Stochastic Differential Models With Applications To Physiology (Ph.D. dissertation)

Department of Statistics, Probability and Applied Statistics

Ph.D. Dissertation

Stochastic Differential Models With Applications To Physiology

Umberto Picchini Supervisor: Prof. Enzo D’Arcangelo

University of Rome “La Sapienza” November 2006

Ringraziamenti Il presente lavoro rappresenta il risultato di tre anni di ricerca. Sono stati anni caratterizzati da un impegno molto intenso, e se il prodotto di questi sforzi ha dei pregi, i meriti sono anche di tutte le persone che di seguito tento di ringraziare, anche se, in questi casi, il linguaggio formale risulta limitante e insufficiente. L’avvio della mia attività di ricerca coincide con l’inizio della collaborazione con il Laboratorio di Biomatematica (BioMatLab) dell’Istituto di Analisi dei Sistemi ed Informatica “A. Ruberti” del CNR. Grazie a questa collaborazione ho avuto non solo modo di accrescere la mia esperienza di ricercatore, venendo a contatto con un ambiente scientificamente vivo e radicato in un contesto di ricerca sia teorico che applicativo, ma anche di effettuare le mie ricerche con tranquillità e valido supporto. Per questo ringrazio coloro che non posso sterilmente definire “colleghi”, dal momento che hanno dimostrato essere molto di più: anzitutto il direttore di ricerca del Laboratorio, Dr. Andrea De Gaetano, per avermi fattivamente sostenuto e voluto come collaboratore in questi anni, sapendomi indirizzare e consigliare in un settore di ricerca non sempre affine a quello statistico. In ordine rigorosamente alfabetico ringrazio gli altri “non-colleghi”, che hanno reso questa esperienza una pagina importante della mia vita, non solo professionale, grazie alla loro quotidiana amicizia, consiglio e fruttuosa collaborazione: Mariella Galasso, Alessia Mammone, Lorenzo Marchesi, Simona Panunzi e Valeria Poli. Ringrazio quindi: il Dipartimento di Statistica, Probabilità e Statistiche Applicate dell’Università degli Studi di Roma “La Sapienza”, per aver reso possibile la collaborazione di cui sopra; il relatore di questa tesi, Prof. Enzo D’Arcangelo, per la cordiale supervisione, per la contagiosa passione scientifica che mi ha trasmesso, e per la stima che mi ha sempre dimostrato; il Prof. Enzo Orsingher per i fruttuosi consigli. Sincera gratitudine va ancora al Dr. Andrea De Gaetano ed alla Prof.ssa Susanne Ditlevsen, per la sincera amicizia e per aver supervisionato e suggerito l’argomento di questa tesi, le equazioni differenziali stocastiche. Si è trattato di una scelta quantomai felice, essendo questo un settore di ricerca ricco di opportunità, soprattutto nel suo impiego in campo biomedico. Li ringrazio nuovamente, e con loro la Dr.ssa Simona Panunzi, per avermi concesso il privilegio di essere coautore di diverse pubblicazioni, e mi auguro che il nostro rapporto di amicizia e ricerca possa continuare a lungo. Alla Prof.ssa Susanne Ditlevsen va ulteriore gratitudine per aver reso fruttuosa e piacevole la mia esperienza di ricerca a Copenhagen. Ringrazio tutti coloro che, inviando articoli e suggerimenti, hanno aiutato lo sviluppo di questa tesi: Pamela Marion Burrage, Bo Martin Bibby, Kenneth A. Lindsay, João Nicolau, Isao Shoji, Michael Sørensen, Andreas Rößler, Nigel J. Newton, Leif Kristoffer Sandal, Hermann Singer,

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Andreas Velsing Groth, Arturo Kohatsu-Higa. Last but not least, ringrazio la mia famiglia, per l’affetto e per aver sempre incoraggiato le mie scelte.

Roma, Novembre 2006

Umberto Picchini [email protected] http://www.biomatematica.it/Pages/Picchini.html

Acknowledgments The present work represents the result of three years of research. These have been years featured by an intense commitment, and if the outcome of these efforts has some value, the credit goes also to all those persons who in the following paragraph I will try to thank, even if, in a case of this kind, formal language appears limitative and insufficient. The beginning of my research activity coincides with the beginning of my collaboration with the Laboratory of Biomathematics (BioMatLab) of the Institute of Systems Analysis and Computer Science of the Italian National Council of Research (IASI - Istituto di Analisi dei Sistemi ed Informatica “A. Ruberti” del Consiglio Nazionale delle Ricerche). Thanks to this collaboration I had the opportunity not only of increasing my experience as a researcher, in contact with an environment scientifically alive and deeply rooted in a theoretical and applicatory research context, but also of performing research in peaceful surroundings and with valid support. For this I wish to thank those whom I cannot define as “colleagues” in a sterile way, as they have proved to be much more: first of all the Research Director of the Laboratory Dr. Andrea De Gaetano, for having positively supported me and for having asked me to collaborate with him during these years, orienting me and advising me in a field of research not always akin to the statistical one. In a rigorously alphabetical order I wish to thank the other “non-colleagues”, who have contributed to make this experience an important page of my life, not only professionally, but thanks also to their constant friendship, advice and fruitful collaboration: Mariella Galasso, Alessia Mammone, Lorenzo Marchesi, Simona Panunzi and Valeria Poli. I also thank the Department of Statistics, Probability and Applied Statistics of the University of Rome “La Sapienza”, for having made the above-mentioned collaboration possible; the Supervisor of this thesis, Prof. Enzo D’Arcangelo, for his friendly supervision, for his contagious scientific passion which has transferred me, and for the consideration he has always shown me; Prof. Enzo Orsingher for his fruitful advice. Sincere gratitude goes again to Dr. Andrea De Gaetano and Prof. Susanne Ditlevsen, for their true friendship and for having supervised and suggested the subject of this thesis: stochastic differential equations. It has been a very successful choice, since this subject is rich in opportunities, especially where applied to the biomedical field. I thank them once more, and along with them Dr. Simona Panunzi, for having given me the privilege of being coauthor of various publications. I hope that our friendship and research collaboration may continue for a long time. To Prof. Susanne Ditlevsen goes my further gratitude for having made my research experience in Copenhagen fruitful and pleasant. I also wish to thank all those who, by sending me articles and exchanging advice and ideas,

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have contributed to the development of this thesis: Pamela Marion Burrage, Bo Martin Bibby, Kenneth A. Lindsay, João Nicolau, Isao Shoji, Michael Sørensen, Andreas Rößler, Nigel J. Newton, Leif Kristoffer Sandal, Hermann Singer, Andreas Velsing Groth, Arturo Kohatsu-Higa. Last but not least, I wish to thank my family for their affection and for having always encouraged my choices.

Rome, November 2006

Umberto Picchini [email protected] http://www.biomatematica.it/Pages/Picchini.html

Sintesi In questa tesi viene considerata la classe dei modelli definiti da equazioni differenziali stocastiche (SDEs) e vengono illustrate applicazioni di SDEs alla modellizzazione di problemi metabolici che, nel corso di questi anni, il candidato ha avuto occasione di affrontare e che sono state pubblicate su riviste di biomatematica (Picchini et al. (2005, 2006a)). Il principale contributo statisticometodologico è contenuto nel capitolo 3 (considerato anche in Picchini et al. (2006b)), dove si propone un nuovo metodo di stima dei parametri di modelli SDE con effetti misti. La struttura della tesi consta di due parti: la Parte 1, intitolata Stochastic Differential Models, comprende i capitoli 1 (Stochastic differential equations), 2 (Parameter estimation of SDE models) e 3 (Stochastic differential mixed-effects models) di contenuto metodologico. La Parte 2, intitolata Papers, contiene gli articoli pubblicati dal candidato aventi contenuto rilevante per la tesi, ossia Picchini et al. (2005, 2006a). Nel capitolo 1 viene fornito un background relativo alla teoria delle SDEs: qui si illustrano i motivi del crescente interesse della comunità scientifica verso i modelli SDE, quindi si considerano alcuni fondamentali costrutti relativi al calcolo stocastico ed ai metodi numerici di soluzione delle SDE. La trattazione non ha pretese di completezza: vengono discussi risultati utili ad una comprensione generale del problema e metodi effettivamente utilizzati nel corso della tesi. Alcuni importanti argomenti sono stati inseriti in un’apposita appendice, per mantenere la fluidità della trattazione. Il capitolo 2 ha una connotazione più propriamente statistica: viene considerato il problema della stima di massima verosimiglianza dei parametri di SDEs, nel caso di processi Markoviani e non-Markoviani, nonchè metodi diagnostici di bontà di adattamento (quest’ultimo essendo un tema raramente affrontato nei modelli di SDEs). Viene infine presentata un’applicazione relativa alla dinamica del sistema glucosio/insulina in soggetti umani sottoposti all’esperimento del clamp: per facilitarne la lettura, tale applicazione è stata sintetizzata per quanto riguarda i risultati e gli argomenti di interesse fisiologico (illustrati in maniera completa nel Paper 2 della Parte II della tesi), mentre ne vengono dettagliate le problematiche matematico/statistiche. Tale applicazione è risultata essere molto più che una mera esemplificazione dei metodi matematici esposti nei capitoli 1 e 2. È stato infatti definito un nuovo modello SDE (derivato da un modello DDE precedentemente considerato in Picchini et al. (2005)) per la dinamica glucosio/insulina (il primo modello non banale per lo specifico problema affrontato), che è stato pubblicato dal Journal of Mathematical Biology (Picchini et al. (2006a)). Tale risultato ha richiesto notevole impiego di tempo per diversi motivi: (i) per la difficoltà nell’identificare un modello innovativo, a fronte di decenni di letteratura dedicati al problema, che potesse essere di reale interesse per le comunità biomatematica e diabetologica,

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e (ii) per i notevoli tempi di calcolo e di risorse hardware richiesti nello stimare i parametri delle SDEs. Inoltre (iii) per gestire problemi di stima dei parametri di SDEs, non esistendo all’uopo un software standard. A questo scopo è stata creata un’apposita libreria di funzioni MATLAB, capaci di interfacciarsi con funzioni C laddove necessario per accelerare i processi di calcolo. Tali motivazioni giustificano l’ampio spazio riservato nella tesi alle applicazioni proposte. Nel capitolo 3 sono presentati i contributi del candidato di natura prettamente statisticometodologica, che riguardano una nuova classe di modelli differenziali, i modelli differenziali stocastici ad effetti misti (SDME). Un modello di SDME è una SDE in cui alcuni parametri sono espressi come combinazione di una componente costante (parametro di popolazione) e di una componente aleatoria (variabile da esperimento a esperimento), al fine di poter effettuare stima di popolazione anche nell’ambito dei modelli differenziali stocastici. Il modello che ne risulta è estremamente interessante per le sue caratteristiche di flessibilità: vengono combinate le ben note proprietà dei modelli ad effetti misti (variabilità totale esplicitamente scissa in variabilità tra-gruppi, ad es. tra-soggetti, ed entro-gruppi) con la possibilità di perturbare la dinamica del sistema introducendo un appropriato processo stocastico (noise di sistema, da cui deriva una SDE). La tematica è molto recente, ed è stata affrontata solamente in tre recenti pubblicazioni (Overgaard et al. (2005), Tornøe et al. (2005), Ditlevsen and De Gaetano (2005a)), ma limitatamente alla stima dei parametri di modelli SDME con diffusione costante ed effetti casuali aventi distribuzione normale o lognormale. Nel capitolo 3, invece, si propone un metodo (approssimato) di stima di massima verosimiglianza per i parametri di SDME con diffusione anche non-costante, dove i parametri casuali possono avere una qualsiasi distribuzione; tale metodo risulta di pratica attuazione nonostante i problemi computazionali insiti nella problematica trattata. Il metodo proposto sfrutta un risultato sull’espansione in serie Hermitiana della densità di transizione pX del (K) processo X considerato (Aït-Sahalia (2001, 2002b)), che risulta in un’espressione pX esplicita, e quindi consente, quando applicabile, una diminuzione esponenziale dei tempi di calcolo rispetto ad approssimazioni di p basate, ad esempio, su simulazioni Monte Carlo. Al momento non sono disponibili risultati teorici sulle proprietà degli stimatori dei parametri ottenuti, ma l’evidenza sperimentale suggerisce che questi rappresentino una buona approssimazione ai veri stimatori di massima verosimiglianza. Si ritiene che il contenuto di tale capitolo possa risultare molto utile per le future applicazioni delle SDME: infatti, l’utilizzo di modelli di SDE (ed in particolare di SDME, sebbene recentemente introdotti) è stato sinora limitato dall’enorme dispendio di risorse di calcolo necessarie per la stima dei parametri. La ricerca è stata svolta in collaborazione con il Laboratorio di Biomatematica CNR–IASI di Roma (www.biomatematica.it).

Summary The present work considers the notion of stochastic differential equation (SDE) and shows some applications of SDEs to those metabolic problems which the candidate had occasion to tackle during the doctoral years, and which have been published on biomathematical journals (Picchini et al. (2005, 2006a)). The main contribution to statistical methodology is to be found in chapter 3 (see also Picchini et al. (2006b)), where a new parameter estimation method for SDE models including mixed-effects is proposed. This thesis is structured in two parts: Part I, Stochastic Differential Models, includes chapters 1 (Stochastic differential equations), 2 (Parameter estimation of SDE models) and 3 (Stochastic differential mixed-effects models), of methodological content. Part II, Papers, includes articles published by the candidate which are relevant to the thesis, i.e. Picchini et al. (2005, 2006a). In chapter 1, a background for SDEs theory is offered; the reasons for the growing interest of the scientific community for SDE models are exposed, and some fundamental achievements regarding stochastic calculus and numerical methods of solution of SDEs are considered. The exposition does not pretend to be complete: results useful to a general comprehension of the problem and of effectively used methods within this thesis are discussed. Some important arguments were inserted in an appendix in order to maintain fluidity. Chapter 2 has a more properly statistic connotation: the problem of the maximum likelihood estimation of parameters of SDEs comes under consideration in the case of Markovian and nonMarkovian processes, as well as goodness-of-fit methods (the latter being an issue rarely addressed in SDE models). Finally, an application regarding the dynamics of the glucose/insulin system in human subjects undergoing the clamp experiment is presented: to facilitate the reading, this application was summarized where the results and arguments of physiological interest are concerned (they are illustrated more completely in Paper 2 of Part II), while the mathematical/statistic aspects are addressed in detail. This application turned out to be more than a mere exemplification of the mathematical methods exposed in chapters 1 and 2. Actually a new SDE model (derived from a DDE model previously considered in Picchini et al. (2005)) was defined for the dynamics of glucose/insulin (the first non-trivial model for the specific problem addressed), which was published in the Journal of Mathematical Biology (Picchini et al. (2006a)). Such a result required an outstanding commitment in terms of time for different reasons: (i) for the difficulty in identifying an innovative model, after decades of literature devoted to the problem, which might be of real interest to the biomathematical and diabetological communities, and (ii) for the considerable calculation times and hardware resources required for estimating the parameters of the SDEs, since a specific software standard does not exist. To this end a special MATLAB functions library,

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capable of interfacing with C functions when necessary to accelerate the calculus processes, was created. These reasons justify the ample space devoted in the thesis to the proposed applications. In chapter 3 the strictly methodological contributions of the candidate, focused on a new class of differential models named Stochastic Differential Mixed-Effects (SDME) models, are presented. A SDME model is an SDE in which given parameters are expressed as a combination of a constant component (population parameter) and of a random component (which varies from experiment to experiment), in order to perform population estimation procedures also into the stochastic differential framework. The resulting model is very interesting for its flexibility: it combines the well-known Mixed-Effects models properties (total variation is splitted into its withinand between-experiment components) with the possibility of perturbing the system dynamics by introducing a proper stochastic process (system noise, giving rise to a SDE). This is a very recent theme which has been previously considered in only three papers (Overgaard et al. (2005), Tornøe et al. (2005), Ditlevsen and De Gaetano (2005a)) focused on SDME models with constant diffusion term and normal- or lognormal-distributed random effects. In chapter 3 a new (approximated) maximum likelihood procedure for the parameters of SDME models is proposed, even with a nonconstant diffusion term, where random parameters may follow any distribution; this method results of practical actuation despite the computational problems inborn in the issue under consideration. The proposed method exploits a result concerning the Hermitian expansion of the transition density pX of the considered X process (Aït-Sahalia (2001, 2002b)), which re(K) sults in a explicit expression pX , thus allowing, when applicable, an exponential decrease of the calculus times with respect to p approximations based, for example, on Monte Carlo simulations. At present theoretical results on the properties of the obtained parameter estimators are not available, but experimental evidence suggests that these may represent a good approximation of the true maximum likelihood estimators. The contents of this chapter may be very useful for future applications of SDME: in fact the use of models of SDE (and in particular of SDME, although recently introduced) has been until now limited by the enormous deploy of calculus resources necessary for estimating the parameters. The research was performed in collaboration with the Biomathematics Laboratory (BioMatLab) of CNR-IASI in Rome, Italy (www.biomatematica.it).

Contents Sintesi

7

Summary

9

I

STOCHASTIC DIFFERENTIAL MODELS

1 Stochastic Differential Equations

13 15

1.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

1.2

SDEs: motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

1.3

Stochastic integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

1.4

Numerical solution of SDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

1.4.1

21

Euler–Maruyama and stochastic Taylor approximations . . . . . . . . . . .

2 Parameter Estimation of SDE Models

27

2.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

2.2

Estimation method for partially observed diffusion processes . . . . . . . . . . . . .

29

2.3

Closed-form likelihood expansion for discretely sampled diffusions . . . . . . . . . .

30

2.4

SDE model diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

2.5

An application: modeling glycemia and insulinemia dynamics by SDEs . . . . . . .

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2.5.1

The physiological problem . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

2.5.2

A deterministic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

2.5.3

A stochastic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

2.5.4

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 Stochastic Differential Mixed-Effects Models

53

3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

3.2

Formulation of Stochastic Differential Mixed-Effects Models . . . . . . . . . . . . .

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3.3

Exact and Approximated Maximum Likelihood Estimation . . . . . . . . . . . . .

56

3.4

Closed-form transition density expansion and likelihood approximation . . . . . . .

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3.4.1

Transition density expansion for SDE models . . . . . . . . . . . . . . . . .

57

3.4.2

Likelihood approximation for SDME models . . . . . . . . . . . . . . . . . .

57

3.5

Implementation issues and numerical applications . . . . . . . . . . . . . . . . . . .

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3.6

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3.7

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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II

CONTENTS

PAPERS

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A Mathematical Model Of The Euglycemic Hyperinsulinemic 3.8 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9.1 Subjects . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9.2 Experimental protocol . . . . . . . . . . . . . . . . . . 3.9.3 Modelling . . . . . . . . . . . . . . . . . . . . . . . . . 3.9.4 Statistical analysis . . . . . . . . . . . . . . . . . . . . 3.10 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.11 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.12 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modeling The Euglycemic Hyperinsulinemic Equations 3.13 Introduction . . . . . . . . . . . . . . . . . . 3.14 Material and methods . . . . . . . . . . . . 3.14.1 Subjects . . . . . . . . . . . . . . . . 3.14.2 Experimental protocol . . . . . . . . 3.14.3 Deterministic model . . . . . . . . . 3.14.4 Stochastic model . . . . . . . . . . . 3.14.5 SDE estimation . . . . . . . . . . . . 3.15 Results . . . . . . . . . . . . . . . . . . . . . 3.15.1 Deterministic differential model . . . 3.15.2 Stochastic differential model . . . . . 3.16 Discussion . . . . . . . . . . . . . . . . . . . 3.17 Appendix . . . . . . . . . . . . . . . . . . .

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Clamp By Stochastic Differential . . . . . . . . . . . .

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Conclusioni

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Conclusions

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Appendix

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A Some Results in Probability Theory 119 A.1 Continuity and nowhere differentiability of the Wiener process . . . . . . . . . . . 121 A.2 Transition Densities, Kolmogorov Equations and Diffusion Processes . . . . . . . . 123 A.3 Itô and Stratonovich SDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 Bibliography

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Part I

STOCHASTIC DIFFERENTIAL MODELS

Chapter 1

Stochastic Differential Equations 1.1

Introduction

The area of deterministic differential equations (ordinary (ODE), partial (PDE), or delay (DDE)) is a rich one, well–researched with plenty of software packages and tools available for the numerical solution of such systems. Deterministic differential equations arise as a description of a model of a physical system and are solved in order to provide answers to such questions as how the system is changing or developing, when change might occur, what effect a different starting point may have on the solution, and so on. Until recently, many of the models developed to describe physical phenomena have ignored stochastic effects because of the difficulty in solution, both because of the lack of suitable numerical methods and also because of the nonavailability of sufficiently powerful computers. Deterministic models can represent idealized situations and can often be improved by including stochastic elements, but specifically designed methods for handling stochastic differential equations (SDEs) are required, and work in this area is far less advanced than for deterministic differential equations. Some areas where SDEs are used in modelling include investment finance, turbulent diffusion, population dynamics, polymer dynamics, biological waste treatment, neuronal models and hydrology. The transition from ODE to SDE takes place by incorporating random elements in the differential equation describing the physical system. Randomness can be included in the initial value for the problem; alternatively, the function itself can include random fluctuations, and in this case the differential equation is called a stochastic differential equation. Consider a one-dimensional ODE with given initial value x0 , dx(t) = f (x(t)), dt

x(t0 ) = x0 ,

f :R→R

where x(t) represents the (deterministic) state of the physical system at time t, hypothesizing that we are interested in the time dynamics of the system; it is evident that no random term is present. Consider now the following equation dx(t) = f (x(t))dt + g(x(t))dW (t),

f ∈ R, g ∈ R, W ∈ R

(1.1)

16

Stochastic Differential Equations

which introduces a one-dimensional stochastic process W perturbing x(·) with an intensity factor g(·): equation (1.1) is a typical representation of a stochastic differential equation. In the following sections the element involved in the definition of a SDE are further specified, and some basic probabilistic results are reported in Appendix A. General references to SDE and stochastic calculus are Karatzas and Shreve (1991), Kloeden and Platen (1992), Øksendal (2000), Rogers and Williams (1987, 1994) and Stroock and Varadhan (1979). Practical implementation of numerical techniques for SDE can be found in Kloeden and Platen (1992), Kloeden et al. (1994), Higham (2001), Burrage and Burrage (1996), Burrage (1999), Rößler (2003), Rümelin (1982).

1.2

SDEs: motivations

Fix a point x0 ∈ R and consider the ODE dx(t) = f (x(t)), dt

x(t0 ) = x0 ,

where x(·) : [t0 , ∞) → R is the trajectory of the ODE solution (e.g. the one reported in Figure 1.1). In many applications, however, the experimentally measured trajectories of systems modeled 11

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Figure 1.1:

Trajectory of an ODE’s solution.

by ODEs do not in fact behave as predicted (see Figure 1.2), and this is often attributed only to measurement error, which does not influence the course of x(·); another interpretation is to allow the possibility of including random effects (“noise”) disturbing the system, above and beyond measurement error. The justification of this choice lies in that the noise is supposed to model the collective influence of many individually neglected effects on the x(·) dynamics (e.g. effects not explicitly represented in f (·)). It is reasonable to look for some stochastic process ξ(t), to represent the noise term, so that dx(t) = f (x(t)) + g(x(t))ξ(t), dt

x(t0 ) = x0 ,

(1.2)

1.2 SDEs: motivations

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Figure 1.2:

Sample path of an irregular (random) process.

and g(·) is an intensity factor. Based on many situations, one is led to assume that ξ(t) has these properties: 1. ξ(t1 ) and ξ(t2 ) are independent for t1 6= t2 ; 2. ξ(t) is stationary, i.e. the joint distribution of {ξ(t1 + t), ..., ξ(tk + t)} does not depend on t; 3. E(ξ(t)) = 0 for all t. However, there does not exist any ordinary stochastic process with continuous paths satisfying both the first and the second property (Øksendal (2000)). Nevertheless it is possible to represent ξ(t) as a generalized stochastic process named the white noise process. Thus, we can equivalently express equation (1.2) as dx(t) = f (x(t))dt + g(x(t))ξ(t)dt,

x(t0 ) = x0 ,

(1.3)

with a deterministic or averaged drift term perturbed by a noisy, diffusive term g(x(t))ξ(t) (see also section A.2). Following Øksendal (2000), it is possible to rewrite (1.3) in a form that suggests a replacement of ξ(t) by a proper stochastic process: let 0 = t0 < t1 < · · · < tn = t and consider a discrete version of (1.3): x(tk+1 ) − x(tk ) = f (x(tk ))∆tk + g(x(tk ))ξ(tk )∆tk where ∆tk = tk+1 − tk . Now replacing ξ(tk )∆tk by ∆V (tk ) = V (tk+1 ) − V (tk ), where {V (t)}t≥0 is some suitable stochastic process, the desired properties 1–3 on ξ(t) suggest that V (t) should have stationary independent increments with mean zero. It turns out that the only such process with continuous paths is the Wiener process W (t), which is a random process with independent

18

Stochastic Differential Equations

and normal distributed increments (see Definition A.7). Thus, if we put V (t) ≡ W (t), we obtain

x(tk ) = x0 +

k−1 X

f (x(tj ))∆tj +

j=0

k−1 X

g(x(tj ))∆W (tj ).

j=0

Now, if the limit of the right hand side of the previous equation exists, when ∆tj → 0, then we should obtain Z t Z t g(x(s))dW (s) (1.4) f (x(s))ds + x(t) = x0 + 0

0

or dx(t) = f (x(t))dt + g(x(t))dW (t)

(1.5)

and we would adopt as a convention that (1.2) means that x(t) = x(t, ω) is a stochastic process satisfying (1.5) for each path labelled by ω ∈ Ω. Therefore, to get from (1.2) to (1.5) we formally just replace the white noise ξ(t) by dW (t)/dt in (1.2) and multiply by dt. The problem is that a Wiener process W (·) is nowhere differentiable (see Theorem A.2), so strictly speaking the white noise ξ(t) does not exist as a conventional function of t. Worse still, the continuous sample paths of a Wiener process are not of bounded variation on any bounded time interval (Theorem A.2) – and from Definition A.7 V ar(W (t)) = t so it increases with time even though the mean stays at 0 – so the second integral in (1.4) cannot be interpreted as a Riemann-Stieltjes integral for each sample path. In the next section two alternative approaches to the Riemann-Stieltjes integral will be considered: they will give a solution to the second integral in (1.4).

1.3

Stochastic integrals

Rt In the previous section it was stated that t0 g(s, x(s, ω))dW (s, ω) cannot be interpreted as a Riemann-Stieltjes integral: here some different interpretations for this stochastic integral are considered (see e.g. Rogers and Williams (1987), Karatzas and Shreve (1991), Kloeden and Platen (1992)). Consider the time instants 0 ≤ t0 < t1 < · · · < tn . Approximate the stochastic integral by the sum n X g(τi , x(τi ))(W (ti ) − W (ti−1 )), i=1

then it converges in the mean square sense to different values for different τi in the interval [ti−1 , ti ]. Rb To determine a value for a W (t)dW (t) approximate W (t) by the function ϕλn (t) (n)

(n)

(n)

ϕλn (t) = λW (tk ) + (1 − λ)W (tk−1 ),

(n)

tk−1 ≤ t ≤ tk

for any λ ∈ [0, 1], and then approximate the integral by the sums Z a

b

ϕλn (t)dW (t) =

n X k=1

(n)

(n)

ϕλn (tk−1 )(W (tk ) − W (tk−1 )).

1.4 Numerical solution of SDEs

(n)

19

(n)

It is possible to show that when tk − tk−1 → 0 Z a

b

  1 1 2 2 W (t)dW (t) = (W (b) − W (a)) + λ − (b − a), 2 2

thus for any choice of λ there is a different result; in particular, if λ = 0 (which is equivalent to choosing τi = ti−1 , the left-hand endpoint of the interval) the integral is known as the Itô integral, while if λ = 1/2 (⇒ τi = 12 (ti + ti−1 ), the midpoint of the interval) then the resulting integral is the Stratonovich integral (where the symbol ◦ is employed), and the Stratonovich calculus follows the same rules as for the Riemann–Stieltjes calculus. So, the integral evaluations are Z

b

W (t)dW (t) = a

and Z

1 1 (W 2 (b) − W 2 (a)) − (b − a) 2 2

b

W (t) ◦ dW (t) = a

1 (W 2 (b) − W 2 (a)) 2

for the Itô or Stratonovich calculus, respectively. Rt For the Itô stochastic integral t0 X(t)dW (t) to be defined (X(t) is a real-valued process), it is necessary that X(·) and W (·) are both defined on the same probability space (Ω, A, P ); and that X(·) is non-anticipating, i.e. X(t) does not depend on events occurring after time t (this is equivalent to λ = 0). Furthermore, the Itô integral forms a martingale, which follows from the fact that W (t) is a martingale (Remark A.3): only for the non-anticipating Itô case (λ = 0) does the martingale property hold. There has been much discussion about whether to use the Itô or Stratonovich interpretation of the stochastic integral, and while both approaches are correct, the choice depends on the modelling process that leads to the SDE formulation. The Stratonovich calculus obeys the ordinary (Riemann-Stieltjes) rules of integration and differentiation. On the other hand, a Stratonovich SDE does not satisfy the martingale property, so Itô SDEs are often used when martingale and non-anticipatory properties are needed. A discussion of some of the controversies concerning the Itô or Stratonovich interpretation is contained in van Kampen (1981), Burrage (1999), Kloeden and Platen (1992) and Braumann (2005).

1.4

Numerical solution of SDEs

In the previous sections we have sketched the problem of constructing an SDE, but we haven’t exhibited any specific analytic technique to obtain the relative solution: analytic strategies can be found, for example, in Kloeden and Platen (1992) and Øksendal (2000). On the other hand, we will often make use of numerical methods for the solution of SDEs, so here we illustrate some general techniques suitable for computer implementation (see also Kloeden and Platen (1992), Kloeden et al. (1994), Burrage and Burrage (1996), Burrage (1999), Higham (2001), Rößler (2003)). Very few specific SDEs have explicitly known solutions, these being mainly reducible by appropriate transformation to the solution of linear SDEs; the computation of important characteristics such as moments or sample paths for a given SDE is thus crucial for the effective practical application of SDEs. The visualization of sample paths of SDEs can be achieved by use of discrete time

20

Stochastic Differential Equations

strong approximation methods: these provide algorithms in which the SDE is discretized in time, with values at intermediate times being obtained through interpolation. An approximate sample path of the SDE is thus obtained for a given sample path of the driving Wiener process, which is usually simulated with an appropriately chosen pseudo-random number generator (e.g. listing A.1 on page 119). When considering a numerical solution of a differential equation (ODE or SDE), we must restrict our attention to a finite subinterval [t0 , T ] of the time-interval [t0 , +∞) and, in addition, it’s necessary to chose an appropriate discretization 0 ≤ t0 < t1 < · · · < tn < · · · < tN = T of the time-interval [t0 , T ], because of computer limitations. The other crucial problem is simulating a sample path from the Wiener process over the discretization of [t0 , T ] (see Higham (2001)): so considering an equally-spaced discretization, i.e. tn − tn−1 = (T − t0 )/N = h, n = 1, . . . , N , where h is the integration stepsize, from Definition A.7 we have the following (independent) random increments W (tn ) − W (tn−1 ) ∼ N (0, h) n = 1, . . . , N of the Wiener process {W (t), t0 ≤ t ≤ T }, where W (0) = 0. Moreover, the sampling of normal variates to approximate the Wiener process in the SDE is achieved by computer generation of pseudo-random numbers. However, the use of a pseudo-random number generator needs to be evaluated in terms of statistical reliability; when a batch of independent pseudo-random numbers is generated and used in the numerical solution, the errors from the random sample no longer have an impact on the numerical calculation and the expected order of accuracy is confirmed. Nevertheless, most commonly used pseudo-random number generators have been found to fit their supposed distribution reasonably well, but the generated numbers often seem not to be independent as they are supposed to be: this is not surprising since, for congruential generators at least, each number is determined exactly by its predecessor (Kloeden et al., 1994). The two most reliable methods for generating independent standard Gaussian distributed pseudo-random numbers are the Box-Muller and the Polar Marsaglia methods (Kloeden et al. (1994)), so we used the built-in MATLAB randn function, which is a refinement of the Polar Marsaglia method that implements the ziggurat method (Marsaglia and Tsang, 2000). In Appendix A a simulation of Wiener sample paths is reported (see Figure A.1 and the MATLAB Listing A.1 on page 120); other implementations are available in Kloeden et al. (1994) and Higham (2001). Before we start illustrating numerical techniques for SDEs it is useful to introduce the concepts of “strong” and “weak” approximations and to give conditions for the existence and uniqueness of a solution to a SDE. In fact, to judge the quality of a discrete time approximation, a criterion must be specified: in the area of SDEs there are two ways of measuring this accuracy, namely strong and weak convergence. For strong approximations a single sample path trajectory is calculated and it is important that the trajectory of the numerical approximation is close to the exact solution (i.e. remains inside some region), while for weak approximations many trajectories are computed and the various statistical measures can be applied: in fact, when using weak approximations interest focuses on approximating expectations of functionals of the process, and this is relevant because, often, such functionals cannot be determined analytically. Definition 1.1. Let yN be the numerical approximation to x(tN ) after N steps with constant stepsize h = (tN − t0 )/N ; then y is said to converge strongly to x with order p if ∃C > 0 (independent

1.4 Numerical solution of SDEs

21

of h) and δ > 0 such that E(|yN − x(tN )|) ≤ Chp ,

h ∈ (0, δ)

Definition 1.2. The discrete time approximation y(tN ) is said to converge weakly with order p to x if for each polynomial g (which is 2(p+1) times continuously differentiable) ∃C > 0 (independent of h) and δ > 0 such that |E(g(yN )) − E(g(x(tN )))| ≤ Chp ,

h ∈ (0, δ)

The computational requirements for the simulation of moments, probabilities or other functionals of the process (⇒ weak convergence) are not as demanding as for the pathwise approximations (⇒ strong convergence), see also Kloeden and Platen (1992). Consider now the scalar (d = m = 1) Itô SDE dx(t) = f (t, x(t))dt + g(t, x(t))dW (t)

(1.6)

then we have the following theorem that gives necessary and sufficient condition for the existence and uniqueness of a solution to (1.6). Theorem 1.1. Let the functions f (·, ·) : [t0 , T ] × R → R and g(·, ·) : [t0 , T ] × R → R be measurable on the interval [t0 , T ], and suppose that they satisfy both the Lipschitz conditions |f (t, x) − f (t, y)| ≤ c|x − y|,

|g(t, x) − g(t, y)| ≤ c|x − y|,

x, y ∈ R, t ∈ [t0 , T ]

and the linear growth conditions |f (t, x)| ≤ c(1 + |x|),

|g(t, x)| ≤ c(1 + |x|),

x ∈ R, t ∈ [t0 , T ]

for some constant c ∈ R. If x0 = x(t0 ) is independent of the Wiener process W (t) for t > 0, then there exists a unique solution of the SDE (1.6) with initial condition x0 on the entire time interval [t0 , T ]. The uniqueness of the solution is pathwise-uniqueness, by which is meant that if X and Y are two solution to the SDE, then  P

 sup |X(t) − Y (t)| = 0 = 1. [t0 ,T ]

1.4.1

Euler–Maruyama and stochastic Taylor approximations

As previously pointed out many SDE systems do not have a (known) analytic solution, so it is necessary to solve these systems numerically: the simplest stochastic numerical approximation is the Euler–Maruyama method. Considering the Itô SDE (1.6) on [t0 , T ] with initial value x0 = x(t0 ), for a given discretization 0 ≤ t0 < t1 < · · · < tn < · · · < tN = T of [t0 , T ], an Euler– Maruyama approximation is a continuous time stochastic process satisfying the iterative scheme

22

Stochastic Differential Equations

yn+1 = yn + hn f (yn ) + g(yn )∆Wn

y0 = x0 ,

n = 0, 1, . . . , N − 1

(1.7)

where yn = y(tn ), hn = tn+1 − tn and ∆Wn = W (tn+1 ) − W (tn ) ∼ N (0, hn ) (the increments ∆Wn can be generated by e.g. the Box–Muller and by the Polar Marsaglia methods in Kloeden et al. (1994), or by the ziggurat method in Marsaglia and Tsang (2000)). The Euler–Maruyama scheme (1.7), as other schemes given later, determines values of the approximating process at the discretization times only: if required, values can be determined at the intermediate instants t ∈ (tn , tn+1 ) , n = 0, 1, . . . , N − 1 by an appropriate interpolation method, for example by the linear interpolation y(t) = ynt +

t − tnt (ynt +1 − ynt ) tnt +1 − tnt

t ∈ [t0 , T ],

where nt is the integer defined by nt = max{n = 0, 1, . . . , N ; tn ≤ t}. We shall concentrate on the values of a discrete time approximation at the given discretization instants; it will not be possible to reproduce the finer structure of sample paths of an Itô process as these inherit the irregularity of the sample paths of the driving Wiener process, in particular their nondifferentiability. The Euler-Maruyama method has strong order of convergence 1/2 (and weak order of convergence 1) and it converges to the Itô solution. Note that a method may have order of accuracy p in general, but this order may be increased for SDEs of a particular type: for example, the Euler–Maruyama method has strong order of accuracy 1 for systems with additive noise (i.e. the diffusion term g is a constant).

Other numerical methods may have a much simpler form when being used to solve additive noise SDEs, and this often leads to a cheaper implementation. As the order of the Euler–Maruyama method is low, the numerical results are inaccurate unless a small stepsize is used, and clearly more efficient methods are needed: one possible improvement is given by the stochastic Heun method or by the stochastic Itô-Taylor approximation (Kloeden and Platen, 1992). The equation (1.6) can be written in integral form Z

t

x(t) = xt0 +

Z

t

f (x(s))ds + t0

g(x(s))dW (s) t0

and if f and g are expanded in an Itô stochastic Taylor series about x0 then a representation as an infinite series for x(t) is obtained; truncating the stochastic Itô-Taylor series at a particular point yields an expression for x(t) of a certain order. For example the Milstein scheme for scalar Itô SDE (Milstein, 1974) consists of the first few terms of the stochastic Itô-Taylor series, and is given by 1 yn+1 = yn + hf (yn ) + g(yn )∆Wn + g(yn )g 0 (yn )((∆Wn )2 − h). 2 This scheme converges with strong order 1 as long as E(x20 ) < ∞, f and g are twice continuously differentiable, and that f , f 0 , g, g 0 and g 00 satisfy a uniform Lipschitz condition. There exists also a Milstein scheme for the corresponding scalar Stratonovich SDE, which is given by 1 yn+1 = yn + hf (yn ) + g(yn )∆Wn + g(yn )g 0 (yn )(∆Wn )2 2

1.4 Numerical solution of SDEs

23

In the general multi-dimensional case d, m = 1, 2, . . . the kth component of the Milstein scheme (k = 1, . . . , d) for Itô SDEs has the form k yn+1 = ynk + hf k +

m X

m d X X

g kj ∆Wnj +

j=1

g kj1

∂g k,j2 I(j1 ,j2 ) ∂xk

g kj1

∂g k,j2 J(j1 ,j2 ) , ∂xk

j1 ,j2 =1 k=1

whereas for Stratonovich SDEs it has the form k yn+1 = ynk + hf¯k +

m X

m d X X

g kj ∆Wnj +

j=1

j1 ,j2 =1 k=1

where f¯ denotes the drift of the SDE in the Stratonovich representation (see also section A.3). I(j1 ,j2 ) and J(j1 ,j2 ) are respectively multiple Itô and Stratonovich integrals given by (j1 , j2 = 1, . . . , m) Z τn+1 Z s1 dWsj21 dWsj12 , (j1 6= j2 ) I(j1 ,j2 ) = J(j1 ,j2 ) = τn

I(j1 ,j1 ) =

τn

1 [(∆W j1 )2 − h] and 2

J(j1 ,j1 ) =

1 (∆W j1 )2 . 2

When j1 6= j2 the multiple integrals can be approximated as suggested in Kloeden and Platen (1992). Obtaining higher order stochastic numerical methods from the Taylor series is straight-forward in derivation but involves considerable complexities in implementation not only in the approximation of the higher order stochastic integrals but also in the calculation of the derivatives of the coefficient functions f and g; it is natural then to look at methods that are derivative free, which are suggested in Burrage and Burrage (1996)-Burrage and Burrage (2002). However in this dissertation we do not consider those latter approaches. Computer implementations of the Euler–Maruyama and Milstein methods are available in Kloeden et al. (1994) and Higham (2001). We now illustrate the importance of choosing an appropriate numerical integration method by means of two examples: it will be clear that the Milstein scheme (strong order 1) is more precise than the Euler-Maruyama scheme (strong order 0.5) by simply looking at the average absolute error (1.8): in the following examples the analytic solutions of the SDEs are known and so the (average absolute) error at time T , depending on the desired number of simulations R, can be computed accurately as proposed in Kloeden et al. (1994) R

ˆ =

1 X |x(T, k) − y(T, k)|, R

(1.8)

k=1

were x(T, k) and y(T, k) denote the value of the analytic solution at time T in the kth trajectory and the value of the numerical solution for the chosen scheme at time T in the kth trajectory respectively. Thus equation (1.8) is a Monte Carlo approximation to the expectation in Definition 1.1 for large values of R. For the calculation of the error, the analytic solution and the numerical solution have been computed on the same path (i.e. on the same sequence of pseudo-random numbers).

24

Stochastic Differential Equations

Example 1.1. The following Stratonovich SDE dx(t) = (1 + x2 (t))dt + (1 + x2 (t)) ◦ dW (t),

x(0) = x0

has actual solution (Kloeden and Platen, 1992) x(t) = tan(t + W (t) + arctan(x0 )) The initial value is x0 = 0, the stepsize is h = 0.001, and to avoid unboundedness of the actual solution the integration is carried out from 0 to T = 0.1; the ˆ values and the corresponding standard deviations are reported in Table 1.1 for the Euler-Maruyama and the Milstein method over 1000 trajectories. In Figure 1.3 the plots of the Euler-Maruyama and the Milstein approximations Method Error at T SD at T Table 1.1:

Euler-Maruyama 1.87 · 10−2 2.41 · 10−2

Milstein 5.25 · 10−4 1.43 · 10−3

Numerical results from Example 1.1.

with the analytic solution for one trajectory are included in the same chart. The Milstein and the actual solutions are so close that they result practically indistinguishable. 0.2

0.1

0

−0.1

X(t) −0.2

−0.3

−0.4

−0.5

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

t

Figure 1.3:

Euler-Maruyama approximation (dotted line), Milstein approximation (dashed line) and analytic solution (solid line) for a single trajectory from Example 1.1.

Example 1.2. The following Stratonovich SDE dx(t) = (x2 (t) − 1)dt + 2(1 − x2 (t)) ◦ dW (t),

x(0) = x0

1.4 Numerical solution of SDEs

25

has solution (Kloeden and Platen, 1992) x(t) =

(1 + x0 )e−2t+4W (t) + x0 − 1 (1 + x0 )e−2t+4W (t) + 1 − x0

The initial value is x0 = 0, the stepsize is h = 0.01, and the integration is from 0 to T = 1; the numerical results are reported in Table 1.2 for the Euler-Maruyama and Milstein method over 1000 trajectories. Method Error at T SD at T Table 1.2:

Euler-Maruyama 4.36 · 10−1 5.76 · 10−1

Milstein 2.95 · 10−2 4.99 · 10−2

Numerical results from Example 1.2

. In Figure 1.4 the plots of the Euler-Maruyama and the Milstein scheme with the analytic solution for one trajectory are included in the same chart. 0.6

0.4

0.2

0

X(t)−0.2 −0.4

−0.6

−0.8

−1

Figure 1.4:

0

0.1

0.2

0.3

0.4

0.5

t

0.6

0.7

0.8

0.9

1

Euler-Maruyama approximation (dotted line), Milstein approximation (dashed line) and analytic solution (solid line) for a single trajectory from Example 1.2.

Chapter 2

Parameter Estimation of SDE Models 2.1

Introduction

It is often convenient to model the time evolution of dynamic phenomena by using a diffusion process, i.e. a Markov process with transition density (see section A.2), which is characterized by a stochastic differential equation. Parameters in these stochastic differential equations are crucial for the characterization of dynamic phenomena being considered. It is often the case that these parameters are not known accurately, while sample data for the particular dynamic phenomena are available. Naturally, researchers are interested in obtaining better estimates of the parameters using the observation data. In practical situations the available data are discrete time series data sampled over some time interval, whereas SDEs are almost surely continuous processes; this introduces estimation problems. Thus, the parameter estimation for discretely observed diffusion processes is non-trivial and during the past decade it has generated a great deal of research effort (e.g. Aït-Sahalia (2001, 2002b), Alcock and Burrage (2004), Bibby et al. (2005), Bibby and Sørensen (1995), Brandt and Santa-Clara (2002), Dacunha-Castelle and FlorensZmirnou (1986), Ditlevsen and De Gaetano (2005a), Durham and Gallant (2002), Elerian et al. (2001), Gallant and Long (1997), Gouriéroux et al. (1993), Hurn and Lindsay (1999), Hurn et al. (2003), Nicolau (2002), Pedersen (1995, 2001), Shoji and Ozaki (1998), Sørensen (2000a), Sørensen (2000b), Yoshida (1992)). The vast majority of parameter estimation methods were developed for fully-observed diffusion processes (all the d coordinates of the multivariate X process are observed at each sampled timepoint) driven by a Wiener process W , since in this case X is Markovian as a consequence of the fact that W is Markovian (Theorem A.3; section 4.6 in Kloeden and Platen (1992)), and thus the transition densities can be computed or at least numerically approximated. Nevertheless, some variables (coordinates of the d-dimensional process) may not be fully observed for all the observation time-points, or may not be observed at all. This situation is typical in biomedical applications, and in this case the process of the observations is non-Markovian. In the latter situation we have to consider a different approach, as described in section 2.2, which has been

28

Parameter Estimation of SDE Models

employed in Paper 2 of this dissertation. Further, the collected observations of the dependent variables are often assumed to be generated without measurement error from the stochastic process X, which is modelized as an SDE. This is a handy and comfortable assumption that preserve the Markovian property of the observations: otherwise, if we assume that they were inflated with measurement error, they would not be Markovian anymore. In fact, many applications of SDEs can be found in financial models, where the modelized variables are assumed not perturbed by measurement error. Anyway, as motivated in Paper 2 of this dissertation, it still makes sense to build an SDE model in situations were measurement error exists (e.g. in biomedicine), since the latter can be regarded as a perturbing term which is assumed to be small with respect to the stochastic-noise term, represented by the diffusion part of the SDE. The stochastic noise can be interpreted as the action of many factors, each with a small individual effect, which are not explicitly represented in the deterministic part of the model (that is, in the drift term of the SDE), and which instantaneously affect the response variables. Therefore, in the stochastic differential models the collective influence of many individually neglected effects is added to the average drift term, which, on the other side, represents the most relevant and generally well-recognized factors affecting the dependent variables. The general framework is given by the following d-dimensional system of (Itô) SDEs dXt = f (t, Xt ; θ)dt + g(t, Xt ; θ)dWt ,

X0 = x0 ,

t≥0

(2.1)

where W is an m-dimensional standard Wiener process, f : [0, +∞) × Rd × Θ → Rd and g : [0, +∞) × Rd × Θ → Rd×m are known functions depending on an unknown finite-dimensional parameter vector θ ∈ Θ, and we write Xt ≡ X(t, θ). We assume that the initial value x0 is fixed and that x0 , x1 , . . . , xn is a sequence of historical observations from the diffusion process X sampled at non-stochastic discrete time-points 0 = t0 < t1 < · · · < tn , except for section 2.2, where the observations contaminated with measurement error are denoted with y0 , y1 , ..., yn . If X is Markovian, then the maximum likelihood estimator (MLE) of θ can be calculated if the transition densities p(xt ; xs , θ) of X are known, s < t (see also section A.2). The log-likelihood function of θ is given by n X log p(xi ; xi−1 , θ) (2.2) ln (θ) = i=1

and the maximum likelihood estimator θˆ can be found by maximizing (2.2) with respect to θ. Under mild regularity conditions, θˆ is consistent, asymptotically normally distributed and asymptotically efficient as n tends to infinity (Dacunha-Castelle and Florens-Zmirnou (1986)). The difficulty with the MLE is that the transition density function of the underlying diffusion process is often unknown. One response to this problem is to compute an approximation to the transition density function numerically. There are essentially three ways to do this: 1. solving numerically the Kolmogorov partial differential equations satisfied by the transition density (Lo (1988)); 2. deriving a closed-form Hermite expansion to the transition density (Aït-Sahalia (2001, 2002b), Egorov et al. (2003), see section 2.3 of this dissertation); 3. simulating R times the process to Monte-Carlo integrate the transition density (e.g. Peder-

2.2 Estimation method for partially observed diffusion processes

29

sen (1995), Brandt and Santa-Clara (2002), Durham and Gallant (2002), Hurn and Lindsay (1999), Hurn et al. (2003), Nicolau (2002)): this methodology is known as simulated maximum likelihood (SML). Each of these three techniques have been successfully implemented by the aforementioned authors, but each has their limitations. Aït-Sahalia (2002b) notes that methods 1 and 3 above are computationally intense and poorly accurate. In response Durham and Gallant (2002) build on their importance sampling ideas in order to improve the performance of Pedersen’s (1995) (or equivalently Brandt and Santa-Clara’s (2002)) method and point out that method 2 above, while accurate and fast, is only available for a small number of models. Our opinion is that a method should be not only accurate and fast, but also “practicable”, that is the simulation-based methods (e.g. the one proposed in section 2.2 for non-Markovian models, or the references in method 3 above) are highly time-consuming and have proved less accurate than e.g. method 2 (Jensen and Poulsen (2002)). On the other hand, they are a very simple and general tool, and have proved to be applicable over a wide range of SDE models. In general method 2 (see section 2.3) should be the tool of choice, but computing the Hermite expansion of the transition density could be a very difficult task, especially if the SDE is multivariate and non-linear. We apply method 2 in Chapter 3, were we propose an estimation method for the parameters of SDE models including random effects.

2.2

Estimation method for partially observed diffusion processes

As outlined above, when some coordinate of a multi-dimensional system of SDEs is not observed over subsets of {t0 , t1 , . . . , tn } the process of the observations is non-Markovian. Moreover, in most applications the variables are observed with non-negligible measurement error. In such case an error-model can be defined by Yi = hi (ti , Xti ; ψ) + εi ,

i = 0, 1, . . . , n

(2.3)

where the Xti ’s are generated from (2.1), hi : [0, +∞] × Rd × Ψ → Rmi are known functions which may depend on an unknown finite-dimensional parameter vector ψ and ε0 , ε1 , . . . , εn are stochastically independent unknown measurement errors (and independent of the Wiener process W ) with marginal distributions εi ∼ Nmi (0, Σi (η)), and η is an unknown finite-dimensional parameter vector that parametrizes the non-singular error covariances matrices. We suppose that at time ti the vector Yi has mi ≤ d components, therefore, if mi = d this means that at time ti we have observed all the state variables of the system (2.1), otherwise one or more components of the system are unobserved at ti . We also suppose that the observed value yi is generated by Yi (i = 0, 1, . . . , n). The problem is to estimate θ, ψ, η and possibly x0 (if x0 is non-fixed) given y0 , y1 , ..., yn . Since in this case the observations are non-Markovian, the transition densities are not appropriate to specify the likelihood, and for each observation we must consider the past “history” of

30

Parameter Estimation of SDE Models

the process, i.e. the statistical inference is based on the likelihood function L(x0 , θ, ψ, η) = p0 (y0 ; x0 , θ, ψ, η)

n Y

pi|i−1 (yi ; y0 , yi , . . . , yi−1 , x0 , θ, ψ, η)

i=1

where p0 denotes the density function of Y0 and pi|i−1 denotes the conditional density function of Yi given Y0 , Y1 , . . . , Yi−1 . Since measurement errors are stochastically independent, and independent of the diffusion process, the likelihood function can be rewritten as (Pedersen (2001)) L(x0 , θ, ψ, η)

Z "Y n

=

# ϕi (yi ; xi , ψ, η) λ(x1 , . . . , xn ; x0 , θ)dx1 , . . . , dxn

i=0 n Y

= Ex0 ,θ

ϕi (yi ; Xti ; ψ, η)

i=0

where λ denotes the joint density function of Xt1 , . . . , Xtn , Ex0 ,θ denotes expectation with respect to the distribution of Xt1 , . . . , Xtn for the indicated parameter values, and −1/2

ϕi (yi ; xi , ψ, η) = |2πΣi (η)|

 0    1 −1 exp − yi − hi (ti , xi ; ψ) Σi (η) yi − hi (ti , xi ; ψ) 2

(2.4)

is the density function of the mi -dimensional normal distribution with expectation and covariance matrix hi (ti , xi ; ψ) and Σi (η) respectively. Here |A| and A0 denotes, respectively, the determinant and transpose of a matrix A. If X r (x0 , θ) (r = 1, ..., R) are stochastically independent random vectors, each distributed as (Xt1 , . . . , Xtn ) for the indicated parameter values, then it follows from the strong law of large numbers that R

L(x0 , θ, ψ, η) '

n

1 XY ϕi (yi ; Xtri (x0 , θ), ψ, η) R r=1 i=0

(2.5)

so the right hand expression may be used as a Monte Carlo approximation of the likelihood function for a large value of R. The Xtri can be simulated using a standard algorithm (e.g. EulerMaruyama, Milstein or higher order methods, see section 1.4.1) and the independent Wiener increments should be simulated, initially, and kept fixed in all subsequent calculations of the Monte Carlo approximation of the likelihood function. This estimation method is highly time consuming, because the number of simulations R should be chosen large enough for the estimation variability due to different sets of random increments is negligible for practical purposes. Once an expression for the right-hand side of equation (2.5) is available, it can be maximized and the estimates derived can be used as approximate MLE of θ, ψ and η. An application of this method is proposed in section 2.5 and in Paper 2 of this dissertation.

2.3

Closed-form likelihood expansion for discretely sampled diffusions

In the introduction to this chapter we stated that the method proposed in Aït-Sahalia (2001, 2002b) should be the tool of choice in the estimation of SDE models of Markovian processes,

2.3 Closed-form likelihood expansion for discretely sampled diffusions

31

since it is accurate and generally faster than other methods (Jensen and Poulsen (2002)). Here we briefly review the methodology for one-dimensional Markovian processes and adapt the more compact notation used in the multidimensional case (Aït-Sahalia (2001)) to the one-dimensional situation and state the asymptotical properties of the derived estimates. In Chapter 3 we employ the transition density approximation strategy considered here. Consider the following one-dimensional SDE dXt = µ(Xt , θ)dt + σ(Xt , θ)dWt

X(t0 ) = x(t0 )

(2.6)

where θ is the vector of unknown parameters for which we want an estimate. In the remaining of this section we drop any reference to θ when not strictly necessary, that is we write f (x) instead of f (x, θ) for a given function f . Suppose that n + 1 observations x0 , x1 , ..., xn generated from Pi model (2.6) are available at non-stochastic time-points {t = j=1 ∆j |i = 1, ..., n} ∪ {t0 }, where ∆j > 0 is the deterministic time interval between the (j − 1)th and the j th observations. If we denote with pX (x|xs ) the transition density of X, that is the conditional density of Xs+∆j given Xs = xs induced by equation (2.6), then the log-likelihood of θ can be computed as ln (θ) =

n X

ln pX (xi |xi−1 ).

i=1

Under mild regularity conditions on µ(·) and σ(·), the logarithm of the transition density ln pX (x|xs ) can be expanded in closed form using a order J = +∞ Hermite series whose coefficients can be approximated by a Taylor expansion up to a order K, thus obtaining (−1)

C 1 1 (K) ln pX (x|xs ) = − ln(2π∆j ) − ln(σ 2 (x)) + Y 2 2 (k)

where the coefficients CY defined by

K

∆kj (γ(x)|γ(xs )) X (k) + CY (γ(x)|γ(xs )) (2.7) ∆j k! k=0

are given in Theorem 2.1 below and γ(·) is the Lamperti transform, Z Yt ≡ γ(Xt ) =

Xt

du σ(u)

(2.8)

where the lower bound of integration is an arbitrary point in the interior of the dominion of the diffusion X (i.e. the constant of integration is irrelevant). Then, using the Itô’s formula (Øksendal (2000)), Yt is the solution to the following SDE dYt = µY (Yt )dt + dWt where µY (·) is given by µY (Yt ) =

µ(γ −1 (Yt )) 1 ∂σ −1 − (γ (Yt )). σ(γ −1 (Yt )) 2 ∂x

By taking the exponential of (2.7) we get an expression for the corresponding transition density (K) approximation pX .

Theorem 2.1 (Aït-Sahalia). For given values y and ys of the Y process (2.8) the coefficients

32

Parameter Estimation of SDE Models

of the log-density (2.7) are given by (−1)

CY (0) CY (y|ys )

1 (y|ys ) = − (y − ys )2 2 Z

1

µY (ys + u(y − ys ))du

= (y − ys ) 0

and, for k ≥ 1, Z

(k)

CY (y|ys ) = k The functions

(k) GY

(1) GY (y|ys )

1

0

(k)

GY (ys + u(y − ys )|ys )uk−1 du.

are given by

 (0) 2 (0) (0) ∂CY (y|ys ) 1 ∂ 2 CY (y|ys ) 1 ∂CY (y|ys ) ∂µY (y) + =− − µY (y) + ∂y ∂y 2 ∂y 2 2 ∂y

and for k ≥ 2 (k−1)

(k)

GY (y|ys )

(k−1)

(y|ys ) 1 ∂ 2 CY (y|ys ) + ∂y 2 ∂y 2  (h) k−1  (k−1−h) (y|ys ) 1 X k − 1 ∂CY (y|ys ) ∂CY + . 2 h ∂y ∂y

= −µY (y)

∂CY

h=0

(k)

(K)

Once the coefficients CY are obtained by means of Theorem 2.1, an expression for ln pX (x|xs ) is available and it is possible to approximate the log-likelihood ln (θ) with ln(K) (θ) =

n X

(K)

ln pX (xi |xi−1 , θ).

i=1 (K) So the approximated log-likelihood ln can be maximized in order to obtain an estimate θˆ(K) of θ such that θˆ(K) → θˆ in probability (as K → +∞), where θˆ is the true (but uncomputable) MLE of θ (Aït-Sahalia, 2002b); thus for K large θˆ(K) inherits the same asymptotic properties of ˆ Optimizing ln(K) (θ) is often a very fast procedure, since no evaluation nor the unknown MLE θ. numerical integration of the SDE model (2.6) is required. An application of this transition density approximation strategy is considered in Chapter 3 of this dissertation, in the framework of SDE models including random effects.

2.4

SDE model diagnostics

In any statistical analysis it is important to be able to validate the assumed statistical model, on which the whole analysis relies. In Pedersen (1994, 2001) it has been shown how to calculate uniform distributed residuals; the residuals can then be analyzed by standard techniques such as goodness of fit tests and quantile plots. With reference to the notation used in section 2.2, let Yi ∈ Rmi denote the vector of observable

2.4 SDE model diagnostics

33

random variables at time ti (i = 0, 1, ..., n) and suppose that we want to validate the parametric model for the variables Y0 , Y1 , ..., Yn given by the parametrized family of probability measures {Pϑ : ϑ ∈ Θ ⊆ Rp }. Let Fi (·|y0 , y1 , ..., yi−1 ; ϑ) denote the conditional distribution function of Yi given {Y0 = y0 , Y1 = y1 , ..., Yi−1 = yi−1 } for the given parameter. The distribution function is assumed to have a absolutely continuous density function; then the random variables Ui (ϑ) = Fi (·|Y0 , Y1 , ..., Yi−1 ; ϑ),

i = 1, ..., n

are stochastically independent and uniformly distributed in [0, 1] under Pϑ ; we call {Ui (ϑ)}ni=1 the uniform residuals.

For ease of notation suppose mi = d = 1 (i = 0, 1, ..., n), and assume the model is given by (2.1) and (2.3), with ϑ = (θ, ψ, η) and εi ∼ N (0, η 2 ); we write Y (i) for (Y0 , Y1 , ..., Yi ), and likewise for y (i) , and we write X (i) for (X1 , ..., Xi ), and likewise for x(i) . The density function of Z is denoted by fZ (·). By definition on conditional probabilities we have Fi (yi |Y (i−1) ; x0 , ϑ)

Now Z yi

fY (i) (y (i−1) , y; x0 , ϑ)dy

Z

yi

= P (Yi ≤ yi |Y (i−1) ; x0 , ϑ) R yi f (y (i−1) , y; x0 , ϑ)dy −∞ Y (i) . = fY (i−1) (y (i−1) ; x0 , ϑ)

Z

=

−∞

Ri

−∞

Z

Z

fY (i) |X (i) (y (i−1) , y|x(i) ; ψ, η)fX (i) (x(i) ; x0 , θ)dx(i) dy

yi

=

ϕi (y|xi ; ψ, η)dy Ri

−∞

i−1 Y

ϕj (yj |xj ; ψ, η)fX (i) (x(i) ; x0 , θ)dx(i)

j=0

  i−1   yi − hi (ti , Xti ; ψ) Y ϕj (yj |Xtj ; ψ, η) = Ex0 ,θ Φ η j=0 where Φ denotes the standard normal distribution function, and ϕj (·) is given by (2.4) with Σj (η) = η 2 . Likewise we have fY (i−1) (y

(i−1)

; x0 , ϑ) = Ex0 ,θ

 i−1 Y

 ϕj (yj |Xtj ; ψ, η) .

j=0

Finally we obtain the following expression for the uniform residuals U0

=

Ui

=

  y0 − h0 (t0 , x0 ; ψ) Φ η     yi −hi (ti ,Xti ;ψ) Qi−1 Ex0 ,θ Φ j=0 ϕj (yj |Xtj ; ψ, η) η   , Qi−1 Ex0 ,θ ϕ (y |X ; ψ, η) tj j=0 j j

i = 1, ..., n.

Hence the uniform residuals can be calculated by Monte Carlo methods in a similar way to the

34

Parameter Estimation of SDE Models

simulated likelihood function (2.5) as

ˆi = U

1 R

    yi −hi (ti ,Xtr (x0 ,θ);ψ) Qi−1 r i r=1 Φ j=0 ϕj (yj |Xtj (x0 , θ); ψ, η) η   , PR Qi−1 1 r r=1 j=0 ϕj (yj |Xtj (x0 , θ); ψ, η) R

PR

i = 1, ..., n

where Xtri (x0 , θ) is as in (2.5). The model fit can then be evaluated by testing whether the uniform residuals seem to be distributed uniformly in the unit interval, e.g. with goodness-of-fit test statistics or quantile plots. Obviously if Y is a Markov process then the theory is simplified, since Fi (yi |Y (i−1) ; x0 , ϑ) = Fi (yi |yi−1 ; x0 , ϑ), i = 1, ..., n. An application of this diagnostics is proposed in section 2.5 and in Paper 2 of this dissertation.

2.5

An application: modeling glycemia and insulinemia dynamics by SDEs

This section gives a summary of the application published in Picchini et al. (2006a) (Paper 2 of this dissertation): it is focused on the mathematical and statistical aspects of the application, so it can be considered as a useful introduction to the original article, since physiological issues are simply sketched in section 2.5.1 and can be skipped in first reading. For those interested on a complete and organic description of the problem, as well as on a detailed discussion of the physiological results, refer to Picchini et al. (2005, 2006a), the former being Paper 1 of this dissertation, that concerns of purely deterministic modelization, which has also been considered in Paper 2. Therefore, in this section we only summarize the Paper 2 contents. Here we modelize glycemia and insulinemia dynamics in fifteen human subjects by ODEs and SDEs, and we estimate the parameters of the SDEs using the method for partially observed processes outlined in section 2.2. Finally, we apply the diagnostic measures considered in section 2.4 to validate our model. In section 2.5.1 we briefly review the physiological aspects of our problem, then in sections 2.5.2-2.5.3 we focus our attention on the mathematical and statistical part of our application. The present application has two main goals: on one hand it purports to determine whether, in a particular physiological situation, system error (or “system noise”) is identifiable and necessary to explain observations, above and beyond commonly accepted levels of measurement error. The second goal is to show, by means of a practically occurring experimental situation, that SDE models are physiologically relevant and that their parameters can be numerically estimated using commonly available resources.

2.5.1

The physiological problem

With the growing epidemiological importance of insulin resistance states (like obesity and type 2 diabetes mellitus) and with the increasing clinical recognition of the impact of the so-called metabolic syndrome (an association of high insulin levels, central obesity, cholesterol abnormalities

2.5 An application: modeling glycemia and insulinemia dynamics by SDEs

35

and/or high blood pressure), the assessment of insulin sensitivity has become a very relevant issue in metabolic research. Insulin is a hormone that is produced by the beta-cells, which are cells that are scattered throughout the pancreas. The insulin produced is released into the blood stream and travels throughout the body. Insulin has many actions within the body: most of its actions are directed at metabolism (control) of carbohydrates (sugars and starches), lipids (fats), and proteins; insulin is also important in regulating the cells of the body including their growth. Insulin resistance is a condition in which the cells of the body become resistant to the effects of insulin, that is, the normal response to a given amount of insulin is reduced. As a result, higher levels of insulin are needed in order for insulin to have its effects. One of the actions of insulin is to cause the cells of the body, particularly the muscle and fat cells, to remove and use glucose from the blood: this is one way in which insulin controls the level of glucose concentration in blood, which is contained into a narrow range (between 3-5 mM, 1mM = 1 mmole/L = 18 mg/100ml). As long as the pancreas is able to produce enough insulin to overcome insulin resistance, blood glucose levels remain normal. When the pancreas can no longer produce enough insulin, the blood glucose levels begin to rise, initially after meals when glucose levels are at their highest and more insulin is needed, but eventually in the fasting state too; at this point, type 2 diabetes is present. A major feature of type 2 diabetes is a lack of sensitivity to insulin by the cells of the body (particularly fat and muscle cells). Thus, larger quantities of insulin are produced as an attempt to get these cells to recognize that insulin is, in fact, present. Finally, the liver in these patients continues to produce glucose through a process called gluconeogenesis, despite elevated glucose levels. This control of gluconeogenesis becomes compromised. The interdependence between glucose concentrations (glycemia) and insulin concentrations (insulinemia) is represented in Figure 2.1 by a negative feedback-loop: here an essential representation of this interaction is reported, including the “actors” playing the leading roles, i.e. plasma glucose and insulin compartments 1 , liver, muscle tissues and pancreas. Solid arrows represent “stimulation” whereas dashed arrows represent “inhibition”: thus, liver produces glucose, which enters the glucose compartments and promotes insulin secretion by acting on the pancreatic betacells; the insulin enters the insulin compartment and either inhibits the production of new glucose, by acting on the liver (that’s the negative feedback), and promotes the glucose tissue absorption, which is consequently removed by the glucose compartment. The experimental procedures currently employed to gather information on the degree of insulin resistance of a subject are the Oral Glucose Tolerance Test (OGTT), the Intra-Venous Glucose Tolerance Test (IVGTT), the Euglycemic Hyperinsulinemic Clamp (EHC), the Hyperglycemic Clamp, the insulin-induced hypoglycemia test (KIT T ), and less commonly used methods based on tracer administration. Of these, the EHC (DeFronzo et al. (1979)) is considered the tool of choice in the diabetological community, in spite of its labor-intensive execution. In the next section we modelize glycemia and insulinemia dynamics recorded from the EHC procedure: we spend only few words for the description of the EHC, since it is not necessary to expose its implementation in detail (see DeFronzo et al. (1979) for a complete description). Essentially in this procedure: 1 The concept of “compartment” has received several definition attempts, e.g. (see Segre (1986) for further definitions): 1) an anatomical region in which a drug is uniformly distributed; 2) an ideal volume in which each molecule or particle of the substance has equal probability of leaving.

36

Parameter Estimation of SDE Models

Figure 2.1:

The glucose/insulin negative feedback loop.

1. insulin concentrations are rapidly raised to a high value by means of an insulin bolus injection, and maintained at this level during the experiment (2-5 hrs) by means of a constant insulin infusion; 2. glucose concentrations are maintained close to a “target” (basal, i.e. the glycemia value recorded before the procedure starts) level by means of variable rate glucose infusions; 3. both glycemia and insulinemia are sampled during the experiment. Thus, at the end of the experiment, a series of glycemia and insulinemia values from each subject are available, together with a set of covariates (sex, age, BMI, bodyweight, etc.) and the externally infused glucose and insulin rates.

2.5.2

A deterministic model

We firstly hypothesize a system of ODEs explaining glycemia and insulinemia dynamics (inspired by a previously published deterministic model of the EHC, Picchini et al. (2005), Paper 1 of this dissertation), and obtain the corresponding parameter estimates by numerically fitting the model to observed data. The (deterministic) model is defined by: dG(t) (Tgx (t − τg ) + Tgh (t)) G(t) = − Txg − KxgI G(t)I(t) dt Vg 0.1 + G(t) dI(t) (TiG G(t) + Tix (t)) = − Kxi I(t) dt Vi Tgh (t) = Tghmax exp(−λG(t)I(t)) where G(0) = Gb ,

I(0) = Ib ,

Tgh (0) = Tghb = Tghmax exp(−λGb Ib ),

(2.9) (2.10) (2.11)

2.5 An application: modeling glycemia and insulinemia dynamics by SDEs

Tgx (s) = 0 ∀s ∈ [−τg , 0]

and

37

Tix (0) = Tixb

The model is diagrammatically represented in Figure 2.2. Equations (2.9) and (2.10) express the variations of plasma glucose (G) and plasma insulin (I) concentrations. Equation (2.11) represents the rate of (net) hepatic glucose output. Tgx (t) and Tix (t) are (input or forcing) state variables of which the values are known at each time, representing the externally infused glucose and insulin loads; the state variables and the parameters are defined in Table 2.1 and Table 2.2 and, for ease of reading, we do not comment upon model (2.9)-(2.11), since its functional form is dictated by physiological reasons (see Paper 2 for details). Variables t [min] G(t) [mM ] I(t) [pM ] Tgx (t) [mmol/min/kgBW ] Tix (t) [pmol/min/kgBW ] Tgh (t) [mmol/min/kgBW ] Table 2.1:

time from insulin infusion start plasma glucose concentration at time t serum insulin concentration at time t glucose infusion rate at time t insulin infusion rate at time t net Hepatic Glucose Output (HGO) at time t Definitions of the state variables.

Txg

Tgx (t-τ g )

Tgh (t)

G(t) n

K xgI G(t)I(t)

Vg

n

G(t) 0.1+G(t)

TiG G(t)

K xi I(t)

I(t) Tix (t)

Vi stimulation inhibition

Figure 2.2:

Schematic representation of the model (2.9)-(2.11).

Steady-state conditions are used to decrease the number of free parameters to be estimated: at steady state, before the start of the EHC (G = Gb , I = Ib , Tgx = Tix = 0), we have Tghb = Tghmax exp(−λGb Ib )

0 + Tghb Txg Gb 0= − − KxgI Ib Gb ⇒ Txg = Vg 0.1 + Gb



 Tghb (0.1 + Gb ) − KxgI Ib Gb Vg Gb

(2.12)

(2.13)

Parameter Estimation of SDE Models 38

Parameters Gb [mM ] Ib [pM ] Txg [mM/min] KxgI [min−1 /pM ] Kxi [min−1 ] TiG [pM/min/mM ] Tixb [pmol/min/kgBW ] Tghmax [mmol/min/kgBW ] Tghb [mmol/min/kgBW ] Vg [L/kgBW ] Vi [L/kgBW ] τg [min] λ [mM −1 pM −1 ] σ [pM −1 min−1/2 ]

Definitions of the parameters.

basal glycemia basal insulinemia maximal insulin-independent rate constant for glucose tissue uptake insulin-dependent apparent first-order rate constant for glucose tissue uptake at insulinemia I apparent first-order rate constant for insulin removal from plasma apparent zero-order net insulin synthesis rate at unit glycemia (after liver first-pass effect) basal insulin infusion rate, which is given by the measured value of Tix at time zero according to DeFronzo and Ferrannini (1991) maximal Hepatic Glucose Output at zero glycemia, zero insulinemia basal value of Tgh volume of distribution for glucose volume of distribution for insulin discrete (distributional) delay of the change in glycemia following glucose infusion rate constant for Hepatic Glucose Output decrease with increase of glycemia and insulinemia diffusion coefficient

Table 2.2:

2.5 An application: modeling glycemia and insulinemia dynamics by SDEs

0=

TiG Gb + 0 Kxi Ib Vi − Kxi Ib ⇒ TiG = Vi Gb

39

(2.14)

Therefore the parameters Tghb , Txg , and TiG are completely determined by the values of the other parameters. Individual estimates The system (2.9)–(2.11) has been numerically integrated by means of a fourth–order Runge–Kutta scheme with constant stepsize equal to 0.5 min. In order to distinguish among the n observations (and corresponding predictions) between glucose and insulin, the indices j for glucose and k for insulin are used as follows: j ∈ J, k ∈ K, J ∩ K = {∅}, J ∪ K = {1, ..., n}. We indicate ˆ θ) ≡ G(t) ˆ ˆ θ) ≡ I(t) ˆ the (numerically integrated) solutions of equations (2.9)– with G(t, and I(t, (2.11) for parameter θ at time t. The solutions have been fitted by Iteratively Re-Weighted Least Squares (IRWLS, see e.g. chapter 2 in Davidian and Giltinan (1995) for details) separately on each subject’s glycemia and insulinemia time-points, estimating only the free parameters θ = (Gb , Ib , KxgI , Kxi , Tghmax , Vg , Vi , τg , λ) by minimizing the following loss function (y − yˆ)0 Ω(y − yˆ) where y is the n × 1 array containing both glycemias and insulinemias, observed at times 0 = t1 ≤ t2 ≤ · · · ≤ tn ; yˆ is the array of corresponding predictions obtained by numerical integration of ˆ j ) ∀j ∈ J, yˆ(tk ) = yˆk = I(t ˆ k ) ∀k ∈ K; Ω is an n × n the system (2.9)-(2.11), yˆ(tj ) = yˆj = G(t 0 diagonal matrix of weights. Here Z denotes the transpose of the matrix Z. The statistical weight associated with a generic glucose concentration point yj has been defined as 1/(ˆ yj CVG )2 , where CVG is the coefficient of variation for glucose. Similarly the statistical weight associated with a generic insulin concentration point has been defined as 1/(ˆ yk CVI )2 , where CVI is the coefficient of variation for insulin. Thus we assume the following error models yj

= yˆj + εj ,

j∈J

yk

= yˆk + εk ,

k ∈ K,

where the ε’s are independent measurement errors, with E(εj ) V ar(εj ) Cov(εj εj 0 )

= E(εk ) = E(εj εj 0 ) = E(εk εk0 ) = E(εj εk ) = 0, = CVG2 yˆj2 ,

V ar(εk ) = CVI2 yˆk2

= Cov(εk εk0 ) = Cov(εj εk ) = 0

∀j, j 0 ∈ J, ∀k, k 0 ∈ K

∀j ∈ J, ∀k ∈ K ∀j, j 0 ∈ J, ∀k, k 0 ∈ K

since individual data that follow a nonlinear model often exhibit response variation that changes systematically with the level of the response and, in pharmacokinetic data, response variation is widely acknowledged to be related systematically to mean response (Davidian and Giltinan (1995)). The IRWLS estimator θˆIRW LS is such that θˆIRW LS ∼ N (θ, Ψ),

n → +∞

40

Parameter Estimation of SDE Models

where Ψ−1 = Ψ−1 (θ) = J 0 (θ)Ω−1 J(θ) and J(θ) is the n × p matrix with ith row equal to ∂ yˆi /∂θ (i = 1, ..., n) and p is the length of θ. From this result, an approximate 100(1 − α)% confidence interval for θ(r) , the rth element in θ (r = 1, ..., p), is (r) ˆ 1/2 θˆIRW LS ± tα/2,n−p Ψ r,r ,

where tα/2,n−p is the 100(1 − α/2) percentage point of the t distribution with n − p degrees of ˆ ˆ 1/2 ˆ freedom and Ψ r,r is the (r, r)th element in Ψ = Ψ(θIRW LS ). For each subject, IRWLS parameter estimates of θ were obtained for several different values of the coefficients of variation CVG and CVI , namely2 : (CVG , CVI ) ∈

{(0.015, 0.07), (0.02, 0.10), (0.03, 0.10), (0.03, 0.15), (0.04, 0.15), (0.05, 0.15), (0.15, 0.30)}.

These sets of coefficients of variation were used in order to conduct a sensitivity analysis on the diffusion coefficient (defined in section 2.5.3) by considering different values for the variance of the measurement error, which is assumed proportional to the square of the coefficient of variation. Results for subjects 1, 2, 6 and 10 are reported in Table 2.3 and Figure 2.3 (see Paper 2 for the complete results collected on all subjects for all the considered combinations of CVG and CVI ). Confidence intervals can be obtained only for the set of free parameters θ, and they are thus reported in Table 2.4 for the case (CVG , CVI ) = (0.015, 0.07). From Table 2.4 we see that, though some parameters are very well identified, others have very wide confidence intervals (KxgI , λ) for some subjects. This may be due to a number of reasons, e.g. the model could be overparametrized or incorrectly specified. ˆb G

Iˆb

ˆ xgI K

ˆ xi K

1 2 6 10

3.662 4.535 3.755 5.500

42.07 217.16 27.31 257.45

6.05E-5 4.07E-5 2.48E-5 26.56E-5

0.018 0.010 0.033 0.163

1 2 6 10

3.695 4.433 3.794 5.516

39.42 215.92 21.52 254.06

6.07E-5 4.36E-5 2.48E-5 26.49E-5

0.020 0.010 0.037 0.165

1 2 6 10

3.807 4.616 3.841 5.550

38.69 239.06 21.55 250.52

6.16E-5 4.25E-5 2.44E-5 26.57E-5

0.022 0.012 0.041 0.166

Subject

Tˆghmax

Vˆg

Vˆi

τˆg

CVG = 0.015, CVI = 0.07 0.016 0.464 0.829 0.54 0.012 0.266 0.990 2.50 0.044 0.719 0.535 0.50 0.055 0.141 0.229 5.49 CVG = 0.05, CVI = 0.15 0.016 0.472 0.779 0.50 0.011 0.254 0.990 0.50 0.024 0.726 0.483 0.71 0.054 0.138 0.222 5.04 CVG = 0.15, CVI = 0.30 0.011 0.471 0.730 1.45 0.014 0.290 0.990 0.35 0.034 0.738 0.443 0.91 0.053 0.137 0.216 5.50

ˆ λ

Tˆghb

Tˆxg

TˆiG

7.91E-3 9.33E-8 31.06E-3 0.03E-3

0.005 0.012 0.002 0.053

1.08E-3 5.99E-3 0.02E-7 2.62E-7

0.17 0.45 0.13 1.75

9.06E-3 2.30E-7 34.45E-3 0.04E-3

0.004 0.011 0.001 0.051

1.60E-12 0.01E-3 3.15E-12 1.30E-7

0.16 0.49 0.10 1.69

6.41E-3 0.02E-4 37.88E-3 0.03E-3

0.004 0.014 0.001 0.051

0.02E-3 0.70E-3 0.1E-7 1.36E-3

0.16 0.61 0.10 1.62

Table 2.3:

IRWLS parameter estimates of all the structural parameters for the ODE model (2.9)–(2.11) for subjects 1,2,6,10 and for different combinations of CVG and CVI ; the notation E±p is used for 10±p .

Population estimates The starting point to fix reasonable values for CVG and CVI was suggested in Bergman et al. (1979) where it had been found (CVG , CVI ) = (0.015, 0.07); nevertheless, since these values refer to in vitro estimates of the variance of repeated laboratory measurements on the same preparation, it 2 Estimates

of the non-free parameters Tghb , Txg and TiG were obtained using steady-state relations (2.12)-(2.14).

0.464 [0.419, 0.510] 0.266 [0.207, 32.446] 0.719 [0.627, 0.812] 0.141 [0.098, 0.183]

3.771] 4.649] 3.868] 5.664] 0.829 0.990 0.535 0.229

ˆ xgI K

6.05 [5.40, 6.69]E-5 4.07 [79.2, 87.3]E-5 2.48 [2.15, 2.82]E-5 26.56 [-68.25, 121.37]E-5 τˆg [0.787, 0.871] 0.54 [0.54, 0.541] [0.839, 1.141] 2.50 [2.5, 2.50] [0.485, 0.585] 0.50 [0.499, 0.502] [0.163, 0.295] 5.49 [5.487, 5.487] CVG = 0.015, CVI = 0.07

42.07 [36.21, 47.92] 217.16 [217.155, 217.156] 27.31 [27.31, 27.31] 257.45 [257.45, 257.452] Vˆi

Iˆb [0.017, [0.008, [0.030, [0.112, ˆ λ

0.019] 0.012] 0.037] 0.209]

7.91 [-282, 298]E-3 9.33E-8 [-18, 17.8]E-3 31.06 [-452, 514]E-3 0.03 [-2.5, 2.60]E-3

0.018 0.010 0.033 0.163

ˆ xi K

0.016 [-1.261, 1.294] 0.012 [0.005, 0.019] 0.044 [-0.061, 0.15] 0.055 [0.032, 0.078]

Tˆghmax

IRWLS estimates of θ (and 95% confidence intervals) for the ODE model (2.9)–(2.11) for subjects 1,2,6,10 when CVG = 0.015 and CVI = 0.07; the notation E±p is used for 10±p .

Table 2.4:

[3.552, [4.421, [3.642, [5.335, Vˆg

1 2 6 10 Subject 1 2 6 10

3.662 4.535 3.755 5.500

ˆb G

Subject

2.5 An application: modeling glycemia and insulinemia dynamics by SDEs 41

42

Parameter Estimation of SDE Models

Clamp: Subject 1, plot 1

Clamp: Subject 1, plot 2 800

8 700

7

600

Plasma Insulin (pM)

Plasma Glucose (mM)

500 6

5

4

400

300

200

100 3

0

50

100

150

200

250

0

300

0

50

100

Time (min)

150

200

250

300

200

250

300

200

250

300

200

250

300

Time (min)

(a) Subject 1

(b) Subject 1

Clamp: Subject 2, plot 1

Clamp: Subject 2, plot 2 800

8 700

7

600

Plasma Insulin (pM)

Plasma Glucose (mM)

500 6

5

4

400

300

200

100 3

0

50

100

150

200

250

0

300

0

50

100

Time (min)

150 Time (min)

(c) Subject 2

(d) Subject 2

Clamp: Subject 6, plot 1

Clamp: Subject 6, plot 2 800

8 700

7

600

Plasma Insulin (pM)

Plasma Glucose (mM)

500 6

5

4

400

300

200

100 3

0

50

100

150

200

250

0

300

0

50

100

Time (min)

150 Time (min)

(e) Subject 6

(f) Subject 6

Clamp: Subject 10, plot 1

Clamp: Subject 10, plot 2 800

8 700

7

600

Plasma Insulin (pM)

Plasma Glucose (mM)

500 6

5

4

400

300

200

100 3

0

50

100

150

200

Time (min)

(g) Subject 10

250

300

0

0

50

100

150 Time (min)

(h) Subject 10

Figure 2.3: ODE model - left (right) panel: observed (◦) and predicted (solid line) glycemia (insulinemia) corresponding to the IRWLS estimates for the case (CVG , CVI ) = (0.05, 0.15).

2.5 An application: modeling glycemia and insulinemia dynamics by SDEs

43

could be more realistic to re-estimate CVG and CVI from data. To this aim we adopted a General Least Squares approach (GLS, chapter 5 in Davidian and Giltinan (1995)), to obtain subjectspecific regression parameters and population estimates of CVG and CVI . The GLS is a two-stage method: (stage 1) at first individual estimates for each subject i (i = 1, ..., 15) were obtained; then (stage 2) these estimates were used as building blocks to construct the population estimates of CVG and CVI . Suppose that yi and θi represent the ni -dimensional array of recorded data and the array of (structural) individual parameters for subject i respectively (i = 1, ..., 15), i.e. θi contains the values of the (free) parameters θ = (Gb , Ib , KxgI , Kxi , Tghmax , Vg , Vi , τg , λ) entering the model (2.9)–(2.11) for subject i. Consider now the model yi = fi (θi ) + εi such that E(εi |θi ) = 0,

Cov(εi |θi ) = Ωi (θi , ξ)

with fi (·) representing the numerical solution of the system (2.9)–(2.11) for subject i, and assuming that the functional form of Ωi (·, ·) and the intra-individual covariance parameter ξ = (CVG , CVI ) are the same across individuals. If we denote with G and I the state variable Glucose and Insulin respectively, the covariance matrix Ωi (θi , ξ) in the present application has the structure of an ni × ni block-diagonal matrix Ωi (θi , ξ) =

Ωi,G 0

0 Ωi,I

! i = 1, ..., 15

where   2 (θi , ti,1 ) 0 · · · 0 CVG2 fiG   Ωi,G =  ··· ··· ··· ··· , 2 2 0 ··· 0 CVG fiG (θi , ti,niG )   2 (θi , ti,1 ) 0 · · · 0 CVI2 fiI   Ωi,I =  ··· ··· ··· ···  2 2 0 ··· 0 CVI fiI (θi , ti,niI ) with fiG (θi , ti,j1 ) and fiI (θi , ti,j2 ) representing the predicted glycemia and insulinemia values at times ti,j1 and ti,j2 respectively (j1 = 1, ..., niG ; j2 = 1, ..., niI ; niG + niI = ni ). Then the GLS algorithm is given by the following scheme: (p) 1. in m = 15 separate regressions, obtain preliminary estimates θˆi for each individual, i = 1, ..., m;

2. use residuals from these preliminary fits to estimate ξ by minimizing the following functional m X i=1

(p) P Li (θˆi , ξ) =

m X

(p) (p) ˆ(p) ˆ(p) log |Ωi (θˆi , ξ)| + (yi − fi (θˆi ))0 Ω−1 i (θi , ξ)(yi − fi (θi ))

i=1

where P Li is the pseudolikelihood of ξ for the i th individual. Form estimated weight matrices (p) based on the estimate ξˆ obtained from this procedure, along with the preliminary θˆi , to

44

Parameter Estimation of SDE Models

form ˆ ˆ i (θˆ(p) , ξ) Ω i 3. using the estimated weight matrices from step 2, re-estimate the θi ’s by m separate minimizations: for individual i, minimize in θi ˆ −1 (yi − fi (θi )) (yi − fi (θi ))0 Ω i Treating the resulting estimators as new preliminary estimators, return to step 2. The algorithm should be iterated at least once to eliminate the effect of potentially inefficient preliminary estimates in step 1. Using the GLS approach we have estimated simultaneously the individual structural parameters as well as the population parameters CVG and CVI : in this way d G = 0.0710 and CV d I = 0.1702, which are too high to be compatible with we found that CV measurement error, especially if compared with commonly accepted values (e.g. (CVG , CVI ) = (0.015, 0.07) in Bergman et al. (1979)). This finding has prompt us to consider an additional source of noise, besides measurement error, explaining the variation of the observations around the predicted curve, as motivated in section 2.5.3.

2.5.3

A stochastic model

The simply deterministic model does not accommodate random variations of metabolism. In fact a deterministic model assumes that: (i) the mathematical process generating the observed glycemias is smooth (continuous and continuously differentiable) in the considered time-frame; and (ii) the variability of the actual measurements is due only to observation error, which does not influence the course of the underlying process. An alternative, stochastic, approach would result from the hypothesis that the underlying mathematical process itself is not smooth, at least when considered at the practicable time resolution. The glucose metabolizing organs and tissues are in fact subject to a variety of internal and external influences, which change over time (e.g. blood flow, energy requirements, hormone levels, the cellular metabolism of the tissues themselves) and which may affect the instantaneous glycemias. We may thus imagine that the insulin-dependent glucose disposal rate may be subject to moment-by-moment variations and that the rate constant KxgI is likely to exhibit substantial irregular oscillations over time. We therefore assume that some degree of randomness is present in the glucose disposition process, and that observational error is superimposed to it. Thus we define two sources of noise: a dynamic noise term, which is a part of the process, such that the value of the process at time t depends on this noise up to time t, and a measurement noise term, which does not affect the process itself, but only its observations. We therefore define an SDE model by adding a suitable system variability to the simple deterministic model. We allow the parameter KxgI to vary randomly as (KxgI −ξ(t)), where ξ(·) is a gaussian whitenoise process. Then the system noise ξ(t)dt can be written as σdW (t) (see section 1.2, Ditlevsen and De Gaetano (2005b), Kloeden and Platen (1992) and Øksendal (2000)), where σ ≥ 0 represents the (unknown) diffusion coefficient and W (·) is the Wiener process. By incorporating the KxgI

2.5 An application: modeling glycemia and insulinemia dynamics by SDEs

45

variation into the deterministic model, we obtain the following (Itô) SDE: 

 (Tgx (t − τg ) + Tgh (t)) G(t) dG(t) = − Txg − KxgI G(t)I(t) dt Vg 0.1 + G(t) +σG(t)I(t)dW (t),   (TiG G(t) + Tix (t)) dI(t) = − Kxi I(t) dt, Vi Tgh (t) = Tghmax exp(−λG(t)I(t))

(2.15) (2.16) (2.17)

with G(0) = Gb , I(0) = Ib and Tgh (0) = Tghb = Tghmax exp(−λGb Ib ). Notice that this formulation has the theoretical advantage of never becoming negative in any of the coordinates. A variety of methods for statistical inference in discretely observed diffusion processes have been developed during the past decades, as mentioned in the Introduction to this chapter. A natural approach would be maximum likelihood inference, but it is rarely possible to write the likelihood function explicitly. In our case it becomes further complicated since we are dealing with partially observed state variables. The estimation approach we follow is to first estimate by IRWLS the parameters of the ODE system (2.9)–(2.11), which represents the deterministic part (drift) of the SDE model and the mean of the corresponding stochastic process. We then use the MonteCarlo approximation to the unknown likelihood function as suggested in section 2.2, in order to estimate σ by keeping fixed the previously obtained drift parameters estimates, since estimating all the parameters together is computationally unattainable with commonly available resources. We make recourse to the method in Pedersen (2001) since plasma glucose concentrations G and serum insulin concentrations I are not observed at the same time-points (every 5 min. and every 20 min. respectively), and we therefore deal with partially observed state variables, and further because the concentrations are observed with measurement error. In the following, the application of the method to the problem under investigation is detailed; for ease of notation we denote with Zi the generic variable Z(ti ) at time ti . Consider the model (2.15)–(2.17) and observation times 0 = t1 ≤ t2 ≤ · · · ≤ tn : all the observations y are collected in a single array and distinguished using the following label-variable ( χi =

G, I,

if the observation at time ti ref ers to glucose if the observation at time ti ref ers to insulin

i = 1, ..., n

We consider the error-model Yi = Hχi + εi , where

( Hχi =

Gi , Ii ,

if χi = G if χi = I

(2.18)

i = 1, ..., n

and the εi ’s are independent normal variables with mean 0 and variance σχ2 i representing the measurement errors. We assume that (i) ( σχ2 i

=

(CVG Gi )2 , (CVI Ii )2 ,

if χi = G if χi = I

i = 1, ..., n

46

Parameter Estimation of SDE Models

where CVG and CVI represent the coefficient of variations for the glucose and the insulin concentrations respectively, and that (ii) the measurement errors are independent of the process W (·). Equation (2.18) and the SDE system (2.15)-(2.17) provide a representation of the error-structure in our problem. Denote with yi the observed value of Yi at time ti , then the likelihood function of σ can be written as Z L(σ)

=

Y n

 gi (yi |Hχi ; σ) f (Hχ3 , ..., Hχn ; Hχ1 , Hχ2 , σ)dHχ3 · · · dHχn

Rn−2 i=1 n Y


gi yi |Hχi ; σ ,

= Eσ

i=1

where Hχ1 = Gb and Hχ2 = Ib are the initial conditions of G and I respectively, f denotes the (unknown) joint density function of Hχ3 , ..., Hχn given (Hχ1 , Hχ2 , σ), Eσ denotes expectation w.r.t. the distribution of Hχ3 , ..., Hχn for the indicated value of σ and gi (yi |Hχi ; σ) = 2πσχ2 i


−1/2

 
2 1 exp − 2 yi − Hχi 2σχi

is the normal density function with expectation Hχi and variance σχ2 i . If H r (r = 1, ..., R) are stochastically independent random vectors, each distributed as (Hχ3 , ..., Hχn ), then it follows from the strong law of large numbers that the likelihood function can, for large values of R, be approximated by R n 1 XY L(σ) ' gi (yi |Hχr i ; σ) (2.19) R r=1 i=1 In practice the approximation is obtained by simulating the Hχr i ’s (see Kloeden and Platen (1992)) for a large finite number R. We have initially simulated R = 1000, 2000 and 4000 trajectories of the process according to the Euler-Maruyama scheme (see section 1.4.1) with an integration step-size of 0.1 min.: given a σ of the order of 10−5 , the step-size ensures a standard deviation for dG smaller than 0.02 in each integration step, which is very small compared to the order of magnitude of the glucose concentrations. The simulated likelihood functions did not appreciably change the location of their maximum when increasing the number of trajectories beyond 2000 (see Figure 2.4 for an example based on 2000 trajectories). Therefore the reported estimates of σ were obtained by maximizing the approximated likelihood (2.19), based on R = 2000 trajectories, when keeping fixed the parameters entering the drift part of the model and using different combinations of levels of CVG and CVI (see section 2.5.2), in order to explore the sensitivity of the obtained estimates σ ˆ to mis-specification of the observation error variance. The stochastic model (2.15)–(2.17) was adapted to our data and σ was estimated. The estimates of σ corresponding to the different sets of coefficients of variation are reported in Table 2.5 for each subject. In this table we notice that the σ estimates are stable when considered in a reasonable region of the coefficient of variations values, that is when considered in (CVG , CVI ) ∈ [0.02, 0.05] × [0.10, 0.15]. At the smallest level (CVG , CVI ) = (0.015, 0.07) the σ estimate results numerically unidentifiable for subject two, and is thus marked with an ‘NA’. Theoretical results on the asymptotic properties of the σ estimates are unavailable, therefore means

2.5 An application: modeling glycemia and insulinemia dynamics by SDEs

(a) Subject 6

Figure 2.4:

47

(b) Subject 10

Simulated likelihoods of σ for subjects 6 and 10 using 2000 trajectories for the case CVG = 0.03 and

CVG = 0.15.

and 95% confidence intervals of the estimates were produced by a parametric bootstrap procedure (Efron and Tibshirani (1993)): for each subject, one hundred artificial data sets were simulated from model (2.15)–(2.17) using the Euler-Maruyama scheme and the obtained estimates of θ and σ then, for each artificial data set, σ was estimated anew. For each subject, the sample mean, the empirical 95% confidence intervals (from the 2.5th to the 97.5th percentile) and measures of symmetry (skewness and kurtosis3 ) from the 100 obtained estimates are reported in Table 2.6. This bootstrap procedure is highly time consuming, therefore it was performed only for the cases (CVG , CVI ) = (0.015, 0.07) and (CVG , CVI ) = (0.03, 0.15). For illustration purposes, graphical results of the fitting only for the case (CVG , CVI ) = (0.05, 0.15) are shown in Figure 2.5, only for the glycemia values since the insulin curves are almost identical to those produced by the deterministic model. For each subject Figure 2.5 reports the observed glycemias and the empirical mean of R = 2000 simulated trajectories of the G(t) process, their empirical 95% confidence limits (from the 2.5th percentile to the 97.5th percentile) and one simulated trajectory; from this figure we get pictorial evidence of the diffusion coefficient magnitude. We are able to check the plausibility of our stochastic model by simulating uniform residuals, as suggested in section 2.4; the q-q plots of the simulated uniform residuals are reported in figures 2.6 and 2.7, where the residuals are plotted against percentiles from the U (0, 1) distribution. The caption of each subfigure also reports the p-value from the two-tailed Kolmogorov-Smirnov goodness-of-fit test; if p < 0.05 the simulated residuals do not conform to the hypothesis of U (0, 1) distribution at a 5% confidence level. The tests have not been subjected to correction for simultaneous inference (Bonferroni or similar) in order to be more conservative. All tests had p > 0.05, except for the glycemia residuals for subject 6.

2.5.4

Conclusions

In the present application, a (simple) deterministic model of the clamp procedure is studied first. The main result of this study is that the level of error around the predicted curve is very large, in particular it is much larger than the (0.015, 0.07) commonly accepted levels of measurement error in in vitro repeated testing of the same laboratory preparation. This result would theoretically 3 Recall

that, for the normal distribution, skewness=0 and kurtosis=3.

48

Parameter Estimation of SDE Models

Clamp: Subject 2, plot 1 8

7

7

6

6

5

5

Plasma Glucose (mM)

Plasma Glucose (mM)

Clamp: Subject 1, plot 1 8

4

3

4

3

2

2

1

1

0

0

50

100

150

200

250

0

300

0

50

100

Time (min)

(a) Subject 1 Clamp: Subject 6, plot 1

250

300

200

250

300

Clamp: Subject 10, plot 1 8

7

7

6

6

5

5

Plasma Glucose (mM)

Plasma Glucose (mM)

200

(b) Subject 2

8

4

3

2

4

3

2

1

0

150 Time (min)

1

0

50

100

150

200

250

300

0

0

50

100

Time (min)

150 Time (min)

(c) Subject 6

(d) Subject 10

Figure 2.5: SDE model: a simulated trajectory of G(t), empirical mean curve of the G(t) process (smooth solid lines), empirical 95% confidence limits of the mean process (dashed lines) for the case (CVG , CVI ) = (0.05, 0.15) and glycemia observations.

1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.0

0.0

0.1

0.3

0.5

0.7

0.1

0.9

0.3

0.5

0.7

0.9

Uniform Distribution

Uniform Distribution

(a) Subject 1, p = 0.658

(b) Subject 2, p = 0.2545

1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.0

0.0

0.1

0.3

0.5

0.7

0.9

Uniform Distribution

(c) Subject 6, p = 0.0452

0.1

0.3

0.5

0.7

0.9

Uniform Distribution

(d) Subject 10, p = 0.2676

Figure 2.6: SDE model: q-q plots of the simulated glycemia residuals in the case (CVG , CVI ) = (0.05, 0.15) and p-values from the Kolmogorov-Smirnov goodness-of-fit test. If p < 0.05 the residuals do not conform the hypothesis of U (0, 1) distribution at a 5% confidence level.

2.5 An application: modeling glycemia and insulinemia dynamics by SDEs

1.0

1.0

0.8

0.8

49

0.6 0.6

0.4 0.4 0.2 0.2 0.0 0.0 0.1

0.3

0.5

0.7

0.1

0.9

0.3

(a) Subject 1, p = 0.0922

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.3

0.5

0.7

0.9

(b) Subject 2, p = 0.0657

0.8

0.1

0.5 Uniform Distribution

Uniform Distribution

0.7

0.9

0.1

Uniform Distribution

0.3

0.5

0.7

0.9

Uniform Distribution

(c) Subject 6, p = 0.2847

(d) Subject 10, p = 0.8462

Figure 2.7:

SDE model: q-q plots of the simulated insulinemia residuals in the case (CVG , CVI ) = (0.05, 0.15) and p-values from the Kolmogorov-Smirnov goodness-of-fit test. If p < 0.05 the residuals do not conform the hypothesis of U (0, 1) distribution at a 5% confidence level.

Subjects 1 2 6 10

σ ˆ (1)

σ ˆ (2)

σ ˆ (3)

σ ˆ (4)

σ ˆ (5)

σ ˆ (6)

σ ˆ (7)

1.60E-5 NA 2.72E-5 3.08E-5

1.78E-5 1.38E-5 2.65E-5 3.04E-5

2.25E-5 1.47E-5 2.68E-5 3.04E-5

1.59E-5 1.38E-5 2.71E-5 3.04E-5

2.10E-5 1.38E-5 2.72E-5 3.00E-5

2.25E-5 1.15E-5 2.73E-5 2.88E-5

0 2.88E-7 5.29E-8 4.25E-7

Table 2.5: Estimates of σ in the cases (CVG , CVI ) = (0.015, 0.07), (CVG , CVI ) = (0.02, 0.10), (CVG , CVI ) = (0.03, 0.10), (CVG , CVI ) = (0.03, 0.15), (CVG , CVI ) = (0.04, 0.15), (CVG , CVI ) = (0.05, 0.15) and (CVG , CVI ) = (0.15, 0.30) given by σ ˆ (1) , σ ˆ (2) , σ ˆ (3) , σ ˆ (4) , σ ˆ (5) , σ ˆ (6) and σ ˆ (7) respectively. The notation E±p is used for 10±p . Subjects 1

2

6

10

Table 2.6:

estimates Skewness Kurtosis estimates Skewness Kurtosis estimates Skewness Kurtosis estimates Skewness Kurtosis

(mean [95% CI])

(mean [95% CI])

(mean [95% CI])

(mean [95% CI])

σ ˆ (1) · 10−5

σ ˆ (4) · 10−5

1.60 (1.27 [0.61, 1.85]) -0.2684 2.216 NA NA NA 2.72 (2.26 [0.89, 3.01]) -0.676 3.142 3.08 (1.92 [1.10, 3.17]) 0.787 3.084

1.59 (1.39 [0.45, 2.63]) 0.6751 4.552 1.38 (1.25 [0.39, 2.32]) 0.434 3.521 2.71 (2.25 [1.15, 3.27]) -0.273 2.670 3.04 (2.06 [0, 4.08]) 0.286 3.961

Estimates of σ, simulated mean and empirical 95% confidence intervals, in the cases (CVG , CVI ) = (0.015, 0.07) and (CVG , CVI ) = (0.03, 0.15)

50

Parameter Estimation of SDE Models

be compatible with either one of the following alternatives: the model is mis-specified; the in vivo measurement error is in reality much larger than (0.015, 0.07); or, there is some additional source of noise, besides measurement error, which substantially impacts observations. From an examination of Figure 2.3, it would not seem that the average model prediction is systematically wrong (see also Paper 2 of this dissertation for the complete results). Similarly, the coefficient of variations estimated by GLS around the deterministic prediction are much too large to be compatible with measurement error. The idea that glucose absorption by tissues varies in time is, on the other hand, rather natural: it seems evident that, subject to variable hormonal concentrations, variable stress levels, even to minor posturali changes, muscle uptake and liver output of glucose may vary from moment to moment. What remains to be seen is if a mathematical model incorporating this idea would be supported by the actual observations. A random fluctuation in the net tissue glucose uptake rate is a reasonable approximation to the effect of a host of the poorly controlled, additive influences mentioned. When considering this random fluctuations as well, the original deterministic model (2.9)–(2.11) is thus transformed into the SDE model (2.15)–(2.17). Essentially, it has been shown that: (i) for any reasonable level of observation error, the estimated diffusion has more or less the same value. For “reasonable” it is here meant larger than pure measurement error and smaller than the total error around the expected trajectory as estimated by GLS. Adopting the lowermost observation error level (0.015, 0.07) would be equivalent to stating that the same variability exists on repeated laboratory measurements on the same sample as on repeated sampling/measurement procedures at the same actual glycemia and insulinemia, disregarding further potential sources of variation accruing to the sampling procedure itself (volume of blood vs. volume of anticoagulant, degree of coagulation, variation in spinning time etc.): this seems extreme in one direction. In the other direction, the largest error level considered (0.15, 0.30) is much higher than the total observation variability actually estimated with the GLS procedure around the deterministic prediction (0.0710, 0.1702), and for this reason should also be discarded. Having excluded these extreme cases, it can be seen that, in the present situation, the estimation of the diffusion is very robust to changes in the likely value of the observation error, as similarly robust are the estimates of the structural (drift) parameters. From Table 2.6 we see that the σ estimates are contained into the simulated 95% confidence intervals, though the mean values of the simulated estimates (from bootstrap procedure) are different from the original estimates. Further (ii) we showed that the diffusion coefficient estimates are generally strictly positive: this means that the dynamical process which most likely represents the glycemia time-course (given the estimated deterministic differential model) is a stochastic process with a non-negligible system noise, whose intensity factor is represented by the diffusion coefficient. Pictorial evidence of the diffusion coefficient magnitude is given in Figure 2.5. This system noise represents the additive action of many factors, each with a small individual effect, which are not explicitly represented in the deterministic model (that is in the drift term of the SDE), and which instantaneously affect glucose uptake rate. Therefore, in the stochastic differential model the collective influence of many individually neglected effects is added to the average drift term, which, on the other side, represents the most relevant and generally well-recognized factors affecting glycemia. The particular behavior of the estimated diffusion (Table 2.5), for the different choices of

2.5 An application: modeling glycemia and insulinemia dynamics by SDEs

51

coefficient of variation values ξ ≡ (CVG , CVI ) could seem counterintuitive: one would expect that as the observation error is assumed to increase, the estimated system error should decrease (this actually happens for macroscopically exaggerated values of the observation error). This unexpected result may be due to the estimation method we adopted: since the array of free structural parameters θ for the drift part of the stochastic model is estimated anew for different ˆ are depending on these levels of error, so we can write θˆ ≡ θˆξ . As a consequence values of ξ, the θ’s of that, the (numerical) solution of the SDE system (2.15)–(2.17) and the measurement error ε are not independent and, by means of equation (2.18), we can write V ar(Yi )

= V ar(Hχi (θˆξ )) + V ar(εi ) + 2Cov(Hχi (θˆξ ), εi ) = V ar(Hχi (θˆξ )) + σχ2 i + 2E(Hχi (θˆξ )εi ).

Since H(·) is unknown, so is E(Hχi (θˆξ )εi ) and we cannot compute the covariance analytically: however it is generally not zero, and the variance of the observations is not the simple sum of the variance of the trajectories and of the variance of the measurement error.

Chapter 3

Stochastic Differential Mixed-Effects Models This chapter has been published as: Umberto Picchini, Andrea De Gaetano and Susanne Ditlevsen (2006). “Parameter estimation in stochastic differential mixed-effects models”. Research Report 06/12, Department of Biostatistics, University of Copenhagen.

3.1

Introduction

Studies in which repeated measurements are taken on a series of individuals or experimental animals play an important role in biomedical research. It is often reasonable to assume that responses follow the same model form for all experimental subjects, but model parameters vary randomly among individuals. The increasing popularity of Mixed-Effects models lies in their ability to model total variation, splitting it into its within- and between-individual components. This often leads to more precise estimation of population parameters, which is especially useful in pharmacokinetic/pharmacodynamic (PK/PD) modeling, where enhanced precision of estimation translates into considerable savings both in resources and in human or animal discomfort. Dynamical biological processes are usually modeled by means of systems of deterministic differential equations (ordinary (ODE), partial (PDE), or delay (DDE)). These however do not account for the noisy components of the system dynamics often present in biological systems. System error (or system noise) represents the cumulative effect on the actual state of the system of a host of mechanisms which cannot be individually included in the model description (like hormonal oscillations, variations of the stress level, variable muscular activity etc.). Noise in the differential equations describing the behavior of the system requires an extension to the class of stochastic differential equation (SDE) models. The theory for Mixed-Effects models is well developed for deterministic models (without system error), both linear and non-linear (Lindstrom and Bates (1990), Breslow and Clayton (1993), Vonesh and Chinchilli (1997), Diggle et al. (2002)), and standard software for model fitting is available, see e.g. Pinheiro and Bates (2002) and references therein. Early and important references in the pharmacokinetic field are Sheiner and Beal (1980, 1981). On the other hand, to

54

Stochastic Differential Mixed-Effects Models

our knowledge there is practically no theory at present for SDE models with random effects. The problem here is that estimating parameters in SDE models is not straightforward, except for simple cases. A natural approach would be likelihood inference, but the transition densities of the process are rarely known, and thus it is usually not possible to write the likelihood function explicitly. In Jelliffe et al. (2000) methods for PK/PD population modeling are reviewed, but the authors regret that system noise is not considered since it is difficult to estimate. In Overgaard et al. (2005) and Tornøe et al. (2005) a SDE model with log-normal distributed random effects and a constant diffusion term is treated, but this constrains the class of models to be SDEs with additive noise. In Ditlevsen and De Gaetano (2005a) the likelihood function for a simple SDE model with normal distributed random effects is calculated explicitly, but generally the likelihood function is unavailable. Eventually, as SDE models are more commonly applied to biomedical data, there will be an increasing need for developing a general theory for parameter estimation including mixed-effects. In the present chapter an estimation method for the parameters of an SDE model incorporating random effects is proposed: these models may be called stochastic differential mixed effects (SDME) models. We consider SDME models whose drift and diffusion terms can depend linearly or nonlinearly on state variables and random effects following any continuous distribution, and an approximation to the likelihood function is computed. The likelihood can seldom be obtained in closed form since it involves explicit knowledge of the process transition density, which is often unavailable, and thus exact parameter estimators are also unavailable. It is therefore necessary to approximate the transition density numerically. To our knowledge, three ways have been proposed to do this (see also section 2.1): 1. solving numerically the Kolmogorov partial differential equations satisfied by the transition density (Lo (1988)); 2. deriving a closed-form Hermite expansion to the transition density (Aït-Sahalia (2001, 2002b)); 3. simulating the process in order to Monte-Carlo-integrate the transition density (e.g. Pedersen (1995), Brandt and Santa-Clara (2002), Durham and Gallant (2002), Hurn and Lindsay (1999), Hurn et al. (2003), Nicolau (2002)): this methodology is known as simulated maximum likelihood (SML). Each of these three techniques has been successfully implemented by the aforementioned authors, but each has its limitations. Aït-Sahalia (2002b) notes that methods 1 and 3 above are computationally intense and poorly accurate. Conversely, Durham and Gallant (2002) build on their importance sampling ideas in order to improve the performance of Pedersen’s (1995) (or equivalently Brandt and Santa-Clara’s (2002)) method, and point out that method 2 above, while accurate and fast, is only available for a small number of models. We choose to employ the transition density approximation method suggested in Aït-Sahalia (2001, 2002b) for time-homogeneous SDE, since it is the fastest and the most accurate among the available methods (Durham and Gallant (2002), Jensen and Poulsen (2002)). This is a desirable condition to make the parameter estimation procedure proposed here effective and reliable. We thus derive an approximation to the likelihood function and estimate the parameters of a SDME model by (approximated) maximum likelihood.

3.2 Formulation of Stochastic Differential Mixed-Effects Models

55

Here attention is restricted to time-homogeneous SDEs, but the method proposed can be applied to more general SDME models, since also for time-inhomogeneous SDEs (which depend directly on time t, not only through the process values) the transition density can be expanded in closed form (Egorov et al. (2003)). Evidence of the accuracy of the estimation method is given by simulation results, where exact and approximated parameter estimates are compared for SDME models of a Brownian motion with drift, of Geometric Brownian Motion and of the Ornstein-Uhlenbeck process. The estimates obtained are close to the true parameter values, and this result is achieved using moderate values of M (the number of experimental units, e.g. the number of subjects) and n (the number of observation for a given experimental unit). This is relevant for applications of these methods in situations where large data sets are unavailable, e.g. in biomedical applications, where MixedEffects theory is broadly applied.

3.2

Formulation of Stochastic Differential Mixed-Effects Models

Consider a d-dimensional (Itô) SDE model for some continuous process evolving in M different experimental units (e.g. subjects) randomly chosen from a theoretical population: dXti = µ(Xti , θ, bi )dt + σ(Xti , θ, bi ) dWti ,

X0i = xi0

i = 1, . . . , M

(3.1)

where θ ∈ Θ ⊆ Rp is a p-dimensional fixed effects parameter (the same for the entire population) and bi ≡ bi (Ψ) ∈ B ⊆ Rq is a q-dimensional random effects parameter (subject specific) whose density function in the population pB is parametrized by an r-dimensional parameter Ψ ∈ Υ ⊆ Rr . The Wti are standard (L × 1)-dimensional Brownian motions. The Wti,l and bj are assumed mutually independent for all 1 ≤ i, j ≤ M , 1 ≤ l ≤ L, and independent of X0i . The drift and the diffusion coefficient functions µ(·) : E × Θ × B → Rd and σ(·) : E × Θ × B → S are assumed known up to the parameters, and are assumed sufficiently regular to ensure a unique solution (Øksendal (2000)), where E ⊆ Rd denote the state space of Xti and S denotes the set of the d × L positive definite matrices. Assume that the distribution of Xti given (bi , θ) and Xsi = xs , s < t, has a strictly positive density w.r.t. the Lebesgue measure on E, which we denote by x → pX (x, t − s|xs , bi , θ) > 0, x ∈ E.

(3.2)

Assume that subject i is observed at (ni +1) discrete time points (ti0 , ti1 , . . . , tini ) for each coordinate k of the process (k = 1, ..., d; i = 1, ..., M ). Let xi be the d(ni +1)-dimensional vector containing the i,d i,d i,k i i,1 model (3.1) responses for the i’th subject, xi = (xi,1 0 , . . . , xni , ..., x0 , . . . , xni ), where x (tj ) = P M i,k i,k xti = xj , and let x = (x1 , ..., xM ) be the N -dimensional total response vector, N = i=1 d(ni + j

1). Write tij − tij−1 = ∆ij for the distance between observation j − 1 and j for subject i. We wish to estimate (θ, Ψ) given x, and we call (3.1) a stochastic differential mixed-effects (SDME) model.

56

3.3

Stochastic Differential Mixed-Effects Models

Exact and Approximated Maximum Likelihood Estimation

To obtain the marginal density of xi , we integrate the conditional density of the data given the non-observable random effects bi with respect to the marginal density of the random effects, using the fact that Wti,l and bj are independent (1 ≤ i, j ≤ M , 1 ≤ l ≤ L). This yields the likelihood L(θ, Ψ)

=

M Y

i

p(x |θ, Ψ) =

i=1

M Z Y i=1

pX (xi |bi , θ) pB (bi |Ψ) dbi

(3.3)

B

where p(·), pX (·) and pB (·) are density functions. Notice that p(xi |·) and pX (xi |·) are in general different: the former being the density of xi given (θ, Ψ), and the latter being the product of the transition densities for a given realization of the random effects and for a given θ : pX (xi |bi , θ)

=

ni Y

pX (xij , ∆ij |xij−1 , bi , θ),

(3.4)

j=1

where the transition densities pX (·) are as in (3.2). The distribution of the random effects is often assumed to be normal, but pB (·) could be any density function. Solving the integral in (3.3) yields the marginal likelihood of the parameters, independent of the random effects bi ; by maximizing the resulting expression (3.3) with respect to θ and Ψ we obtain the corresponding maximum ˆ likelihood estimators θˆ and Ψ. In simple cases we can find an explicit expression for the likelihood function, and even find explicit estimating equations for the maximum likelihood estimators (see Example 1). However, in general it is not possible to find an explicit solution for the integral, and thus exact maximum likelihood estimators are unavailable, i.e. when: (i) pX (xij , ·|xij−1 , ·) is known but we are unable to analytically solve the integral, and (ii) pX (xij , ·|xij−1 , ·) is unknown. In (i) we have to numerically evaluate the integral to obtain an approximation of the likelihood (3.3) and then, by maximizing the resulting expression, approximate maximum likelihood estimators are obtained. In (ii) we can approximate pX (xij , ·|xij−1 , ·), then numerically solve the integral in (3.3) and get the corresponding approximated maximum likelihood estimators. In situation (ii) there exist several strategies to approximate the density pX (xij , ·|xij−1 , ·), e.g. by simulating a large number of process sample paths (e.g. Pedersen (1995), Brandt and SantaClara (2002), Nicolau (2002), Hurn and Lindsay (1999)), or by solving numerically the Kolmogorov partial differential equations satisfied by the transition density (Lo (1988)). However, these techniques are computationally expensive. We propose to approximate the transition density as suggested in Aït-Sahalia (2001, 2002b), where the approximation is obtained in closed-form, using a Hermite expansion as reviewed in section 3.4.1. Then, using this expression, the likelihood function is approximated, thus deriving approximated maximum likelihood estimators of θ and Ψ, as suggested in section 3.4.2.

3.4 Closed-form transition density expansion and likelihood approximation

3.4

57

Closed-form transition density expansion and likelihood approximation

3.4.1

Transition density expansion for SDE models

For ease of reading, here we review the transition density expansion of a scalar (d = L = 1) time-homogeneous SDE as suggested in Aït-Sahalia (2002b), which has already been considered in section 2.3. A generalization to multidimensional SDEs can be found in Aït-Sahalia (2001), and we adapt the more compact notation used in the multidimensional case to the one-dimensional situation. The extension to time-inhomogeneous processes (i.e. the SDE depends directly on t, not only through the state variable Xt ) is given in Egorov et al. (2003). In the remaining of this section we drop the reference to θ when not necessary, that is, we write f (x) instead of f (x, θ) for a given function f . Consider the following one-dimensional time-homogeneous SDE dXt = µ(Xt )dt + σ(Xt )dWt ,

X(t0 ) = x0

(3.5)

where we want to approximate pX (xj , ∆j |xj−1 ), the conditional density of Xtj given Xj−1 = xj−1 , where ∆j = tj − tj−1 . Under mild regularity conditions (Aït-Sahalia (2001)) the logarithm of the transition density can be expanded in closed form using an order J = +∞ Hermite series, and approximated by a Taylor expansion up to order K: (−1)

(K)

ln pX (xj , ∆j |xj−1 )

C 1 1 = − ln(2π∆j ) − ln(σ 2 (xj )) + Y 2 2 +

K X

(k)

CY (γ(xj )|γ(xj−1 ))

k=0 (k)

The coefficients CY

∆kj . k!

(γ(xj )|γ(xj−1 )) ∆j (3.6)

are given in Theorem 2.1 and γ(·) is the Lamperti transform, defined by Z Yt ≡ γ(Xt ) =

Xt

du σ(u)

(3.7)

where the lower bound of integration is an arbitrary point in the interior of E (i.e. the constant of integration is irrelevant). Then Yt is the solution to the SDE dYt = µY (Yt )dt + dWt where µY (·) is given by µY (Yt ) =

3.4.2

µ(γ −1 (Yt )) 1 ∂σ −1 − (γ (Yt )). σ(γ −1 (Yt )) 2 ∂x

Likelihood approximation for SDME models (k)

For scalar time-homogeneous SDME models, the coefficients CY can be obtained in the same way as suggested in Theorem 2.1, by considering (θ, bi ), ∆ij and (xij , xij−1 ) instead of θ, ∆j and (K) (xj , xj−1 ), respectively. Once the coefficients are obtained, an expression for pX is available and

58

Stochastic Differential Mixed-Effects Models

it is possible to approximate the likelihood of (θ, Ψ) for the SDME model (3.1) by substituting the unknown transition density in (3.4) with its approximation, thus obtaining a sequence of approximations to the likelihood function L(K) (θ, Ψ) =

M Z Y i=1

B

(K)

pX (xi |bi , θ) pB (bi |Ψ) dbi ,

(3.8)

where (K)

pX (xi |bi , θ)

=

ni Y

(K)

pX (xij , ∆ij |xij−1 , bi , θ)

(3.9)

j=1 (K)

and pX is given by equation (3.6). By maximizing (3.8) with respect to (θ, Ψ), we obtain the corresponding approximated maximum likelihood estimators θ(K) and Ψ(K) . The method can be extended to time-inhomogeneous and/or multidimensional SDME models, by extensions of the density expansion method, which are given by Aït-Sahalia (2001) for the multidimensional time-homogeneous case, and by Egorov et al. (2003) for the one-dimensional time-inhomogeneous case.

3.5

Implementation issues and numerical applications

This section reports applications of our estimation method to some famous SDE models that we perturb with random effects: Brownian motion with drift, the Geometric Brownian Motion and the Ornstein-Uhlenbeck process. The main goals are to show the feasibility and effectiveness of the proposed estimation method for SDME models, and to show that accurate results can be obtained when using a “reasonable” and “realistic” data-set, i.e. when handling a limited amount of data (say M = 10, ..., 50 subjects and n = 10, ..., 50 observations collected on each subject), instead of considering large data-sets that are often unavailable, especially in biomedical applications. For numerical optimization reasons, the approximated estimators are always obtained by minimizing the negative log-likelihood function, e.g. when using the density expansion method we minimize Z M X (K) (K) (3.10) log pX (xi |bi , θ) pB (bi |Ψ) dbi , − log L (θ, Ψ) = − i=1

B

and we denote with (θ(K) , Ψ(K) ) the resulting estimator (θ(K) , Ψ(K) ) = arg min(− log L(K) (θ, Ψ)). θ,Ψ

It has been shown that K = 1 or 2 (Aït-Sahalia, 1999, 2001, 2002b) is often sufficient to approximate the transition density to obtain accurate estimates. We use either K = 1 or 2 order density expansion depending on the model, which seems to be sufficient for the considered applications (in particular, for the Brownian motion with drift and the Geometric Brownian Motion, a K = 1 order (k) expansion gives the exact density expression). The coefficients CY for the considered models are (k) given in section 3.7 (notice that, in general, the CY can be calculated using a symbolic calculus software). The integral appearing in (3.10) is numerically evaluated using the trapezoidal rule on

3.5 Implementation issues and numerical applications

59

a grid of two-hundred bi values, except for Example 3 where B is assumed to be a square grid of 100 × 100 values. For each example, parametric bootstrap was performed (Efron and Tibshirani (1993)) to obtain means of the parameters estimates and their 95% confidence intervals. More specifically, for each SDME model two hundred data sets, of dimensions n×M each, were generated using different sets of parameters and different values of M and n, and the corresponding (exact and/or approximated) parameter estimates were obtained. For each parameter, the sample mean and the empirical 95% confidence intervals (from the 2.5th to the 97.5th percentile) from the obtained estimates are reported in Table 3.1–3.5 together with measures of symmetry (skewness and kurtosis). In Example 1 the exact expression for the log-likelihood function of the SDME model is available, so we can graphically compare the shape of the surface of a chosen profile-loglikelihood with the corresponding surface obtained by numerical integration, as given in Figure 3.1. The same comparison is conducted in Example 2, but here only the transition density is known whereas the exact expression of the log-likelihood is unavailable in closed form, so we compare the surface of a profile of the logarithm of expression (3.19) (where the integral is numerically evaluated) with the surface of its K = 2 order approximation, as given in Figure 3.2. Finally, we want to stress the usefulness of using the method considered in section 3.4.1 to approximate pX . In fact, using e.g. the SML-like approaches (see the Introduction), for each iteration of an optimization algorithm maximizing (3.3), the numerical simulation of thousands of trajectories of the process can be required to approximate pX (xij , ∆ij |xij−1 , ·). Then, expression (3.4) must be evaluated and the integral in (3.3) must be numerically computed for the given subject. Finally, repeating the procedure for all the M subjects, we get the likelihood approximation for the current iteration of the optimization algorithm. From a computational point of view this is a highly expensive procedure, essentially because of the necessary large number of simulations of trajectories. Worse still, the larger the dimensions of θ and Ψ the slower the optimization procedure convergence, and obviously the computational time increases for large values of M and n. Instead, using the closed-form density expansion, simulating process trajectories is not required, and the likelihood approximation (3.8) can be evaluated more rapidly (Jensen and Poulsen (2002)).

Example 1: Brownian Motion with drift and Geometric Brownian Motion with one random effect Consider a Brownian motion with drift: dZt

=

(β − σ 2 /2)dt + σdWt ,

Zt

= Z0 + (β − σ 2 /2)t + σWt .

Z0 = z0 ,

with solution (3.11)

Assume an experiment is conducted on M different subjects. We are interested in estimating the parameters in the population, but expect individual differences in the processes, and would

60

Stochastic Differential Mixed-Effects Models

therefore consider a random effect in β, which leads to the SDME model: dZti = (β + β i − σ 2 /2)dt + σdWti ,

Z0i = z0i ,

i = 1, ..., M

and we assume β i ∼ N (0, σβ2 ). The latter model has solution given by Zti

= Z0i + (β + β i − σ 2 /2)t + σWti ,

i = 1, ..., M.

(3.12)

In this simple example we have bi = β i , θ = (β, σ 2 ) and Ψ = σβ2 . We wish to estimate (β, σ 2 , σβ2 ) given a set z = (z 1 , ..., z M ) of observations from model (3.12). The log-likelihood function is (Ditlevsen and De Gaetano (2005a))

log L(θ, Ψ)

 2  ! M M σ N −M 1X σ2 2 i ni i = log 2 − log(2πσ ) − log (∆ ) T + 2 2 σβ 2 2 i=1 σβ −1  P P i 1 σ2 i i i 2 i i 2 i T + σ2 i,j ∆ij (yj − yj−1 − α∆j ) − i (yni − y0 − αT ) β − (3.13) 2 2σ

Q 1 Pni ni i ni where, for ease of notation, we define α = β − σ 2 /2, ∆i = ∆ and T i = j=1 ∆ij . The j j=1 last sum is simply the length of the observation interval for the i’th subject. Assume equidistant observations and that each subject has the same number of observations, that is, assume ∆ij = ∆ and ni = n for all 1 ≤ i ≤ M , 1 ≤ j ≤ ni . The maximum likelihood estimators are given by (Ditlevsen and De Gaetano (2005a)):

σ ˆ

2

σ ˆβ2

= =

1 M

PM Pn i=1

i j=1 (zj

i − zj−1 −α ˆ ∆)2 −

∆ MT

PM

i i=1 (zn

− z0i − α ˆ T )2

T −∆ PM i  PM Pn 1 i 2 i ˆ T ) − i=1 j=1 (zji − zj−1 −α ˆ ∆)2 i=1 (zn − z0 − α MT T −∆

σ ˆ2 βˆ = α ˆ+ 2 where α ˆ=

PM

i i=1 (zn

(3.14) (3.15) (3.16)

− z0i )/(M T ) and T = T i = n∆.

Now consider the transformed process Xt = exp(Zt ), which leads to a SDME model of the Geometric Brownian motion dXti = (β + β i )Xti dt + σXti dWti ,

X0i = xi0 ,

i = 1, ..., M

with β i ∼ N (0, σβ2 ) and Itô solution Xti = X0i exp((β + β i − σ 2 /2)t + σWti ),

i = 1, ..., M.

(3.17)

The process is relevant e.g. in pharmacokinetics for the metabolism of a compound in plasma following first order kinetics where we expect β < 0, or as a growth model, e.g. the initial growth of bacterial or tumor cell populations, where we expect β > 0. See e.g. Braumann (2002) for generalizations of this model.

3.5 Implementation issues and numerical applications

61

The exact estimators (3.14)–(3.16) can be used as a benchmark to test the effectiveness of the (k) estimation method. In this example CY (·) = 0 for all k ≥ 2, and thus the order K = 1 density expansion results in the exact transition density of the process, see section 3.7 for details. We therefore compare the exact maximum likelihood estimators with the approximated estimators, the only difference being that the integral in (3.3) is solved analytically or numerically. For different sets of parameter values and for different choices of M and n, 200 data sets were generated from (3.12) and the parameters were estimated using (3.14)–(3.16) (see Table 3.1). Then, 200 data sets were generated from (3.17) and the approximated estimators were obtained by minimization of the numerical solution of (3.10); results from the latter approach are reported in Table 3.2. In all simulations we fixed Z0i = log(100), X0i = 100 for all i and T = 100. From Table 3.1 it is seen that the true parameter values are well identified using the exact maximum likelihood estimators (3.14)-(3.16) and, in particular, the cases (M, n) = (50, 10) produce better estimates of β and σβ than the cases (M, n) = (10, 50) as expected, since M is the sample size of draws from the distribution of β. The same apply to the approximated estimates in Table 3.2, but here the cases (M, n) = (10, 50) produce much worse estimates whereas cases (M, n) = (50, 10) produce estimates comparable in quality to the exact ones. Obviously, the approximated estimators suffer the bias induced by the numerical integration in expression (3.10), and a finer integration grid (see section 3.5) should improve the performance of the method at the cost of increasing computational time. In all cases σ is well determined and does not seem affected by the numerical integration. Finally, in Figure 3.1 the contour plots of the shapes of the profiles of the exact log-likelihoods (3.13) (for fixed σβ2 = 0.02) and the shapes of the corresponding approximations are compared for different values of M and n: the exact log-likelihood is conditioned on observations generated from model (3.12) with (β, σ 2 , σβ2 ) = (−0.2, 0.2, 0.02), whereas the approximated log-likelihood is conditioned on observations generated from model (3.17) with the same parameter values. By looking at Figure 3.1, we see that the exact and approximated surfaces are quite similar, and the approximation improves for increasing values of M . Differences in contour values are imputable to the models used to generate observations: model (3.12) for the exact log-likelihood and model (3.17) for the corresponding approximation. This implies proportional surfaces with similar shapes: differences in the shape are due to the numerical evaluation of the integral in (3.3).

Example 2: Ornstein-Uhlenbeck process with one random effect Consider the Ornstein-Uhlenbeck process, defined by the following scalar SDE (d = L = 1)  dXt =

 Xt − + µ dt + σdWt ; τ

X0 = x0 = 0

where µ ∈ R, τ > 0 and σ > 0 (see later for a different parametrization). This model is the simplest mean-reverting SDE, and has been widely used e.g. in neuronal modeling, biology, physics, engineering and finance, see e.g. Ditlevsen et al. (2005). Consider the following SDME model dXti

 =

 Xti i − + µ + µ dt + σdWti ; τ

X0i = xi0 = 0,

i = 1, ..., M

(3.18)

62

Stochastic Differential Mixed-Effects Models

and assume µi ∼ N (0, σµ2 ). Here bi = µi and we want to estimate θ = (µ, τ, σ) and Ψ = σµ2 given a set of observations x from model (3.18). The conditional mean and variance of the Xti process are E(Xti |X0i = x0 , µ, τ, σ, µi ) V ar(Xti |X0i = x0 , µ, τ, σ, µi )

= x0 e−t/τ + (µ + µi )τ (1 − e−t/τ ) σ2 τ = (1 − e−2t/τ ) 2

and the transition density is normal and given by pX (xij , ∆ij |xij−1 , µ, τ, σ, µi )

 
−1/2 2 (−2∆ij /τ ) = πσ τ 1 − e
2  i i  xij − xij−1 e−∆j /τ − (µ + µi )τ (1 − e−∆j /τ ) . × exp − i σ 2 τ (1 − e−2∆j /τ )

Thus, the likelihood of (θ, Ψ) is given by

L(θ, Ψ)

=

ni M  Y Y i (πσ 2 τ )−ni /2 (1 − e−2∆j /τ )−1/2 (2πσµ2 )−1/2 i=1

j=1


2  i i X ni  xij − xij−1 e−∆j /τ − (µ + µi )τ (1 − e−∆j /τ ) − exp i σ 2 τ (1 − e−2∆j /τ ) R j=1   (µi )2 dµi . 2σµ2

Z × −

(3.19)

We have no closed-form solution to the integral in (3.19), so exact estimators of θ and Ψ are unavailable. We first consider a numerical integration approach, and the resulting estimators are ˜ Ψ) ˜ = arg minθ,Ψ (− log L(θ, Ψ)). As a second attempt, we ignore the fact that denoted with (θ, the exact transition density expression is already available, and we compute the approximated estimator (θ(K) , Ψ(K) ) by approximating in closed-form the transition density of model (3.18) with K = 2. The estimation results, obtained on 200 artificial data sets generated by (3.18) using the Euler-Maruyama scheme with integration stepsize of 0.01 (Kloeden and Platen (1992)), are reported in Table 3.3 and Table 3.4 for the first and the second estimation approach respectively. For both the strategies we fixed ni = n for all i and T = 100. From Tables 3.3 and 3.4 we notice that the true parameter values are correctly identified using both the likelihood (3.19) and the corresponding order K = 2 approximation, though in the second approach we notice that n should be larger than 10 in order to get satisfactory results. Surface shapes of the log-likelihood profiles (for fixed (σ, σµ2 ) = (1, 1)) are reported in Figure 3.2, and compare the numerical evaluation of the logarithm of expression (3.19) with the corresponding order K = 2 expansion, both conditioned on observations generated from model (3.18) with (µ, τ, σ, σµ2 ) = (1, 10, 1, 1). The comparison is satisfactory and, as suggested above, values of n larger than 10 produce better results.

Many readers will be more familiar with a different parametrization of the Ornstein-Uhlenbeck

3.6 Conclusions

63

process, i.e. dXt = −β(Xt − α)dt + σdWt ;

X0 = x0 = 0

with (α, β, σ) ∈ R × R+ × R+ . The relations between µ, τ , α and β are obviously given by τ = 1/β and µ = αβ. If we consider the following SDME model dXti = −β(Xti − α − αi )dt + σdWti ;

X0 = x0 = 0,

i = 1, ..., M

with αi ∼ N (0, σµ2 /β 2 ), we have µi = αi β. Thus, it is straightforward to obtain the coefficients of the transition density expansion with respect to this parametrization, by substituting τ = 1/β, µ = αβ and µi = αi β into the expressions given in section 3.7.

Example 3: Ornstein-Uhlenbeck process with two random effects Reconsider the Ornstein-Uhlenbeck model with both µ and τ perturbed by random effects µi and τ i , respectively. The following SDME model results dXti

 =

 Xti i − + µ + µ dt + σdWti ; τ + τi

X0i = xi0 = 0,

i = 1, ..., M

(3.20)

where µi ∼ N (0, σµ2 ) and τ i has exponential pdf with parameter λ > 0. The latter distribution is 0 chosen to ensure that τ + τ i > 0. Assume µi and τ i independent for any i, i0 = 1, ..., M . Here bi = (µi , τ i ) and we want to estimate θ = (µ, τ, σ) and Ψ = (σµ2 , λ) given a set of observations x from model (3.20). Now we only consider the estimation approach based on the transition density expansion, i.e. we optimize (3.10), where Z B

(K)

pX (xi |bi , θ)pB (bi |Ψ)dbi =

and p(µi |σµ2 ) =

Z

+∞

−∞

Z

+∞

0

exp(−(µi )2 /(2σµ2 )) √ , σµ 2π

(K)

pX (xi |µi , τ i , θ)p(µi |σµ2 )p(τ i |λ)dµi dτ i ,

p(τ i |λ) = λ exp(−λτ i ).

The estimation results with K = 2, obtained on 200 artificial data sets generated by (3.20) using the Euler-Maruyama scheme with stepsize 0.01, are reported in Table 3.5. We fixed ni = n for all i and T = 100. Also in this example estimates are satisfactory, especially for increasing n values; only the true λ value is not well identified in the case (µ, τ, σ, σµ2 , λ) = (2, 12, 3, 0.5, 6). That is natural considering the large variance of the exponential distribution. However, it has to be noticed that, for ease of computations in the bootstrap procedure, a coarse grid has been chosen for the numerical integration of the likelihood (see section 3.5), so it is likely that better results can be achieved using a finer grid.

3.6

Conclusions

In the present chapter an approximated maximum likelihood estimator for the parameters of stochastic differential mixed-effects models has been proposed. SDE models incorporating random

64

Stochastic Differential Mixed-Effects Models

effects have been considered in few recent works (Overgaard et al. (2005); Tornøe et al. (2005); Ditlevsen and De Gaetano (2005a)) focused on models with constant diffusion and normal or lognormal distributed random effects. The proposed estimation method can be applied to models having non-constant diffusion term, with random effects following any continuous distribution and can be extended to multidimensional SDMEs. The method is based on the construction of a sequence of approximations L(K) to the true likelihood function L, which is obtained by expanding the process transition densities in closed-form to order K, thus obtaining an expression which can be rapidly evaluated. For SDME models more complex than the ones here considered, the likelihood approximation can be obtained by taking advantage of any software with symbolic calculus capabilities. Simulation results for the considered models show that the estimates obtained by minimizing − log L(K) , with K = 1 or 2, are close to the true parameter values, and this result can be achieved using moderate values of M (the number of experimental units, e.g. the number of subjects) and n (the number of observation for a given experimental unit). This is relevant for applications in situations where large data sets are unavailable, e.g. in biomedical applications, where MixedEffects theory is broadly applied. The method suffers some limitations, e.g. it may be difficult (though theoretically possible, see Aït-Sahalia (2001)) to obtain the transition density expansion for some multidimensional SDME systems with irreducible or non-commutative noise (Kloeden and Platen (1992)). Moreover, it may be difficult to numerically evaluate the integral in (3.3) when the dimension of B increases, and efficient numerical algorithms are needed. Finally, the models used e.g. in biomedical applications are often more complicated than the simple examples illustrated here, and it is still needed to see the applicability of the method in more realistic settings. In conclusion, we propose a parameter estimation method for SDE models incorporating random effects, which at least for the models considered here is reliable and effective and can be easily applied using commonly available computational resources. We believe that such a class of models will undergo increasing popularity, since it combines the nice features of the Mixed-Effects theory (total variation is split in within-subject and between-subject variation) with the possibility of considering random variability into the within-subject process dynamics, thus providing a very flexible modeling approach.

3.7

Appendix

Here we report the explicit expressions for the coefficients of the log-density expansion for both the Geometric Brownian Motion and the Ornstein-Uhlenbeck SDME models.

Geometric Brownian Motion: order K = 1 density expansion coefficients For model (3.17) we have: Yt = γ(Xt ) = then µY (Yt ) =

log(Xt ) , σ

σ β + βi − σ 2

3.7 Appendix

65

i and for given values yji and yj−1 of the Y process, we have

σ β + βi − = − σ 2   2 1 σ2 (1) i CY (yji |yj−1 ) = − 2 β + βi − 2σ 2

(0) i CY (yji |yj−1 )

(k)

i CY (yji |yj−1 )



(yji

=

i yj−1 )



  log(xij ) − log(xij−1 ) σ2 i β+β − = σ2 2

k≥2

0,

which yields the exact transition density (1) pX (xij , ∆ij |xij−1 )

=

 log(xij ) − log(xij−1 ) − (β + β i − q exp − 2σ 2 ∆ij xij 2πσ 2 ∆ij 1


σ2 i 2 2 )∆j

= pX (xij , ∆ij |xij−1 ).

Ornstein-Uhlenbeck process with one random effect: order K = 2 density expansion coefficients For model (3.18) we have: Yt = γ(Xt ) = Xt /σ then µY (Yt ) = −Yt /τ + ρ, i where ρ = (µ + µi )/σ, and for given values yji and yj−1 of the Y process, we have

 i yji + yj−1 = − ρ− 2τ i 2 i i i i 3τ − (yj ) − yj yj−1 − (yj−1 )2 + 3ρτ (yji + yj−1 ) − 3ρ2 τ 2 (1) i ) = CY (yji |yj−1 6τ 2 1 (2) i CY (yji |yj−1 ) = − 2 6τ

(0) i CY (yji |yj−1 )

(yji



i yj−1 )

and (2)

pX (xij , ∆ij |xij−1 )

=

+

 (xij − xij−1 )2 exp − + C˜ (0) (xij |xij−1 ) + C˜ (1) (xij |xij−1 )∆ij 2 ∆i 2σ i 2 j 2πσ ∆j  (∆ij )2 (2) i i C˜ (xj |xj−1 ) 2 1

q

i

i

(k) x x where C˜ (k) (xij |xij−1 ) = CY ( σj | j−1 σ ),

k = 0, 1, 2.

Ornstein-Uhlenbeck process with two random effects: order K = 2 density expansion coefficients

66

Stochastic Differential Mixed-Effects Models

For model (3.20) we have: Yt = γ(Xt ) = Xt /σ then µY (Yt ) = −Yt /(τ + τ i ) + ρ, i where ρ = (µ + µi )/σ, and for given values yji and yj−1 of the Y process, we have

  i yji + yj−1 i (yji − yj−1 ) ρ− 2(τ + τ i ) i i i 3(τ + τ i ) − (yji )2 − yji yj−1 − (yj−1 )2 + 3ρ(τ + τ i )(yji + yj−1 ) − 3ρ2 (τ + τ i )2 (1) i CY (yji |yj−1 ) = 6(τ + τ i )2 1 (2) i CY (yji |yj−1 ) = − 6(τ + τ i )2 (0)

i CY (yji |yj−1 )

=

and (2)

pX (xij , ∆ij |xij−1 )

   (xij − xij−1 ) (xij − xij−1 )2 (xij + xij−1 ) 1 q + exp − ρ − σ 2σ(τ + τ i ) 2σ 2 ∆ij 2πσ 2 ∆ij  (∆ij )2 (2) i i C˜ (xj |xj−1 ) + C˜ (1) (xij |xij−1 )∆ij + 2 =

i

i

(k) x x where C˜ (k) (xij |xij−1 ) = CY ( σj | j−1 σ ),

k = 1, 2.

3.7 Appendix

Parameter values 2 β σ2 σβ

ˆ β

-0.2

0.2

0.02

Mean [95% CI] Skewness Kurtosis

-0.203 [-0.291, -0.112] 0.065 2.780

-0.2

0.2

0.02

Mean [95% CI] Skewness Kurtosis

-0.198 [-0.245, -0.152] 0.070 2.890

-0.02

0.02

0.02

Mean [95% CI] Skewness Kurtosis

-0.023 [-0.103, 0.061] 0.093 2.688

-0.02

0.02

0.02

Mean [95% CI] Skewness Kurtosis

-0.018 [-0.062, 0.022] -0.051 2.909

67

σ ˆ2 M = 10, n = 50 0.201 [0.173, 0.222] -0.301 3.540 M = 50, n = 10 0.199 [0.171, 0.226] -0.025 2.630 M = 10, n = 50 0.020 [0.017, 0.022] -0.301 3.540 M = 50, n = 10 0.020 [0.017, 0.023] -0.025 2.630

2 σ ˆβ 0.018 [0.005, 0.038] 0.590 3.048 0.019 [0.012, 0.029] 0.394 2.717 0.018 [0.006, 0.036] 0.600 3.057 0.019 [0.012, 0.028] 0.302 2.675

Table 3.1:

Brownian Motion with drift: exact maximum likelihood estimates (and 95% empirical confidence intervals) from simulations of model (3.11).

Parameter values 2 σ2 σβ

β (1)

β

Table 3.2:

(σ (1) )2 M = 10, n = 50 0.200 [0.176, 0.221] 0.161 2.295 M = 50, n = 10 0.199 [0.171, 0.226] -0.025 2.630 M = 10, n = 50

(σ

(1) 2 ) β

-0.2

0.2

0.02

Mean [95% CI] Skewness Kurtosis

-0.135 [-0.197, -0.083] -0.069 2.761

0.008 [0.001, 0.019] 0.887 4.329

-0.2

0.2

0.02

Mean [95% CI] Skewness Kurtosis

-0.198 [-0.247, -0.158] -0.209 2.817

-0.02

0.02

0.02

Mean [95% CI] Skewness Kurtosis

-0.038 [-0.101, −10−4 ] -0.564 2.580

0.020 [0.017, 0.022] 0.072 3.772 M = 50, n = 10

0.014 [0.005, 0.026] 0.482 2.958

-0.02

0.02

0.02

Mean [95% CI] Skewness Kurtosis

-0.020 [-0.061, −10−4 ] -0.715 2.938

0.020 [0.017, 0.023] -0.023 2.632

0.019 [0.012, 0.028] 0.164 2.539

0.019 [0.011, 0.028] 0.066 2.488

Geometric Brownian Motion: maximum likelihood estimates (and 95% empirical confidence intervals), from simulations of model (3.17), solving the integral numerically.

68

µ

Stochastic Differential Mixed-Effects Models

Parameter values 2 τ σ σµ

µ ˜

τ ˜

σ ˜

M = 10, n = 50 10.084 [8.085, 12.082] 0.990 [0.919, 1.047] 0.240 -0.268 3.025 3.694 M = 50, n = 10

2 σ ˜µ

1

10

1

1

Mean [95% CI] Skewness Kurtosis

0.980 [0.380, 1.576] 0.104 2.594

1

10

1

1

Mean [95% CI] Skewness Kurtosis

1.019 [0.693, 1.317] -0.117 2.802

9.943 [8.852, 10.949] 0.947 [0.875, 1.022] -0.105 -0.047 2.760 2.795 M = 10, n = 50

0.991 [0.553, 1.471] 0.395 3.423

2

12

0.1

0.25

Mean [95% CI] Skewness Kurtosis

2.021 [1.783, 2.213] -0.385 4.896

11.996 [11.840, 12.168] 0.099 [0.091, 0.104] -0.213 -0.395 3.534 3.665 M = 50, n = 10

0.240 [0.076, 0.444] 0.698 4.456

2

12

0.1

0.25

Mean [95% CI] Skewness Kurtosis

2.018 [1.869, 2.212] -0.536 3.570

11.995 [11.917, 12.068] -0.011 2.205

0.251 [0.154, 0.370] 0.268 2.871

0.094 [0.088, 1.100] -0.065 2.902

0.915 [0.304, 1.935] 0.982 4.257

Table 3.3:

Ornstein-Uhlenbeck process: approximated maximum likelihood estimates (and 95% empirical confidence intervals) from simulations of model (3.18), using the exact transition density.

Parameter values µ

τ

σ

2 σµ

µ(2)

τ (2)

σ (2)

M = 10, n = 50 10.182 [8.196, 12.174] 1.000 [0.928, 1.057] 0.250 -0.268 3.033 3.680 M = 50, n = 10

(2) (σµ )2

1

10

1

1

Mean [95% CI] Skewness Kurtosis

0.972 [0.377, 1.562] 0.105 2.597

1

10

1

1

Mean [95% CI] Skewness Kurtosis

0.866 [0.585, 1.130] -0.101 2.848

11.820 [10.887, 12.727] 0.994 [0.920, 1.058] 0.052 -0.160 2.736 2.742 M = 50, n = 50

0.711 [0.403, 1.041] 0.309 3.026

1

10

1

1

Mean [95% CI] Skewness Kurtosis

1.006 [0.691, 1.283] -0.084 2.972

10.077 [9.221, 10.848] 1.000 [0.970, 1.028] 0.042 -0.112 2.789 2.832 M = 10, n = 50

0.962 [0.556, 1.362] 0.327 2.963

2

12

0.1

0.25

Mean [95% CI] Skewness Kurtosis

2.017 [1.761, 2.248] -0.180 3.729

12.058 [11.901, 12.230] 0.106 [0.099, 0.111] -0.240 -0.222 3.529 3.012 M = 50, n = 10

0.239 [0.075, 0.475] 0.618 3.039

2

12

0.1

0.25

Mean [95% CI] Skewness Kurtosis

1.813 [1.671, 1.945] -0.076 2.904

13.366 [13.294, 13.433] 0.375 [0.349, 0.400] -0.012 -0.031 2.179 2.838 M = 50, n = 50

0.197 [0.123, 0.284] 0.226 2.626

2

12

0.1

0.25

Mean [95% CI] Skewness Kurtosis

2.010 [1.854, 2.134] -0.573 3.272

12.057 [11.977, 12.133] 0.008 2.361

0.245 [0.151, 0.344] 0.042 2.396

Table 3.4:

0.106 [0.103, 0.108] -0.172 2.923

0.898 [0.298, 1.898] 0.972 4.214

Ornstein-Uhlenbeck process: approximated maximum likelihood estimates (and 95% empirical confidence intervals), from simulations of model (3.18), using an order K = 2 density expansion.

10

10

12

12

12

5

5

2

2

2

3

3

3

1

1

1

0.5

0.5

0.5

1

1

1

6

6

6

1

1

1

λ

Mean [95% CI] Skewness Kurtosis

Mean [95% CI] Skewness Kurtosis

Mean [95% CI] Skewness Kurtosis

Mean [95% CI] Skewness Kurtosis

Mean [95% CI] Skewness Kurtosis

Mean [95% CI] Skewness Kurtosis

2.061 [1.728, 2.373] 0.047 2.594

1.904 [1.633, 2.162] -0.108 2.519

2.027 [1.363, 2.820] 0.284 2.809

4.973 [4.627, 5.309] -0.084 3.012

2.926

4.427 [4.132, 4.735] −10−4

4.944 [4.336, 5.617] 0.196 2.754

µ(2)

9.841 [7.123, 14.127] 0.621 3.293

11.520 [8.555, 16.126] 1.049 4.995

11.330 [5.086, 25.388] 1.441 5.114

9.924 [9.317, 10.511] -0.090 3.362

11.655 [10.939, 12.683] 0.772 4.251

10.029 [8.448, 11.638] 0.234 2.999

τ (2)

σ (2)

2.999 [2.905, 3.088] -0.101 2.838

2.960 [2.748, 3.170] -0.162 2.454 M = 50, n = 50

3.003 [2.786, 3.182] -0.281 3.642 M = 50, n = 10

1.001 [0.971, 1.029] -0.112 2.803 M = 10, n = 50

1.336 [1.266, 1.398] -0.135 2.594 M = 50, n = 50

1.002 [0.931, 1.053] -0.366 3.454 M = 50, n = 10

M = 10, n = 50

0.686 [0.330, 1.058] 0.152 2.650

0.640 [0.317, 1.061] 0.388 2.982

0.583 [0.001, 1.510] 0.665 3.309

1.208 [0.736, 1.671] 0.119 2.859

1.131 [0.711, 1.552] 0.152 2.631

1.101 [0.261, 2.230] 0.427 2.798

(2) (σµ )2

9.510 [5.939, 14.336] 0.354 3.050

9.646 [4.682, 14.757] -0.204 3.848

8.509 [5.276 · 10−6 , 19.897] 0.071 2.195

1.261 [0.729, 1.975] 0.472 3.279

1.019 [0.001, 1.630] -0.596 3.318

1.165 [1.983 · 10−6 , 2.507] 0.168 2.732

λ(2)

Ornstein-Uhlenbeck process: approximated maximum likelihood estimates (and 95% empirical confidence intervals), from simulations of model (3.20), using an order K = 2 density expansion.

Table 3.5:

10

Parameter values 2 τ σ σµ

5

µ

3.7 Appendix 69

70

Stochastic Differential Mixed-Effects Models

σ2=0.02

σ2β=0.02

β

0.28

0.28

0.26

0.26

2187.6

σ2

75

82

9

2203.641

0.2

2202.496

8

0.18

972

−484.7

09 −486.

0.12

0.1 −0.3

95

22

0.12

0.1 −0.3

−0.28

−0.26

−0.24

−0.22

64

.20

00

0.14

0.14

−0.2

β

−0.18

−0.16

−0.14

−0.28

−0.26

−0.24

−0.22

−0.2

−0.12

−0.18

−0.16

−0.14

−0.12

β

(a) M = 10, n = 50.

(b) M = 10, n = 50. σ2β=0.02

σ2β=0.02 0.28

−1033 .1484 −102 2.373 7 −101 5.19 06 −100 8.00 75

0.28

0.24

0.22

σ2

σ2

0.22

0.26

0.2

0.2

65

75 8.

4774.0839

06 5. 19

47

.4

37

4770.4

50

37

546

02

2.

0.12

−1

0.12

47

00 −1

0.14

−1

−1002.6202

57 .7 5

00

0.16

01

0.14

.0

4775.8985

−997.2328

0.16

47

0.18

10 8

0.18

93 8

0.24

23

0.26

0.1 −0.3

−0.28

−0.26

−0.24

−0.22

−0.2

β

−0.18

−0.16

−0.14

−0.28

−0.26

−0.24

−0.22

−0.2

−0.12

−0.18

−0.16

(c) M = 50, n = 10.

σ2=0.02

β

β

0.28

0.26

0.26

0.24

0.24

0.22

0.22

0.2

0.18

631

6746

95

0.14

8

49

950

0.12

2

0.6

11

42.

−1730.3043

11494.

58.8

0.16

−17

−1726.2755

66.5

−17 5

0.16

114

0.44

8

0.18

18

0.2

1147 0.80

σ2

0.28

0.12

114

11486.71

7

0.1 −0.3

−0.28

−0.26

−0.24

−0.22

−0.2

−0.18

β

(e) M = 50, n = 26.

Figure 3.1:

−0.12

(d) M = 50, n = 10.

σ2=0.02

0.14

−0.14

β

−0.16

−0.14

−0.12

0.1 −0.3

654

0.1 −0.3

σ2

2183

2192

0.16

0.16

.028 9

.190

48

−501.72

−483.49

0.18

−496.51

σ2

0.2

2

088

0.22

−491.3

−510.8438

−496.5182 −500.4252

0.22

096

0.24

0.24

−0.28

−0.26

−0.24

−0.22

−0.2

−0.18

−0.16

−0.14

−0.12

β

(f) M = 50, n = 26.

Example 1 - Contour plots of the exact log-likelihood profiles (left panels) and the corresponding approximations by numerical integration (right panels) for fixed σβ2 = 0.02, given observations generated from model (3.12) (left) and (3.17) (right) with (β, σ 2 , σβ2 ) = (−0.2, 0.2, 0.02).

3.7 Appendix

71

2

2

σ = 1, σµ = 1 15

14

14

13

13

12

12

11

11

10

τ

τ

σ = 1, σµ = 1 15

9

−85 8

10

74 4.53

9

−862.2707

55 −8

8

−863.2293

28 .17 1 08 5.8 85

−

7

6

5

−864.1879

7

6

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

5

2

0

0.2

0.4

0.6

0.8

1

µ

(a) M = 10, n = 50.

14

14

13

13

12

12

11

11

10

10

9

1.8

2

1.6

1.8

2

1.6

1.8

2

9

0942 −1276. 1856 −1279. 2769 −1282. .3682 −1285

8

8

−1

7

4 22

7

64

6

0

0.2

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−1213.1855

.97

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1

−1219.081 1.2

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µ

(c) M = 50, n = 10.

(d) M = 50, n = 10.

σ = 1, σ2µ = 1

σ = 1, σ2µ = 1

15

15

14

14

13

13

12

12

11

11

10

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τ

1.6

σ = 1, σ2µ = 1 15

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σ = 1, σ2µ = 1

9

9

7571

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−2590.

7

−2595.0672

8

4334

−2524.

7

−2529.402

−2599.3773 6

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(b) M = 10, n = 50.

15

5

1.2

µ

−2603.6875 0

0.2

0.4

0.6

0.8

1

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µ

(e) M = 50, n = 26.

Figure 3.2:

1.4

1.6

1.8

2

5

7

719 −2534.3

6

0

0.2

0.4

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(f) M = 50, n = 26.

Example 2 - Contour plots of the profiles of the numerically approximated log-likelihood (3.19) (left 2 = 1, given panels) and the corresponding order K = 2 approximations (right panels) for fixed σ = 1 and σµ 2 ) = (1, 10, 1, 1). observations generated from model (3.18) with (µ, τ, σ, σµ

Part II

PAPERS

75

Paper 1 A Mathematical Model of the Euglycemic Hyperinsulinemic Clamp Umberto Picchini†, Andrea De Gaetano†, Simona Panunzi†, Susanne Ditlevsen§ and Geltrude Mingrone‡

† CNR-IASI BioMatLab, Largo A. Gemelli 8 - 00168, Rome, Italy; § Department of Biostatistics, University of Copenhagen, Denmark; ‡ Istituto di Medicina Interna e Geriatria, Divisione di Malattie del Ricambio, Università Cattolica del Sacro Cuore, Policlinico Universitario “A. Gemelli”, Rome, Italy.

Publication details: published in Theoretical Biology and Medical Modelling, 2005 Nov. 3;2(1):44 [DOI: 10.1186/1742-4682-2-44]. This publication does not contain SDE modeling but only deterministic differential modeling. However this can be regarded as a useful introduction to a physiological issue that has been reconsidered for stochastic modelization in Paper 2 of this dissertation.

Abstract Background: The Euglycemic Hyperinsulinemic Clamp (EHC) is the most widely used experimental procedure for the determination of insulin sensitivity, and in its usual form the patient is followed under insulinization for two hours. In the present study, sixteen subjects with BMI between 18.5 and 63.6 kg/m2 were studied by long-duration (five hours) EHC. Results: From the results of this series and from similar reports in the literature it is clear that, in obese subjects, glucose uptake rates continue to increase if the clamp procedure is prolonged beyond the customary 2 hours. A mathematical model of the EHC, incorporating delays, was fitted to the recorded data, and the insulin resistance behavior of obese subjects was assessed analytically. Obese subjects had significantly less effective suppression of hepatic glucose output and higher pancreatic insulin secretion than lean subjects. Tissue insulin resistance appeared to be higher in the obese group,

76

but this difference did not reach statistical significance. Conclusion: The use of a mathematical model allows a greater amount of information to be recovered from clamp data, making it easier to understand the components of insulin resistance in obese vs. normal subjects.

3.8

Background

With the growing epidemiological importance of insulin resistance states such as obesity and Type 2 Diabetes Mellitus, T2DM, and with increasing clinical recognition of the impact of the so-called metabolic syndrome, the assessment of insulin sensitivity has become highly relevant to metabolic research. The experimental procedures currently employed to gather information on the degree of insulin resistance of a subject are the Oral Glucose Tolerance Test (OGTT), the Intra-Venous Glucose Tolerance Test (IVGTT), the Euglycemic Hyperinsulinemic Clamp (EHC), the Hyperglycemic Clamp, the insulin-induced hypoglycemia test (KIT T ), and less commonly used methods based on tracer administration (Ferrannini and Mari (1998), Wallace and Matthews (2002), Starke (1992)). Of these, the EHC is considered the tool of choice in the diabetological community, in spite of its labor-intensive execution, because it is usually considered that the results obtained can be interpreted simply (DeFronzo et al. (1979), Zierler (1999)). The favor with which the EHC is viewed in this context stems in part from the belief that while mathematical models of the glucose insulin system make untenable assumptions, the EHC approach is relatively assumption free, or model-independent. In general, insulin resistance expresses an imbalance between the amount of pancreatic insulin secreted in response to a glucose load and the levels of plasma glucose attained. In other words, in order to obtain the same plasma glucose concentration, higher levels of plasma insulin are necessary in insulin-resistant subjects than in normal controls (Felber et al., 1993). The clamp, as usually employed, yields easy-to-compute indices, which are commonly used as measures of insulin resistance. The M value (DeFronzo et al., 1979) is defined as the average glucose infusion rate over a period of 80-120 minutes from the start of the insulin infusion. The M/I ratio is the ratio of the M value to the average plasma insulin concentration during the same period. If a two-step clamp is performed (though see negative comments Zierler (1999)) the ∆M/∆I ratio is defined as the increment of M produced by raising the insulin infusion rate over the corresponding increment of I. The use of these indices, however, makes two fundamental assumptions: first, that at the end of 120’ of insulin infusion the experimental subject is at steady state with regard to glucose uptake rate; and second, that the glucose uptake rate increases linearly with increasing insulinemia, either throughout the insulin concentration range (when using the M/I index for characterizing the subject’s response) or between successive insulin concentrations reached in the two-step clamp (when using the ∆M/∆I index). These assumptions are, however, only a first approximation to the real state of things. On the one hand, it has already been shown that if a clamp experiment is continued beyond the customary 2 hours “[...] glucose utilization increases progressively through(out) five hours of moderate hyperinsulinemia.” (Doberne et al., 1981). On the other hand (Thiebaud et al., 1982), carefully measured average glucose uptake rates at two hours are nonlinearly related to increasing levels of plasma insulin, and from the reported data, glucose uptake may approach a maximal value asymptotically as insulinemia increases. In spite of these observations, the vast majority of experimental diabetologists (Ferrannini et al.

3.9 Methods

BM I[kg/m2 ] BSA[m2 ] Gf ast [mM ] If ast [pM ] Imax [pM ]

Lean subjects (nlean = 7) 20.0 [18.5, 22.7] 1.55 [1.49, 1.73] 3.67 [3.4, 5.4] 27.8 [13.9, 49.4] 482.14 [464.5, 526.9]

Overweight and Obese subjects (no = 8) 37.0 [27.8, 63.6] 2.1 [1.83, 2.38] 5.2 [4.61, 5.9] 123.7 [79.2, 152.9] 606.3 [497.3, 683.2]

77

p 0.001 0.001 0.024 0.001 0.004

Table 3.6:

Anthropometric and metabolic characteristics for lean (BM I = 25) and overweight or obese (BM I > 25) subjects. Values are median [min, max]. All comparisons were performed by the Mann-Whitney U-test. BSA is the Body Surface Area [m2 ] calculated via the DuBois formula (BSA = 0.20247 · height0.725 [m] · weight0.425 [kg]).

(1985), Zierler (1999), Ferrannini et al. (1997)) consider the EHC the procedure of choice and many studies have already been conducted using it. It would be interesting to be able to reinterpret this vast mass of observations using a more explicitly quantitative approach. The goal of the present work is to formulate a model of the EHC and fit it to EHC data recorded from human subjects. The structure of the model we have developed allows us to discuss the mechanisms whereby a sufficiently long insulin infusion might be able to increase glucose uptake progressively, and to explore the possible implications of the commonly observed insulin resistance pattern in obese subjects.

3.9 3.9.1

Methods Subjects

Sixteen subjects were enrolled in the study, 8 normal volunteers and 8 patients from the Obesity Outpatient Clinic of the Department of Internal Medicine at the Catholic University School of Medicine. For one normal subject the recorded glycemia values were accidentally lost and this subject was therefore discarded from the following mathematical analysis. The subjects had widely differing BMIs (from 18.5 to 63.6 kg/m2 ). All subjects were clinically euthyroid, had no evidence of diabetes mellitus, hyperlipidemia, or renal, cardiac or hepatic dysfunction and were undergoing no drug treatments that could have affected carbohydrate or insulin metabolism. The subjects consumed a weight maintaining diet consisting of at least 250 g of carbohydrate per day for 1 week before the study. Table 3.6 reports the main anthropometric and metabolic characteristics of the subjects. The study protocol followed the guidelines of the Medical Ethics Committee of the Catholic University of Rome Medical School; written informed consent was obtained from all subjects.

3.9.2

Experimental protocol

Each subject was studied in the postabsorptive state after a 12-14 h overnight fast. Subjects were admitted to the Department of Metabolic Diseases at the Catholic University School of Medicine in Rome the evening before the study. At 07.00 hours on the following morning, the infusion catheter was inserted into an antecubital vein; the sampling catheter was introduced in the contralateral dorsal hand vein and this hand was kept in a heated box (60 ◦ C) in order to obtain arterialized blood. A basal blood sample was obtained in which insulin and glucose levels were measured. At 08.00 hours, after a 12-14 h overnight fast, the Euglycemic Hyperinsulinemic glucose Clamp was

78

Variables t [min] G(t) [mM ] I(t) [pM ] Tgx (t) [mmol/min/kgBW ] Tix (t) [pmol/min/kgBW ] Tgh (t) [mmol/min/kgBW ] Table 3.7:

time from insulin infusion start; plasma glucose concentration at time t; serum insulin concentration at time t; glucose infusion rate at time t; insulin infusion rate at time t; net Hepatic Glucose Output (HGO) at time t Definitions of the state variables

performed according to DeFronzo et al. (1979). A priming dose of short-acting human insulin was given during the initial 10 min in a logarithmically decreasing manner so that the plasma insulin was raised acutely to the desired level. During the five-hour clamp procedure, the glucose and insulin levels were monitored every 5 min and every 20 min respectively, and the rate of infusion of a 20% glucose solution was adjusted during the procedure following the published algorithm DeFronzo et al. (1979). Because serum potassium levels tend to fall during this procedure, KCl was given at a rate of 15-20 mEq/h to maintain the serum potassium between 3.5 and 4.5 mEq/l. Serum glucose was measured by the glucose oxidase method using a Beckman Glucose Analyzer II (Beckman Instruments, Fullerton, Calif., USA). Plasma insulin was measured by microparticle enzyme immunoassay (Abbott Imx, Pasadena, Calif., USA).

3.9.3

Modelling

In order to explain the oscillations of glycemia occurring in response to hyperinsulinization and to continuous glucose infusion at varying speeds, we hypothesized the following system:

dG(t) dt

(Tgx (t − τg ) + Tgh (t)) G(t) − − Txg Vg 0.1 + G(t)   Z +∞ − KxgI ω(s)I(t − s)ds G(t)

=

(3.21)

0

dI(t) dt Tgh (t)

(TiG G(t) + Tix (t)) − Kxi I(t) Vi   Z +∞ = Tghmax exp −λG(t) ω(s)I(t − s)ds

=

(3.22) (3.23)

0

where G(0) = Gb ,

I(t) = Ib

ω(s) = α2 se−αs ,

∀t ≤ 0,

Tgh (0) = Tghb = Tghmax exp(−λGb Ib )

Tgx (s) = 0 ∀s ∈ [−τg , 0]

and

Tix (0) = Tixb

Tgx (t) and Tix (t) are (input or forcing) state variables of which the values are known at each time; the state variables and the parameters are defined in Table 3.7 and Table 3.8. The model is diagrammatically represented in Figure 3.3. Equations (3.21) and (3.22) express the variations of plasma glucose and plasma insulin concentrations. Equation (3.23) represents the rate of net Hepatic Glucose Output, starting at maximal

3.9 Methods

79

Txg

Tgx (t-τ g )

G(t)

Tgh (t)

+∞

Vg

n

G(t) 0.1+G(t)

G(t)K xgI ∫ ω(s)I(t-s)ds 0

TiG G(t)

K xi I(t)

I(t) Tix (t)

Vi stimulation inhibition

Figure 3.3:

Schematic representation of the model (3.21)–(3.23).

HGO at zero glucose and zero insulin and decaying monotonically with increases in both glucose and effective insulin concentrations in the plasma. The variation of glucose concentration in its distribution space may be attributed to the external glucose infusion rate, liver glucose output and delayed-insulin-dependent as well as insulin-independent glucose tissue uptake. Infused glucose raises glycemia after a delay τg due to the time required to equilibrate the intravenously infused quantity throughout the distribution space. The net HGO is assumed to be equal to Tghb at the beginning of the experiment and to decrease toward zero as glycemia or insulinemia levels increase. Serum insulin, after a delay depending on its transport to the periphery and the subsequent activation of cellular membrane glucose transporters, affects glucose clearance through equation (3.21) and the glucose synthesis rate through equation (3.23). We hypothesize that ω(s) represents the density of the metabolic effect at time t for unit serum insulin concentration at time t − s (s ≤ t). We could choose ω(s) as a single function or as a linear combination of functions (with positive coefficients adding up to unity) from the family of Erlang-functions: ω (k) (s) =

αk sk−1 e−αs , (k − 1)!

α ∈ R+ , k ∈ N, s ∈ R+0

The first two functions of the family are ω (1) (s) = αe−αs ,

ω (2) (s) = α2 se−αs

We note that while ω (1) (s) is monotonically decreasing, ω (2) (s) increases to a maximum at s = 1/α, then decreases monotonically and asymptotically to zero. We choose the second Erlangfunction as our kernel because it is the simplest member of the family with a peak. This embodies the concept that, in order to produce its metabolic effect, insulin has to reach the tissues and activate intracellular enzymatic mechanisms (hence its maximal action on glucose metabolism is delayed) and that natural breakdown of insulin induces a progressive loss of effect of increased concentrations of the hormone as they become more distant in the past. A high á value determines

80

a concentrated kernel corresponding to a fast-rising, fast-decaying effect of insulin on peripheral tissues. We therefore set Z

+∞

Z ω(s)I(t − s)ds =

+∞

α2 s−αs I(t − s)ds

0

0

R +∞

and we define ρ = 0 s(α2 s−αs )ds = 2/α as the average time for the metabolic effect of insulin in changing glycemia. The insulin-independent glucose tissue uptake process is modelled as a Hill function rapidly increasing to its (asymptotic) maximum value Txg ; thus for glycemia values near 2mM the insulin-independent glucose tissue uptake is already close to its maximum. This formulation is intended to represent the aggregated apparent zero-order (fixed) glucose utilization mechanism at rest (mainly the brain and heart, see Olson and Pessin (1996) and Sacks (1969) p. 320), with the mathematical and physiological requirement that glucose uptake tends to zero as glucose concentration in plasma approaches zero. The variation of insulin concentration in its distribution space (equation (3.22)) may be thought of as due to the external insulin infusion, glucose dependent pancreatic insulin secretion and the apparently first-order insulin removal from plasma. We use steady-state conditions to decrease the number of free parameters to be estimated: at steady state, before the start of the clamp (G = Gb , I = Ib , Tgx = Tix = 0), we have Tghb = Tghmax exp(−λGb Ib ) Txg Gb 0 + Tghb − − KxgI Ib Gb ⇒ Txg = 0= Vg 0.1 + Gb 0=



 Tghb (0.1 + Gb ) − KxgI Ib Gb Vg Gb

TiG Gb + 0 Kxi Ib Vi − Kxi Ib ⇒ TiG = Vi Gb

Therefore the parameters Tghb , Txg , and TiG are completely determined by the values of the other parameters (and ρ is determined from α).

3.9.4

Statistical analysis

The system (3.21)–(3.23) has been numerically integrated by means of a fourth–order Runge– Kutta scheme with constant stepsize equal to 1 min. In order to distinguish among the n observations (and corresponding predictions) between glucose and insulin, the indices j for glucose and k for insulin are used as follows: j ∈ J, k ∈ K, J ∩ K = {∅}, J ∪ K = {1, ..., n}. We ˆ θ) ≡ G(t) ˆ ˆ θ) ≡ I(t) ˆ the (numerically integrated) solutions of equaindicate with G(t, and I(t, tions (3.21)–(3.23) for parameter θ at time t. The solutions have been fitted by Iteratively Re-Weighted Least Squares (IRWLS, see e.g. (Davidian and Giltinan, 1995, chapter 2)) separately on each subject’s glycemia and insulinemia time-points, estimating only the free parameters θ = (Gb , Ib , KxgI , Kxi , Tghmax , Vg , Vi , α, τg , λ) by minimizing the following loss function (y − yˆ)0 Ω(y − yˆ) where y is the n × 1 array containing both glycemias and insulinemias, observed at times 0 = t1 ≤ t2 ≤ · · · ≤ tn ; yˆ is the array of corresponding predictions obtained by numerical integration

3.10 Results

81

ˆ j ) ∀j ∈ J, yˆ(tk ) = yˆk = I(t ˆ k ) ∀k ∈ K; Ω is an of the system (3.21)-(3.23), yˆ(tj ) = yˆj = G(t 0 n × n diagonal matrix of weights. Here Z denotes the transpose of the matrix Z. The statistical weight associated with a generic glucose concentration point yj has been defined as 1/(ˆ yj CVG )2 , where CVG = 0.015 is the coefficient of variation for glucose (Bergman et al., 1979). Similarly the statistical weight associated with a generic insulin concentration point has been defined as 1/(ˆ yk CVI )2 , where CVI = 0.07 is the coefficient of variation for insulin (Bergman et al., 1979). In order to highlight possible physiological differences among subjects depending on their BM I, two groups were defined: a group consisting of lean subjects (BM I ≤ 25) and a group consisting of overweight or obese subjects (BMI > 25). Comparisons of anthropometric characteristics, metabolic characteristics and model parameter values between these groups were performed by the Mann-Whitney U-test owing to the small number of subjects in each group. Comparisons within groups were performed by the Wilcoxon test for matched pairs.

3.10

Results

Table 3.6 shows anthropometric characteristics (BM I, BSA), measured plasma glucose and insulin concentrations (Gf ast , If ast ) in the two groups immediately before the clamp, and the average levels of insulin after 80’ of clamp insulinization (Imax ). All differences in the characteristics were highly significant, with the median values in the obese/overweight group markedly higher than those in the lean group. Even though there was a significant difference in fasting glycemia between the groups, average levels remained within the norm. However, fasting insulinemia was more than four-fold higher in the obese/overweight group, consistent with what is usually observed in this patient population. For each parameter fitted and determined, the median, minimum and maximum from the sample of values obtained are reported in Table 3.9. The predicted basal glycemia and insulinemia values (Gb , Ib ) were close to the observed fasting values and were significantly different between groups (respectively p = 0.001 and p = 0.002). Lean subjects have a greater ability (about 3-fold higher) to reduce hepatic glucose output when glycemia and insulinemia increase (expressed by the parameter λ, p = 0.037). The parameter TiG (glucose-dependent pancreatic secretion of insulin) is also significantly different between groups (p = 0.011) and the insulin synthesis rate in obese/overweight subjects is about three-fold higher than in lean subjects. The delay coefficient τg is of the order of 3 to 5 minutes, which seems a reasonable time for glucose infused through an arm vein to be distributed throughout the body, equilibrate, and be detected by sampling through the arterialized contralateral arm vein. In Table 3.10 the measured values of the M/I index over the time periods 80’-120’ and 260’-300’ are shown for normal and obese/overweight subjects: as expected, the rate of glucose uptake per unit plasma insulin concentration is significantly higher in lean subjects in both the 80’-120’ (p = 0.001) and the 260’-300’ periods (p = 0.015). However, whereas in lean subjects the M/I value remains stable between the two periods (p = 0.6), in the obese/overweight group it increases significantly (p = 0.02). Figure 3.4 show the time course of observed and predicted glycemia, observed and predicted insulinemia and glucose infusion rate for four experimental subjects (two lean and two obese).

82

Composite graph, subject 6: Glucose, Insulin and Tgx 9

8

8

Glucose [mM], Insulin [pM] / 300, Tgx [mmol/min/kgBW] * 100

Glucose [mM], Insulin [pM] / 300, Tgx [mmol/min/kgBW] * 100

Composite graph, subject 2: Glucose, Insulin and Tgx 9

7

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150 Time (min)

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(b) Subject 6

Composite graph, subject 9: Glucose, Insulin and T

Composite graph, subject 10: Glucose, Insulin and T

gx

gx

9

9

8

8

Glucose [mM], Insulin [pM] / 300, Tgx [mmol/min/kgBW] * 100

Glucose [mM], Insulin [pM] / 300, Tgx [mmol/min/kgBW] * 100

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Figure 3.4:

200

250

300

0

50

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150 Time (min)

200

(d) Subject 10

Composite plot for subject 2 (BMI = 35.9), subject 6 (BMI = 19.33), subject 9 (BMI = 63.6) and subject 10 (BMI = 18.6). Observed (black ♦) and predicted (· · · ) glycemia; observed (◦) and predicted (dashed line) insulinemia; glucose infusion rate (solid line). For ease of comparison, the insulin concentrations and the glucose infusion rates are divided by factors of 300 and 0.01 respectively.

3.11 Discussion

3.11

83

Discussion

It was shown in the early ’80s (Doberne et al., 1981) that a significant increase of glucose tissue uptake during the euglycemic hyperinsulinemic clamp could be obtained in obese subjects by waiting for up to 4-6 hours. This basic observation, confirmed by the series of obese subjects studied in the present work, challenges the assumption that steady state is attained after 2 hours of the clamp, at least in one patient subpopulation of great metabolic interest. Nolan et al. (1997), while performing an isoglycemic hyperinsulinemic clamp, also demonstrated a marked delay in activation of whole-body glucose disposal rate, arteriovenous glucose difference and leg glucose uptake in seven subjects with Type 2 Diabetes Mellitus and in seven obese non-diabetic subjects, as compared to healthy controls. The concept of insulin resistance as a decreased effect of the hormone on whole body glucose uptake can be made more specific: on the one hand we might wish to measure the speed with which a given level of metabolic response is attained; on the other, we might wish to quantify the maximal response attainable by a suitably raised insulin plasma concentration. It is clear now that when using the classical two-hour clamp, subpopulations of subjects respond within different time frames. Concentrating on the level of response at 2 hours would label subjects with a residual metabolic capacity as insulin-resistant: this may or may not be appropriate depending on the mode of insulin resistance that the physiologist is interested in, whether the speed or the capacity of response. The case of the obese subject represents this ambiguity very well: if by insulin resistance we mean the result of the EHC at 2 hours, that is to say a decreased effect of insulin on whole body glucose uptake under hyperinsulinization with respect to a specific and short time frame, then obese subjects can be adequately diagnosed by the clamp as being generally insulin resistant. If, on the other hand, we abandon the time frame requirement and address the maximal ability to respond to the hormone, then the standard clamp procedure is not adequate since it fails to allow slowly-responding subjects to develop a complete response. A way out of this ambiguity for diagnostic purposes could be to use the parameters of a mathematical model of the metabolic response during the clamp. Hopefully, this model would be able to quantify both the maximal response obtainable by the subject and the rate at which this response is generated. Hence the diabetologist would be offered separate, independent and complementary items of information on which to base the diagnosis. Given the above considerations, the approach followed in the present work was therefore to construct a deterministic mathematical model of the time course of glucose uptake rate during a clamp experiment. A series of studies (Yang et al. (1989), Bergman et al. (1990), Ader et al. (1992)) demonstrated that insulin-stimulated glucose uptake correlates with the appearance of insulin in lymph fluid, a marker for interstitial insulin, rather than with the appearance of insulin in the circulatory stream. Whether trans-endothelial passage of insulin from the circulation to the interstitial space is the sole or the main mechanism for the delay is debatable, even though it may be rate-limiting in the activation of glucose uptake, since the pancreatic response to glucose should be fast and since, once insulin is in the interstitial space, further endocellular steps are very rapid. In any case, out of the many models we tried in order to explain the observed insulin and glucose concentration time courses, the model that best explains the data includes a delay in the action of plasma insulin in correcting glycemia. Of the many alternative explicit representations of such delay that could have been used, one of the simplest was chosen, a Erlang-function kernel, to

84

simplify the model’s mathematical treatment. It has been shown (Nolan et al., 1997) that Hepatic Glucose Output (HGO) suppression after step insulinization is not immediate, HGO decreasing towards 0 in an approximately exponential manner from its pre-insulinization level. In the present work, HGO was not independently measured by tracer techniques. The model proposed here assumes that the variable representing HGO (identified with the symbol Tgh ) falls progressively to a new equilibrium value as delayed insulin increases progressively to its new equilibrium level after a step increase in plasma insulin. In this, our model agrees with Nolan’s observation. Further, in the model proposed in the present work, equilibrium Tgh falls exponentially (with parameter λ) as equilibrium insulin increases from baseline to full insulinization levels. The two parameters Tghmax and KxgI express respectively the maximum Hepatic Glucose Output and the sensitivity of glucose uptake to insulin concentration. Neither was significantly different between lean and obese subjects. However, Tghmax was higher and KxgI was lower in obese subjects, and both these changes would point to a decreased insulin sensitivity in this patient group. While the observed lack of significance may well be a consequence of the limited power of the present study, given the small number of subjects considered, the fact that these two parameters were not much changed in obese subjects while λ was significantly lower again indicates a relative slowness in mounting an appropriate response rather than a relative incapacity to mount a sustained response eventually. From the modelling point of view, the present study prompts two considerations. The first is that a clamp that is medically very successful (i.e. during which the physician manages to clamp glycemia effectively to within a narrow range) may be less informative about the actual subject’s compensation mechanisms than a clamp where imprecise correction of glycemia gives rise to oscillations. The second is that, especially for subjects such as the one reported in Figure 3.4(d), where sustained oscillations are produced, random perturbations of the system may give rise to accidental phase shifts. This makes it very hard or impossible to follow the oscillations unless for the model can accommodate random variations of metabolism. Future efforts in modelling the clamp will have to consider this feature.

3.12

Conclusion

In conclusion, the present paper describes a possible deterministic modelling of the EHC, which may prove useful for studying obese subjects who show delayed expression of their maximal increase of glucose uptake under insulinization. Considering the amplitude of response independently of the time factor, the whole body capacity of glucose uptake in obese subjects does not appear to be decreased with respect to lean subjects.

Tghmax [mmol/min/kgBW ] Tghb [mmol/min/kgBW ] Vg [L/kgBW ] Vi [L/kgBW ] α []] τg [min] λ [mM −1 pM −1 ] ρ []]

Parameters Gb [mM ] Ib [pM ] Txg [mM/min] KxgI [min−1 /pM ] Kxi [min−1 ] TiG [pM/min/mM ] Tixb [pmol/min/kgBW ]

Table 3.8:

Definitions of the parameters.

basal glycemia; basal insulinemia; maximal insulin-independent rate constant for glucose tissue uptake; insulin-dependent apparent first-order rate constant for glucose tissue uptake at insulinemia I; apparent first-order rate constant for insulin removal from plasma; apparent zero-order net insulin synthesis rate at unit glycemia (after liver first-pass effect); basal insulin infusion rate, which is given by the measured value of Tix at time zero according to DeFronzo and Ferrannini (1991); maximal Hepatic Glucose Output at zero glycemia, zero insulinemia; basal value of Tgh ; volume of distribution for glucose; volume of distribution for insulin; time constant for the insulin delay kernel ω(·); discrete (distributional) delay of the change in glycemia following glucose infusion; rate constant for Hepatic Glucose Output decrease with increase of glycemia and insulinemia; average delay of insulin effect

3.12 Conclusion 85

86

Estimated parameters Gb [mM ] Ib [pM ] KxgI [min−1 /pM ] Kxi [min−1 ] Tghmax [mmol/min/kgBW ] Vg [L/kgBW ] Vi [L/kgBW ] α []] τg [min] λ [mM −1 pM −1 ] Determined Parameters Tghb [mmol/min/kgBW ] Txg [mM/min] TiG [pM/min/mM ] ρ []]

Lean subjects (nlean = 7)

Overweight and obese subjects (no = 8)

p

0.36 0.203 0.011 0.908

0.001 0.002 0.132 0.203 0.105 0.643 0.487 0.908 0.917 0.037 [0.009, 0.117] [0.012, 0.397] [0.128, 0.668] [42.1, 267.7]

5.11 [4.52, 5.97] 121.05 [61.55, 256.41] 6.34 [0, 13.3] · 10−6 0.029 [0.021, 0.045] 0.128 [0.026, 0.274] 0.47 [0.25, 0.67] 0.39 [0.21, 0.65] 0.024 [0.008, 0.048] 5.14 [0.50, 9.00] 3.1 [0.2, 4] · 10−3 0.019 0.046 0.267 83.6

3.67 [2.80, 4.36] 17.91 [8.59, 63.41] 9.94 [7.1, 21.2] · 10−6 0.039 [0.022, 0.057] 0.069 [0.05, 0.12] 0.49 [0.33, 0.90] 0.4 [0.36, 0.78] 0.017 [0.015, 0.082] 3.00 [1.00, 11.50] 8.9 [1.2, 21.3] · 10−3 0.042 [0.028, 0.052] 0.085 [0.057, 0.126] 0.096 [0.031, 0.29] 115.4 [24.3, 136.2]

Table 3.9: Estimated and determined parameter values for lean (BM I ≤ 25) and overweight or obese (BM I > 25) subjects. Comparisons were performed by the Mann-Whitney U-test. Values are expressed as median [min, max].

3.12 Conclusion

M/I (800 − 1200 ) M/I (2600 − 3000 ) p (Wilcoxon)

Lean subjects (nlean = 7) 9.75 [6.97, 11.42] · 10−5 8.9 [4.9, 13.2] · 10−5 0.6

Overweight and obese subjects (no = 8) 2.66 [1.57, 5.2] · 10−5 3.86 [2.54, 7.44] · 10−5 0.02

87

p (M-W U) 0.001 0.015

Table 3.10: M/I index values for lean and overweight or obese subjects measured over the 80’-120’ and on the 260’-300’ time periods. Comparisons between groups were performed by the Mann-Whitney U-test. Comparisons within groups were performed via the Wilcoxon test for matched pairs. Values are expressed as median [min, max].

3.12 Conclusion

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Paper 2 Modeling the Euglycemic Hyperinsulinemic Clamp by Stochastic Differential Equations Umberto Picchini†, Susanne Ditlevsen§ and Andrea De Gaetano†

† CNR-IASI BioMatLab, Largo A. Gemelli 8 - 00168, Rome, Italy; § Department of Biostatistics, University of Copenhagen, Denmark.

Publication details: Journal of Mathematical Biology, 53(5), 771-796 Nov. 2006 [DOI: 10.1007/s00285-006-0032-z]. This publication defines a stochastic differential model of the same experiment considered in Paper 1 of this dissertation: it employs the estimation method for the parameters of partially observed diffusions, as described in section 2.2, and diagnostic measures as suggested in section 2.4 of this dissertation.

Abstract The Euglycemic Hyperinsulinemic Clamp (EHC) is the most widely used experimental procedure for the determination of insulin sensitivity. In the present study, sixteen subjects with BMI between 18.5 and 63.6 kg/m2 have been studied with a long-duration (five hours) EHC. In order to explain the oscillations of glycemia occurring in response to the hyperinsulinization and to the continuous glucose infusion at varying speeds, we first hypothesized a system of ordinary differential equations (ODEs), with limited success. We then extended the model and represented the experiment using a system of stochastic differential equations (SDEs). The latter allow for distinction between (i) random variation imputable to observation error and (ii) system noise (intrinsic variability of the metabolic system), due to a variety of influences which change over time. The stochastic model of the EHC was fitted to data and the system noise was estimated by means of a (simulated) maximum likelihood procedure, for a series of different hypothetical measurement error values. We showed that, for the whole range of reasonable measurement error values: (i) the system noise estimates are non-negligible; and (ii) these estimates are robust to changes in the likely value of

90

the measurement error. Explicit expression of system noise is physiologically relevant in this case, since glucose uptake rate is known to be affected by a host of additive influences, usually neglected when modelling metabolism. While in some of the studied subjects system noise appeared to only marginally affect the dynamics, in others the system appeared to be driven more by the erratic oscillations in tissue glucose transport rather than by the overall glucose-insulin control system. It is possible that the quantitative relevance of the unexpressed effects (system noise) should be considered in other physiological situations, represented so far only with deterministic models.

Keywords: mathematical models, dynamical systems, glucose, insulin, parameter estimation, Monte Carlo methods, simulated maximum likelihood.

3.13

Introduction

With the growing epidemiological importance of insulin resistance states (like obesity and Type 2 Diabetes Mellitus, T2DM) and with the increasing clinical recognition of the impact of the socalled metabolic syndrome, the assessment of insulin sensitivity has become a very relevant issue in metabolic research. The experimental procedures currently employed to gather information on the degree of insulin resistance of a subject are the Oral Glucose Tolerance Test (OGTT), the Intra-Venous Glucose Tolerance Test (IVGTT), the Euglycemic Hyperinsulinemic Clamp (EHC), the Hyperglycemic Clamp, the insulin-induced hypoglycemia test (KIT T ), and less commonly used methods based on tracer administration (Ferrannini and Mari (1998), Starke (1992), Wallace and Matthews (2002)). Of these, the EHC is considered the tool of choice in the diabetological community, in spite of its labor-intensive execution, due to the simple interpretation which is usually attributed to the obtained results (DeFronzo et al. (1979), Zierler (1999)). The favor with which the EHC is viewed in this context stems in part from the belief that while mathematical models of the glucose insulin system make untenable assumptions, the EHC approach is relatively assumption-free, or model independent. In the present work we study the dynamic behavior of glycemia and insulinemia recorded from human subjects during a EHC procedure. We firstly hypothesize a system of ODEs explaining this dynamics (inspired by a previously published deterministic model of the EHC Picchini et al. (2005)), and obtain the corresponding parameter estimates by numerically fitting the model to observed data. This simply deterministic model, however, does not accommodate random variations of metabolism. In fact a deterministic model assumes that: (i) the mathematical process generating the observed glycemias is smooth (continuous and continuously differentiable) in the considered time-frame; and (ii) the variability of the actual measurements is due to observation error, which does not influence the course of the underlying process. An alternative, stochastic, approach would result from the hypothesis that the underlying mathematical process itself is not smooth, at least when considered at the practicable time resolution. The glucose metabolizing organs and tissues are in fact subject to a variety of internal and external influences, which change over time (e.g. blood flow, energy requirements, hormone levels, the cellular metabolism of the tissues themselves) and which may affect the instantaneous glycemias. This second approach would maintain that some degree of randomness is already present in the glucose disposition process

3.14 Material and methods

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itself, and that observational error is superimposed to it. A natural extension of the deterministic model is given by a stochastic differential equations (SDEs) model (see e.g. Kloeden and Platen (1992) and Øksendal (2000)). We therefore define an SDE model by adding a suitable system variability to the simple deterministic model. While SDE models are currently being employed in different applied fields (e.g. finance, engineering, physics), they are rarely used in biomedicine (except for specific fields like neuronal modelling, population growth models and recent contributions, e.g. Tornøe et al. (2004b) for another approach to SDE modeling of the EHC), even though it is generally recognized that biological data are fraught with many sources of intrinsic (system) error. This large amount of variability is often attributed to observation error exclusively, giving rise to inaccurate, but manageable, modelling representations of physiological phenomena. The present work has two main goals: on one hand it purports to determine whether, in a particular physiological situation, system error is identifiable and necessary to explain observations, above and beyond commonly accepted levels of measurement error. The second goal is to show, by means of a practically occurring experimental situation, that SDE models are physiologically relevant and that their parameters can be numerically estimated using commonly available resources.

3.14 3.14.1

Material and methods Subjects

Sixteen subjects were enrolled in the study, 8 normal volunteers and 8 patients from the Obesity Outpatient Clinic of the Department of Internal Medicine at the Catholic University School of Medicine. For one normal subject the recorded glycemia values were accidentally lost and this subject was therefore discarded from the following considerations. The subjects had widely differing BMIs (from 18.5 to 63.6 kg/m2 ). All subjects were clinically euthyroid, had no evidence of diabetes mellitus, hyperlipidemia, or renal, cardiac or hepatic dysfunction and were undergoing no drug treatments that could have affected carbohydrate or insulin metabolism. The subjects consumed a weight-maintaining diet consisting of at least 250 g. of carbohydrate per day for 1 week before the study. Table 3.11 reports anthropometric characteristics (BM I, BSA) of the subjects, measured plasma glucose and insulin concentrations (Gf ast , If ast ) immediately before the EHC procedure, and the average levels of insulin after 80 min. of clamp insulinization (Imax ). The study protocol followed the guidelines of the Medical Ethics Committee of the Catholic University of Rome Medical School; written informed consent was obtained from all subjects.

3.14.2

Experimental protocol

Each subject was studied in the postabsorptive state after a 12-14 h. overnight fast. Subjects were admitted to the Department of Metabolic Diseases at the Catholic University School of Medicine in Rome the evening before the study. At 07.00 hours on the following morning, the infusion catheter was inserted into an antecubital vein; the sampling catheter was introduced in the contralateral dorsal hand vein and this hand was kept in a heated box (60 ◦ C) in order to obtain arterialized blood. A basal blood sample was obtained in which insulin and glucose levels were measured. At

92

08.00 hours, after a 12-14 h. overnight fast, the EHC was performed according to DeFronzo et al. (1979). A priming dose of short-acting human insulin was given during the initial 10 min. in a logarithmically decreasing manner so that the plasma insulin was raised acutely to the desired level. During the five-hour clamp procedure, the glucose and insulin levels were monitored every 5 min. and every 20 min. respectively, and the rate of infusion of a 20% glucose solution was adjusted during the procedure following the published algorithm DeFronzo et al. (1979). Because serum potassium levels tend to fall during this procedure, KCl was given at a rate of 15-20 mEq/h to maintain the serum potassium between 3.5 and 4.5 mEq/l. Serum glucose was measured by the glucose oxidase method using a Beckman Glucose Analyzer II (Beckman Instruments, Fullerton, Calif., USA). Plasma insulin was measured by microparticle enzyme immunoassay (Abbott Imx, Pasadena, Calif., USA).

3.14.3

Deterministic model

In order to explain the oscillations of glycemia occurring in response to hyperinsulinization and to continuous glucose infusion at varying speeds, we hypothesized the following system (inspired by a previously published deterministic model of the EHC Picchini et al. (2005)):

(Tgx (t − τg ) + Tgh (t)) dG(t) G(t) = − KxgI G(t)I(t) − Txg dt Vg 0.1 + G(t) dI(t) (TiG G(t) + Tix (t)) = − Kxi I(t) dt Vi Tgh (t) = Tghmax exp(−λG(t)I(t))

(3.24) (3.25) (3.26)

where G(0) = Gb ,

I(0) = Ib ,

Tgh (0) = Tghb = Tghmax exp(−λGb Ib ), Tgx (s) = 0 ∀s ∈ [−τg , 0]

and

Tix (0) = Tixb

Tgx (t) and Tix (t) are (input or forcing) state variables of which the values are known at each time; the state variables and the parameters are defined in Table 3.12 and Table 3.13. The model is diagrammatically represented in Figure 3.5. Equations (3.24) and (3.25) express the variations of plasma glucose and plasma insulin concentrations. The variation of glucose concentration in its distribution space is attributed to the external glucose infusion rate, to liver glucose output and to insulin-dependent as well as insulinindependent glucose tissue uptake. Infused glucose raises glycemia after a delay τg due to the time required to equilibrate the intravenously infused quantity throughout the distribution space. The insulin-independent glucose tissue uptake process is modelled as a Hill function rapidly increasing to its (asymptotic) maximum value Txg ; thus for glycemia values appreciably larger than 0.1 mM the insulin-independent glucose tissue uptake is already close to its maximum. This formulation is intended to represent the aggregated apparent zero-order (fixed) glucose utilization mechanism at rest (mainly by the brain and heart Olson and Pessin (1996) and Sacks (1969) p. 320), with the mathematical and physiological requirement that glucose uptake tends to zero as glucose concen-

3.14 Material and methods

93

tration in plasma approaches zero. The variation of insulin concentration in its distribution space (equation (3.25)) may be thought of as due to the external insulin infusion, to glucose-dependent pancreatic insulin secretion and to the apparently first-order insulin removal from plasma. Equation (3.26) represents the rate of net Hepatic Glucose Output, starting at maximal HGO at zero glucose and zero insulin and decaying monotonically with increases in both glucose and effective insulin concentrations in the plasma. The net HGO is assumed to be equal to Tghb at the beginning of the experiment and to decrease toward zero as glycemia or insulinemia levels increase. Serum insulin affects glucose clearance through equation (3.24) and the glucose synthesis rate through equation (3.26). Steady-state conditions are used to decrease the number of free parameters to be estimated: at steady state, before the start of the clamp (G = Gb , I = Ib , Tgx = Tix = 0), we have Tghb = Tghmax exp(−λGb Ib ) 0=

0 + Tghb Txg Gb − − KxgI Ib Gb ⇒ Txg = Vg 0.1 + Gb 0=



 Tghb (0.1 + Gb ) − KxgI Ib Gb Vg Gb

TiG Gb + 0 Kxi Ib Vi − Kxi Ib ⇒ TiG = Vi Gb

Therefore the parameters Tghb , Txg , and TiG are completely determined by the values of the other parameters.

Deterministic model estimation The system (3.24)–(3.26) has been numerically integrated by means of a fourth–order Runge– Kutta scheme with constant stepsize equal to 0.5 min. In order to distinguish among the n observations (and corresponding predictions) between glucose and insulin, the indices j for glucose and k for insulin are used as follows: j ∈ J, k ∈ K, J ∩ K = {∅}, J ∪ K = {1, ..., n}. ˆ θ) ≡ G(t) ˆ ˆ θ) ≡ I(t) ˆ We indicate with G(t, and I(t, the (numerically integrated) solutions of equations (3.24)–(3.26) for parameter θ at time t. The solutions have been fitted by Iteratively Re-Weighted Least Squares (IRWLS, see e.g. (Davidian and Giltinan, 1995, chapter 2)) separately on each subject’s glycemia and insulinemia time-points, estimating only the free parameters θ = (Gb , Ib , KxgI , Kxi , Tghmax , Vg , Vi , τg , λ) by minimizing the following loss function (y − yˆ)0 Ω(y − yˆ) where y is the n × 1 array containing both glycemias and insulinemias, observed at times 0 = t1 ≤ t2 ≤ · · · ≤ tn ; yˆ is the array of corresponding predictions obtained by numerical integration of ˆ j ) ∀j ∈ J, yˆ(tk ) = yˆk = I(t ˆ k ) ∀k ∈ K; Ω is an n × n the system (3.24)-(3.26), yˆ(tj ) = yˆj = G(t diagonal matrix of weights. Here Z 0 denotes the transpose of the matrix Z. The statistical weight associated with a generic glucose concentration point yj has been defined as 1/(ˆ yj CVG )2 , where CVG is the coefficient of variation for glucose. Similarly the statistical weight associated with a generic insulin concentration point has been defined as 1/(ˆ yk CVI )2 , where CVI is the coefficient of variation for insulin. For each subject IRWLS parameter estimates of θ were obtained for several

94

different values of the coefficients of variation CVG and CVI , namely: (CVG , CVI ) ∈

{(0.015, 0.07), (0.02, 0.10), (0.03, 0.10), (0.03, 0.15), (0.04, 0.15), (0.05, 0.15), (0.15, 0.30)}.

These sets of coefficients of variation were used in order to conduct a sensitivity analysis on the diffusion coefficient (defined in section 3.14.4 below) by considering different values for the variance of the measurement error, which is assumed proportional to the square of the coefficient of variation. The starting point to fix reasonable values for CVG and CVI was suggested in Bergman et al. (1979) where it had been found (CVG , CVI ) = (0.015, 0.07); nevertheless, since these values refer to in vitro estimates of the variance of repeated laboratory measurements on the same preparation, it could be more realistic to re-estimate CVG and CVI from data. To this aim we adopted a General Least Squares approach (GLS (Davidian and Giltinan, 1995, chapter 5 ), detailed in the appendix of this paper).

3.14.4

Stochastic model

As an alternative to the above deterministic model, we may assume that the underlying tissue glucose uptake process is not smooth, subject as it is to a variety of metabolic and hormonal influences, which change over time. In fact, tissue glucose uptake is determined not only by the varying concentrations of certain hormones (e.g. cortisol or growth hormone) and by the rhythm of food intake, events which take place over periods of hours, but also by sudden changes in physical activity or emotional stresses induced by thought processes. We may thus imagine that the insulin-dependent glucose disposal rate may be subject to moment-by-moment variations and that the rate constant KxgI is likely to exhibit substantial irregular oscillations over time. Thus we define two sources of noise: a dynamic noise term, which is a part of the process, such that the value of the process at time t depends on this noise up to time t, and a measurement noise term, which does not affect the process itself, but only its observations. We therefore allow the parameter KxgI to vary randomly as (KxgI − ξ(t)), where ξ(·) is a gaussian white-noise process. Then the system noise ξ(t)dt can be written as σdW (t) (see e.g. Ditlevsen and De Gaetano (2005b), Kloeden and Platen (1992) and Øksendal (2000)), where σ ≥ 0 represents the (unknown) diffusion coefficient and W (·) is the Wiener process (Brownian motion), which is a random process whose increments are independent and normally distributed with zero mean and with variance equal to the length of the time interval over which the increment take place. By incorporating the KxgI variation into the deterministic model, we obtain the following (Itô) SDE: 

 (Tgx (t − τg ) + Tgh (t)) G(t) dG(t) = − Txg − KxgI G(t)I(t) dt Vg 0.1 + G(t) +σG(t)I(t)dW (t),   (TiG G(t) + Tix (t)) dI(t) = − Kxi I(t) dt, Vi Tgh (t) = Tghmax exp(−λG(t)I(t))

(3.27) (3.28) (3.29)

3.14 Material and methods

95

with G(0) = Gb , I(0) = Ib and Tgh (0) = Tghb = Tghmax exp(−λGb Ib ). Notice that this formulation has the theoretical advantage of never becoming negative in any of the coordinates.

3.14.5

SDE estimation

While the model estimation procedure for systems of ODEs is well established, estimating parameters in SDE models is not straightforward, except for simple cases. A variety of methods for statistical inference in discretely observed diffusion processes have been developed during the past decades (e.g. Aït-Sahalia (2002b), Bibby et al. (2005), Bibby and Sørensen (1995), DacunhaCastelle and Florens-Zmirnou (1986), Elerian et al. (2001), Gallant and Long (1997), Gouriéroux et al. (1993), Hurn et al. (2003), Pedersen (1995), Pedersen (2001), Shoji and Ozaki (1998), Sørensen (2000a), Yoshida (1992)). A natural approach would be maximum likelihood inference, but it is rarely possible to write the likelihood function explicitly. In our case it becomes further complicated since we are dealing with partially observed state variables. The estimation approach we follow is to first estimate by IRWLS the parameters of the ODE system (3.24)–(3.26) (as explained in section 3.14.3), which represents the deterministic part (drift) of the SDE model and the mean of the corresponding stochastic process. We then use the Monte-Carlo approximation to the unknown likelihood function as suggested in Pedersen (2001), in order to estimate σ by keeping fixed the previously obtained drift parameter estimates. We make recourse to the method in Pedersen (2001) since plasma glucose concentrations G and serum insulin concentrations I are not observed at the same time-points (every 5 min. and every 20 min. respectively), and we therefore deal with partially observed state variables, and further because the concentrations are observed with measurement error. In the following the application of the method to the problem under investigation is detailed; for ease of notation we denote with Zi the generic variable Z(ti ) at time ti . Consider the model (3.27)–(3.29) and observation times 0 = t1 ≤ t2 ≤ · · · ≤ tn : all the observations y are collected in a single array and distinguished using the following label-variable ( χi =

G, I,

if the observation at time ti ref ers to glucose if the observation at time ti ref ers to insulin

i = 1, ..., n

We consider the error-model Yi = Hχi + εi , where

( Hχi =

Gi , Ii ,

if χi = G if χi = I

(3.30)

i = 1, ..., n

and the εi ’s are independent normal variables with mean 0 and variance σχ2 i representing the measurement errors. We assume that (i) ( σχ2 i

=

(CVG Gi )2 , (CVI Ii )2 ,

if χi = G if χi = I

i = 1, ..., n

where CVG and CVI represent the coefficient of variations for the glucose and the insulin concentrations respectively, and that (ii) the measurement errors are independent of the process W (·). Equation (3.30) and the SDE system (3.27)-(3.29) provide a representation of the error-structure

96

in our problem. Denote with yi the observed value of Yi at time ti , then the likelihood function of σ can be written as Z L(σ)

=

Y n

Rn−2 i=1 n Y

 gi (yi |Hχi ; σ) f (Hχ3 , ..., Hχn ; Hχ1 , Hχ2 , σ)dHχ3 · · · dHχn

gi yi |Hχi ; σ

= Eσ




i=1

where Hχ1 = Gb and Hχ2 = Ib are the initial conditions of G and I respectively, f denotes the (unknown) joint density function of Hχ3 , ..., Hχn given (Hχ1 , Hχ2 , σ), Eσ denotes expectation w.r.t. the distribution of Hχ3 , ..., Hχn for the indicated value of σ and gi (yi |Hχi ; σ) = 2πσχ2 i


−1/2

 
2 1 exp − 2 yi − Hχi 2σχi

is the normal density function with expectation Hχi and variance σχ2 i . If H r (r = 1, ..., R) are stochastically independent random vectors, each distributed as (Hχ3 , ..., Hχn ), then it follows from the strong law of large numbers that the likelihood function can, for large values of R, be approximated by R n 1 XY gi (yi |Hχr i ; σ) L(σ) ' (3.31) R r=1 i=1 In practice the approximation is obtained by simulating the Hχr i ’s (see Kloeden and Platen (1992)) for a large finite number R. We have initially simulated R = 1000, 2000 and 4000 trajectories of the process according to the Euler-Maruyama scheme Kloeden and Platen (1992) with an integration step-size of 0.1 min.: given a σ of the order of 10−5 , the step-size ensures a standard deviation for dG smaller than 0.02 in each integration step, which is very small compared to the order of magnitude of the glucose concentrations. The simulated likelihood functions did not appreciably change the location of their maximum when increasing the number of trajectories beyond 2000. Therefore the reported estimates of σ were obtained by maximizing the approximated likelihood (3.31), based on R = 2000 trajectories, when keeping fixed the parameters entering the drift part of the model and using different combinations of levels of CVG and CVI (see section 3.14.3), in order to explore the sensitivity of the obtained estimates σ ˆ to mis-specification of the observation error.

3.15 3.15.1

Results Deterministic differential model

We first estimated by IRWLS, separately on each subject, the structural (free) parameters Gb , Ib , KxgI , Kxi , Tghmax , Vg , Vi , τg , λ, and computed the corresponding structural (determined) parameters Tghb , Txg and TiG entering the deterministic model (3.24)–(3.26): the estimates corresponding to three different choices for the coefficients of variation (see section 3.14.3) are reported in tables 3.14, 3.15 and 3.16, whereas graphical results of the IRWLS fitting are shown in figures 3.6 and 3.7 only for the case (CVG , CVI ) = (0.05, 0.15) (in fact these values are very similar to

3.16 Discussion

97

the GLS population-estimates of CVG and CVI , as explained below), since in our problem the predicted curves do not vary substantially with different measurement error values. Secondarily, we estimated the individual structural parameters, as well as the population parameters CVG and CVI , simultaneously by GLS (Table 3.17): in this way we found that d G = 0.071 and CV d I = 0.1702. CV

3.15.2

Stochastic differential model

The stochastic model (3.27)–(3.29) was adapted to our data and σ was estimated as described in section 3.14.5. The estimates of σ corresponding to the different sets of coefficients of variation are reported in Table 3.18 for each subject. In this table we notice that the σ estimates are stable when considered in a reasonable region of the coefficient of variations values, that is when considered in (CVG , CVI ) ∈ [0.02, 0.05] × [0.10, 0.15]. At the smallest level (CVG , CVI ) = (0.015, 0.07) the σ estimates result numerically unidentifiable for three subjects, and are thus marked with an ‘NA’. While theoretically we could have tried to estimate simultaneously all parameters appearing in the SDE model (drift parameters, diffusion coefficient determining the system noise variance, observation error variance), from a purely computational point of view this proved exceedingly expensive and we had to be content with a sequential estimation approach. In this way, several combinations of observation error levels were hypothesized, the drift parameters were estimated under each hypothesis, and the corresponding diffusion was then estimated in each case. For illustration purposes, graphical results of the fitting only for the cases (CVG , CVI ) = (0.05, 0.15) and (CVG , CVI ) = (0.03, 0.15) are shown in figures 3.8 and 3.9 respectively, only for the glycemia values since the insulin curves are almost identical to those produced by the deterministic model. For each subject figures 3.8 and 3.9 report the observed glycemias and the empirical mean of R = 2000 simulated trajectories of the G(t) process, their empirical 95% confidence limits (from the 2.5th percentile to the 97.5th percentile) and one simulated trajectory. According to Pedersen (1994)-Pedersen (2001) we are able to check the plausibility of our stochastic model by simulating uniform residuals; the q-q plots of the simulated uniform residuals are reported in figures 3.10 and 3.11, where the residuals are plotted against percentiles from the U (0, 1) distribution. The caption of each subfigure also reports the p-value from the two-tailed Kolmogorov-Smirnov goodness-of-fit test; if p < 0.05 the simulated residuals do not conform to the hypothesis of U (0, 1) distribution at a 5% confidence level. The tests have not been subjected to correction for simultaneous inference (Bonferroni or similar) in order to be more conservative. All tests had p > 0.05, except for the glycemia residuals for subjects 4 and 6 and for the insulinemia residuals for subjects 8 and 15.

3.16

Discussion

The Euglycemic Hyperinsulinemic Clamp is the procedure most commonly employed by research diabetologists in their quest for the determination of the degree of insulin sensitivity (or resistance) exhibited by a given experimental subject. In its common usage, after having “clamped”, i.e. stabilized, the subject’s glycemia to pre-insulinization levels, the average rate of glucose infusion necessary to maintain euglycemia is measured, and is directly employed (once normalized by the subject’s body mass) as an index of insulin sensitivity.

98

As an alternative to the above, a model may be drawn to describe the mass flow of glucose into and out the central (sampling) plasma compartment of the subject, explicitly representing the physiological mechanisms known to intervene in the process. In the present work, a (simple) deterministic model of the clamp procedure is studied first. The main result of this study is that the level of error around the predicted curve is very large, in particular it is much larger than the (0.015,0.07) commonly accepted levels of measurement error in in vitro repeated testing of the same laboratory preparation. This result would theoretically be compatible with either one of the following alternatives: the model is mis-specified; the in vivo measurement error is in reality much larger than (0.015,0.07); or, there is some additional source of noise, besides measurement error, which substantially impacts observations. From an examination of Figure 3.6, it would not seem that the average model prediction is systematically wrong. Similarly, the coefficient of variations estimated by GLS around the deterministic prediction are much too large to be compatible with measurement error. The idea that glucose absorption by tissues varies in time is, on the other hand, rather natural: it seems evident that, subject to variable hormonal concentrations, variable stress levels, even to minor posturali changes, muscle uptake and liver output of glucose may vary from moment to moment. What remains to be seen is if a mathematical model incorporating this idea would be supported by the actual observations. A random fluctuation in the net tissue glucose uptake rate is a reasonable approximation to the effect of a host of the poorly controlled, additive influences mentioned. When considering this random fluctuations as well, the original deterministic model (3.24)–(3.26) is thus transformed into the SDE model (3.27)–(3.29). The approach followed for the estimation of the relevant quantities of the model (structural parameters and diffusion) is motivated by the computing–intensive algorithms necessary for the estimation of the diffusion, which require the simulation of thousands of possible trajectories of the process for every evaluation of the merit function. Essentially, it has been shown that: (i) for any reasonable level of observation error, the estimated diffusion has more or less the same value. For “reasonable” it is here meant larger than pure measurement error and smaller than the total error around the expected trajectory as estimated by GLS. Adopting the lowermost observation error level (0.015,0.07) would be equivalent to stating that the same variability exists on repeated laboratory measurements on the same sample as on repeated sampling/measurement procedures at the same actual glycemia and insulinemia, disregarding further potential sources of variation accruing to the sampling procedure itself (volume of blood vs. volume of anticoagulant, degree of coagulation, variation in spinning time etc.): this seems extreme in one direction. In the other direction, the largest error level considered (0.15,0.30) is much higher than the total observation variability actually estimated with the GLS procedure around the deterministic prediction (0.071,0.1702), and for this reason should also be discarded. Having excluded these extreme cases, it can be seen that, in the present situation, the estimation of the diffusion is very robust to changes in the likely value of the observation error, as similarly robust are the estimates of the structural (drift) parameters. Further (ii) we showed that the diffusion coefficient estimates are generally strictly positive: this means that the dynamical

3.16 Discussion

99

process which most likely represents the glycemia time-course (given the estimated deterministic differential model) is a stochastic process with a non-negligible system noise, whose intensity factor is represented by the diffusion coefficient. Pictorial evidence of the magnitude of diffusion is given in figures 3.8 and 3.9. This system noise represents the additive action of many factors, each with a small individual effect, which are not explicitly represented in the deterministic model (that is in the drift term of the SDE), and which instantaneously affect glucose uptake rate. Therefore, in the stochastic differential model the collective influence of many individually neglected effects is added to the average drift term, which, on the other side, represents the most relevant and generally well-recognized factors affecting glycemia. It is interesting to note from figures 3.8 and 3.9 that when the process average (which also represents the ordinary differential model solution) fits well the observed glycemias, its 95% confidence band is narrow (e.g. subjects 1, 2, 3, 10 and 12). On the other hand, when the process average itself is not able to meaningfully capture the general trend of the observations, the corresponding confidence band is much larger (e.g. subjects 4, 5 and 9). In these last cases, the system is driven more by the erratic oscillations in tissue glucose transport rather than by the smooth dynamics of the overall actual system, of course under the hypothesis that the proposed deterministic term is correctly specified. This finding would prompt us to re-consider the quantitative relevance of the unexpressed effects, in other situations, represented so far only with deterministic models, especially when less than perfectly satisfactory fits to data have been obtained. The particular behavior of the estimated diffusion (Table 3.18), for the different choices of coefficient of variation values ξ ≡ (CVG , CVI ) could seem counterintuitive: one would expect that as the observation error is assumed to increase, the estimated system error should decrease (this actually happens for macroscopically exaggerated values of the observation error). This unexpected result may be due to the estimation method we adopted: since the array of free structural parameters θ for the drift part of the stochastic model is estimated anew for different ˆ are depending on these levels of error, so we can write θˆ ≡ θˆξ . As a consequence values of ξ, the θ’s of that, the (numerical) solution of the SDE system (3.27)–(3.29) and the measurement error ε are not independent and, by means of equation (3.30), we can write V ar(Yi )

= V ar(Hχi (θˆξ )) + V ar(εi ) + 2Cov(Hχi (θˆξ ), εi ) = V ar(Hχi (θˆξ )) + σχ2 i + 2E(Hχi (θˆξ )εi ).

Since H(·) is unknown, so is E(Hχi (θˆξ )εi ) and we cannot compute the covariance analytically: however it is generally not zero, and the variance of the observations is not the simple sum of the variance of the trajectories and of the variance of the measurement error. There is indeed the possibility that the parameters of the system, or the model structure itself, may be non-stationary over the time course of the experiment. If this non-stationarity is judged to be potentially important, it can be represented by actually modeling the time-course of the parameter value over the duration of the experiment, as a function of other (meta-) parameters. The statistical significance of such meta-parameters would then indicate whether the influence of this non-stationarity is relevant. Similarly for a possible variation of the model structure over time.

100

It is of interest to note that even when glycemia is allowed to vary stochastically, responding to the Wiener process introduced in equation (3.27), the corresponding oscillations in insulinemia are very small, and the insulin process does not substantially differ from its own expected value. This may very well be explained when considering the relative inertia of the pancreatic insulin secretion mechanism, coupled with the large volume of distribution of the hormone, which combine to minimize oscillations in insulin concentrations in response to rapidly varying glucose concentrations. We conclude therefore that the stochastic differential model (3.27)–(3.29) is statistically robust, physiologically meaningful and represents well the glucose metabolism occurring during a clamp study. More generally, it can be concluded that stochastic differential equations are theoretically useful and practically applicable, and deserve to be considered more often as a valuable addition to the biomedical modeller’s toolbox.

Acknowledgements: The authors are grateful to Prof. G. Mingrone (Università Cattolica del Sacro Cuore, Policlinico Universitario “A. Gemelli”, Rome, Italy) for having provided the original data sets and commented on the EHC procedure. The work was supported by grants from the Danish Medical Research Council and the Lundbeck Foundation to S. Ditlevsen.

3.17

Appendix

To obtain subject-specific regression parameters and population estimates of CVG and CVI the GLS method was performed (Davidian and Giltinan, 1995, chapter 5). The GLS is a two-stage method: (stage 1) at first individual estimates for each subject i (i = 1, ..., 15) were obtained; then (stage 2) these estimates were used as building blocks to construct the population estimates of CVG and CVI . Suppose that yi and θi represent the ni -dimensional array of recorded data and the array of (structural) individual parameters for subject i respectively (i = 1, ..., 15), i.e. θi contains the values of the (free) parameters θ = (Gb , Ib , KxgI , Kxi , Tghmax , Vg , Vi , τg , λ) entering the model (3.24)–(3.26) for subject i. Consider now the model yi = fi (θi ) + εi such that E(εi |θi ) = 0,

Cov(εi |θi ) = Ωi (θi , ξ)

with fi (·) representing the numerical solution of the system (3.24)–(3.26) for subject i, and assuming that the functional form of Ωi (·, ·) and the intra-individual covariance parameter ξ = (CVG , CVI ) are the same across individuals. If we denote with G and I the state variable Glucose and Insulin respectively, the covariance matrix Ωi (θi , ξ) in the present application has the

3.17 Appendix

101

structure of an ni × ni block-diagonal matrix Ωi (θi , ξ) =

Ωi,G 0

0 Ωi,I

! i = 1, ..., 15

where   2 CVG2 fiG (θi , ti,1 ) 0 · · · 0   Ωi,G =  ··· ··· ··· ··· , 2 2 0 ··· 0 CVG fiG (θi , ti,niG )   2 CVI2 fiI (θi , ti,1 ) 0 · · · 0   Ωi,I =  ··· ··· ··· ···  2 0 ··· 0 CVI2 fiI (θi , ti,niI ) with fiG (θi , ti,j1 ) and fiI (θi , ti,j2 ) representing the predicted glycemia and insulinemia values at times ti,j1 and ti,j2 respectively (j1 = 1, ..., niG ; j2 = 1, ..., niI ; niG + niI = ni ). Then the GLS algorithm is given by the following scheme: (p) 1. in m = 15 separate regressions, obtain preliminary estimates θˆi for each individual, i = 1, ..., m;

2. use residuals from these preliminary fits to estimate ξ by minimizing the following functional m X i=1

(p) P Li (θˆi , ξ) =

m X

(p) (p) ˆ(p) ˆ(p) log |Ωi (θˆi , ξ)| + (yi − fi (θˆi ))0 Ω−1 i (θi , ξ)(yi − fi (θi ))

i=1

where P Li is the pseudolikelihood of ξ for the i th individual. Form estimated weight matrices (p) based on the estimate ξˆ obtained from this procedure, along with the preliminary θˆi , to form ˆ ˆ i (θˆ(p) , ξ) Ω i 3. using the estimated weight matrices from step 2, re-estimate the θi ’s by m separate minimizations: for individual i, minimize in θi ˆ −1 (yi − fi (θi )) (yi − fi (θi ))0 Ω i Treating the resulting estimators as new preliminary estimators, return to step 2. The algorithm should be iterated at least once to eliminate the effect of potentially inefficient preliminary estimates in step 1.

102

Table 3.11:

Anthropometric and metabolic characteristics for the normal and obese (*) subjects; BSA is the Body Surface Area [m2 ] calculated via the DuBois formula (BSA = 0.20247 · height0.725 [m] · weight0.425 [kg]).

Subject

BM I [kg/m2 ]

BSA [m2 ]

Gf ast [mM ]

If ast [pM ]

Imax [pM ]

20.20 35.93 27.77 38.10 20.03 19.33 48.07 18.51 63.57 18.59 42.19 22.59 31.35 27.91 22.68

1.60 2.38 2.04 2.11 1.49 1.55 2.16 1.55 2.08 1.46 2.02 1.71 2.11 1.83 1.73

3.61 4.83 5.39 5.11 4.83 3.67 5.39 4.06 5.94 5.44 5.28 3.39 4.94 4.61 3.50

36.84 108.42 79.23 139.00 15.29 21.55 139.00 13.90 152.90 49.34 139.00 32.66 79.23 83.40 27.80

472.00 607.30 683.18 625.76 506.89 464.54 592.02 527.00 522.25 482.42 497.31 469.22 605.22 679.62 482.14

1 2* 3* 4* 5 6 7* 8 9* 10 11* 12 13* 14* 15

Table 3.12:

Definitions of the state variables.

Variables t [min] G(t) [mM ] I(t) [pM ] Tgx (t) [mmol/min/kgBW ] Tix (t) [pmol/min/kgBW ] Tgh (t) [mmol/min/kgBW ]

time from insulin infusion start plasma glucose concentration at time t serum insulin concentration at time t glucose infusion rate at time t insulin infusion rate at time t net Hepatic Glucose Output (HGO) at time t

Gb [mM ] Ib [pM ] Txg [mM/min] KxgI [min−1 /pM ] Kxi [min−1 ] TiG [pM/min/mM ] Tixb [pmol/min/kgBW ] Tghmax [mmol/min/kgBW ] Tghb [mmol/min/kgBW ] Vg [L/kgBW ] Vi [L/kgBW ] τg [min] λ [mM −1 pM −1 ] σ [pM −1 min−1/2 ]

Parameters

Definitions of the parameters.

basal glycemia basal insulinemia maximal insulin-independent rate constant for glucose tissue uptake insulin-dependent apparent first-order rate constant for glucose tissue uptake at insulinemia I apparent first-order rate constant for insulin removal from plasma apparent zero-order net insulin synthesis rate at unit glycemia (after liver first-pass effect) basal insulin infusion rate, which is given by the measured value of Tix at time zero according to DeFronzo and Ferrannini (1991) maximal Hepatic Glucose Output at zero glycemia, zero insulinemia basal value of Tgh volume of distribution for glucose volume of distribution for insulin discrete (distributional) delay of the change in glycemia following glucose infusion rate constant for Hepatic Glucose Output decrease with increase of glycemia and insulinemia diffusion coefficient

Table 3.13:

3.17 Appendix 103

104

3.662 4.535 6.153 5.764 5.043 3.755 5.547 4.227 6.645 5.500 5.589 3.146 5.332 5.136 3.629

42.07 217.16 588.73 141.49 15.52 27.31 141.74 15.64 219.74 257.45 140.83 34.14 84.34 246.11 48.01

6.05E-5 4.07E-5 9.94E-5 0.20E-8 2.16E-5 2.48E-5 0.75E-5 1.85E-5 0.95E-5 26.56E-5 0.64E-5 3.57E-5 0.63E-5 1.97E-5 2.06E-5

0.018 0.010 0.028 0.040 0.026 0.033 0.030 0.019 0.010 0.163 0.029 0.019 0.025 0.010 0.017

0.016 0.012 1.000 0.320 0.002 0.044 0.033 0.005 0.012 0.055 0.007 0.004 0.025 0.646 0.010

0.464 0.266 0.990 0.990 0.988 0.719 0.988 0.990 0.854 0.141 0.874 0.990 0.990 0.948 0.990

0.829 0.990 0.990 0.274 0.578 0.535 0.341 0.714 0.990 0.229 0.480 0.731 0.422 0.990 0.758

0.54 2.50 0.13 0.00 17.00 0.50 0.00 0.00 8.00 5.49 0.00 0.00 0.38 12.49 0.50

7.91E-3 9.33E-8 19.18E-8 3.28E-3 0.09E-3 31.06E-3 2.20E-3 20.00E-3 0.01E-3 0.03E-3 0.06E-3 0.06E-3 4.91E-3 1.25E-8 5.81E-3

0.005 0.012 0.999 0.022 0.002 0.002 0.006 0.001 0.012 0.053 0.004 0.004 0.003 0.646 0.004

1.08E-3 5.99E-3 659.84E-3 22.69E-3 2.01E-7 0.02E-7 0.09E-7 0.01E-7 11.80E-7 2.62E-7 4.79E-7 0.40E-7 1.92E-7 0.66 0.11E-7

0.17 0.45 2.63 0.27 0.05 0.13 0.26 0.05 0.33 1.75 0.35 0.15 0.17 0.48 0.17

Table 3.14: IRWLS parameter estimates for the ODE model (3.24)–(3.26) when CVG = 0.015 and CVI = 0.07; the notation E±p is used for 10±p . ˆ ˆb ˆ xgI ˆ xi Subject G Iˆb K K Tˆghmax Vˆg Vˆi τˆg λ Tˆghb Tˆxg TˆiG 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

3.695 4.433 7.101 5.758 5.112 3.794 5.574 4.318 6.661 5.516 5.600 3.209 5.368 5.648 3.687

39.42 215.92 490.98 141.16 15.65 21.52 140.98 14.32 220.06 254.06 139.75 34.12 79.21 467.41 28.97

6.07E-5 4.36E-5 52.05E-5 4.66E-9 2.23E-5 2.48E-5 0.75E-5 1.94E-5 0.91E-5 26.49E-5 0.77E-5 3.66E-5 0.63E-5 6.21E-5 2.18E-5

0.020 0.010 0.019 0.040 0.029 0.037 0.034 0.022 0.012 0.165 0.032 0.023 0.03 0.021 0.020

0.016 0.011 0.976 0.319 0.073 0.024 0.031 0.004 0.013 0.054 0.006 0.004 0.026 0.969 0.010

0.472 0.254 0.130 0.990 0.990 0.726 0.988 0.976 0.967 0.138 0.905 0.990 0.990 0.943 0.980

0.779 0.990 0.990 0.270 0.532 0.483 0.308 0.646 0.912 0.222 0.432 0.663 0.359 0.985 0.681

0.50 0.50 3.08 0.01 16.50 0.71 0.50 0.49 2.01 5.04 0.24 0.02 0.36 22.00 0.49

9.06E-3 2.30E-7 0.16E-3 3.29E-3 4.66E-7 34.45E-3 2.11E-3 20.48E-3 0.03E-3 0.04E-3 0.06E-3 0.02E-4 5.42E-3 0.03E-3 13.5E-3

0.004 0.011 0.554 0.022 0.073 0.001 0.006 0.001 0.013 0.051 0.006 0.004 0.003 0.902 0.002

1.60E-12 0.01E-3 247.88E-3 22.60E-3 73.70E-3 3.15E-12 3.17E-8 4.07E-6 1.10E-7 1.30E-7 0.15E-3 0.04E-3 1.80E-10 806.10E-3 0.20E-3

0.16 0.49 1.30 0.27 0.05 0.10 0.26 0.05 0.36 1.69 0.35 0.16 0.16 1.68 0.11

Table 3.15: IRWLS parameter estimates for the ODE model (3.24)–(3.26) when CVG = 0.05 and CVI = 0.15; the notation E±p is used for 10±p . ˆ ˆb ˆ xgI ˆ xi Subject G Iˆb K K Tˆghmax Vˆg Vˆi τˆg λ Tˆghb Tˆxg TˆiG

3.17 Appendix 105

106

3.807 4.616 6.943 5.752 5.193 3.841 5.600 4.448 6.825 5.550 5.612 3.335 5.414 5.546 3.890

38.69 239.06 507.86 140.18 15.33 21.55 139.77 13.96 154.79 250.52 139.41 32.91 80.64 447.67 27.66

6.16E-5 4.25E-5 24.88E-5 9.0E-10 2.27E-5 2.44E-5 0.75E-5 1.96E-5 0.70E-5 26.57E-5 0.76E-5 3.67E-5 0.63E-5 5.68E-5 2.27E-5

0.022 0.012 0.021 0.041 0.035 0.041 0.038 0.027 0.024 0.166 0.035 0.029 0.035 0.020 0.028

0.011 0.014 0.814 0.308 0.002 0.034 0.026 0.005 0.008 0.053 0.006 0.676 0.024 0.996 0.007

0.471 0.290 0.908 0.990 0.990 0.738 0.990 0.990 0.990 0.137 0.923 0.989 0.990 0.990 0.990

0.730 0.990 0.990 0.266 0.449 0.443 0.276 0.558 0.433 0.216 0.404 0.539 0.309 0.990 0.532

1.45 0.35 0.00 0.49 14.52 0.91 0.50 0.37 0.50 5.50 0.02 1.05 0.24 12.00 0.50

6.41E-3 0.02E-4 0.01E-3 3.28E-3 0.02E-3 37.88E-3 1.91E-3 21.27E-3 0.06E-3 0.03E-3 0.04E-3 0.01E-4 5.00E-3 0.01E-3 9.67E-3

0.004 0.014 0.796 0.022 0.002 0.001 0.006 0.001 0.007 0.051 0.006 0.676 0.003 0.962 0.002

0.02E-3 0.70E-3 0.99E-7 0.02 0.14E-3 0.1E-7 0.05E-7 0.1E-9 0.24E-5 1.36E-3 0.1E-3 0.700 0.029E-9 0.85 0.16E-6

0.16 0.61 1.52 0.26 0.05 0.10 0.26 0.05 0.24 1.62 0.35 0.16 0.16 1.63 0.11

Table 3.16: IRWLS parameter estimates for the ODE model (3.24)–(3.26) when CVG = 0.15 and CVI = 0.30; the notation E±p is used for 10±p . ˆ ˆb ˆ xgI ˆ xi Subject G Iˆb K K Tˆghmax Vˆg Vˆi τˆg λ Tˆghb Tˆxg TˆiG 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Subject

3.662 4.737 6.183 5.764 5.118 3.754 5.547 4.227 6.759 5.498 5.600 3.247 5.332 5.168 3.629 Population estimates d G = 0.071, CV d I = 0.1702 CV

42.07 238.23 588.73 141.49 15.49 27.31 141.74 15.64 219.60 257.37 140.83 33.93 84.34 247.46 48.01

6.40E-5 4.16E-5 27.65E-5 0.52E-10 2.28E-5 2.51E-5 0.76E-5 1.95E-5 0.74E-5 21.07E-5 0.73E-5 3.68E-5 0.63E-5 2.13E-5 2.18E-5

0.019 0.011 0.039 0.040 0.033 0.035 0.031 0.020 0.016 0.171 0.033 0.028 0.025 0.013 0.018

0.016 0.340 1.000 0.321 1.000 0.045 0.033 0.005 0.013 0.059 0.006 0.860 0.025 1.000 0.010

0.464 0.297 0.990 0.990 0.990 0.720 0.988 0.990 0.990 0.140 0.899 0.990 0.990 0.990 0.982

0.829 0.990 0.849 0.274 0.471 0.518 0.338 0.713 0.706 0.215 0.421 0.569 0.422 0.921 0.758

0.54 2.50 0.13 0.00 17.00 0.50 0.00 0.00 8.00 5.49 0.00 0.00 0.38 12.49 0.50

8.19E-3 3.68E-8 1.02E-6 3.28E-3 7.61E-8 31.03E-3 2.19E-3 19.40E-3 0.13E-3 0.25E-3 0.14E-3 2.61E-8 4.90E-3 5.45E-9 5.69E-3

0.005 0.340 0.996 0.022 1.000 0.002 0.006 0.001 0.011 0.042 0.005 0.860 0.003 1.000 0.004

0.57E-7 1.12 5.53E-7 0.02 1.03 0.11E-7 0.95E-7 0.12E-7 12.83E-7 1.58E-7 0.11E-7 0.89 0.79E-7 1.00 0.08E-7

Table 3.17: GLS individual and population parameter estimates for the ODE model (3.24)–(3.26); the notation E±p is used for 10±p . Individual estimates ˆ ˆb ˆ xgI ˆ xi G Iˆb K K Tˆghmax Vˆg Vˆi τˆg λ Tˆghb Tˆxg

0.18 0.57 3.16 0.27 0.05 0.13 0.27 0.05 0.38 1.73 0.35 0.17 0.17 0.57 0.18

TˆiG

3.17 Appendix 107

108

Table 3.18: Estimates of σ in the cases (CVG , CVI ) = (0.015, 0.07), (CVG , CVI ) = (0.02, 0.10), (CVG , CVI ) = (0.03, 0.10), (CVG , CVI ) = (0.03, 0.15), (CVG , CVI ) = (0.04, 0.15), (CVG , CVI ) = (0.05, 0.15) and (CVG , CVI ) = (0.15, 0.30) given by σ ˆ (1) , σ ˆ (2) , σ ˆ (3) , σ ˆ (4) , σ ˆ (5) , σ ˆ (6) and σ ˆ (7) respectively. The notation E±p is used for 10±p . σ ˆ (1)

σ ˆ (2)

σ ˆ (3)

σ ˆ (4)

σ ˆ (5)

σ ˆ (6)

σ ˆ (7)

1.60E-5 NA 2.39E-5 NA 1.83E-5 2.72E-5 0.80E-5 0.72E-5 NA 3.08E-5 0.62E-5 1.44E-5 1.23E-5 1.73E-5 1.87E-5

1.78E-5 1.38E-5 4.55E-5 1.00E-5 1.97E-5 2.65E-5 0.80E-5 0.76E-5 2.42E-5 3.04E-5 0.59E-5 0.98E-5 0.82E-5 1.65E-5 1.23E-5

2.25E-5 1.47E-5 5.71E-5 1.00E-5 2.00E-5 2.68E-5 0.80E-5 0.73E-5 2.65E-5 3.04E-5 3.68E-8 1.36E-5 0.86E-5 1.64E-5 1.44E-5

1.59E-5 1.38E-5 2.54E-5 1.00E-5 1.93E-5 2.71E-5 0.80E-5 0.72E-5 2.50E-5 3.04E-5 3.68E-8 1.41E-5 0.74E-5 1.62E-5 1.86E-5

2.10E-5 1.38E-5 3.95E-5 1.00E-5 1.93E-5 2.72E-5 2.35E-8 0.60E-5 2.60E-5 3.00E-5 2.35E-8 1.47E-5 0.84E-5 1.56E-5 1.47E-5

2.25E-5 1.15E-5 2.58E-5 0.95E-5 1.93E-5 2.73E-5 2.35E-8 0.42E-5 2.69E-5 2.88E-5 3.68E-8 1.53E-5 0.87E-5 0.12E-5 1.50E-5

0 2.88E-7 0 3.68E-8 0.91E-5 5.29E-8 1.47E-7 2.12E-7 4.77E-7 4.25E-7 4.77E-7 8.47E-7 2.12E-7 7.21E-8 0.84E-5

Subjects 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Txg

Tgx (t-τ g )

Tgh (t)

G(t) n

Vg

n

G(t) 0.1+G(t)

K xgI G(t)I(t)

TiG G(t)

K xi I(t)

I(t) Tix (t)

Vi stimulation inhibition

Figure 3.5:

Schematic representation of the model (3.24)–(3.26).

3.17 Appendix

Clamp: Subject 2, plot 1

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(h) Subject 8

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ODE model: observed (◦) and predicted (solid line) insulinemia corresponding to the IRWLS estimates for the case (CVG , CVI ) = (0.05, 0.15) (see Table 3.15).

3.17 Appendix

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Figure 3.8: SDE model: a simulated trajectory of G(t), empirical mean curve of the G(t) process (smooth solid lines), empirical 95% confidence limits of the mean process (dashed lines) for the case (CVG , CVI ) = (0.05, 0.15) and glycemia observations.

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3.17 Appendix

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Figure 3.10: SDE model: q-q plots of the simulated glycemia residuals in the case (CVG , CVI ) = (0.05, 0.15) and p-values from the Kolmogorov-Smirnov goodness-of-fit test. If p < 0.05 the residuals do not conform the hypothesis of U (0, 1) distribution at a 5% confidence level.

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Conclusioni Nel presente lavoro è stato affrontato il problema della modellizzazione di sistemi dinamici attraverso equazioni differenziali stocastiche (SDE) e della stima dei relativi parametri. Il crescente interesse della comunità scientifica verso modelli SDE (che da ambiti di applicazione consolidata come quello, ad esempio, finanziario si diffondono verso la biomedicina) dipende dalla maggiore consapevolezza che la determinazione del rumore di sistema è indispensabile alla comprensione del fenomeno in oggetto. Le SDE sono state utilizzate come un utile strumento di analisi di processi dinamici, per i quali viene supposto che le osservazioni delle variabili modellizzate siano realizzazioni di un processo stocastico e che siano, eventualmente, perturbate da errore di misurazione. Per contro, i modelli differenziali deterministici, per loro natura, non sono in grado di rappresentare adeguatamente le osservazioni di processi caratterizzati da una stocasticità intrinseca nella loro dinamica. A questo proposito, nel presente lavoro sono state considerate applicazioni di modellizzazione di dinamiche fisiologiche (sistema glucosio/insulina), affrontate prima con modelli deterministici (ODE, DDE), quindi con nuovi modelli stocastici definiti da SDE. Costruendo un opportuno modello di errore, è stato inoltre possibile rappresentare la struttura di variabilità delle osservazioni, distinguendo la variabilità intrinseca (di sistema) da quella derivante da errore di misurazione. Tali applicazioni sono già state pubblicate su riviste di biomatematica (Picchini et al. (2005) su Theor. Biol. Med. Model., Picchini et al. (2006a) su J. Math. Biol.). Il principale contributo statistico-metodologico di questa tesi è stato l’introduzione di un nuovo metodo di stima per i parametri di modelli SDE con effetti misti (Picchini et al. (2006b)): è stata considerata una nuova classe di modelli matematici, qui denominata equazioni differenziali stocastiche ad effetti misti (SDME) (precedentemente studiata in soli tre lavori del 2005), in cui alcuni parametri sono variabili deterministiche, mentre altri variano stocasticamente. È stato proposto un nuovo metodo approssimato di stima di massima verosimiglianza per i parametri di SDME, mostrando attraverso simulazioni che il metodo proposto è efficace, a dispetto delle difficoltà computazionali intrinseche nel trattare tale classe di modelli. Le SDME, vista la loro natura che ingloba SDE e modelli ad effetti misti, possono essere utili specialmente in campo biomedico, dove i modelli ad effetti misti vengono ampiamente utilizzati, ma anche, in generale, in qualsiasi processo dinamico definito da SDE aventi alcuni parametri che variano stocasticamente tra diversi esperimenti (ad esempio tra diversi soggetti). Il risultato generale del lavoro è stato, quindi, fornire metodi e modelli per l’utilizzo di SDE, proponendo soluzioni alle difficoltà di tipo implementativo ed enfatizzando le problematiche più prettamente statistiche.

Conclusions In the present work dynamical systems modeling by means of stochastic differential equations (SDE) has been considered, as well as SDE parameter estimation methods. The increasing attention of the scientific community for SDE models which, starting from well-established fields such as the financial one, diffuse to biomedicine, depends on the awareness that system noise determination is necessary to understand the phenomenon under study. SDE has been used as a useful tool for the analysis of dynamical processes, whose observations of the modelized variables are supposed to represent realizations of a stochastic process, and that they may be perturbed by measurement error. On the opposite side, deterministic differential models, because of their nature, cannot adequately represent observations from dynamical processes characterized by an intrinsic stochasticity. In the present work, applications of physiological dynamics modeling (glucose/insulin system) have been considered: initially deterministic models (ODE, DDE) were considered then new stochastic models defined through SDE were applied. By constructing a suitable error-model, it has been possible to represent the variability structure of the observations, and distinguish between intrinsic (system) variability and measurement error variability. These applications have already been published on biomathematical journals (Picchini et al. (2005) on Theor. Biol. Med. Model., Picchini et al. (2006a) on J. Math. Biol.). The principal contribution of this thesis to statistical methodology is the introduction of a new estimation method for the parameters of SDE models including mixed-effects (Picchini et al. (2006b)): it has been studied a new class of mathematical models, here named stochastic differential mixed-effects models (SDME) (previously considered in only three works of 2005), whose parameters can be deterministic or stochastic. A new approximated maximum likelihood estimation method for SDME parameters has been proposed, showing through simulations that the proposed method is effective, in spite of the intrinsic computational difficulties in treating this models class. SDME models, given their capability of combining SDE and mixed-effects models, can be useful especially in biomedicine, where mixed-effects models are broadly applied, but also in any dynamical process defined through an SDE whose parameters vary stochastically among different experiments (e.g. among different subjects). Therefore the general result of this work resulted in providing methods and models to employ SDE, suggesting solutions to implementation difficulties and emphasizing statistical issues.

Appendix A

Some Results in Probability Theory In this appendix a fundamental stochastic process for the definition of a standard SDE is introduced: the Wiener process W . Necessary topics regarding the theory of stochastic processes are also treated. For further reading see Karatzas and Shreve (1991), Kloeden and Platen (1992), Øksendal (2000), Rogers and Williams (1987, 1994) and Stroock and Varadhan (1979). Definition A.1. If Ω is a given set of elements, then a collection A of subsets of Ω is a σ-algebra if 1. Ω ∈ A; 2. if A ∈ A then Ac = {ω ∈ Ω|ω ∈ / A} ∈ A; 3. for any sequence {An } ⊆ A,

S∞

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Definition A.2. A triple (Ω, A, P ) is called a probability space provided Ω is any set, A is a σ-algebra of subsets of Ω, and P is a probability measure on A. Remark A.1. The smallest σ-algebra containing all the opens subsets of R is called the Borel σ-algebra. Definition A.3. A stochastic process is a family of random variables X(t, ω), t ∈ T, ω ∈ Ω on a common probability space (Ω, A, P ), which assumes real values and is P -measurable as a function of ω for each fixed t. Thus X(t, ·) is a random variate defined on (Ω, A, P ) ∀t ∈ T , whereas X(·, ω) : T → R is a trajectory (or realization or sample path) of the stochastic process for any given ω ∈ Ω. In Figure A.1 three realizations of a stochastic process W are reported with W (0, ω) = 0 ∀ω ∈ Ω (it is the Wiener process, which is defined later on), generated by the MATLAB code in Listing A.1. Listing A.1. % Simulations of Brownian paths on [T0,T] %--- Settings ----------------------------------------------------% randn(’state’,500) % fix the initial state to get repeatable results T0 = 0; T = 1; N = 500;

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1

Three realizations of the Wiener process.

h = (T - T0) / N; % the stepsize NUMSIM = 3; % the number of desired simulations %-------------------------------------------------------------------% NORMRAND = [zeros(1,NUMSIM);randn(N,NUMSIM)]; % an (N+1)x(NUMSIM) matrix... % of (pseudo)random entries... % from N(0;1) for(i=1:NUMSIM) dW = sqrt(h)*NORMRAND(:,i); % the Wiener increments (ith simulation) W = cumsum(dW); % the Brownian path (ith simulation) plot([T0:h:T],W,’k-’), hold on xlabel(’t’,’Fontsize’,13) ylabel(’W(t)’,’Fontsize’,13,’Rotation’,0) end

We want to introduce the concept of martingale, an important property that characterize the Wiener process: for this purpose it is necessary to introduce the concept of conditional expectation. Let (Ω, A, P ) be a probability space and let X : Ω → R be a random variable such that E(|X|) < ∞. If H ⊂ A is a σ-algebra, then the conditional expectation of X given H, denoted by E(X|H), is defined as follows: Definition A.4. E(X|H) is the (a.s. unique) function from Ω to R satisfying: 1. E(X|H) is H-measurable; 2.

R H

E(X|H)dP =

R H

XdP,

∀H ∈ H

The existence and uniqueness of E(X|H) comes from the Radon-Nikodym theorem: let µ be the measure on H defined by Z µ(H) = XdP, H∈H H

A.1 Continuity and nowhere differentiability of the Wiener process

121

then µ is absolutely continuous with respect to P |H, so there exists a P |H-unique H-measurable function F on Ω such that Z µ(H) = F dP, ∀H ∈ H H

Thus if we define F = E(X|H) this function does the job and F is unique a.s. with respect to the measure P |H. Definition A.5. Let X(·) be a real valued stochastic process; then the σ-algebra A(t) generated by the random variables X(s) for 0 ≤ s ≤ t is called the “history” of the process until (and including) time t ≥ 0. Definition A.6. Let X(·) be a stochastic process such that E(|X(t)|) < ∞ ∀t ≥ 0; if E(X(t)|A(s)) = X(s)

a.s.

0≤s≤t

then X(·) is called a martingale. So a martingale is a process whose expected value at time t equals the process value at time s: thus, events occurring between s and t do not influence the process expected value at t. Now we introduce the Wiener process, which is crucial in the definition of a SDE. In 1828 the Scottish botanist Robert Brown observed that pollen grains suspended in liquid performed an irregular motion. The motion was later explained by the random collision with the molecules of the liquid. This erratic motion has been named Brownian, but its mathematical description is due to N. Wiener, who defined the mathematical properties of this motion: Definition A.7. A real valued stochastic process W (t), t ∈ [0, +∞) is named Wiener process (or Brownian motion) if 1. W (0) = 0 a.s.; 2. W (t + h) − W (t) ∼ N (0, h) ∀t, h > 0; 3. the increments W (t1 ) − W (t0 ), . . . , W (tn ) − W (tn−1 ) are independent for 0 ≤ t0 < t1 < · · · < tn . Furthermore we have E(W (t)) = 0,

E(W 2 (t)) = t,

E(Ws Wt ) = min(s, t)

V ar(W (t) − W (s)) = t − s

0 ≤ s ≤ t ∀t

0 ≤ s ≤ t.

Remark A.2. W (t) is a gaussian process, i.e. for all 0 ≤ t0 ≤ t1 ≤ · · · ≤ tn the random variable (W (t0 ), . . . , W (tn )) ∈ Rn+1 has a (multi)normal distribution. Remark A.3. Let W (·) be a one-dimensional Wiener process, then W (·) is a martingale.

A.1

Continuity and nowhere differentiability of the Wiener process

In this section we want to consider the following questions:

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Some Results in Probability Theory

1. is t → W (t, ω) continuous for almost all ω? 2. is t → W (t, ω) differentiable for almost all ω? Definition A.8. (i) Let 0 < γ ≤ 1: a function f : [0, T ] → R is called uniformly Hölder continuous with exponent γ if there exists a constant K such that |f (t) − f (s)| ≤ K|t − s|γ

∀s, t ∈ [0, T ]

(ii) We say f (·) is Hölder continuous with exponent γ at the point s if there exists a constant K such that |f (t) − f (s)| ≤ K|t − s|γ ∀t ∈ [0, T ] A good general theorem to prove Hölder continuity is the following: Theorem A.1 (Kolmogorov’s continuity theorem). Let X(·) be a stochastic process with continuous sample paths a.s., such that E(|X(t) − X(s)|β ) ≤ C|t − s|1+α for constants β, α > 0, C > 0, and for all s, t ≥ 0; then for each 0 < γ < α/β, T > 0, and almost every ω, there exists a constant K = K(ω, γ, T ) such that |X(t, ω) − X(s, ω)| ≤ K|t − s|γ

0 ≤ s, t ≤ T

Hence the sample path t → X(t, ω) is uniformly Hölder continuous with exponent γ on [0, T ]. Example A.1 (Application to Wiener process). Consider X(·) = W (·); we have for all integers m = 1, 2, . . . 2m

E(|W (t) − W (s)|

Z |x|2 1 2m − 2r |x| e ) = dx for r = t − s > 0 (2πr)1/2 R Z √ |y|2 1 = rm |y|2m e− 2 dy (y = x/ r) 1/2 (2π) R = Crm = C|t − s|m .

(A.1)

Thus the hypotheses of Kolmogorov’s theorem hold fo β = 2m, α = m − 1; the process W (·) is thus Hölder continuous a.s. for exponents 0<γ<

α 1 1 = − β 2 2m

∀m

Thus for almost all ω and any T , the sample path t → W (t, ω) is uniformly Hölder continuous on [0, T ] for each exponent 0 < γ < 1/2. The following theorem prove that sample paths of Wiener process are a.s. nowhere Hölder continuous with γ > 1/2, and thus are nowhere differentiable. Theorem A.2. (i) For each 12 < γ ≤ 1 and almost every ω, t → W (t, ω) is nowhere Hölder continuous with exponent γ; (ii) in particular, for almost every ω, the sample path t → W (t, ω) is nowhere differentiable and is of infinite variation on each subinterval.

A.2 Transition Densities, Kolmogorov Equations and Diffusion Processes

123

It is possible to explain the nowhere differentiability of W (·) by using the Markov property. Definition A.9. An R-valued stochastic process X(·) is called a Markov process if P (X(t) ∈ B|A(t)) = P (X(t) ∈ B|X(s))

a.s.

for all 0 ≤ s ≤ t and all Borel subset B ⊂ R. So, given the value X(s) you can predict the probabilities of future values of X(t) just as well as if you knew the entire history of the process before time s; so the process only “knows” its value at time s and does not “remember” how it got there. Theorem A.3. Let W (·) be a Wiener process. Then W (·) is a Markov process and P (W (t) ∈ B|W (s)) =

1 (2π(t − s))1/2

Z

e−

|x−W (s)|2 2(t−s)

dx

a.s.

B

for all 0 ≤ s < t and Borel sets B. So the Markov property partially explains the nondifferentiability of sample paths of Wiener process: if W (s, ω) = a, say, then the future behavior of W (t, ω) depends only upon this fact and not how W (t, ω) approached the point a as t → s− ; thus the path “cannot remember” how to leave a in such a way that W (·, ω) will have a tangent here.

A.2

Transition Densities, Kolmogorov Equations and Diffusion Processes

The transition density of a process is a central concept for the maximum likelihood estimation of the parameters of a stochastic differential model, as motivated in section 2.1. In this section we consider some properties of the transition densities of a Markovian process and define the diffusion processes (see e.g Kloeden and Platen (1992) for further details). Consider a Markov process X = {X(t), t ≥ 0} (Definition A.9), then we write its transition probabilities as P (s, x; t, B) = P (X(t) ∈ B|X(s) = x) for all Borel subsets B of R and s < t. Under this assumptions it has a density p(s, x; t, ·), called a transition density, so Z P (s, x; t, B) = p(s, x; t, y)dy B

for all B ∈ B, the σ-algebra of Borel subsets of R. A Markov process with transition densities p(s, x; t, y) is called a diffusion process if the following three limits exists for all ε > 0, s ≥ 0 and x ∈ R: Z 1 lim p(s, x; t, y)dy = 0, t↓s t − s |y−x|>ε Z 1 lim (y − x)p(s, x; t, y)dy = f (s, x), t↓s t − s |y−x|<ε

124

Some Results in Probability Theory

and lim t↓s

1 t−s

Z

(y − x)2 p(s, x; t, y)dy = g 2 (s, x)

|y−x|<ε

where f and g are well-defined functions. The first condition prevents a diffusion process from having instantaneous jumps; the quantity f (s, x) is the drift of the diffusion process and g(s, x) its diffusion coefficient at time s and position x. This implies that f (s, x) = lim t↓s

and g 2 (s, x) = lim t↓s

1 E(X(t) − X(s)|X(s) = x) t−s

1 E((X(t) − X(s))2 |X(s) = x) t−s

so the drift f (s, x) is the instantaneous rate of change in the mean of the process given that X(s) = x, and the squared diffusion coefficient denotes the instantaneous rate of change of the squared fluctuations of the process given that X(s) = x. When the drift and diffusion coefficient of a diffusion process are moderately smooth functions, then its transition density p(s, x; t, y) also satisfies partial differential equations: these are the Kolmogorov forward equation (commonly called the Fokker-Planck equation) ∂ 1 ∂2 2 ∂p + (f (t, y)p) − (g (t, y)p) = 0, ∂t ∂y 2 ∂y 2

(s, x) f ixed,

and the Kolmogorov backward equation ∂p ∂p 1 2 ∂2p − f (s, x) − g (s, x) 2 = 0, ∂s ∂x 2 ∂x

(t, y) f ixed

with the former giving the forward evolution with respect to the final state (t, y) and the latter giving the backward evolution with respect to the initial (s, x).

A.3

Itô and Stratonovich SDEs

We can write an d-dimensional SDE in Itô form as dx(t) = f (x(t))dt + g(x(t))dW (t),

x ∈ Rd

(A.2)

or in Stratonovich form as dx(t) = f (x(t))dt + g(x(t)) ◦ dW (t),

x ∈ Rd

(A.3)

where f (·) : Rd → Rd is called the drift of the SDE, g(·) : Rd → Rd×m is called the diffusion of the SDE, and W (t) is an m-dimensional process having independent scalar Wiener process components (t ≥ 0). It is possible to convert from one interpretation to the other in order to take advantage of one of the approaches as appropriate: in the scalar case (d = m = 1), if the Itô SDE

A.3 Itô and Stratonovich SDEs

125

is as given in (A.2) then the Stratonovich SDE is given by dx = f¯(t, x)dt + g(t, x) ◦ dW where

1 ∂g (t, x)g(t, x) f¯(t, x) = f (t, x) − 2 ∂x

In other words (A.2) and (A.3), under different rules of calculus, have the same solution: for example, dx(t) = ax(t)dt + bx(t)dW (t) has solution x(t) = exp((a − 21 b2 )t + bW (t))x0 as does dx(t) = (a − 21 b2 )x(t)dt + bx(t) ◦ dW (t), while dx(t) = ax(t)dt + bx(t) ◦ dW (t) has solution x(t) = exp(at+bW (t))x0 , where x(0) = x0 . Obviously, in the case of additive noise (g independent of x ⇒ ∂g/∂x ≡ 0) the Itô and Stratonovich representations are equivalent. In vector form, the relationship between the two integrals has d

m

1 XX ∂gik f¯i (t, x) = fi (t, x) − gjk (t, x) (t, x), 2 j=1 ∂xj k=1

i = 1, . . . , d.

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