desired to use the Wilcoxon detector to test the hypothesis. Since the Wilcoxon detector cannot be directly applied to the above observations due to the presence of the zero, the zero is simply eliminated from consideration and the test performed on the remaining 12 observations. Pratt calls this the “reduced sample procedure.” After dropping the zero, the signed ranks 1, 2, 3, 4, 5 , 6, 7, 8, 9, 10, 11, and - 12 result, which is a total negative rank sum of 12. When the hypothesis is true, the probability of a negative rank sum of 11 or less out of 12 observations is 55/2’*. Therefore, for any false alarm probability a < 55/212, the reduced sample procedure would accept the hypothesis, since for these false alarm probabilities, the total positive rank sum will not exceed the threshold. It seems only logical now that any set of observations more negative than the above set would also cause the hypothesis to be accepted for the same false alarm probabilities. If, however, all of the earlier observations are decreased by an amount < 1, the signed ranks of the modified observations are 0,2,3,4,6,7,8,9,11,14,15,17,

- 1,2,3,4,5,6,7,8,9, 10,11,12, -13 which yield a negative rank sum of - 14. Since the probability of a negative rank sum of 14 or less in 13 observations is lO9/2I3 when the hypothesis is true, the Wilcoxon detector would accept the alternative for any a > 109/2? Therefore, for lO9/2I3 < a < 55/212 = 1lO/2I3, the reduced sample procedure operating on the first set of observations would reject the alternative, while a more negative set of observations would result in the alternative being accepted by the standard Wilcoxon detector. The dropping of the zero thus yields a reduction in the power of the test. The reduced sample procedure can also cause the alternative to be accepted when the standard Wilcoxon detector applied to a more positive set of observations would cause the alternative to be rejected. Pratt’s proposal for solving this problem is to rank all of the observations

178

7 Tied Observations

including the zeros, and then drop the ranks of the zeros without changing the ranks of the remaining observations. For the one specific example considered by Pratt in his paper, this approach does not produce the inconsistencies previously noted to occur with the reduced sample procedure. Pratt also considers the case of nonzero ties in the Wilcoxon detector. He advocates using the procedure of averaging the ranks of all observations that are tied in absolute value before affixing the signs. Pratt clearly points out, however, that this approach changes the probability distribution under the hypothesis and gives examples to show that the average rank procedure can cause the alternative to be accepted even though by increasing the observations appropriately, the standard Wilcoxon detector will accept the hypothesis. An excellent discussion of the average rank procedure and other procedures is available in Pratt’s paper [ 19591. Hajek [1969] gives examples of the effects of breaking ties for several nonparametric detectors and Chanda [ 19631 considers the efficiency of a modified Mann-Whitney detector in the presence of ties. Kendall [ 19621 gives techniques for handling tied observations for use with the rank correlation coefficient and other closely related test statistics. 7.4

Summary

The treatment of ties in nonparametric detectors is clearly a problem and is also of great importance since the choice of techniques for treating ties can markedly affect the significance level, distribution-free properties, and power of a detector. Whether one procedure is better than another depends on the specific application, but the dropping of zero observations and the use of midranks or averaged ranks seem to be the most widely used and most thoroughly investigated techniques. It is clear, however, that many of the techniques mentioned in Section 7.2 have not been thoroughly investigated and considerable work remains to be done in this area. The occurrence of ties and zero observations generally causes a loss of power regardless of the technique used to treat them, and thus experimental procedures should be carefully planned to avoid ties and zero observations if possible.

Problems Assume that the 20 independent, identically distributed observations listed below have been taken, and it is desired to use the one-input Wilcoxon detector to test H , versus K , (test for a positive location shift of

z 79

Problems

the probability density function). The observations are as follows: 8.1 2.4 3.0 7.1

- 0.2

1.1 4.9 4.2

-

1.4 6.2 0.4 0.6

- 4.2

6.5

7.7 4.2

- 0.1 5.4 5.2 3.9

Suppose further that observation numbers 7, 10, and 14 (in italics) should not be tied but are actually given by the values -4.4, 4.1, and 4.3, respectively, and that the ties occurred due to an imprecise measuring instrument. The desired false alarm probability is a = 0.1. The “correct” answer to this problem can be found using the Gaussian approximation to the distribution of the Wilcoxon test statistic and the given value of a to find the threshold T. The true value of the test statistic can then be computed by replacing the tied observations with their given true values and a decision reached by comparing the calculated value of the test statistic to the threshold. This problem, without the tied observations, is Problem 3-1 (Chapter 3, Problem 1) and should be worked for comparison purposes prior to attempting the following problems. Use Eqs. (3.4-10) and (3.4-15) for the mean and variance of the test statistic. However, note that the variance expression usually must be modified to account for ties [see Problem 111. 1. Compute the value of the Wilcoxon test statistic using midranks on the set of observations including ties. Decide whether H , or K, is true by comparing the resulting value of the test statistic to the threshold obtained in Problem 3- 1. 2. Use randomization to break the ties between the previously given observations. (For example, toss three coins simultaneously and assign the highest rank to the first coin that turns up different from the other two. Then, toss two coins to assign ranks between the remaining two tied observations.) Compare this value of the test statistic to the threshold from Problem 3-1 and arrive at a decision. 3. Resolve the ties in all possible ways and compute the test statistic for each case. Assume that the probability density of the input observations is symmetric. Find the average value of these statistics and use the average as the final test statistic to decide whether H , or K, is true. 4. Repeat Problem 3 but use the median of the statistics as the final statistic. 5. Compare the results of Problems 3 and 4. Is this an intuitive result? Is this a general result? 6. Use the technique of simply dropping or ignoring all tied observations and compute the value of the Wilcoxon test statistic. Again use the normal approximation and arrive at a decision. Has dropping the observations had any effect?

I80

7 Tied Observations

7. Use the method recommended by Bradley [1968] to account for the tied observations. Resolve the ties in the way that is most likely to cause the hypothesis to be accepted and in the way most likely to cause the alternative to be accepted. Find the probability of occurrence of each test statistic value using the normal approximation. If both probabilities fall below a, accept K. If both fall above a,accept H. 8. Compare and discuss the results obtained in Problems 1-7. 9. Given the 15 independent and identically distributed observations which contain three zero-valued observations, use the Wilcoxon detector to test for a positive location shift of a symmetric probability density. Use the normal approximation and compute the test statistic in two ways: (1) Drop the 0 values from consideration and use the reduced sample size to test the hypothesis; and (2) assign signs to the 0 value observations at random. Use a = 0.2 and let the hypothesis and alternative being considered be H, and K,. 3.4 - 2.1 0. 1.6 1.1 - 1.5 - 0.6 2.3 - 3.6 - 1.9 4.2 0. - 2.8 1.4 0.

10. Repeat Problem 9 using the sign test and the exact distribution of the test statistic. 11. In all of the above problems, no modification to the variance calculation was used to account for the effect of the tied observations. However, it is generally recommended that the variance be modified for tied observations to [Hollander and Wolfe, 19731 var,,,( U ) =

[

8

n(n

+ 1)(2n + 1) - t 2 ri(ri i- I

- l)(ri + 1)

1

where g denotes the number of tied groups and ti is the number of tied samples in group i. Rework Problem 1 using this modified variance expression.

CHAPTER

8

Dependent Sample Performance

8.1 Introduction

In most of the discussions, calculations, and comparisons thus far, the input observations to the nonparametric and parametric detectors have been assumed to be independent. This assumption is justified by the fact that if the samples are taken at a very slow rate, approximately equal to the reciprocal of the input filter bandwidth, the observations can be considered to be independent. In some radar detection applications, such as the case of a ground-based surveillance radar that scans the horizon or the case of an air-to-air radar operating at high altitudes, the assumption of independent samples can be validly applied if many radar returns or pulses are received from the target to be detected and one sample is taken from each radar return pulse. Pulse-to-pulse correlation may be present, however, whenever radars are operated within close proximity to the earth. Many times it may be desirable or necessary to obtain several observations from each pulse, such as when only a few radar returns are expected from a target or when the message in a communication system cannot be repeated; the observations obtained in this manner clearly cannot be assumed to be independent. Practicality and real-time data processing also dictate that the input data be sampled at a high rate since the more observations available, the more accurate the detector’s decision is likely to be and too, a slow sampling rate discards a large portion of the available input information. Of course, the higher the sampling frequency, the greater the possibility of correlation between input observations to the detector.

182

8 Dependent Sample Performance

From the preceding discussion it can be seen that the assumption of independent input observations is not always a valid one, and that the performance of nonparametric detectors when the input samples are dependent or correlated is of great practical importance. Nonparametric or distribution-free hypothesis testing procedures were originally developed by applied statisticians and used in fields such as psychology, agriculture, and manufacturing. In these applications the sampling method for obtaining the required observations can generally be structured to insure that the samples are independent. Since dependent samples in nonparametric detection do not occur often in nonengineering applications, relatively little work has been done in this area, but the work that has been done has produced some quite interesting, although not too surprising, results. Section 8.2 discusses the results of theoretical analyses concerning the effects of dependent input observations on the constant false alarm rate property of nonparametric detectors. This section is followed in Section 8.3 by a presentation of simulation and theoretical results obtained by several researchers on the relative efficiency of nonparametric tests when the input observations are dependent. 8.2 Dependence and the Constant False Alarm Rate Property

Perhaps the most detrimental effect that correlated observations can have on a nonparametric detector is that the false alarm rate a may no longer be constant for a fixed threshold. Nonparametric detectors may therefore lose their nonparametric or distribution-free properties when the input samples are correlated. Studies of the effect of dependent observations on the distribution-free properties of the two most common nonparametric detectors, the sign detector and the Wilcoxon detector, have been conducted and are discussed here. Wolff et al. [1967] have computed the mean and variance of the sign detector test statistic given by Eq. (3.3-13), which will be called S,,, when the observations have a dependence of the form (8.2-1) where w, is the input observation at time t, p is the correlation coefficient < JpI< 1, and n, is stationary white noise sampled at time t. They derive the mean and variance of the test statistic for the case where the noise density is symmetric with respect to the origin and then calculate the mean and variance for two specific examples, one where w has a Laplace distribution and the other where w is Gaussian distributed. The mean

0

8.2 Dependence and the Comtant False Alarm Rate Properv

183

value is the same for both cases and is given by E{S,,} = n/2

(8.2-2)

The variance of the test statistic when the input observations w, have a Laplace or double exponential density is given by 2P(l - P " )

1

(8.2-3)

which as the number of samples becomes large, asymptotically approaches var{S,,} 1 p lim =(8.2-4)

+

n-m

n/4

1 - P

When the input observations have a Gaussian distribution with zero mean and unit variance, the variance of the test statistic is

where ( u ) = ~ a(a + l)(a approaches

+ 2 ) . . . ( a + k - 1).

As n

00,

Eq. (8.2-5)

Since the derivations of these equations were not repeated here, the results seem somewhat complex and meaningless; however, the behavior of the mean and variance for different values of p and n can yield much information concerning the distribution-free properties of the sign detector. This is because if the mean or variance of a detector test statistic depends on the background probability density, then a fixed threshold cannot be found which maintains a constant false alarm rate. The mean value of the sign detector is clearly independent of the density of the input observations for the present case, as can be seen from Eq. (8.2-2). If p is chosen to be zero (independent samples), the variance in both cases is n / 4 , which is likewise independent of the background noise density. If however, Ip( > 0, the variances given by both Eqs. (8.2-3) and (8.2-5) depend on p, and hence, are a function of the probability density of the input observations w. The false alarm rate therefore does not remain constant as p changes, but for a specified correlation coefficient, the false alarm rate will remain fixed as other parameters of the background noise density are

184

8 Dependent Sample Performance

varied. A very disturbing result which is evident from Eqs. (8.2-4) and (8.2-6) is that the false alarm rate is still a function of p when the limit is taken as the number of observations becomes large. Armstrong [1965] has derived the mean and variance of a test statistic closely related to the sign detector, which is called the median detector. The test statistic of the median detector is given by (8.2-7) where the wi are the input samples and M is the known median of the noise density. The mean and variance of the median detector under no signal conditions with input observations that are pairwise bivariate Gaussian distributed with zero mean are given by Armstrong [ 19651 as E { S L } = 1/2

(8.2-8)

and

Again, the expected value is independent of the input noise density, but the variance is a function of the correlation coefficient p. The false alarm rate for the median detector therefore will not remain constant as the correlation coefficient changes. Davisson and Thomas [ 19691have shown, enroute to finding the efficacy of the Wilcoxon detector, that both the mean and variance of the oneinput Wilcoxon test statistic depend on the correlation coefficient p when the input observations are bivariate Gaussian distributed with zero mean and correlation coefficient p. The actual expressions are not reproduced here due to their complexity. Since the mean and variance of the Wilcoxon detector test statistic are functions of p, the false alarm rate is again dependent on the correlation coefficient of input samples, and the Wilcoxon detector thus will not be nonparametric when the input observations have a fluctuating correlation. From the discussion thus far, it is clear that the sign, median, and Wilcoxon detectors may lose their nonparametric property for dependent input observations. Since the sign detector is probably the simplest of all nonparametric detectors and the Wilcoxon detector is one of the most basic and simple rank detectors, it seems reasonable to conjecture or at least be concerned that most, if not all, other nonparametric detectors may fail to maintain a constant false alarm rate for a fixed threshold when the input observations are correlated. How does this result constrain the usefulness of nonparametric detectors? The answer is, of course, not a simple one; however, the discussion in the following section points out

8.3 Nonparametric and Parametric Detector Performance

185

some very interesting comparisons of the performance of parametric and nonparametric detectors for dependent input samples. From all of the previous developments, it is clear that for correlated input observations the selection of a detector threshold in order to insure that a desired false alarm rate is achieved may no longer be possible. This is because in general the false alarm rate will now be a. function of the input probability density, and hence, a constant threshold cannot be found to guarantee a specified false alarm rate. Of course, some relief from this problem may be available if the correlation between samples is relatively constant over the time interval of interest. In this case, both the mean and variance of the test statistics given earlier in this section will not fluctuate even though the variance depends on p, and thus it may be possible to determine the threshold to yield the required a. This is obviously contingent upon one being able to derive the distribution of the test statistic for nonzero correlation between samples. One point that may have been overlooked by the reader during this development is that if the input observations have a fluctuating, unknown correlation between them, the false alarm rate of the optimum parametric detectors for the problem being considered also will not remain fixed. Since some nonparametric detectors are invariant to certain nonlinear and nonstationary operations on the input data, nonparametric detectors should be less affected by dependent input samples than their parametric competitors.

8.3 Nonparametric and Parametric Detector Performance for Correlated Inputs Now that the effect of correlated input observations on detector false alarm probability has been at least partially addressed, it is necessary to investigate the detection performance or relative efficiency of nonparametric tests for correlated input samples. Results of theoretical and system simulation analyses that have been conducted by several authors on nonparametric detector performance with correlated inputs are summarized and discussed in this section. Armstrong [1965] has conducted a fairly extensive study of the performance of the median detector for correlated input samples. The median detector, whose test statistic is given by Eq. (8.2-7), is compared in this study to the optimum parametric detector for detecting a positive dc signal with independent observations taken from a Gaussian noise density which has the test statistic .

Ln

=

1

n

--

k- 1

wk

(8.3-1)

186

8 Dependent Sample Performance

and to the optimum detector for the same problem when the noise has a bivariate Gaussian density with correlation coefficient p whose test statistic has the form (8.3-2) where QU is the inverse covariance matrix, the wi are the input samples, and O is the signal to be detected. The test statistic given by Eq. (8.3-,1) is called the sample mean detector and is equivalent to the linear detector in Eq. (3.2-6). The ARE of the median detector with respect to both parametric detectors is computed by Armstrong for four different correlation functions, the decaying exponential, the damped cosine, the sin x / x , and the Gaussian, for various sampling rates. The simple case of the decaying exponential correlation function yields some quite interesting and intuitively appealing results. The ARE of the median detector with respect to the sample mean detector for this case varies from a value of 0.64, when the samples are essentially independent, to a value of 0.91 when the samples are highly correlated. The ARE of the median detector with respect to the optimum detector €or dependent samples varies from 0.68 for independent samples to 0.78 for highly correlated samples. These results show that if the input observations are highly correlated, the median detector is 91% as efficient as the optimum detector for independent samples and 78% as efficient as the optimum detector for dependent samples. The ARE of the median detector with respect to the sample mean detector and the optimum detector for correlated samples is more erratic for the other three correlation functions as the sampling frequency is varied; but, in general, the ARE is larger when the samples are highly correlated than when the observations are independent. This result is obviously very encouraging since it indicates that the median detector, and hence the sign test, is even more efficient in comparison to the optimum parametric detectors when the input samples are highly correlated than when the samples are independent. It is also noted that for the above comparisons the background noise density was assumed to be Gaussian and therefore the parametric detector design criteria is exactly satisfied. If the noise density deviates from the assumed Gaussian form, the nonparametric detector could in fact, as before, outperform the optimum parametric detector for Gaussian noise. Since the validity of the ARE as a performance indicator in engineering problems can justifiably be questioned, parametric and nonparametric detector performance evaluation must also be made for the case when a small number of input observations is available for the decision process. Davisson and Thomas [1969] have investigated the effect of dependent

-

8.3 Nonparametric and Parametric Detector Performance

187

observations on the performance of the Wilcoxon rank sum detector for both the large and small sample cases. Their dependence model is the same as that given by Eq. (8.2-1), and the detection problem they consider is the detection of a binary signal of amplitude +A, where A is the ratio of signal amplitude to noise standard deviation. For Gaussian noise and a correlation coefficient p = 0.9, the Wilcoxon detector with only five dependent input observations is shown to have almost exactly the same probability of error (P,)as the optimum parametric detector for various values of A. For A = 2.0, p = 0.5, and a Cauchy noise density, the Wilcoxon detector has a smaller P, than the optimum detector for Gaussian noise for four or more input observations. Computations of the ARE of the Wilcoxon detector with respect to the sample mean detector when p is allowed to take the form p(At) = exP(- IAtl)

(8.3-3)

where At is the sampling interval, have also been carried out by Davisson and Thomas [ 19691. As was shown earlier for independent samples, At >> 2.0, and Gaussian noise, the Wilcoxon detector is 95.5% as efficient as the sample mean detector. As At becomes smaller, the samples become more correlated, and the ARE of the Wilcoxon detector with respect to the sample mean detector takes the value 0.975 for At = 2.0, 0.98 for At = 1.0, and 0.9875 for almost continuous sampling. The Wilcoxon detector thus 1 with respect to the sample mean detector for Gaussian has an ARE of noise when the input observations are highly correlated. Thus far only the performance of one-input detectors has been considered when the input observations are correlated. Davisson and Thomas [ 19691 also calculated the ARE of both the two-input Mann-Whitney detector and the two-input Kendall 7 test. For the Mann-Whitney detector, the input observations satisfy Eq. (8.2- 1) and the correlation coefficient is given by Eq. (8.3-3). The ARE of the Mann-Whitney detector with respect to the two-input version of the sample mean detector given by Eq. (6.4-1) varies from 0.955 for independent samples to 0.964 for At = 2.0, 0.978 for At = 1.0, and 0.986 for perfectly correlated input observations. Since the Kendall 7 detector is a test of association rather than a test of location shift, a different correlation model is required. Davisson and Thomas [ 19691 assume that the noise in the two channels has a correlation coefficient given by Eq. (8.3-3), but that the signal samples are correlated according to the function

-

p,(At) = (1

+ At) exp( - IAtl)

(8.3-4)

The results presented by Davisson and Thomas on the Kendall 7 test are actually slightly confusing. In fact, the ARE values plotted for nonzero

I88

8 Dependent Sample Performance

+

signal and noise correlation seem to be in error by a factor of ,just as the ARE value quoted for independent input samples and attributed to Kendall and Stuart (19671 is in error by a factor of +. If their results are modified by multiplying by 2, the ARE values become more consistent and very interesting. The comparison test used is what they call the energy detector and is the detector given by Eq. (5.3-9), which is the optimum Neyman-Pearson detector for testing Hi, versus K;,.The ARE values of the Kendall 7 with respect to the energy detector vary from 0.97 for essentially perfect positive correlation of the input observations to 0.95 for A t = 2.0. The limiting value as A t + 00, that is, for independent input samples, as given in Chapter 6, is 0.912. The two-input Kendall 7 detector thus performs exceptionally well for correlated input observations, considering the fact that the test is designed to detect dependence between the two simultaneous inputs. The research results discussed in this section can probably best be described as encouraging and mildly surprising. This is because the nonparametric detectors become even more efficient with respect to parametric detectors when the input samples are correlated.

- -

8.4

Summary

From the information presented in this chapter, it seems that the most damaging effect that dependent input observations can have on nonparametric detectors is that the false alarm rate may not remain constant for a fixed threshold as the input observation correlation or background noise density varies. One possible way that the test threshold can be selected in order to maintain a desired false alarm probability is by system simulation. If the input observations are correlated but the correlation between samples is known to be relatively constant, it may be possible to determine the necessary constant threshold setting to achieve the specified false alarm rate if the probability density of the detector test statistic can be calculated under the dependent sample assumption. The invariance properties of nonparametric detectors imply some advantage over parametric detectors in this area since the false alarm rate of parametric detectors is also greatly affected by varying degrees of input sample correlation and changing background noise densities. Computations of the ARE of the median, Wilcoxon, Mann-Whitney, and Kendall 7 detectors with respect to the optimum or asymptotically optimum parametric detectors indicate that as

189

Problems

the samples become more highly correlated, the efficiency of the nonparametric detectors increases substantially. The validity of the ARE as a performance criterion for engineering applications may of course be questioned; however, the probability of error calculations conducted by Davisson and Thomas illustrate that the Wilcoxon detector performs well when compared to the optimum detector even when the input sample size is small. Nonparametric detectors should therefore, in general, be very efficient and quite useful when the input observations are correlated; care must be exercised in their application however, since their false alarm rate may exceed the significance level when fluctuations in the input probability density occur. Problems Derive an expression for the variance of the sign detector test statistic, Iu(xi), given that the input observations are dependent samples taken from a stationary random process. Express the answer in terms of E [u(x~)u(x,)]= R [ ( i - j ) 7 ] = R [ k ~ ] . Hint: See Lee [1960, pp. 282-2831 and Armstrong [1965]. 2. Since R ( k 7 ) = E [ u ( x i ) u ( x i + , ) ] is nonzero only for xi > 0 and xi+k 1.

> 0,

R ( k ~= ) P[xi

>0

and xi+& > 01

= L w L w L , x i + k ( x i 9 Xi+&) dxi dxi+k

Given that the joint probability density function of Xi and Xi+&is bivariate Gaussian with zero means, L , x , + &(xi,

xi+&) =

1 2 n 0 2 4 7

- exp( -

[xi' - 2PXiXi+& 2uy1 -

+ Xi'+&]

p2)

I

find R (h). [Armstrong, 19651. 3. Combine the results of Problems 1 and 2 to obtain Eq. (8.2-9). 4. Calculate the mean and variance of the linear detector for dependent input observations when a signal of amplitude 8 is present. [Davisson and Thomas, 19691. 5. Using the results of Problem 4, obtain an expression for the efficacy of the linear detector for dependent samples.

190

8 Dependent Sample Performance

6. Given that the efficacy of the Wilcoxon detector for dependent observations is [Davisson and Thomas, 19691

GW =

[

W

o2

2

2sin-' 2

n--w

show that the ARE of the Wilcoxon detector with respect to the linear detector is n--w

2

2 m

n--03

sin-' )p(nAt)

by using the efficacy of the linear detector calculated in Problem 5. 7. Assume that

*

0.8, n = 1 1, n =O 0, otherwise and compute the ARE of the Wilcoxon detector with respect to the linear detector using the ARE expression in Problem 6. 8. Show that the ARE of the PCC detector with respect to the correlation detector in Eq. (6.2-1) for correlated input observations is p(nAt) =

{

by assuming that the two test statistics are asymptotically Gaussian and by following the procedure used in Sections 6.2 and 6.3. Let the alternative K indicate that a signal is present and let H indicate that no signal is present [Carlyle, 19681. Note that the test statistics will be asymptotically Gaussian if they satisfy what are called strong mixing conditions [Rosenblatt, 19611.

CHAPTER

9

Engineering Applications

9.1 Introduction

With the exception of some mention of system complexity and the discussion of tied and dependent input observations, practical considerations of applying the detectors developed in Chapters 3 and 5 of the text have been essentially ignored. This approach can be easily defended since a more detailed investigation of system complexity, say, for example, on a logic gate by logic gate or power requirement basis, is highly dependent on the specific application and cannot be validly addressed here. One facet of applying nonparametric or distribution-free detectors to realistic engineering problems is sufficiently general to be discussed here, however, and has its basis in the fact that nonparametric detectors essentially “grew-up” in the statistical literature, and hence some modifications of the test statistics may be necessary before applying these techniques to engineering problems. For instance, in the case of a pulsed radar system, the input signal to the radar receiver is a radio frequency (rf) waveform and therefore this signal must be processed in some manner before applying it to the detector input. The required signal processing methods transform the receiver input waveform so that the noise probability density at the detector input may be considerably different than the noise density at the input to the receiver. Since considerable filtering is also accomplished in the receiver, the detector input is narrow-band. It seems clear then that one or all of these signal processing effects may require some modification to the detector test statistrics.

191

192

9 Engineering Applications

In this chapter, mathematical analyses and systems simulation results, previously reported by researchers in the literature, on the performance of nonparametric detectors which have been appropriately modified to account for signal transformations in realistic engineering applications are presented and discussed. Such a treatment, of course, falls short of discussing what may be called “true applications,” however, the reader may be surprised at the number of nonparametric detectors already being applied to physical problems. In these cases, the study of nonparametric detectors is important since such a study may provide the basis for mathematical analyses of existing systems. Furthermore, as any systems engineer knows, no system can be built in hardware without some analytical investigations being performed to establish, at the very least, performance bounds. The research results presented here should provide the necessary background for both of the studies mentioned above. Most of the applications of nonparametric detectors which have been considered are applications to the radar detection problem. This is probably because nonparametric detectors fit nicely into the Neyman-Pearson detection theory, which is the design basis for most realistic radar systems. Because of this strong emphasis on radar system applications, a separate section, Section 9.2, is devoted to this topic. The other main section in this chapter, Section 9.3, discusses several engineering applications that could also pertain to radar systems, but are included in this section rather than in Section 9.2 in the interest of generality.

9.2 Radar System Applications Radar system applications of nonparametric detectors have received the greatest attention by engineers. Perhaps this is true because radar systems may have a wider variation in background probability densities than some other communications systems due to the presence of clutter and possibly jamming signals. Further, radar systems are usually designed on the basis of the Neyman-Pearson criterion, which is well suited for use with nonparametric detectors. One other possibility is, of course, that there are simply more radar systems engineers in the world than other types of communications systems engineers! This conjecture has not been validated by the authors. For whatever reason, many nonparametric detectors have been applied to various radar detection problems, and several of these applications are discussed here. Hansen [197Oa] has studied the application of several one- and two-input nonparametric detectors to a pulsed search radar. The initial discussion below considers only the one-input detectors. Hansen limits consideration to only a single range resolution element, and in the case of coherent

9.2 Radar System Applications

193

detection, only a single Doppler resolution element. A matched filter is assumed for all input pulses, and the radar target returns are assumed to occur at a constant range delay and to have a constant signal-to-noise ratio over all observations. The performance of the various nonparametric detectors is determined and compared for the case where the background noise density is Gaussian. Gaussian noise is chosen because although fluctuations of the background noise density which would cause the false alarm rate in parametric detectors to vary do occur frequently, the background noise may generally be assumed to be predominantly Gaussian. For the case of coherent detection, but with the initial phase angle unknown, the output of the matched filter can be written in complex notation and the observations taken on this waveform to yield the hypothesis and alternative H : x i = xic jxis = n, = niC j n , (9.2-1) K : xi = Si + ni = Sic+ n,, + j ( S i , + nis) for i = 1, 2, . . . , n where the real and imaginary parts may be interpreted to be the in-phase and quadrature components, and all observations are assumed to be statistically independent. The signal has the form

+

+

(9.2-2) si = A & f ~ + * )= [ A cos + + j~ sin +l.’f~ where f, is the phase term due to the Doppler frequency shift, which can be dropped since only a single Doppler resolution cell is being considered here. The resulting signal is Si = Sic + j S i , = A cos + + j A sin + which when substituted into the expression for the alternative yields

K : xi = ( A cos

+ + nit) + j ( A sin + + nis)

(9.2-3)

If the real and imaginary parts of Eq. (9.2-3) are squared and summed, the result is A ’(cos’ + + sin’ +) 2A (nit cos n, sin +) + ni: + ni:

+

++ = A ’ + 2A (nic cos + + n,, sin +) + n,,2 + ni:

(9.2-4)

If no signal is present, A = 0, and Eq. (9.2-4) reduces to ni2 + n,:. If a signal is present, however, A’ will be the dominant term, and therefore, if the square root of Eq. (9.2-4) is taken, the result is approximately equal to A . A reasonable test statistic is therefore one which squares the in-phase and quadrature components of the matched filter output, adds them together, takes the square root of this sum, and compares the result to a threshold. Actually, the detector just described is the optimum detector for the detection problem being considered and Gaussian noise [Carlyle, 19681.

194

9 Engineering Applications

Several nonparametric detectors may be adapted to this form, and the three treated by Hansen are the Wilcoxon, normal scores, and sign detectors. In the formulation of his test statistics, Hansen utilizes the signum function given by

instead of the unit step function previously used when discussing the nonparametric detectors in this book. The use of the signum function only affects the setting of the detector threshold and does not change any of the basic detector properties. The test statistic for each case is formed by applying the basic nonparametric test to the in-phase and quadrature components separately, and then squaring the two resulting statistics, adding them together, and taking the square root to obtain the final test statistic. The test statistics for the Wilcoxon, normal scores, and sign detectors are given in Table 9.2-1. The R, in the partial statistic expressions for the Wilcoxon detector are the ranks of the lxicl or lxisl within each set of the in-phase or quadrature observations. In the normal scores detector partial statistics, E ( y i ) is the expected value of the ith order statistic in a sample of n observations taken from a standardized Gaussian distribution with zero mean and unit variance. Hansen obtains the ARE of the three detectors with respect to the optimum parametric detector for Gaussian noise as 0.955 for the Wilcoxon, 1.0 for the normal scores, and 0.64 for the sign detector, all of which are the same as the ARE values previously Table 9.2-1 Nonparametric detector narrow-band test statistics Detector

Partial statistics

Test statistic

n

ScW

=

s,

=

Wilwxon

Sign

2 R,sgn(x,c)

i- 1

z R,

sw

n

i- I

SdXi3

=Imx

9.2 Radar System Applications

195

obtained. Hansen [ 197Obl also defines an asymptotic loss by the expression SNR.( B. a. nl I

(9.2-5)

where SNR{ P, a,n} is the signal-to-noise ratio required to obtain the desired values of a and P. Using this expression, the values given below for the Wilcoxon, normal scores, and sign detectors can be found. The asymptotic loss of the normal scores detector is zero since it has an ARE of 1.0 when compared to the optimum parametric detector for a location shift and Gaussian noise. Detector Wilcoxon Normal scores Sign

Asymptotic loss (dB) 0.2 0.0 1.9

Hansen [ 1970al also defines a vector sign test which utilizes the statistic (9.2-6) which is the vector sum of the normalized observations taken from the matched filter output. The vector sign test defined by Eq. (9.2-6) is a discrete form of the Dicke-fix radar signal processing technique and has an ARE of 0.785 and an asymptotic loss of 1.0 dB. Hansen conducted a Monte Carlo simulation to determine the performance of the narrow-band Wilcoxon detector and the vector sign test for a finite number of observations. Since the data were not available in the literature, the probability density of the Wilcoxon detector test statistic had to be generated utilizing an importance sampling technique in a Monte Carlo simulation. The false alarm probability versus threshold curves were then found by direct integration. The importance sampling technique is described in detail in one of Hansen’s papers [1970b]. The results of this study are reproduced here in Fig. 9.2-1 [Hansen, 1970a1, which shows the signal-to-noise ratio (SNR) loss in decibels of the narrowband Wilcoxon and vector sign detectors when compared to the optimum parametric detector for different numbers of observations. The false alarm and detection probabilities for these results are a = and P, = 1 - p = 0.5. Figure 9.2-1 shows that at least 16 to 32 observations are required for the loss to be generally acceptable, and as the number of observations decreases below 16, the loss increases rapidly. A requirement for 50 observations to detect a target is not too stringent for most present-day radar systems, and therefore the loss suffered from the use of the narrowband Wilcoxon or vector sign detectors should be more than offset by their constant false alarm rate (CFAR) properties. The results in Fig. 9.2-1

1%

9 Engineering Applications

Pd VECTOR SIGN NARROW- BAND WILCOXON

0.5

a = lo*

\

I

0.1 I

2

4

8 16 32 64 NUMBER OF OBSERVATIONS

128

1

256

Fig. 9.2-1. Signal-to-noise ratio loss in dB of the narrow-band Wilcoxon and vector sign [Hansen, detectors compared to the optimum parametric detector. P, = 0.5; 01 = 1970al.

should be used with caution however, since probabilities of detection much greater than 0.S are necessary in most search radars. If the probability of detection were increased to the more realistic region of 0.9-1.0, the number of observations required to hold the loss in SNR to a reasonable level might increase significantly. Another interesting point that is illustrated by Fig. 9.2-1 is that the loss in SNR for the vector sign detector is less than the loss for the Wilcoxon detector when the number of observations is less than 30, even though the Wilcoxon detector has a higher ARE and has a smaller loss when the number of observations is > 30. This comparison illustrates the point that asymptotic results in general, and the ARE in particular, do not always clearly respresent the efficiency of detectors for small samples. A somewhat unique adaption of a nonparametric detector to the radar detection problem has been treated by Hansen [1970a, b]. The detector is based on the Kendall 7 test and has several practical advantages, one of which is the fact that incoherent signal processing may be employed, thus eliminating any requirement for a Doppler filter bank. The nonparametric detector in this application utilizes the fact that in a scanning pulsed radar system, the received pulse train is amplitude-modulated due to the gain pattern of the antenna. When a target is illuminated by the radar, the received observations should have a characteristic rank order. Hansen describes a digital implementation of a detector based on the above concept; the block diagram of the detector is reproduced here in Fig. 9.2-2 [Hansen, 197Obl. The input waveform to the detector is passed through a matched filter, envelope detected, and the resulting amplitudes converted

-

9.2 Radar System Applications

197

FROM PRE-AMP

MATCHED FILTER

COMPARATOR

T

Fig. 9.2-2. Block diagram of the Kendall T radar detector [Hansen, 1970bl.

into digital form. The observations in a single range resolution element are sent to an n-bit shift register which is divided into two halves. The Kendall 7 test is applied to each half, and the smaller of the two test statistic values is selected by a modified type of AND gate and compared to the threshold. The two Kendall7 tests utilize a statistic similar to that given by Eq. (3.8-5), but the actual test statistics are modified slightly so that a target echo centered on the complete n-bit shift register will maximize both test statistics. This detector implementation also utilizes the recursive property of the Kendall 7 detector in updating the shift register contents. The performance analysis of the Kendall 7 radar detector does not include the effects of the digital implementation and in this respect, is not directly applicable to the system just described. For instance, if the sampled observations are digitized, the background noise density can no longer be assumed to be continuous, and the possibility of tied observations must be considered; the effects of the modified AND gate must also be considered. The asymptotic loss of the Kendall T detector for a Gaussian-shaped antenna pattern and Gaussian noise, and including the

198

9 Engineering Applications

Fig. 9.2-3. Performance of the Kendall T radar detector and the optimum parametric detector for incoherent processing in Gaussian noise. ---optimum; ---Kendall T [Hansen, 1970bl.

loss due to the AND gate, is 2.8 dB when compared to the optimum parametric detector. The finite sample performance of the Kendall 7 detector was determined by Hansen utilizing a Monte Carlo simulation. A comparison of the Kendall 7 detector with the optimum parametric detector for incoherent processing in Gaussian noise can be made by using Fig. 9.2-3 [Hansen, 197Obl. The loss of the Kendall 7 detector is seen to vary from 3 to 4 dB for n/2 = 64 to 6 to 8 dB for n/2 = 16. By referring back to Fig. 9.2-1, it can be seen that for P, = 0.5, the loss for both the narrow-band Wilcoxon and the vector sign detectors is approximately 1 dB for 64 observations and increases to 5 dB for the Wilcoxon detector and 3 dB for the vector sign detector for 16 Observations. The Kendall 7 detector thus has a greater asymptotic loss than the other four one-input detectors discussed by Hansen and has a greater finite sample loss than the narrow-band Wilcoxon and vector sign detectors. The Kendall 7 test does have the advantage that due to its recursive property, it may be simpler to implement than other rank detectors. Hansen [ 1970al and Hansen and Olsen [ 19711 have investigated the application of two-input nonparametric detectors, or more precisely, oneinput detectors with a set of reference noise samples, to the radar detection problem. Noncoherent processing and a rectangular-shaped antenna beam are assumed, with the observations being taken from the envelope-detected output of a matched filter. To obtain the observations to be tested for a signal being present, one sample is taken per pulse transmission at a specified range (delay), and it is assumed that M such transmissions are possible as the antenna scans across the fixed target azimuth position. The reference noise samples are collected by taking a total of N observations at shorter and longer ranges than the target range. The noise observations are

-

9.2 Radar System Applications

199

assumed to be taken such that they are unaffected by the presence or absence of a target return, and for the purpose of obtaining the quantitative performance of the detectors, the reference noise observations are taken to be independent and identically distributed. The Mann-Whitney detector test statistic for this problem is %w=

M

M

2 2

N

24xkS-x;)

(9.2-7)

k=l j - I i - 1

where the 57 are the reference noise samples and the x i are the constant range samples to be tested for the presence or absence of a signal. The symbol u ( . ) denotes the unit step function defined previously. Besides considering the test statistic in Eq. (9.2-7), Hansen and Olsen also studied what they called the generalized sign detector which has a test statistic

x x +k” M

SGS

=

N

- x;)

(9.2-8)

k-1 j - 1

The Mann-Whitney test statistic totals the number of times the “signalplus-noise” observations are greater than any of the M * N reference noise samples, whereas the test statistic in Eq. (9.2-8) computes the number of times a signal-plus-noise observation in a specified pulse transmission interval exceeds the N noise observations in that same interval. The asymptotic losses of both detectors for white, Gaussian background noise were determined as a function of N and vary from a 0.9 dB advantage for N = 1 to a loss of 0.6 dB for both detectors compared to the optimum detectors at N = 00. Finite sample comparisons by Hansen and Olsen [1971] of the probability of detection as a function of SNR indicate a slight loss in detection probability by the generalized sign detector with respect to the Mann-Whitney detector for M = 4, N = 8, and a = for both fluctuating and nonfluctuating signals. For the M = 16, N = 2, a = case an additional loss occurs, with the loss becoming significant for M = 16, N = 2, and a = Although the results for finite sample sizes are not too encouraging, the asymptotic performance of the generalized sign detector is very good. Further note, as Hansen [1970a] points out, that for N = 1, Eq. (9.2-8) is the test statistic of a two-input version of the sign detector which is perhaps the simplest nonparametric detector. An even more important point for a practical application is that the antenna beam shape is in many cases not rectangular-shaped, and hence the signal-plus-noise samples may be modulated by the antenna beam shape. This could result in a reduced probability of detection for the Mann-Whitney detector if the noise observations are not similarly modulated, or at best, it complicates the

200

9 Engineering Applications

selection of a threshold to achieve a specified false alarm rate while maintaining a high detection probability. The generalized sign detector does not experience this problem since the signal-plus-noise samples are compared only to those noise observations taken during the same pulse transmission period, and therefore with approximately the same antenna gain. Dillard and Antoniak [ 19701 have recently formulated and analyzed a nonparametric detection procedure for application to radars that is a slight modification of the sign detector and, as will be seen as the discussion progresses, is related to the Kendall 7 detector. The detection procedure is based on the concept of a moving window detector which performs binary integration on the input observations [Dillard, 19671. A moving window detector compares each input observation with a threshold and assigns a value 1 to the observation if the threshold is exceeded, and a zero otherwise. The resulting 1’s and 0’s are continuously summed and com: pared to a second threshold. If the sum of the 0’s and 1’s exceeds the second threshold, a target is declared to be present. A slightly modified version of this moving window detector is utilized in the radar detection procedure described here. In the multiple-range-bin case, k 1 observations, spaced at intervals required to yield the desired range resolution, are taken after each transmitted pulse. Instead of each observation being compared to a fixed threshold, each observation is compared to every succeeding observation in the outer range bins, and a 1 is assigned to every range bin whose observation is greater than all of the outer observations, and a 0 is assigned otherwise. After the j t h transmitted pulse, all of the 0’s and 1’s for each range bin are summed and compared to a threshold T; a target is declared to be present in every range bin whose test statistic (sum of 0’s and 1’s) exceeds T. Under the hypothesis that noise only is present, let PH denote the probability that the observation in one range bin is greater than the observations in all outer range bins, then

+

1 PH = k + l

(9.2-9)

since k + 1 is the total number of range bins, and the input observations are independent and identically distributed. Let the test statistic for the ith range bin be denoted by S j , then the probability that Sj = r when noise only is present is given by ‘H

{ Si

=

‘1

=

( )( k+l )‘( 1

-

k+l ) 1

n--r

(9.2- 10)

9.2 Radar System Applications

201

where n is the total number of observations across the moving window detector. Equation (9.2-10) can be rewritten as

and used to obtain an expression for the false alarm probability

k

(9.2- 1 1) r= T

For a specified false alarm probability a = ao, the threshold setting T can be determined from Eq. (9.2-1 1). As was found in discussing the sign detector for fixed k and n, only certain discrete values of a may be exactly obtained unless some method of randomization is applied. Under the alternative that a target is present in the ith range bin and no target is present in the other k range bins, letp, denote the probability that the observation in the ith range bin is greater than the observations in the other range bins, then clearly PK

> l/

( k + l)

and using the probability density

p, { si =

'> = (;)(PKlrQ

- P,)^--'

(9.2-12)

-p,)'--'

(9.2-13)

the probability of detection can be written as

P, = 1 - p

=

2 (;)p;(l

r-

T

where T is found from Eq. (9.2-1 1). To obtain analytical results, the expression for p , given by Eq. (9.2-14) must be utilized, PK

=/-pWl*dG(w)

(9.2- 14)

where F(w) is the cumulative distribution function of the range bins containing noise only, and G(w) is the CDF of the range bin containing a target return. Dillard and Antoniak have calculated the performance of their nonparametric radar detector for the three target plus noise distributions given in Table 9.2-2. The noise only distribution is assumed to be the same for all three cases, namely ~ ( w =) 1 - exp( - w2/2)

(9.2- 15)

202

9 Engineering Applications Table 9.2-2 “Target present” probability density functions

Case

Densitv function

2

Case 1 is called the Rice distribution and is the density of the envelope of a constant amplitude sinewave plus additive narrow-band Gaussian noise. The symbol I) denotes the input SNR to the envelope detector. Case 2, called the Rayleigh density, and Case 3 are the densities of the output of a linear envelope detector when the target models are those known as Swerling’s cases 2 and 4 [Swerling, 19601. Since the input fluctuates from pulse-to-pulse for both cases, I) is the average SNR over all fluctuations at the input to the envelope detector. Figures 9.2-4-9.2-6 are reproduced from Dillard and Antoniak [1970] to illustrate the performance of their radar detector, called the modified sign test, when compared to other detectors for the three target present densities in Table 9.2-2 (N = the number of observations). A discussion of the video integration and binary integration detection schemes is given in Dillard and Antoniak [1970]. In all three cases the loss in SNR is seen to be approximately 2 dB with respect to the video integration, and 1 dB with respect to binary integration. This loss is, of course, almost negligible, especially when it is considered that the video and binary integration procedures would be greatly affected by changes in the background noise density, whereas the modified sign test will maintain a constant false alarm rate for a wide class of noise densities. The modified sign test has several important disadvantages. One is the fact that weak targets at longer ranges than strong targets at the same relative antenna position may not be detected. This situation can be improved by comparing different sets of range bins from scan-to-scan, by comparing range bins in varying orders, or by applying some other similar technique. The advantages of such techniques, however, will in general be paid for in system complexity. The modified sign test also has the disadvantage that highly correlated clutter signals may cause a target to be declared present. The utilization of some filtering or clutter rejection technique could reduce this problem somewhat, but again, only with increased system complexity. The detection performance of the nonparametric Dicke-fix radar detector, also called the vector sign test, and the asymptotically nonparametric

203

9.2 Radar System Applications 1.0-

@ 0.5-

0S N R td0)

Fig. 9.24.

Fig. 92-5.

I.o

Q

0.5

Figs. 9.2A9.M. Comparison of the modified sign test with nondistribution free procedures [Dillard and Antoniak, 19701. The curves in Figs. 4 - 5 are as follows: A , video integration; B, binary integration; and C, modified sign test. For Fig. 6 the curves are: A, binary integration ; B, modified sign test.

-

0 SNR (d0)

Fig. 9.2-6.

Siebert detector has been computed by Hansen and Zottl [ 19711. This comparison is particularly interesting since the two detectors have actually been used in radar systems in the past and since the Siebert detector is a version of the one-input Student’s t test. The results obtained indicate a 1-dB disadvantage for the Dicke-fix receiver for both finite and asymptotically large sample sizes. Again, however, these results must be interpreted with caution since the detection probability used was 0.5. Zeoli and Fong [1971] investigated the performance of the MannWhitney detector with respect to the optimum detector for testing a positive location shift in Gaussian noise, the linear detector, over a wide variety of noise probability densities. The results were, as could have been predicted, that the Mann-Whitney detector maintained a more constant false alarm rate, generally performed well in comparison with the linear detector, and for certain input noise statistics, was clearly superior in performance. One important factor in practical applications that must be emphasized, however, is that the Mann-Whitney detector is considerably more complex than the linear detector. All of the nonparametric detectors considered in this section appear to be quite useful when applied to the radar detection problem. The two most attractive properties of these detectors are their simplicity of implementa-

204

9 Engineering Applications

tion (sometimes) and the maintenance of a constant false alarm rate (always) for a wide range of noise densities. The Wilcoxon and MannWhitney detectors 'are the most efficient of the detectors considered, but the requirement that large amounts of data be rapidly ranked may prohibit their use in real-time applications. The nonparametric detectors based on the sign test are relatively efficient detectors for the detection problems considered and have the advantage of not requiring the ranking of observations, and thus are quite simple to implement. In fact, the practical usefulness of the vector sign detector has already been proven since its continuous analog, the Dicke-fix receiver, has been applied successfully for many years. A detection system already in use by the Federal Aviation Administration, designated the Common Digitizer [ 19701, is strikingly similar to the modified sign test formulated by Dillard and Antoniak. Nonparametric detectors have thus already proven their practical usefulness, and their application to the search radar detection problem is seen to be well justified by the results discussed in this section. 9.3 Other Engineering Applications Less energy has been expended by engineers in trying to apply nonparametric detectors to engineering problems other than various radar detection problems. This does not mean that such applications are not possible. It simply emphasizes the fact that engineers in general have only recently become aware of nonparametric detectors and hence much work remains to be done in applying these detectors to realistic problems. A performance relation which is very useful for system design has been derived and used by Capon [1959b, 19601, and later used by Hancock and Lainiotis [ 19651 for detector performance evaluation. Although the proof is not included here, it can be shown [Capon, 1959bl that for any two-input detector test statistic which satisfies the regularity conditions given in Chapter 2 that are necessary for the efficacy to exist, the performance relation

& O:mn/ ( m + n) = 2{erf-'[l - ~cu,,,,,] + erf-'[ 1 - 2P,,,,,])

2

(9.3-1)

holds for m, n large and el small (the weak signal case). In Eq. (9.3-1), a,,,,, and P,,,,, denote the false alarm and false dismissal probabilities, respectively, & is the efficacy, and

205

9.3 Other Engineering Applications

A similar relation can be obtained for one-input detectors as [Capon, 1959b, 19601

& fl?n = 2{erf-'[ 1 - 2a,]

+ erf-'[

1 - 2P,]}2

(9.3-2)

where the parameters are as defined for Eq. (9.3-1). The performance relations in Eqs. (9.3-1) and (9.3-2) are extremely important for detector performance comparisons and system design since they provide a relatively simple algebraic relation between the detector efficacy, the SNR, the number of observations required, and the false alarm and false dismissal probabilities. It is thus possible, for example, to determine the number of observations required by a detector to achieve a specified a and for a given input signal magnitude if the detector efficacy is known. Of course, the efficacies of most of the more familiar nonparametric detectors have been derived and are available in the literature. A specific example of the utilization of these equations for system design is given in the following application. In many radio communication systems, the magnitude of the input signal to the receiver varies with time even though the transmitted signal power remains constant. This phenomenon is called signal fading and is caused by a wide variety of physical channel effects [Schwartz et al., 19661. Capon [ 1959b] described the application of the Mann-Whitney nonparametric detector to such a communication system problem, which he called the scatter propagation problem. In this problem the received signal power is not constant due to transmission channel effects, but the background noise and reference noise samples are assumed to be stationary and Gaussian with zero mean and unit variance. The noise is also assumed to be additive. The assumption that the noise is stationary even though the signal is fluctuating is valid, since the primary noise sources are present in the receiver itself. A common example of such a noise is thermal noise generated in the receiver front end. The resulting detection problem is therefore one in which it is desired to detect a time-varying signal in a background of stationary noise. If it is assumed that a set of m reference noise observations are available, the hypothesis and alternative to be tested consist of H:

K:

CDF of the xi is F ( x ) , i = 1,2, . . . , m + n (signal absent) CDF of the xi is G@,(x),i = 1,2, . . . , n and the CDF of x,,+~ is ~ ( x )i ,= 1, 2, . . . , m (signal present)

where the xi, i = 1, 2, . . . , n are the input observations to be tested and the xi+,,, i = 1, 2, . . . , m are the reference noise samples. The probability distribution function under the alternative is assumed to differ from

N16

9 Engineering Applications

sample to sample only in terms of the SNR parameter 0,. In order to use the performance relation in Eq. (9.3-1), it is necessary to let m and n be large, and calculate the asymptotic distributions under the hypothesis and alternative. If the observations are independent when H is true, the joint distribution function is just a product of the marginal distributions, and hence is Gaussian. Capon proves, under the restrictions that the limit of m/n exists as m and n + 00 and that F ( x ) and Go,(x)are continuous, that the Mann-Whitney test statistic (actually Capon’s V test which is a slightly modified version of the Mann-Whitney detector and which has been previously considered in Section 5.2) is asymptotically Gaussian when K is true. Further, the means and variances under the hypothesis and alternative are given by E { V I H } = 1/2

+ n + 1)/12mn E { V I K } = (1/2) + (8/6)

var{ V I H } = (m

e

= var{ V I K }

where = Z;-,?/n,and it has been assumed that when K is true, F ( x ) and G8,(x)are related by G,,(x) = ( I -

e , ) q x )+ e p ( . )

Using these results, the efficacy becomes & = 1/3 Letting 8, in Eq. (9.3-1) become performance relation becomes

(9.3-3) (9.3-4)

e and substituting for the efficacy, the +

-e’mn - 6{erf-’(l

- 2a) erf-’(1 - 2p)}’ (9.3-5) m + n Since the reference noise observations are obtained during system “dead time,” it is reasonable to assume that m >> n, and therefore, Eq. (9.3-5) can be written as

ne’ = 6{erf-’(l - 2a)

+ erf-’(1

- Zp)}’

(9.3-6)

To arrive at a specific system design, assume that it is necessary for satisfactory operation that a = p = 0.01, and that the average value of the peak signal to rms noise ratio is known to be 0.1. Substituting these values into Eq. (9.3-6) and solving for n yields

n = 600(erf-’(0.98)

+ erf-’(0.98)}’

= 6500 observations for m >> n To satisfy the assumption that m > n, let m = 10n = 65,000 samples. The

207

9.3 Other Engineering Applications

threshold setting necessary to achieve the false alarm probability of a = 0.01 can be found from 1 -(Y 0.01 = -~a exp[ 2 4

&

ZI,

by letting u: = var{ Y I H }

and

/.to =

I+

(9.3-7)

E{ VIH)

Substituting and solving for the threshold T produces T = 0.5. The probability of an error in this communication system is thus P, = a P { H } P P { K } = O.O1{P{H} + P { K } } = 0.01

+

For this detection problem then, the number of observations required to attain the desired performance level for the weak signal received is fairly large. Additionally, the storage of such a large number of reference observations (65,000) seems prohibitive. This application, however, which is due to Capon [1959b], has illustrated how Eqs. (9.3-1) and (9.3-2) can be used for system design. One possible application of the polarity coincidence correlator is the detection of a weak noise source or random signal in additive background noise. This problem arises in radio astronomy or acoustics, as in an underwater sound detection system. Ekre [ 19631 has compared the performance of the PCC with that of an analog cross-correlator and an analog version of the PCC for three different input signal spectra. The signal and the noises in the two different inputs are all assumed to be independent, zero mean, Gaussian processes with identical normalized power spectra. The SNR is used as the measure of performance. The three different input signals are obtained by passing the white noise signal through an RC low pass filter, a single-tuned RLC filter, and a rectangular band-pass filter which have normalized autocorrelation functions P R d 7 ) = exp( 1' ) (9.3-8) pRLC.7) = exp( -Ao(T~) cos wor PBP(7) =

(9.3-9)

sin Ao7 cos w0r

(9.3- 10)

respectively. The results indicate a loss for broad-band signals of 1-10 dB for the PCC with respect to the analog cross-correlator depending on the sampling frequency. The loss decreases to the 1-4 dB range for narrow-band input signals. The loss of the PCC with respect to the analog PCC is 0.5 dB or less. For relatively high sampling frequencies, the losses fall in the lower

-

208

9 Engineering Applications

end of the given ranges, and therefore, the losses experienced may be negligible in light of the simple implementation requirements of the digital PCC discussed in this text. The following narrow-band version of the Wilcoxon detector was formulated by Helstrom [ 19681 for detecting a narrow-band phase-incoherent signal, and is equivalent to the narrow-band Wilcoxon detector discussed in Section 9.2 and investigated by Hansen [ 1970al for application to radar. A similar test statistic has also been suggested by Carlyle [ 19681. The input signal is assumed to be (9.3-1 1) s ( t ) = A Re{f(t) exp(jwt + j + ) } + n ( t )

+

where A is constant, w is the known carrier frequency, is the unknown phase, and n ( t ) is additive white, Gaussian noise. The input signal is mixed with coswt and sin at to generate the in-phase and quadrature components of s ( t ) . From these components two sets of independent observations are obtained via sampling. These two sets of observations given by { xC1,xC2,. . . , x,,,} and { y,,, ysz, . . , ,y,,} are ranked separately by their absolute values as and lxci,l < lxci21 < . * * < lxciml lxsj,l < Ixsj21< * * * < 1xsi.l where { i l , i,, . . . , in} and { j l , j 2 ,. . . , j n } are some permutations of the first n integers. Using these ranked values, the test statistics

(9.3-12) and (9.3-13) are computed along with

s; = i n ( . s; = in(.

+ 1) - s, + 1) - s,

(9.3- 14) (9.3- 15)

The statistic S: = max{S,, S;} and S," = max{Ss, S ; } are squared and summed, and the result compared to a threshold. Whenever the test statistic is greater than the threshold, a signal is decided to be present. The resulting detector is a two-sided test of the input signal magnitude. For the no-signal present condition, the phase should be random and the test statistics in Eqs. (9.3-12) and (9.3-13) will be close to their mean value, an(, + 1). When a signal is present, however, the phase should remain relatively constant and one or both of the mixer outputs will have a large number of either positive or negative values, thus producing a large value

9.4 Summary

209

of the final test statistic (Sl)2 + (S:)2. Since this statistic is closely related to that narrow-band Wilcoxon version proposed by Hansen, the performance should be the same as the narrow-band Wilcoxon detector discussed in Section 9.2. A few applications of nonparametric detectors to some important communication system configurations have been presented in this section. Other applications of nonparametric detectors to communication system problems are given by Capon [1959b, 19601, and Hancock' and Lainiotis [ 19651. The Mann-Whitney and Wilcoxon detectors' applications could easily have been considered in Section 9.2 rather than here, and reciprocally, the narrow-band forms of the normal scores and sign detectors could have been developed in this section. In any event, it is evident that forms of nonparametric detectors suitable for applications to realistic communication systems can be devised, and therefore, nonparametric or distribution-free detectors provide a viable engineering alternative to parametric detectors for communication system applications. 9.4

S

m

Since nonparametric detectors were originally developed primarily by statisticians for solving nonengineering problems, the test statistics discussed in Chapters 3 and 5 many times are not in a form which allows them to be used for engineering applications. In this chapter, modifications to detector test statistics necessary to provide a form suitable for radar and communication system applications have been presented and performance results of these detectors for their specific applications have been given. The general conclusion that can be reached is that radar and communication system applications of nonparametric detectors are possible, and therefore nonparametric detectors can be used profitably in those systems which require the maintenance of a constant false alarm rate for varying input probability densities. The modified nonparametric detector test statistics seem to retain the encouraging performance characteristics of the original detectors. As engineers become more familiar, and hence, more comfortable with nonparametric detectors, the applications of these detectors to realistic engineering problems is certain to increase.

APPENDIX A

Probability Density Functions

Frequently in the text, a random variable is said to have a certain type of probability density function, such as Gaussian, Cauchy, Laplace, uniform, or others. Since the exact form of the probability density function indicated by these labels is not perfectly standardized and since the reader may be unfamiliar with some of these probability densities, those probability density functions which appear often in the text are listed here for easy reference. Obviously, no attempt has been made at completeness, but rather the goal is to minimize confusion and thus to allow efficient utilization of the textual presentation.

Gaussian or Normal (Fig. A-1)

Parameters: I -

p, u

fX (X

1

P-sCr PQU P-@ P

210

.

x

W WUP+50

Flg. A-1. Example of a Gaussian density function.

ProbabiIity Density Functions

2zz

Uniform or Rectangular (Fig. A-2)

1,

fX(x)

Parameters:

a, b

1fx fl

1 b-a '

for a < x < b otherwise

- -----

'

b-a

,x

Fig. A-2. Example of a density function for a uniform distribution.

Cauchy (Fig. A-3)

Parameters:

A, 0

Fig. A-3.

Example of a Cauchy density function.

Laplace or Double Exponential (Fig. A-4) fX

Parameter:

( x ) = f h e-+l

- 00

< x < 00

A

Exponential (Fig. A-5) fx(x) =

Parameter:

for

A

("7,

for x > o otherwise

Appendix A

212

-2

0

-I

I

Fig. A-4. Example of a Laplace density function.

0

I

I

I

2

Ax

Fig. A-5. Example of a density function for an exponential distribution.

Binomial (Fig. A-6) f x ( x ) = P { X = k } = b(k; n , p ) = (;)p"qn-"

Parameters: n, p Student's t (Fig. A-7)

Parameter: n

for k = 0, 1 , 2 , .

. . , n.

Probability Density Functions

213

Fig. A-7. Example of a density function for a Student's r distribution.

APPENDIX B

Mathematical Tables

Table

Number B- 1 B-2 B-3 B-4

B-5 B-6 B-7 B-8

214

Title Cumulative binomial distribution Cumulative Gaussian distribution Cumulative Student’s t distribution Wilcoxon (one-input) test: Upper tail probabilities for the null distribution Expected normal scores Mann-Whitney (two-input) test: Upper tail probabilities for the null distribution Quantiles of the Kendall 7 test statistic Quantiles of the Spearman p test statistic

Table B-1 Cumulative binomial distribution" ~~

n

r'

2

1 2

- -

3

4

5

6

7

8

9

1

2 3

1 2 3 4

1 2 3 4 5

.10 .15 - 0975 ,1900 ,2775 ,0025 .0100 ,0225

.05

-

~

~~

~

P .25 .3600 4375 ,0400 .0625 .20.

-

-

.30

.35

.40

,5100 ,0900

,5775 ,1225

,1600

.45

.50

- -

,6400 .6975 ,2025

,7500 ,2500

,2710 .3859 ,0280 .0608 ,0010 .0034

.488O ,1040 .0080

,5781 ,1562 ,0156

6570 ,2160 ,0270

.7254 ,7840 ,2818 .3520 ,0429 ,0640

,1855 .0140 .0005

.3439 .0523 .0037 .o001

,4780 ,1095 .0120 ,0005

,5904 ,6836 ,1808 .2617 ,0272 .0508 ,0016 ,0039

.7599 ,3483 ,0837 ,0081

,8215 ,4370 .1265 ,0150

,2262 ,0226 ,012

,4095 ,0815 ,0086 .o005

,5563 ,1648 ,0266 ,0022 .0001

,6723 .2627 ,0579 ,0067 ,0003

,7627 ,3672 .lo35 .0156 ,0010

,8319 .4718 ,1631 ,0308 ,0024

,8840 ,9222 ,5716 .6630 ,2352 ,3174 .0870 ,0540 ,0102 ,0053

,9497 ,7438 .4069 ,1312 ,0185

,8824 ,5798 ,2557 ,0705

,9246

,9533 ,7667 ,4557 ,1792 ,0410

,9723 ,8364

,2553 .0692

.6562 ,3438 .lo94

,1426 ,0072

.ow1

.oooo

.m .m .m

.8704 ,5248 ,1792 .02.58

.a336 .4252 ,0911

.a750

.5000

,1250

,9375 ,9085 ,6090 ,6875 .2415 ,3125 ,0410 ,0625 ,9688

,8125

.5000

,1875 ,0312

1 2 3 4 5

,2649 .0328 ,0022 .0001

,4686 .1143 ,0158 ,0013 ,0001

,6229 .2235 .0473 ,0059 ,0004

,7379 .3447 ,0989 .0170 ,0016

,8220 ,4661 ,1694 ,0376 ,0046

.OlW

,3529 ,1174 ,0223

6

.m .oooo

.m

,0001

,0002

.o007

.0018

.0041

,0083

,0156

1 2 3 4 5

,3017 ,0444

.6794 ,2834 .0738 ,0121 ,0012

.7903 .4233 .1480 .0333 ,0047

,8665 ,5551 ,2436 ,0706 ,0129

,9176 ,6706 ,3529 ,1260 ,0288

,9510 ,7662 ,4677 ,1998 .0556

,9720 ,8414

.9848 .a976 .6836 ,3917 ,1529

,9922 ,9375 ,7734

6 7

.m .m ,0001 .o004 ,0013 ,0038 .0090 ,0188 ,0357 ,0625 .m .m .oooo .m . o001 ,0002 .0006 ,0016 .0037 ,0078

1 2 3 4 5

,3366 ,0572

.m

,0038

,0002

.m

.0058

,0004

,5217 .1497 ,0257 ,0027 .oO02

,5695 .la69 ,0381 ,0050

.m ,0004

,7275 ,3428 ,1052 ,0214 .0029

.8322 ,4967 ,2031 .0563 ,0104

.a999 ,6329 ,3215 ,1138 ,0273

,9424 ,7447 .4482 ,1941 ,0580

,6809

,5801

.2898 ,0963

,5585

.9844 ,8808

.m ,2206

,9681 .9832 ,9916 ,9961 ,8309 ,8936 ,9368 ,9648 .5722 ,6846 .7799 ,8555 ,2936 ,4059 ,5230 ,6367 .lo61 .1737 ,2604 ,3633

.m .m ,0002 .0012 ,0042 .0113 ,0253 ,0498 ,0885 ,1445

6 7 8

.oooo .oooo .m

. o001

,0004 ,0013

,0036

,0085

.0181

,0352

.ooo9

.m .m .ooo .oooo ,0001 ,0002 ,0007 ,0017 .0039

1 2 3 4 5

,3698 ,0712 ,0084 ,0006

.6126 ,2252

.a658 ,5638 .2618

,9249 ,6997 ,3993 ,1657 ,0489

,9793 ,8789 ,6627 ,3911 ,1717

.9899 ,9954 .9295 ,9615 ,7682 .a505 ,5174 ,6386 .2666 ,3786

,9980 .9805 ,9102 ,7461

,0536

,0994 ,1658 ,2539 .0250 .0498 0898 .0038 ,0091 ,0195 .0003 .o008 ,0020

6 7 8 9

.m

,0530

,0083 .o009

.m .0001

,7684 .4005 .1409 ,0339

,0856

,0056

.0196

.OOO6

,0031

,9596 ,8040 ,5372 .2703

,0988

,0100 .0253 .0043 .OOO4 .0000

.oooo .oooo .oooo .0003 .0013 .oooo .oooo .oooo .0000 ,0001 .oooo .oooo .oooo .oooo .oooo

. 0112

,0014 ,0001

.5000

215

Appendix B

216

Table B-1 (Continued)a

n

in

I’

i

2 3 4 5

6 7 8 0

10 11

086 1

0115 0010 0001

m m m m m

10

15

6513 2639 0702 0128 0016

8031 4557 1798

- -

0001

0500 0099

0014 0001

8926 6242 ,3222 ,1209 ,0328

P 25 9437 7560 4744 2241 0781

,0064 .oO09 ,0001

0197 0035 0004

.20

- -

.40

.45

,9718 ,9865 ,8507 ,9140 ,6172 ,7384 ,3504 .4862 ,1503 ,2485

9940 9536 ,8327 ,6177 3669

9975 ,9767 . m 4 ,7340 ,4956

,0473 ,0106 ,0016 ,0001

,0949 ,0280 ,0048 .0005

.1662 ,0548 ,0123 ,0017 .0001

,2616 ,1020 ,0274 .0045 .0003

30

.35

.50

-

.m

,9893 ,9453 8281 ,6230 ,3770 ,1719 ,0547 .0107 .0010

m m

m

m

m

.m oooo .m m

.m .m

6862 3026 0896 0185

8327 5078 2212 0694 0159

,9141 ,6779 ,3826 .lSll .0504

9578 8029 5448 2867 1146

.9802 .a870 .6873 ,4304 ,2103

.9912 ,9964 .9986 .9995 ,9394 ,9698 ,9861 ,9941 ,9348 ,9673 ,7999 ,8811 ,5744 ,7037 ,8089 ,8867 ,3317 ,4672 ,6029 ,7258

0003

0027 0003

,0117

0343 0076 0012 0001

,0782 .0216 .0043

,1487 ,0501 .0122 .o020 .oO02

m o o 0 0

4312 1019 0152 0016 0001

0028

7 8 9 10

m m m m m

m m m m

m .m m .m .ooo8 m .m m .m

,2465 ,0994 ,0293 ,0059 .oO07

,3889 ,1738 ,0610 ,0148 ,0022

,2744 ,1133 ,0327 ,0059

11

m

m

m .m m .m .m .m

,0002

.m5

1 2 3 4 5

4588 1184 0196 0022 0002

,7176 ,3410 ,1109 ,0258 ,0043

9683 8416 8083 3512 1576

,9862 ,9150 ,7472 ,5075 ,2763

6

m ,0005 .0046 ,0194 0544 m .0001 ,0007 .0039 0143 m .m ,0001 ,0006 0028 m .m .m .0001 0004 m .m .m .m oooo

,1178 .03M ,0095 .0017 .o002

1

2 3 4 5 6

12

05 4013

7 8 9

10 11

12

,8578 ,5585 ,2642 ,0922 ,0239

,0020

.9313 ,7251 ,4417 ,2054 ,0726

.m

,9943 ,9576 .a487 ,6533 ,4167

,9978 ,9804 .9lM ,7747 ,5818

,8992 ,9917 ,9579 ,8655 .6956

,9998 ,8888 .8807 .9270 .a062

.2127

,3348 ,1582 .0573 ,0153

,4731 .2607 .1117 .0358 f0079

.6128 .a72 ,1938

,0848

.0255 ,0056 .0008

,0028

.0730

.0n3

m .m .m .m m .m ,0001 ,0003 .0011 .0032 m .m .m .m m .m .m .m ,0001 .0002

-

“From CRC Staruhrd Mathematical Tables, S. M. Selby and B. Girling, editors, Copyright 0 The Chemical Rubber Co., 1965. Used by permission of the Chemical Rubber Company.

Table B-2 Cumulative Gaussian distribution".

.W/ .01

.

.08

z

----------

.o

.5OOO .5398 .5793 .6179 .655

.5040 .5438 .5832 .6217 .6591

.5478 .5871 .6255 .6628

.5517 .5910 .6293 .6664

.5557 .5948 .6331 .6700

.5596 .5987 .6368 .6736

.5636 .5675 .5714 .5753 .6026 .MI64 ,6103 .6141 .6406 .6443 .6480 .6517 .6772 . .6844 .6879

.5 .6 .7

.9

.6915 .7257 .7580 .7881 .8159

.6950 .7291 .7611 .7910 .8186

.6985 .7324 .7642 .7939 .8212

.7019 .7357 .7673 .7967 .8238

.7054 .7389 .7704 .7995 .8264

.7088 .7422 .7734 .8023 .8289

.7123 .7454 .7764 .8051 .8315

1 .o 1.1 1.2 1.3 1.4

.8413 .8643 .8849 .9032 .9192

.8438 .8665 .8869 .go49 .9207

.8461 .a85 .SO8 .8686 .8708 .8729 .8888 .8907 .8925 .9066 .9082 .9099 .9222 .9236 .9251

.8531 .8749 .8944 .9115 .9265

.8554 .8770 .8962 .9131 .9279

1.5 1.6 1.7 1.8 1.9

.9332 .9452 .9554 .9641 .9713

.9345 .9463 .9564 .9649 .9719

.9357 .9474 .9573 .9666 .9726

.9370 .9484 .9582 .9664 .9732

.9382 .9495 .9591 .9671 .9738

.9394 .9505 .9599 .9678 .9744

2.0 2.1 2.2 2.3 2.4

.9772 .9821 .9861 .9893 .9918

.9778 .9826 .9864 .9896 .9920

.9783 .9830 .9868 .9898 .9922

.9788 .9834 .9871 .9901 .9925

.9793 .9838 .9875 .9904 .9927

.9798 .9842 .9878 .9906 .9929

.9941 .99S6 .9967 .9976 .9982

.9943 .9957 .9968 .9977 .9983

.9945 .9959 .9969 .9977 .99a

.9987 .9993 .9993 .9991 .9994 .9995 .9995 .9995 .9997 .9997 .9997

.9988 .9991 .9994 .9996 .9997

.9988 .9992 .9994 .9996 .9997

.1 .2 .3 .4

.8

2.s 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4

:::I

.04

.05

.06

.09

.9803 .9846 .9881 .9909 .9931

1

.7157 .7486 .7794 .8078 .8340

.71 .7517 .7823 .8106 .8365

.7224 .7549 .7852 .8133 .8389

.9418 .9525 .9616 .9693 .9766

.9429 .953S .9625 .9699 .9761

.9441 .9545 .9633 .9706 .9767

.9808 .9812 .9817

.9850 . 9 w .9857 .9884 .9887 ,9890 .9911 .9932

.9986l .998G

1.2821.6451.9602.3262.5763.0903.2913.891 4.417 --------

F(4

- F(z)l

"F(x) =

.03

.5080 .5120 .5160 .5199 ,5239 .S279 .S319 .5359

::ti

2

zI1

.02

J_a,

.90 .95 .975 _____------.20

.99 .995 ,999 .9995 .99995 ,999995

.10 .05 .02 .01

.002 .001

I

.OOO1 .ooOol

e-*/2dt

'From Introduction to the Theory of Statistics by A. M. Mood and F. A. Graybill, 2nd ed., Copyright Q 1963, the McGraw-Hill Book Company, Inc. Used with permission of McGrawHill Book Company. 217

Appendix B

218

Table B-3 Cumulative Student's t distributiona#

'

.75

.90

.95

.975

.9995 -I-

1 2 3 4

1 .Ooo .816 .765 .741 .727

3.078 1.886 1.638 1.533 1.476

6.314 2.920 2.353 2.132 2 015

12.706 4.303 3.182 2.776 2.571

31.821 6.965 4.541 3.747 3.365

63.657 9.925 5.841 4.604 4.032

636.619 31.598 12.941 8.610 6.859

.718 .711 .706 .703 .700

1.440 1.415 1.397 1.383 1.372

1.943 1.895 1.860 1.833 1.812

2.447 2.365 2.306 2.262 2.228

3.143 2.998 2.896 2.821 2.764

3.707 3.499 3.355 3.250 3.169

5.959 5.405 5.041 4.781 4.587

11 12 13 14 15

.697 .695 .694 .692 .691

1.363 1.356 1.350 1.345 1.341

1.796 1.782 1.771 1.761 1.753

2.201 2.179 2.160 2.145 2.131

2.718 2.681 2.650 2.624 2.602

3.106 3.055 3.012 2.977 2.947

4.437 4.318 4.221 4.140 4.073

16 17 18 19 20

.690 .689 .688 .688 .687

1.337 1.333 1.330 1.328 1.325

1.746 1.740 1.734 1.729 1.725

2.120 2.110 2.101 2.093 2.086

2.583 2.567 2.552 2.539 2.528

2.921 2.898' 2.878 2.861 2.845

4.015 3.965 3.922 3.883 3.850

21 22 23 24 25

.686 .686 .685 .685 .684

1.323 1.321 1.319 1.318 1.316

1.721 1.717 1.714 1.711 1.708

2.080 2.074 2.069 2.064 2.060

2.518 2.508 2.500 2.492 2.485

2.831 2.819 2.807 2.797 2.787

3.819 3.792 3.767 3.745 3.725

26 27 28 29

.684 .684

.683

1.315 1.314 1.313 1.311 1.310

1.706 1.703 1.701 1.699 1.697

2.056 2.052 2.048 2.045 2.042

2.479 2.473 2.467 2.462 2.457

2.779 2.771 2.763 2.756 2.750

3.707 3.690 3.674 3.659 3.646

.681 .679 .677 .674

1.303 1.296 1.289 1.282

1.684 1.671 1.658 1.645

2.021 2.000 1.980 1.960

2.423 2.390 2.358 2.326

2.704 2.660 2.617 2.576

3.551 3.460 3.373 3.291

6 6 7 8 9 10

*

.683 .683

30 40

60

120 a

a

F(t)

=/'

(+)!

x) 2

(*+1)/2

dx

-= (Y)!G(l+

bAdapted from Table 111 of Statistical Tables for Biological, Agricultural, and Medical Research, by R. A. Fisher and F. Yates, 6th ed., Longman, 1974. Published by permission.

219

Mathematical Tables Table B-4 Wilcoxon (one-input) test: Upper tail probabilities for the null distribution' n X

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 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 44 45

3

.625 .375 .250 .125

4

.562 .438 .312 .188 .125 .062

5

6

7

8

9

so0

.406 .312 .219 .156 .094 .062 .031

so0

.422 .344 .281 .219 .156 .lo9 .078 .047 .031 .016

.53 1 .469 .406 .344 .289 .234 .188 .148 .lo9 .078

.055

.039 .023 .016 .008

527 .473 .422 .371 .320 .273 .230 .191 .156 .125 .098 .074

.055

.039 .027 .020 .012 .008 .004

.500

.455 .410 ,367 .326 .285 .248 .213 .180 .150 .125 .102 .082 .064 .049 .037 .027 .020 .014 .010 .006 .004 .002

220

Appendix B Table B-4 (Continued)" n ~

X

28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77

10

~~~~

11

12

13

14

15

500

.461 .423 .385 .348 .312 .278 .246 .216 .188 .161 .138 .116 .097

.080

.065 .053 .042 .032 .024 .019 .014 .010 .007

.005

.003 .002 .001

.517 .483 .449 .416 .382 .350 .319 .289 .260 .232 .207 .183 .160 .139 .120 :lo3 .087 .074 .062 .05 1 .042 .034 .027 .021 .016 .012 .009 .007

.005

.003 .002 .001 .001

.ooo

.515 .485 .455 .425 .396 .367 .339 .311 .285 .259 .235 .212 .190 .170 .151 .133 .117 .lo2 .088 .076

.065 .055

.046 .039 .032 .026 .021 .017 .013 .010 .008 .006

.005

.003 .002 .002 .oo 1 .oo 1

.ooo

500 .473 ,446 .420 .393 .368 .342 .318 .294 .271 .249 .227 .207 .188 .170

.153 .137 .122 .lo8 .095 .084

.013 .064 .055

.047 .040

.034 .029 .024 .020 .016 .013

500 .476 .452 .428 .404 .380 .357 .335 .313 .292 ,271 .25 1 .232 .213 .196 .179 .163 .148 .134

.121 .108 .097 .086 .077 .068

.511 .489 .467 .445 ,423 .402 .381 .360 .339 .319 .300 .281 .262 .244 .227 .211

.195 .180

221

Mathematical Tables Table B-4 (Continued)o n X

78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120

10

11

12

13

14

15

.ooo

.011 .009 .007

.059 .05 2 .04 5 .039 .034 .029 .025 .02 1 .O 18

.165 .i51 .138 .I26

.005

.004 .003 .002 .002 .oo 1 .oo 1 .oo 1

.ooo .ooo .ooo

.o 1 5 .o 12 .o 10

.008 .007

.005

.004 .003 .003

.002

.002 .oo 1 .oo 1 .oo 1

.ooo .ooo .ooo .ooo .ooo

.115

. I 04 .094 .084 .076 .068 .060 .053 .04 7 ,042 .036 .032 .028 ,024 .02 1 .O 18 .o 15 .O 13 .01 I .009 .008 .006

.005

.004 ,003 .003 ,002 .002 .oo 1 .oo 1 .001

.oo I .ooo ,000 .ooo .ooo .ooo .ooo ,000

“Adapted from A Nonparametric Introduction to Statistics, by C. H. Kraft and C. van Eeden, Copyright Q 1968, The Macmillan Company. Published by permission.

Appendix B

222

Table B-5 Expected normal scoresa

Y

2

b

1

0.56419 0.84628

2 3 4 5

11 1 2 3 4 5

6 7 8 9

10

1 2

3 4 5

6 7 8 9 10 11 12

13 14 15

3

-

40000

12

13

-

4

5

1.02938 1.16296 0.29701 0.49502

-

14

40000

-

6

-

1.86748 1.40760 1.1 3095 0.92098 .74538

O~QOOOO 0,10259 0.19052

-

-

21

22

23

1.88917 1.43362 1.16047 0.95380 .78150

1.90969 1.45816 1.18824 0.98459 41527

142916 1.48137 1.21445 1.01356 0.84697

8

9

1.42360 1.48501 045222 093330 *47282 .57197 *00000 -15251 .27453 40000

1.26721 1.35218 0.64176 0.75737 .20155 a35271

-

-

-

15

16

1.53875 1.58644 1.62923 1.66799 1.70338 1.73591 1.76599 1@0136 1.06192 1.11573 1.16408 1.20790 1.24794 1.2k474 0.65606 0.72884 0.79284 044983 0.90113 044769 0.99027 .71488 *76317 46176 .37576 *46198 .53684 40285 .12267 .22489 .31225 .38833 .45557 -51570 .57001 -

7

17

18

19

1.79394 1.31878 1.02946 0.80738 .61946

142003 1.35041 1.06573 0.84812 46479

1.84448 1.37994 1.09945 0.88586 .70661

0.26730 0.33530 0.39622 0.45133 0.50158 0.54771 .1653O *23375 .29519 .35084 40164 .07729 .14599 .20774 ,36374 wJo00 .ooOOO .06880 -13072

4oooO 0.08816

-

24

1.94767 1.50338 1.23924 1~04091 0-87682

-

-

40000

25

26

n

28

29

1.96531 1.52430 1.26275 1.06679 0.90501

1.98216 1.54423 1.28511 1.09135 0.93171

1.99827 1.56326 1.30641 1.11471 0.95705

2.01371 1.58145 1.32674 1.13697 0.98115

2.02853 1.59888 1.34619 1.15822 140414

0.59030 042982 0.66667 0.10115 0.73554 0.76405 0.79289 0.82021 0.84615 0.87084 -72508 .75150 -63690 66794 49727 e53157 -56896 60399 .44833 a49148 .64205 61385 ~5,1935 .55267 .58411 .31493 .36203 .40559 .44609 .48391 .50977 .53982 .28579 .32965 .37047 .40860 .44436 .47801 .18696 .23841 .37706 .41096 .44298 *06200 .11836 .16997 .21755 .26163 ,30288 .34105 O~OOOOO 0.05642

-

-

0.108 13

4oooo

-

0.15583 0.20006 0.24128 0.27983 0.31603 0.35013 .05176 .OD953 .14387 .18520 .22389 .26023 ~00000 ,04781 .OD220 .13361 .17240 .OoooO ,04442 .08588

-

-

40000

“Adapted from Table 9 of Biomefrika Tables for Sfafisticiam, Vol. 2, E. S. Pearson and H. 0. Hartley, editors, Cambridge:The University Press, 1972. Published by permission of the Biometrika Trustees and the author, H. L. Harter.

Mathematical Tables

223

Table B-6 Mann-Whitney (two-input) test: Upper tail probabilities for the null distribution" .X

86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110

n=9 .500 .466 .432 .398 ,365 .333 ,302 .273 .245 .218 .I93 .I70 .I49 ,129 .I11 G.095 .081 .068 .057 ,047 .039 .031 .025 .020 .016

m=9 n=lO

x

n-9

n=lO

111

.516 ,484 .452 .42 1 .390 ,360 .330 .302 .274 .248 ,223 .200 .I78 .I58 .I39 ,121 .I06 .09 1 .078 .067 .056

112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 I29 I30 131 132 I33 134 135

,012 .009 ,007 .005

.047 .039 .033 ,027 .022 .017 .014 .01 I .009 .007

.004

,003 .002 ,001 .001 ,001 .Ooo .Ooo .Ooo .Ooo .Ooo

.Ooo

x

105 106

I07 108 109 110 111

112 113 I14

.005

115

.004 .003 .002 .001 .001

118

.00I

.Ooo .Ooo

.Ooo .Ooo .Ooo .Ooo .Ooo .Ooo

116 I17

119 I20 121 122 I23 124 125 126 127 128 I29

m = 10 n-10 x

.515 ,485 ,456 ,427 .398 .370 ,342 ,315 .289 .264 .241 ,218 ,197 .I76 .I57 .I40 .I24 .I03 .095 ,083 .072 .062 .053 ,045 ,038

I30 131 I32 133 134 I35 136 I37 138 I39 140 141 I42 143 144

145 146 147 I48 I49 150 151 I52 I53 I54 155

n*10

,032 ,026 .022 ,018 .014 .012 .009 .007

.006 ,004

.003 .003 .002 .00I .00I .00I .00I .Ooo .Ooo

.Ooo .Ooo

.Ooo .Ooo .Ooo .Ooo

.Ooo

"Adapted from A Nonparametric Introduction to Statistics, by C . H. Kraft and C. van Eeden, Copyright 0 1968, The Macmillan Company. Published by permission.

T d ~ kB-7 Quantiles of the Kendall T test statistie * n

p = .900

.950

.975

.990

.995

4 5 6 7 8 9

4 6 7 9 10 12

4 6 9

14 16

6 8 11 13 16 18

6 8 11 15 18 22

6 10 13 17 20 24

10 11 12 13 14 15

15 17 18 22 23 27

19 21 24 26 31 33

21 25 28 32 35 39

25 29 34 38 41

47

27 31 36 42 45 51

16 17 18 19 20

28 32 35 37

44

40

36 40 43 47 50

50 56 61 65 70

56 62 67 73 78

21 22 23 24 25

42 45 49 52 56

54 59 63 66 70

69 73 78 84

64

76 81 87 92 98

84 89 97 102 108

26 27 28 29 30 31 32 33 34 35

59 61 66 68 73 75 80 84 87 91

75 79 84 88 93 97 102 106 111 1 I5

89 93 98 104 109 I15 I20 126 131 137

105 111 116 124 129 135 I42 I50 155 163

115 123 128 136 143 149 158

11

48 51 55

60

164

173 179

“The values in the table are for the statistic S. The value of T can be found from Eq. (5.3-79, T p

S

$n(n

- 1)

bAdapted from “Tables for Use in Rank Correlation,” by Kaarsemaker and van Winjngaarten,Srarisricu Neerlandica, Vol. 7,pp. 41-54, 1953. Published by permission. 224

Mathematical Tables

225

Table B-8 Quantiles of the Spearman p test statistica IJ

4 5 6 7 8 9

p = .YO0

.950

.Y75

.YYO

.8000 .7000 .6000 s357

.8O00 .8OOO .7714 .6786 .6190 S833

.9OOo .8286 .7450 .7143 .6833

.9OOo

so00 .4667

.995

f999

.8857 .8571 .8095 .7667

.9429 .8929 .8571 .8167

.9643 .9286 .9000

12 13 14 I5 16 17 18 19 20

424 .4182 .3986 .3791 .3626 .3500 .3382 .3260 .3148 .3070 .2977

S273 .4965 .4780 .4593 .4429

.6364 .609 I S804 349 S341 s179

.7333 .7000 .6713 .6429 .6220 .6000

.7818 .7455 .7273 .6978 .6747 .6536

.8667 .8364 .8182 .7912 .7670 .7464

,4265 .4118 .3994 .3895 .3789

so00 .4853 .47 I6 .4579 .445 I

S824 S637 S480 .5333 S203

.6324 .6152 s975 .5825 S684

.7265 .7083 .6904 .6737 .6586

21 22 23 24 25

.2909 .2829 .2767 .2704 .2646

.3688 .3597 .35 I8 .3435 .3362

.4351 .424 1 .4150 .406 I .3977

SO78 .4963 .4852 .4748 A654

5545 S426 3306 3200

.6455

.5100

5962

26 27 28 29 30

.2588 .2540 .2490 .2443 .2400

.3299 .3236 .3175 .31 I3 .3059

.3894 .3822 .3749 .3685 .3620

.4564 .448 I .440 I .4320 .425 I

so02 .4915 4828 .4744 .4665

5856 5757 5660 367 5479

10 I1

.55 15

.6318 .6186

.6070

“Adapted from “Critical Values of the Coefficient of Rank Correlation for Testing the Hypothesis of Independence,” by Glasser and Winter, Biometrika, Vol. 48, pp. 444-448, 1961. Published by permission of the Biometrika Trustees.

Answers to Selected Problems

CHAPTER 3 1. For n = 20, the mean of the Wilcoxon test statistic is 105 and its variance is 717.5. The value of the test statistic is S, = 192. For a = 0.1, the threshold is T = 139.4. Thus, S, = 192 > T = 139.4 accept K , 2.

The power function is 1- p =

210

('k")p*(l - p)'O-*

k-8

where p = 1 - F(0). The false alarm probability is a = 0.0547, and for F(0IK) = 0.1, 1 - p = 0.93 3. The largest value of a such that a < 0.1 is 0.0547, and hence the threshold T = 8. (i) (ii) (iii)

10

2 "(xi) = 6 < 8 *

i= 1

10

2 .(xi)

i- I

=

9

>8 *

acceptH accept K

Cannot use the sign test until Chapter 7 because of the zero-valued observation 4. From Table B-4, the threshold T = 41. The value of the test statistic

226

Answers to Selected Problems

is Sw = 48. Thus, Sw = 48

>

T = 41

227

accept K5

5. The mean and variance of the test statistic for n = 10 are 27.5 and 96.25, respectively. The threshold T = 40. Thus, S, = 48 > T = 40 =$ accept K5 6. For the sign detector, T is found from Table B-1 and

2

0.08

(:)(0.5)*<

k= T

to be T = 9. The exact value of a = 0.073 and 1 - /.? = 0.974, since

p = 0.903 when K is true.

For the optimum detector, the threshold T = 31.41 and the exact value of a = 0.0793. Thus, 1 - /.? = 0.999 7. Using Eq. (3.6-11) where

4

can be found. Using the ranks and signs of the observations, the test statistic value is SVDW = 1.02

8. Using the ranks and signs of the observations and Table B-5, the test statistic is found to be s,, = 1.12 9. Using the ranks and signs of the observations yields

S, = 269 m The false alarm probability is given by a = -, where m is the 2'' * n! number of possible test statistic values that exceed the selected value of the threshold T. Hence a = 0.05 = m / (3.72 X lo9)

so m

= 1.86 X lo8.

10. (i) Exact distribution

n = 8: T = 7, a = 0.0312 n = 10: T = 9, a = 0.0107 n = 12: T = 10, a = 0.0161

Gaussian approximation n = 8: T = 6.33 n = 10: T = 7.61 n = 12: T = 8.85

228

Answers to Selected Problems

The Gaussian approximation is only adequate at best for the sign test and the given values of a and n. (ii) Exact Gaussian approximation

1

T = 29.8 T = 31, a = 0.039 n = 8: a = 0.05 n = 10: T = 45, a = 0.043 n = 10: T = 47.24 n = 12: T = 61, a = 0.046 n = 12: T = 60.04 The Gaussian approximation is excellent for the Wilcoxon detector. n = 8:

CHAPTER 4 1. The ARE is given by Eq. (4.2-22)

E 2 , , = 4uzf2(0)

For the given uniform density, f ( 0 ) = 1 and u2 =

A,thus

E2,,= 3 2. The mean and variance are np and np(1 - p), respectively, where p = P ( X > 0 ) = 1 - F,(O). Therefore,

E [ SSIK] = np = n P [ X

> OIK]

d

and

E [ SSIK] = n f ( 0 )

From Section 2.7 the efficacy is &s, = 4f2(0)

3. For n large, the Student’s t test in Eq. (3.2-18) has a Gaussian distribution with mean p and variance u2 for a general K. Therefore, E[SstIK] =GP / U and var[S,JH or K] = 1. Using the expression for the efficacy yields = l/u2

&, 5.

From Eq. (4.3-12), since u2 = &

E2,,= 1

6. For the sign detector with n = 12 observations, T = 9, a = 0.073, and 1 - /3 = 0.974 from Problem 3-6. The parameters n and T must be selected for the linear detector to achieve a = 0.073 and 1 - /3 = 0.974. Now, E [ S,,,IH] = 31,

var[ SOptlH] = 1/n

a = 0.073 = P

[

Sopt-

l / G

31

>-

T - 31 l/G

H

Answers to Selected Problem

229

and

From Table B-2, and

( T - 31)/ ( l / d i ) = 1.45

( T - 32.3)/ ( l / f i ) = -2.34

Solving simultaneously, n = 8.5. Thus, the relative efficiency is edm, opt = 8.5/ 12 = 0.7 1

CHAPTER 5 1. The mean and variance are given by Eqs. (5.2-47) and (5.2-56),

and

E [ S ' I H ] = 765

var[ S ' I H ] = 2550

Std. Dev.[ S ' I H ] = 50.5 From the Gaussian approximation, a =

P[

SMW

-

50.5

765

>

T'IH]

Since SM, = 817, T' = 1.03, and from Table B-2, a = 0.1515. 4. From Table B-8, TexacI = 0.700 and from Table B-2, Tappro, = 1.282 for 2 = p m . The test statistic is found to be p = 0.7. Therefore, p = TexacI = 0.7, and hence, barely accept K,, with the exact test, but the approximation 2 = 0 . 7 f i = 1.4 > 1.282

clearly accepts K,,. 5. From Table B-7, T,,,,, = 0.6 since in(. - 1) = 10. The value of the test statistic is 7 = 0.6, exactly on the threshold, as obtained with the Spearman p test. 6. From Table B-6, the threshold is T = 115. Using the statistic S in Eq. (5.2-30) yields S =

LO

2 R,=

46

< 115

accept K;

i= 1

or using Eq. (5.2-31) for S' produces 10

S'= i- I

I$,= 116 > 115

accept K.

230

Answers to Selected Problems

7. The threshold can be found from 12

a =

(?)( + ) < 0.10 k

k- T’

as T = 3 for a = 0.073. The value of the test statistic is Spec = 10, so decide “signal present.” 8. The value of the test statistic is p = 0.84 and from Table B-7, T = 0.3986. Thus, decide “signal present.” 9. The value of the test statistic is i = 0.917. From Table B-3, T, = 1.372, thus from Eq. (5.3-44) T; = 0.398 i

>T *

signal present

CHAPTER 6 3. Neither are desirable since both are designed for problems where the scale parameters are identical, and here u: # uz. Both detectors may yield an a more than twice the desired value if applied to this problem. 6. From Problem 6-5, ( a / a r ) E { ~ l K= } 2

/

r

m

so that the efficacy is &, = 18/r2 using the two-input definition of the efficacy. 7. Again using the two-input definition of the efficacy G; = 2. 8. The ARE of the Kendall 7 with respect to the sample correlation coefficient is thus E7.; = 9 / r 2 CHAPTER 7 1. In Eq. (7.2-1), N = 9 and M = 12 so that the midrank = 11. From Problem 3-1 the threshold is T = 139.4, and using the given observations the value of the test statistic is S, = 193. Thus, S , = 193 > T = 139.4, therefore accept 4. 6. The mean and variance of the Wilcoxon test statistic are 76.5 and 446.3, respectively. For a = 0.1, T = 103.6 and the value of the test statistic is found to be S, = 147. Hence S, > T, therefore accept K,.The margin of S, over T is reduced here. 7. The rankings most likely to cause H , to be accepted are R, = 12 and R,, = 11, R,, = 10 or R,, = 10, R,, = 11. This yields a test statistic value

231

Answers to Selected Problems

of s,H = 192. The rankings most likely to cause K, to be accepted are R, = 10, R,, = 12, R,, = 11 or R , , = 11, R,, = 12. For these rankings, the value of the test statistic is S; = 194. From Table B-2,

P[St = 1921H] = 0.0006

and

P [ S ; = 1941H] = 0.0004

and since both are less than a = 0.1, K, is accepted. 10. (1) S!') = 6 and T = 8. Since S!') < T, accept H,. (2) S;*) = 8 = T, thus the test statistic and threshold are equal, hence accept K,. 11. New variance = 717. No change from before due to small number of tied groups. CHAPTER 8 1.

Under dependent sample conditions [Lee, 19601, n

2.

[Armstrong, 19651 R ( k 7 ) = (1/4)

7. ARE = 0.993!

+ (1/27r) sin-'

p(k7)

References

Arbuthnott, J. (1710). An argument for Divine Providence, taken from the constant regularity observed in the births of both sexes, Philos. Tram. 27, 186190. Amstrong, G. L. (1965). The effect of dependent sampling on the performance of nonparametric coincidence detectors. Ph.D. dissertation, Oklahoma State University, Stillwater. Arnold, H. J. (1965). Small sample power of the one-sample Wilcoxon test for nonnormal shift alternatives. Ann. Math. Statist. 36, 1767-1778. Bartlett, M. S. (1937). Properties of sufficiency and statistical tests. Proc. Roy. SOC.Ser. A (London) 160, 268. Blomqvist, N. (1950). On a measure of dependence between two random variables. Ann. Math. Statist. 21, 593. Blyth, C. R. (1958). Note on relative efficiency of tests. Ann. Math. Statist. 29, 898-903. Bradley, J. V. (1963). Studies in research methodology V, irrelevance in the t test’s rejection region. United States Air Force, No. AMRL-TDR-63-109, Wright-Patterson Air Force Base, Ohio. Bradley, J. V. (1968). “Distribution-Free Statistical Tests.” Prentice-Hall, Englewood Cliffs, New Jersey. Capon, J. (1959a). A nonparametric technique for the detection of a constant signal in additive noise. IRE WESCON Conu. Record, Part 4, 92-103. Capon, J. (1959b). Nonparametric methods for the detection of signals in noise. Dept. Electrical Engr. Tech. Report No. T-l/N, Columbia Univ., New York. Capon, J. (1960). Optimum coincidence procedures for detecting weak signals in noise. IRE Internat. Conu. Record, Part 4, 156166. Carlyle, J. W. (1968). Nonparametric methods in detection theory. In “Communication Theory.” (A. V. Balakrishnan, ed.), pp. 293-319. McGraw-Hill, New York. Carlyle, J. W. and Thomas, J. B. (1964). On nonparametric signal detectors. IEEE Tram. Information Theory IT-10,No. 2, 146-152. Chanda, K. C. (1963). On the efficiency of two-sample Mann-Whitney test for discrete populations. Ann. Math. Statist. 34, 612-617. Conover, W. J. (1971). “Practical Nonparametric Statistics.” Wiley, New York. Davisson, L. D., and Thomas, J. B. (1969). Short-time sample robust detection study. United States Air Force, No. RADC-TR-69-24, Griffiss Air Force Base, New York.

232

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Index

A Alternative hypothesis, 3 Antoniak, C. E., 200, 201, 202, 204 A priori probability, 13 Arbuthnott, J., 6 ARE, see Asymptotic relative efficiency Armstrong, G. L., 184, 185, 189 Arnold, H. J., 91 Asymptotically nonparametric, 3 1 Asymptotic loss, 195 Asymptotic relative efficiency, 32-36, see olso Specific detectors Average probability of error criterion, see Ideal observer criterion

B Bayes decision criterion, 12-17 Bayes risk, 14 Binomial distribution, 212-213 table of, 215-216 Birch, J. J., 168 Blomqvist, N., 135 Blyth, C. R., 91 Bradley, J. V., 2, 6, 47, 58, 65, 165, 175, 176, 180 C c , test, see Normal scores test

Capon, J., 33, 94, 111, 115, 116, 204, 205, 206,207, 209

Carlyle, J. W., 87, 141, 190, 193, 208 Cauchy density function, 21 1 Chanda, K. C., 178 Chernoff, H., 88 Circular serial correlation coefficient, 132 Cohn, D. L., 9 Common digitizer, 204 Communication systems applications, 204209 Composite hypothesis, 10, 24 -3 I Conover, W. J., 144, 148 Consistency, 3 1 Correlation coefficient, 123, 134 Correlation detector, 136, 155-156 Craig, A. T., 45, 46, 50, 104, 155 Critical region, 11, 18-20

D Davisson, L. D., 184, 186, 187, 189, 190 Detector comparison techniques, 31-36 Dicke-fix radar detector, see Vector sign test Dillard, G. M., 200, 201, 202, 204 Distribution-free detectors, definition of, 2 Dixon, W. J., 32, 167

E Efficacy, 33-36, see also Specific detectors Ekre, H., 135, 207 Expected cost, 13 Exponential distribution, 21 1-212

237

Index

238 F

False alarm, 11 False alarm probability, see Probability of false alarm Faran, J. J., Jr., 135 Feller, W., 139, 140 Fisher, R. A., 121 Fisher-Yates test, see Normal scores test Fong, T. S., 203 Fraser, D. A. S., 29, 45, 49, 60 G

Gaussian distribution, 2 10 table of, 217 Generalized detector, 3 Generalized sign detector, 199-200 Gibbons, J. D., 143, 146, 169 Girling, B., 55, 114, 142, 143 Granger, C.W. J., 168 Graybill, F. A., 61, 62

H Hajek, J., 53, 71, 110, 112, 178 Hancock, J. C., 34, 204,209 Hansen, V. G., 192, 194, 195, 196, 198, 199, 203, 208 Helstrom, C. W., 26, 208 Hills, R., Jr., 135 Hodges, J. L., 32, 85, 92, 166 Hoeffding, W., 60 Hogg, R. V., 45,46, 50, 104, 155 Hollander, M., 180 Hotelling, H., 6, 143 Hotelling-Pabst test, 143-144

I Ideal observer criterion, 16-17 Inverse normal transformation, 63 Irrelevant. 47

K Kanefsky, M., 135, 159 Karr, P. H., 135 Kendall, M. G., 29, 65, 68, 82, 105, 119, 133, 134, 135, 146, 148, 150, 151, 152, 171, 178, 188

Kendall tau coefficient, 68 Kendall tau detector, 197, 198 one-input, 68-71, 196-198 ARE, 87-90 recursive form of test statistic, 70 table of quantiles, 224 two-input, 68-70, 148-152, 187 ARE, 165 asymptotic distribution, 151 dependent sample performance, 187188 mean, 150-151 relation to Spearman rho, 148, 151152 variance, 150-1 5 1 Klotz, J., 88, 89, 90,91, 93, 167 Kruskal, W. H., 108

L Lainiotis, D. G., 34, 204, 209 Laplace density function, 21 1-212 Least favorable choice, 25 Least favorable distribution, 26 Leaverton, P., 168 Lee, Y.W., 189 Lehmann, E. L., 29, 32, 41, 49, 55, 57, 58, 85, 92, 115, 132, 166, 169 Likelihood ratio, 14 Likelihood ratio test, 14, 18 Linear detector, 41 asymptotic false alarm probability, 76 asymptotic probability of detection, 75-76 efficacy, 82 LMP, see Locally most powerful Locally most powerful, 30

M Mann, H. B., 108, 109, 110, 115 Mann-Whitney detector, 107-1 16, 199200,203, 205-207 ARE, 165-166 ARE calculation, 162-164 ARE with respect to the normal scores test, 166 asymptotic distribution, 115 dependent sample performance, 187 efficacy, 163-164

Index equivalent statistics, 108- 11 mean, 11 1-1 12, 163 small sample performance, 167- 170 table for null distribution, 223 variance. 112-1 15 Medial correlation test, see Polarity coincidence correlator Median detector, 184 performance for correlated input samples, I 85-186 Melsa, J. L., 9, 157 Melton, B. S., 135 Miss, 11, 12 Modified sign test, 200-202 Mood, A. M., 61, 62 Most powerful test, 17-21 Moving window detector, 200

N Neave, H. R., 168 Neyman-Pearson lemma, 17-2 1, 24 Nonparametric detectors, definition of, 2 Nonparametric versus parametric detection, 2 Normal distribution, see Gaussian distribution Normal scores test one-input, 57-61, 64-65 ARE, 87-90 asymptotic loss, 195 locally most powerful, 58-60 narrow-band, 194 small sample performance, 90-91 two-input, 116-122 ARE, 165-166 ARE with respect to the Mann-Whitney detector, 166 asymptotic distribution, 121 locally most powerful, 118 mean, 119 small sample.performance, 167-170 variance, 120-121 Nuisance parameter, 47 Null hypothesis, 3 0

Olsen, B. A., 198, 199

239

P Pabst, M. R., 6, 143 Papoulis, A., 157 Parametric detectors one-input, 39-47 two-input, 97-107, 122-135 asymptotic performance, 155-158 PCC, see Polarity coincidence correlator Pitman, E. J. G., 31, 33 Polarity coincidence correlator, 135-141, 207-208 ARE, 165 ARE calculation, 158-162 asymptotically unbiased, 141 asymptotic distribution, 139-140 consistent, 140-141 nonparametric, 140-141 uniformly most powerful, 141 Power of the test, see Probability of detection Pratt, J. W., 176, 177, 178 Probability of detection, 12 Probability of false alarm, 11-12 Putter, J., 173, 176

R Radar system applications, 192-204 Receiver operating characteristics, 2 1-24 Relative efficiency, 31-32 Robust, see Asymptotically nonparametric ROC, see Receiver operating characteristics Romig, H. C., 138 Rosenblatt, M., 190

s Sage, A. P., 157 Sample correlation coefficient, 126135 asymptotic distribution, 134135 UMP unbiased, 132-133 Savage, I. R., 6, 88 Schwartz, M., 205 Selby, S. M., 55, 114, 142, 143 Sidak, Z., 71 Siebert detector, 203 Sign detector, 47-51 ARE, 87-90

240

Index

ARE calculation, 75-81 asymptotic false alarm rate, 78 asymptotic loss, 195 asymptotic probability of detection, 77 dependent sample performance, 182-183 false alarm rate, 50 invariance properties, 5 1 narrow-band, 194 probability of detection, 50 small sample performance, 91 Significance level, see Probability of false alarm Similar tests, 29 Simple hypothesis, 10 Small sample performance one-input detectors, 90-92 two-input detectors, 167-170 Spearman, C., 141 Spearman rho detector one-input, 65-67 ARE, 87-90 invariance properties, 67 table of quantiles, 225 two-input, 141-148 ARE, 165 asymptotic distribution, 146-147 mean, 144146 variance, 144- 146 Stein, C., 41 Stuart, A., 29, 82, 105, 119, 133, 134, 135, 148, 151, 152, 171, 188 Student's t distribution, 212-213 table of, 218 Student's r test one-input, 4 2 4 7 asymptotically nonparametric, 46 consistent, 46 irrelevance, 47 UMP unbiased, 45-46 two-input, 98-107, 162 asymptotically nonparametric, 107 consistent, 107 efficacy, 162 irrelevance, 107 UMP unbiased, 105-106 Sufficient statistic, 14 Swerling, P., 202

T Terry, M. E., 1'18, 121, 122, 167 Test statistic, 4, 14 Thomas, J. B., 81, 135, 141, 159, 184, 186, 187, 189, 190 Threshold of the test, 4 Tied observations, 172-180 dropping of zeros and ties, 175 general procedures, 173-176 continuous noise denkity case, 173-175 discrete noise density case, 175-176 midrank, 173-174 randomization, 173 specific studies, 176178 Type 1 error, see False alarm Type 2 error, see Miss

U UMP, see Uniformly most powerful Unbiased detector, 28-30 Uniform distribution, 21 1 Uniformly most powerful, 26-28 unbiased, 29 similar, 29

V Van der Waerden, B. L., 61 Van der Waerden's test, 61-65 one-input, ARE, 87-90 Van Trees, H. L., 9 Vector sign test, 195-198 W Whitney, D. R., 108, 109, 110, 115 Wilcoxon detector invariance properties, 55-57 lower bound on ARE, 85-87 one-input, 51-57, 180, 184, 186-187 ARE, 87-90 ARE calculation, 81-87 asymptotic distribution, 53 asymptotic loss, 195 dependent sample performance, 184, 186-187

241

Index efficacy, 84 mean, 53-54 narrow-band, 196198, 208-209 small sample performance, 9&9 I table for null distribution, 219-221 variance, , 5 4 5 5 variance in the presence of ties, 180 tied observations, 177-178 two-input, see Mann-Whitney detector Wilcoxon, F., 52, 108 Witting, H., 168 Wolfe, D. A., 180

A 5 B 6

c 7

o a

c 9 F O

G I

H 2

1 3

1

4

Wolff, S. S., 135, 155, 159, 161, 182

Y Yates, F., 121

2 Zeoli, G. W., 203 Zero-valued observations, see Tied observations Zottl, A. J., 203

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