Recent Advances in Stochastic Modeling and Data Analysis

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RECENTADVANCES IN STOCHASTIC MODELING AND DATA ANALYSIS

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RECENTADVANCES IN STOCHASTIC MODELING AND DATA ANALYSIS 29 May - 1 June 2007

Chania, Greece

editor

Christos H Skiadas Technical University of Crete, Greece

r pWorld Scientific N E W JERSEY

*

LONDON

*

SINGAPORE

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BElJlNG

*

SHANGHAI

*

HONG K O N G

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TAIPEI

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CHENNAI

Published by World Scientific Publishing Co. Re. Ltd. 5 Toh Tuck Link, Singapore 596224 USA ojjice: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601

UK ojjice: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-PublicationData A catalogue record for this book is available from the British Library.

RECENT ADVANCES IN STOCHASTIC MODELING AND DATA ANALYSIS Copyright 0 2007 by World Scientific Publishing Co. Re. Ltd. All rights reserved. This book, or parts thereox may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN-13 978-981-270-968-4 ISBN-10 981-270-968-1

Printed in Singapore by World Scientific Printers (S) Pte Ltd

This volume contains a part of the invited and contributed papers which were accepted and presented at the 12nd International Conference on Applied Stochastic Models and Data Analysis in Chania, Crete, Greece, May 29- June 1, 2007. Since 1981, ASMDA aims to serve as the interface between Stochastic Modeling and Data Analysis and their real life applications particularly in Business, Finance and Insurance, Management, Production and Reliability, Biology and Medicine. Our main objective is to include papers both theoretical and practical, presenting new results having potential for solving real-life problems. Another important objective is to present new methods for solving these problems by analyzing the relevant data. Also, the use of recent advances in different fields will be promoted such as for example, new optimization and statistical methods, data warehouse, data mining and knowledge systems and neural computing. This volume contains papers on various important topics: Stochastic Processes and Models, Distributions, Insurance, Stochastic Modelling for Healthcare Management, Markov and Semi Markov models, Parametric/ Non -Parametric, Dynamical Systems / Forecasting, Modeling and Chaotic Modeling, Sampling and Optimization problems, Data Mining, Clustering and Classification, Applications of Data Analysis and various other applications. The World Scientific had also published the proceedings in two volumes of the 1993 Sixth ASMDA Conference, held also in Chania, Crete, Greece. I acknowledge the valuable support of the Mediterranean Agronomic Institute, Chania, Greece, as well as the IBM France. Sincere thanks must be recorded to those whose contributions have been essential to create the Conference and the Proceedings. Finally, I would like to thank Anthi Katsirikou, Mary Karadima, John Dimotikallis and George Matalliotakis for their valuable support. Chania, July 30,2007 Christos H. Skiadas Editor V

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Contents Preface

V

1 Stochastic Processes and Models

1

An Approach to Stochastic Process using Quasi-Arithmetic Means Etienne Cuvelier and Monique Noirhomme-Fraiture The Quantum Generator of Translations in a Fraction-Dimensional Manifold Paulius MiSkinis Cause of Stock Return Stochastic Volatility: Query by Way of Stochastic Calculus Juho Kanniainen On a Class of Backward Stochastic Differential Equations and Applications to the Stochastic Resonance Romeo Negrea

2 10

18

26

2 Distributions

34

An Application of the Extended Waring Distribution to Model Count Data Variables Jose' Rodriguez Avi, Antonio Conde Sanchez, Antonio Jose' Sa'ez-Castillo and Ma Jose' Olmo Jime'nez Estimation of Simple Characteristics of Samples from Skewed and Heavy-Tailed Distributions Zdengk Fabian Estimating Robustness and Parameter Distribution in Compartmental Models of Neurons Noam Peled and Alon Korngreen On the Stability of Queues with Negative Arrivals Kernane Tewfk Random Multivariate Multimodal Distributions George Kouvaras and George Kokolakis A System Maintained by Imperfect and Perfect Repairs under Phase-Type Distributions Delia Montoro-Gazorla, Rafael Pe'rez-Ocdn and M. Carmen Segovia Asymptotically Robust Algorithms for Detection and Recognition of Signals Veniamin A. Bogdanovich and Aleksey G. Vostretsov

35

vii

43

51

59 68 76

82

viii

Recent Advances in Stochastic Modeling and Data Analysis

Three Parameter Estimation of the Weibull Distribution by Order Statistics Vaida Bartkute and Leonidas Sakalauskas

91

3 Insurance

101

Stochastic Models for Claims Reserving in Insurance Business Tarna's Falukozy, Ildikd Ibolya Vite'z and Miklds Aratd Stochastic Risk Capital Model for Insurance Company Gaida Pettere Measuring Demographic Uncertainty via Actuarial Indexes Mariarosaria Coppola, Ernilia Di Lorenzo, Albina Orlando and Marilena Sibillo Location as Risk Factor. Spatial Analysis of an Insurance Data-Set Ildikd Vite'z A Hierarchical Bayesian Model to Predict Belatedly Reported Claims in Insurances J&os Gyarrnati-Szabd and Lhszld Ma'rkus

102

4 Stochastic Modeling for Healthcare Management

145

Non-Homogeneous Markov Models for Performance Monitoring of Healthcare Sally McClean, Lalit Garg, Brian Meenan and Peter Millard Patient Activity in Hospital using Discrete Conditional Phase-Type (DC-Ph) Models Adele H. Marshall, Louise Burns and Barry Shaw Identifying the Heterogeneity of Patients in an Accident and Emergency Department using a Bayesian Classification Model Louise Burns and Adele H. Marshall Modelling the Total Time Spent in an Accident and Emergency Department and the Associated Costs Barry Shaw and Adele H. Marshall

146

5 Markov and Semi Markov Models

181

Periodicity of the Perturbed Non-Homogeneous Markov System M. A. Syrneonakiand P.-C. G. Vassiliou On the Moments of the State Sizes of the Discrete Time Homogeneous Markov System with a Finite State Capacity G. Vasiliadis and G. Tsaklidis Copulas and Goodness of Fit Tests Pal Rakonczai and Andra's Zernple'ni Discrete Time Semi-Markov Models with Fuzzy State Space Aleka A. Papadopoulou and George M. Tsaklidis An Application of the Theory of Semi-Markov Processes in Simulation Sonia Malefaki and George Iliopoulos

182

114 122

130 137

154

162

172

190

198 206 213

Contents ix

On a Numerical Approximation Method of Evaluating the Interval Transition Probabilities of Semi-Markov Models Dimitrios Bitziadis, George Tsaklidis and Aleka Papadopoulou Markov Property of the Solution of the Generalized Stochastic Equations Khaldi Khaled Partially Markov Models and Unsupervised Segmentation of Semi-Markov Chains Hidden with Long Dependence Noise Je'r6me Lapuyade-Lahorgue and Wojciech Pieczynski

22 1

6 Parametricmon-Parametric

242

x;

Independent Distributed in the Limit Components of Some Chi-Squared Tests Vassilly Voinov, Mikhail Nikulin and Natalie Pya Parametric Conditional Mean and Variance Testing with Censored Data Wenceslao Gonzcilez Manteiga, Ce'dric Heuchenne and Ce'sar Sknchez Seller0 Synthetic Data Based Nonparametric Testing of Parametric Mean-Regression Models with Censored Data Olivier Lopez and Valentin Patilea

229

234

243

25 1

259

7 Dynamical Systems/Forecasting

267

Application of the Single Index Model for Forecasting of the Inland Conveyances Eugene Kopytov and Diana Santalova Development and Application of Mathematical Models for Internet Access Technology Substitution Apostolos N. Giovariis and Christos H. Skiadas Exploring and Simulating Chaotic Advection: A Difference Equations Approach Christos H. Skiadas

268

277 287

8 Modeling and Stochastic Modeling

295

Likelihood Ratio Tests and Applications in 2D Lognormal Diffusions Ramdn Gutie'rrez, Concepcidn Roldan, Ramdn Gutitrrez-Sanchez and Jost Miguel Angulo Cartographical Modeling as a Statistical Method for Monitoring of a Spatial Behaviour of Population Irina Pribytkova Learning and Inference in Switching Conditionally Heteroscedastic Factor Models using Variational Methods Mohamed Saidane and Christian Lavergne

296

304

312

x


Correlation Tests Based NARMAX Data Smoother Validation Li Feng Zhang, Quail Min Zhu and Ashley Lorigden Kernel Based Confidence Intervals for Survival Function Estimation Dimitrios I. Bagkavos, Aglaia Kalamatianou and Dimitrios Ioannides Chaotic Data Analysis and Hybrid Modeling for Biomedical Applications Wlodzimierz Klonowski, Robert Stepien, Marek Darowski and Maciej Kozarski Stochastic Fractal Interpolation Function and its Applications to Fractal Analysis of Normal and Pathological Body Temperature Graphs by Children Anna Sods A Modeling Approach to Life Table Data Sets Christos H. Skiadas and Charilaos Skiadas An Extended Quadratic Health State Function and the Related Density Function for Life Table Data Charilaos Skiadas, George Matalliotakis and Christos H. Skiadas

322 330 338

342

350 360

9 Statistical Applications in Socioeconomic Problems

370

Dumping Influence on a Non Iterative Dynamics Ce'cile Hardouin Firm Turnover and Labor Productivity Growth in the Italian Mechanical Sector Luigi Grossi and Giorgio Gozzi Continuous Sampling Plan under an Acceptance Cost of Linear Form Nicolas Farmakis and Mavroudis Elefheriou A Dynamic Programming Model of a Machine Tool in Flexible Manufacturing Bernard F. Lamond Particle Filter-Based Real-Time Estimation and Prediction of Traffic Conditions Jacques Sau, Nour-Eddin El Faouzi, Anis Ben Aissa and Olivier de Mouzon Probability of Trend Prediction of Exchange Rate by ANFIS George S. Atsalakis, Christos H. Skiadas and Ilias Braimis The Organizational Structure of Greek Libraries: The State of the Art and the Perspective of Team Working Anthi Katsirikou

37 1 382

390 398 406

414 423

10 Sampling and Optimization Problems

433

Applicability of Importance Sampling to Coupled Molecular Reactions Werner Sandmann

434

Contents xi

Bispectrum Estimation for a Continuous-Time Stationary Process from a Random Sampling Karim Benhenni and Mustapha Rachdi Search via Probability Algorithm for Engineering Optimization Problems Nguyen Huu Thong and Tran Van Ha0 Solving the Capacitated Single Allocation Hub Location Problem using Genetic Algorithm Zorica StanimiroviC

442

11 Data Mining and Applications

472

Robust Refinement of Initial Prototypes for Partitioning-Based Clustering Algorithms Sami Ayramo, Tommi Karkkainen and Kirsi Majava Effects of Malingering in Self-Report Measures: A Scenario Analysis Approach Massirniliuno Pastore, Luigi Lombardi and Francesca Mereu The Effect of Agreement Expected by Chance on Some 2 x 2 Agreement Indices Teresa Rivas-Moya Qualitative Indicators of Libraries’ Services and Management of Resources: Methodologies of Analysis and Strategic Planning Aristeidis Meletiou

473

12 Clustering and Classification

511

Languages Similarity: Measuring and Testing Liviu P. Dinu and Denis Encichescu On Clustering Romance Languages Liviu P. Dinu and Denis Encichescu A Clustering Method Associated Pretopological Concepts and k-Means Algorithm T. V. Le, N. Kabachi and M. Lamure Alternatives to the Estimation of the Functional Multinomial Regression Model Manuel Escabias, Ana M. Aguilera and Mariano J. Valderrama A GARCH-Based Method for Clustering of Financial Time Series: International Stock Markets Evidence Jorge Cuiado and Nuno Crato

512

454

464

483

49 1

499

521

5 29 5 37

542

13 Applications of Data Analysis

552

Reliability Problems and Longevity Analysis Anatoli Michalski

553

xii


Statistical Analysis on Mobile Applications among City People: A Case of Bangkok, Thailand Pakavadi Sirirangsi Pollution Sources Detection via Principal Component Analysis and Rotation Marie Chavent, Herve' Gue'gan, Vanessa Kuentz, Brigitte Patouille and J e ' r h e Saracco Option Pricing and Generalized Statistics: Density Matrix Approach Petr Jizba

562 57 1

578

14 Miscellaneous

588

Inference for Alternating Time Series Ursula U. Miiller, Anton Schick and Wolfgang Wefelmeyer Estimation of the Moving-Average Operator in a Hilbert Space Ce'line Turbillon, Jean-Marie Marion and Besnik Pumo Monte Carlo Observer for a Stochastic Model of Bioreactors Marc Joannides, Irbne Larramendy-Valverde and Vivien Rossi Monte Carlo Studies of Optimal Stopping Domains for American Knock Out Options Robin Lundgren SONAR Image Denoising using a Bayesian Approach in the Wavelet Domain Sorin Moga and Alexandru Isar Performance Evaluation of a Tandem Queueing Network Smail Adjabi and Karima Lagha Assessment of Groundwater Quality Monitoring Network Based on Information Theory Malgorzata Kucharek and Wiktor Treichel Improving Type I1 Error Rates of Multiple Testing Procedures by Use of Auxiliary Variables. Application to Microarray Data Maela Kloareg and David Causeur

589

Author Index

597

605 613

62 1

630 636 645

653

CHAPTER 1 Stochastic Processes and Models

An approach t o Stochastic Process using Quasi-Arithmetic Means Etienne Cuvelier and Monique Noirhomme-Fraiture Institut d’Informatique (FUNDP) 21, rue Grandgagnage, 5000 Namur, Belgium (e-mail: e c d i n f 0 . f undp. ac .be, mnoQinf0 . f undp .ac .be)

Abstract. Probability distributions are central tools for probabilistic modeling in data mining. In functional data analysis (FDA) they are weakly studied in the general case. In this paper we discuss a probability distribution law for functional data considered as stochastic process. We define first a new kind of stationarity linked t o the Archimedean copulas, and then we build a probability distribution using jointly the Quasi-arithmetic means and the generators of Archimedean copulas. We also study some properties of this new mathematical tool. Keywords: Functional Data Analysis, Probability distributions, Stochastic Process, Quasi-Arithmetic Mean, Archimedean copulas.

1

Introduction

Probability distributions are central tools for probabilistic modeling in data mining. In functional data analysis , as functional random variable can be considered as stochastic process, the probability distribution have been studied largely, but with rather strong hypotheses , [Cox and Miller, 19651, [Gihman and Skorohod, 19741, [Bartlett, 19781 and [Stirzaker, 20051. Some processes are very famous like Markov process [Meyn and L, 19931. Such a process has the property that present is not influenced by all the past but only by the last visited state. A very particular case is the random walk, which has the property that one-step transitions are permitted only to the nearest neighboring states. Such local changes of state may be regarded as the analogue for discrete states of the phenomenon of continuous changes for continuous states. The limiting process is called the Wiener process or Brownian motion. The Wiener process is a diffusion process having the special property of independent increments. Some more general Markov chain with only local changes of state are permissible, gives also Markov limiting process for continuous time and continuous states. The density probability is solution of a special case of the Fokker-Planck diffusion equation. In preceding work [Cuvelier and Noirhomme-Fraiture, 20051 we used copulas to model the distribution of functional random variables at discrete cutting points. Here, using the separability concept, we can consider the continuous case as the limit of the discrete one. We will use quasi-arithmetic means in order to avoid copulas problem when considering the limit when the number 2

Approach to Stochastic Process using Quasi-Arithmetic Means 3

of cuttings tends to infinit,y. In section 2 we define the concept of distribution of functions and recall the notion of separability. In section 3 we propose to use the Quasi-arithmetic mean in conjunction with an Archimedean generator to build a probability distributions appropriate to the dimensional infiuite nature of the functional data. And in section 4 we study the properties of this new mathematical tool.

2

Distribution of a functional random variable

Let us recall some definitions that will be useful in the following paper.

Definition 1. Let (a,A, P ) a probability space and V a closed real interval. A functional random variable (frv) is any function from V x 0 + R such for any t E V ,X ( t , .) is a real random variable on (a, A, P ) . Each function X ( . , w ) is called a realization. In the following we will write for S ( . , w ) , and for X ( t , .). X,can be considered as a stochastic process.

x,

We study, here, the measurable and bounded functions.

Definition 2. Let 2, a closed real interval, then & ( D ) is the space of real measurable functions u(t) defined on a real interval V such that

-

Definition 3. Let f , g E C 2 ( D ) . The pointwise order between f and g on

V is defined as follows : &E

V >f ( t ) I y ( t )

f

I D 9

(2)

Definition 4. The functional cumulative distribution function (fcdf) of a f r v X on & ( D ) computed at u. E & ( D ) is given by : FX,D(U) = PIX

5 D

I.

(3)

Definition 5 . A frv is called separable if there exists in V an everywhere countable set I of points { t i } and a set N of f 2 of probability 0 such that for an arbitrary open set G c V and an arbitrary closed set F c JR the two sets { w : X ( t , w ) E F, Vt E

G)

{ w : X ( t ,w ) E F, Vt E G n I

}

differ from each other only on the subset N . The set 1 is called the separability set [Gihman and Skorohod, 19741. The space C 2(D ) is a separable Hilbert space. In the following we suppose that any realization of X is in Cz(V).

4 Recent Advances in Stochastic Modeling and Data Analysis

Definition 6. Two frv X , ( t , w ) and X z ( t . w ) (t E V , w E 0)are called stochastically equivalent if for any t E V P(X1(t,w)# X,(t.w))

=0

(4)

The interest of separability comes from the following theorem .

Theorem 1 (J.L. Doob). Let X and J’ be metric spaces, X be separable, y be compact. A n arbitrary random function X ( t , w ) , t E X with values in Y is stochastically equivalent to a certain separable random function.

3

The QAMM and QAMML distributions

In this section we build a sequence of sets that converge toward a separability set of V and at each step we define a probability distribution. Let n E N, and { t y , . . . t;}, n equidistant points of V such that t ; = inf(V) and t; = sup(D), and Vi E (1,. . . , n - 1) we have = A t . Let the two - tll = following sets

n n

A , ( ~ )=

{w 6 R : ~

( t p ,5 ~u(t:)} )

i=l

A(u) = { W E fl : i 5.0 U } We will use the following distribution to approximate the fcdf (3): P[An(u)] = H ( u ( t ; ) >... , ~ ( t : ) )

(5)

where H ( . , . . . , .) is a joint distribution of dimension 11. In previous works (see [Diday, 20021, [Vrac et al., 20011, [Cuvelier and Noirhomme-Fraiture, 20051) the Archimedean copulas were used for the approximation with small value of n. Let us recall the definition and property of copulas.

Definition 7. A copula is a multivariate cumulative distribution function defined on the n-dimensional unit cube [0, 11, such that every marginal distribution is uniform on the interval [0, 11 :

C : [O, 11,

+

[O, 11 ( ~ 1 ,. .. ) z L ~H) c ( ~ 1 ,; . .,

~ n )

The power of copulas comes from the following theorem (see [Nelsen, 19991).

Theorem 2 (Sklar’s theorem). Let H be an n-dimensional distribution function with margins F l , ..., F,. Then there exists an n-copula C such that for all 3: E R” , H ( x ~. ., . , ~ n = ) C ( F l ( x l ) ... ,

)

Fn(Xn)).

(6)

If F1, ...,F, are all continuous, then C is unique; otherwise, C is uniquely determined on Range of F1 x ... xRange of F,.

Approach to Stochastic Process using Quasi-Arithmetic Means 5 Before using copulas, we define a function that gives the distribution of the values of X,for a chosen t E V.

Definition 8. Let

a frv. We define the surface of distributions as follow :

G (tj Y) = PIX,5 Y1

(7)

We can use various methods for determining suitable G for a chosen value of t . Thus for example, if is a Gaussian process with mean value p ( t ) and standard deviation a ( t ) ,then we can use the cdf from N ( p ( t ) ,a ( t ) ) . In other cases we can use the empirical cumulative distribution function to estimate G:

In the following we will alway use this function G in conjunction with a function u of Cz (V): G [t,u ( t ) ] .So, for ease the notations, we will write G [t;u] = G [t,u ( t ) ] .If we use the preceding expression in conjunction with ( 6 ) , then (5) become :

P[A,(u)]= C ( G [ t ; ; u ]..., , G[tt;~])

(9)

An important class of stochastic process is the class of stationary processes. A stochastic process is said to be strictly stationary [Burril, 19721 if its distributions do not change with time; i.e. if for any t l ,..., t , E V and for any h E V ,the multivariate distribution function of . . , Xtn+h)does not depend on h. We propose here a more wide stationary property.

Definition 9. A stochastic process is said copula stationary if V t l , ..., t , E V and for any h E V ,the copula of ...,Xtnth)does not depend on h, i.e. its copula does not change with time. Let us notice that, if we deal with true functional data, realizations of a stochastic process X,we can suppose that there is always the same functional relation between X, and X t for any value s., t E V.If a frv is also a copula stationary stochastic process, then we call it a copula stationary frv. There is an important class of copulas which is well appropriate for copula stationary stochastic processes : the class of Archimedean copulas.

Definition 10. An Archimedean copula is a function from [0, 11, to [ 0 , 1 ]

where

4,called the generator, is a function from [0,1]to [0,ca]such that:

0

4 is a continuous strictly decreasing function,

0

d(0) = co and 4 ( 1 ) = 0,


6

Generator Uom. of 0

mame

Clayton to - 1 -1 - In ea--',+t , Frank , Gumbel-Hougaardl (- lnt)@ I -I

0

= 4-1 is completely monotonic on t in [0, m[ and for all k.

10,

0>0 B >o 0 2 1

m[ i.e.

( - ~ ) ' $ + ( t ) 2 0 for all

Notice that the k-dimension margins of (10) are all the same, and this for any value of 1 5 k 5 n. If X is a copula stationary f r v then expression (9) can be written :

PIAn(u)l =

+

L1

)

4 (G [tl;.I)

(11)

Table 1[Nelsen, 19991 shows three important Archimedean generators for copulas. The distribution (11) with the Clayton generator was already used for clustering of functional data coming from the symbolic data analysis framework (see [Vrac et al., 20011 and [Cuvelier and Noirhomme-Fraiture, 20051). Unfortunately the above limit is almost always null for Archimedean copulas when n + 0;) (see [Cuvelier and Noirhomme-Fkaiture, 2007])! Proposition 1. I f f o r u E &(D) : G ( t ;u ) < 1, Vt E

V ,then

Another objection t o the use of this type of joint distribution is something which we could call volumetric behavior. Definition 11. A function ZL E p , written Q p , if

&(D)is called a functional quantile of value

G(t;Qp) = p , Vt E 2)

(13)

The functional quantile Qp can be seen as the level curve of value p . Now let us remark that for a functional quantile :

P[An(Qp)]= lij

r

C 4(G [tl; i=l

Qpl)

1

= $ .(

' 4( P ) ) < P

+

(14)

And it is easy to see that, if n < m then $J [m#~ (p)] < [nq5(p)],and thus, the more we try to have a better approximation for a functional quantile of value p , the more we move away from reference value p toward zero. A simple way to avoid these two problems is to use the notion of quasi-arithmetic mean, concept which was studied by [Kolmogorov, 19301, [Nagumo, 19301 and [Aczel, 19661.

Approach to Stochastic Process using Quasi-Arithmetic Means

7

Definition 12. Let [ . , b ] be a closed real interval, and n E No. A quasiarithmetic mean is a function M : [a,b]" + [a,b] defined as follows:

where q5 is a continuous strictly monotonic real function. We show below that if we use the generator for Archimedean copulas in (15), we define a cumulative distribution function built from one-dimensional distributions. It's easy to prove the following lemma.

Lemma 1. Let n E No,F be a one dimensional cdf, and q5 a generator of Archimedean copula , then

is also a cdf. In various situations one can apply increasing transformations to the data without destroying the underlying dependence structure. This is classical in multivariate extreme value theory. And for these kind of transformation the copulas does not change. Proposition 2. Let n E No,{Fill5 i 5 ,n} be a set of one dimensional cdf, and q5 a generator of Archimedean copula , then

is a multivariate cdf. Proof. By the above lemma we have that the functions F ; ( z ) are cdf, and as q5 is an "Archimedean generator" so expression (10) is a copula, and thus 3 (Cy=lq5(F;(xi))) is a multivariate cdf. 0 We call the distributions given by the expression (17) the Quasi-Arithmetic Mean of Margins ( Q A M M ) distributions. Now if we use a Q A M M distribution in expression (5) :

then for each n E N we have an approximation, and the limit of the above expression is not always null.

8


Definition 13. Let : X be a frv, u E C z ( D ) ,G its Surface of Distributions and 4 a generator of Archimedean Copulas. We define the Quasi-Arithmetic M e a n of Margins L i m i t ( Q A M M L ) distribution of by :

In fact transformation (16) can be seen like giving an importance to the margins in proportion with the length of an interval [tl,tl+,l]in the approximation of Fxz,(u).

4

QAMML properties

First it is easy to see that the Q A M M L distribution preserves the functional quantiles.

Proposition 3. If I ; ~ C , D ( Q ~=) P

Qp

E

&(D) is a functional quantile of ualue p , t h e n

Now, what is the difference between the quasi-arithmetic mean of margins and the classical mean? Let

p

=

1

L

-

11~Il

G [t;u]d t

and let us define the function tp [t;u] = G [t;u] - p . Thus we can use the following Taylor’s approximation for all t (recall that 0 5 p , G [t;u] 5 1):

4 (G It;74) = 4 (P+ t p ( t I;). = 4(P)

+ 4’(P)tp It;I . + &”(P)-

t;

[t;u]

+ 0 (t; [t;4)

(21)

and then

and so like $ is a decreasing function, 4’’ ( p ) 2 0 and as u a r ( t ) 2 0 we can see that FX ~ ( udecreases ) with the variance of the differences between the function u-knd the quantile function associated to the value of the arithmetic mean of G along u.. And so the Q A M M L distribution is equal to the arithmetic mean only in the case of quantile functions.

Approach to Stochastic Process using Quasi-Arithmetic Means

9

Conclusion I n this paper we do n o t propose a new method in Functional Data Analysis but a new Probabilistic tool. Like in t h e real case, we can hope t h a t this tool can be used for analysis of functional data, like in mixture decomposition, statistical tests, ... Moreover, several ways to improve t h e tool exist. B y example let us note that the QAMML definition (see (19)) uses a uniform distribution over D: other distributions could be considered (see [De Finetti, 19311).

References [Aczel, 19661J Aczel. Lectures o n Functional Equations and Their Applications. Mathematics in Science and Engineering. Academic Press, New York and London, 1966. [Bartlett, 1978lM S Bartlett. An introduction to stochastic processes. Cambridge University Press, Cambridge, 1978. [Burril, 1972lC W Burril. Measure, integration and probability. McGraw-Hill, NewYork, 1972. [Cox and Miller, 1965lD R Cox and HD Miller. The theory of stochastic processes. Methuen, London, 1965. [Cuvelier and Noirhomme-Fraiture, 2005lE. Cuvelier and M. Noirhomme-Fraiture. Clayton copula and mixture decomposition. In ASMDA 2005, pages 699-708, 2005. [Cuvelier and Noirhomme-Fraiture, 2007lE. Cuvelier and M. Noirhomme-Fraiture. Classification de fonctions continues l’aide d’une distribution et d’une densit dfinies dans un espace de dimension infinie. In Extraction et gestion des connaissances EGC’2007, pages 679-690, 2007. [De Finetti, 1931lB De Finetti. Sul concetto di media. Giornale dell’ Instituto Itialiano degli Attuari, 2:369-396, 1931. [Diday, 2002lE. Diday. Mixture decomposition of distributions by copulas. In Classification, Clustering and Data Analysis, pages 297-310, 2002. [Gihman and Skorohod, 197411I Gihman and A V Skorohod. The theory ofstochastic process. Die grundleheren der mathematischen wissenschaften in einzeldarstellungen. Springer, Berlin, 1974. [Kolmogorov, 1930lA Kolmogorov. Sur la notion de moyenne. Rendiconti Accademia dei Lincei, 12(6):388-391, 1930. [Meyn and L, 19931s P Meyn and Tweedie R L. Markov chains and stochastic stability. Communications and Control. Springer-Verlag, New York, 1993. [Nagumo, 1930lM Nagumo. Uber eine klasse der mittelwerte. Japan Journal of Mathematics, 7:71-79, 1930. [Nelsen, 1999lR.B. Nelsen. An introduction to copulas. Springer, London, 1999. [Ramsay and Silverman, 200515 0 Ramsay and B W Silverman. Functional Data Analysis. Springer Series in Statistics. Springer, New-York, 2005. [Stirzaker, 2005lD Stirzaker. Stochastic processes and models. Oxford University Press, Oxford, 2005. [Vrac et al., 2001lMathieu Vrac, Edwin Diday, Alain Chkdin, and Philippe Naveau. Mklange de distributions de distributions, d6composition de melange de copules et application B la climatologie. In Actes du VIIIkme congre‘s de la Socie‘te‘ Francophone de Classijication, pages 348-355, 2001.

The quantum generator of translations in a fraction-dimensional manifold Paulius MiSkinis1>2 NORDITA, Nordic Institute of Theoretical Physics, Blegdamsvej 17, DK-2100 Copenhagen 0, Denmark Vilnius Gediminas Technical University Faculty of Fundamental Sciences Department of Physics Sauletekio Ave.11, LT-10223, Vilnius-40, Lithuania (e-mail: paulius .miskinismfm. vtu.It)

Abstract. In the case of the quantum generalization of random processes with the Hurst index H # 1/2, expression for the quantum Hermitian generator of translations and its eigenfunctions are proposed. The normalization constant has been determined and its relation to the operator of momentum is shown. The interrelation between the momentum and the wave number has been generalized for the processes with a non-integer dimensionality a. Keywords: quantum generator of translations, fractional derivative, non-Markovian process, Hermitian operator.

1

Introduction

The physical basis for the existence of quantum mechanics comprises a series of phenomena described by the mathematical theory of the Wienerian processes. The non-Markovian stochastic process is a natural generalization of the Brownian motion or the Wiener stochastic process [l]. The foundation for this generalization is the theory of stable probability distributions developed by Ldvy [2]. The most fundamental property of such distributions is the stability in respect to addition, in accordance with the generalized central limit theorem. Thus, from the probability theory point of view, the stable probability law is a generalization of the well-known Gaussian law. The nonMarkovian processes are characterized by the Hurst index H , which' takes values 0 < H < 1. At H = 1/2 we have the Gaussian process or the process of the Brownian motion. The non-Markovian stochastic process with stable Ldvy distributions is widely used to model a variety processes such as anomalous diffusion [3], turbulence [4], chaotic dynamics [5], plasma physics [6], financial dynamics [7], biology and physiology [8] (for recent references see e.g. [9-111). 10

The Quantum Generator of Translations 11

The constantly increasing number of experimental facts in various fields of knowledge related to classical non-Wienerian processes evokes a natural desire to “close” the commutative diagram shown in Fig. 1. QM

lh WP

a FQM

lh

a SLP

Fig. 1. Schematic representation of interrelations of Wienerian processes (WP), stable LBvy processes (SLP), quantum mechanics (QM) and fractional quantum mechanics (FQM). and, at least formally, to consider the possible existence of a quantum analogue of a more narrow class of phenomena related to stable Levy processes, the so-called fractional quantum mechanics (FQM) [12-141. Unfortunately, these works are not aimed at a thorough analysis of the properties of the quantum operator of momentum. The lack of such analysis results in some accuracies of even gross blunders while formulating FQM (see Conclusions). The present note offers a brief discussion of one of the crucial issues related to FQM, which is the one-dimensional operator of momentum. Like in usual quantum mechanics (QM), one-dimensional problems are a kind of excess idealization. Nevertheless, they may be used for elucidating the fundamental features of FQM. One-dimensional problems arise while considering the three-dimensional evolutionary equation in which the interaction potential depends on a single coordinate. This fact allows, with the aid of a corresponding factorization, to move to a simpler one-dimensional evolutionary equation. The purpose of this paper is formulation in the explicit form quantum expression of the one-dimensional operator of momentum for the fractional probability processes with a non-integer dimensionality a and the investigation of the interrelation between the quantum generator of translations and the operator of momentum.

2

Fractional quantum operator of momentum

The classical definition of momentum in QM follows from the invariance of the Hamiltonian of the quantum system H with respect to the infinitesimal displacements Sx [15]. Under such transformation, the wave function +(z) turns into the function

+(x here

+ b z ) = +(x) + C5xax+ = (1+Sxa,)

+(z),

axis the differentiation operator over the space variable x.


ax$

However, it may turn out that does not exist, but there exists the so-called fractional derivative a:$ in which the order of the derivative a may be both an integer and a fractional number. For the function determined on the whole real axis R, the right and left derivatives of the order a are derived

where [a]and { a } are the integer and the fractional parts of the parameter a. For the bilateral derivatives to exist, it is sufficient that $ ( x ) E C[a](52), where C[a](f2) is a set of continuously differentiated functions of the order [a]determined on the domain 52 [El. Another peculiarity related to the operator of momentum is the expansion of the wave-function $ ( x ) into a Taylor series by fractional powers [17,18]

c [a1

$(x) =

(x - xO)=+n + Rn(x)

7

(3)

n=O

2'

where are numerical coefficients and R,(x) is he residual term, which provides a better approach to the initial function. n all such cases, determination of the quantum operator of momentum should be specified. It is reasonable to suppose that the momentum operator should be proportional to the fractional derivative:

E

@ = ca;$(x)

(4)

here C is a certain coefficient of proportionality. For a -+ 0, we must obtain a usual quantum operator of momentum, 5 = Thus, in FMQ we always deal with two kinds of limit transitions: 1) ti -+ 0, when we shift to classical mechanics, and 2) a -+ 0, when we turn to usual QM (see Fig. 1). The kind of the coefficient C in the expression for the momentum (4) is best defined if on the whole real axis we consider a flat wave of the form

-%ax.

there fi is the Planck constant and 1, is a certain peculiar scale of the length of the nonlocal process under consideration. Let us impose a requirement for the momentum operator (4) to obey the eigenvalue equation p$ = p $ . Applying the property of the fractional derivatives, d;ienx = neenx (ReK > 0), we obtain that

c = (--i)QfiZga--l.

(6)

For the values observed in QM to be real, the corresponding operators should be Hermitian. It is easy to see that the quantum operator of momentum (4) with the constant C from (6) is non-Hermitian. In order to obtain


a Hermitian operator of momentum, to the type (4) operator we will add a Hermite-conjugated operator $+ ; then, the momentum operator determined in this way

-

here is the symbol of transposition, will be clearly Hermitian. Indeed, the momentum operator (7) fi = ($+ + $ - ) / 2 will be Hermitian because of the idempotency of the operation of Hermitian conjugation ((@+)+= $) and the structure of the operator itself ($- = (@+)+).On the other hand, employing the rule of fractional integration by parts, +m

p* =

1,

'p*$+$dx = 2

/

+m

cp* (@-+@+)$d3: = P

-m

,

(8)

we directly see that the momentum operator is Hermitian for the different functions of state 'p and $. Thus, we obtain that the operator (7) is Hermitian and its eigenvalues on the whole real axis are flat waves of the (5) type. Like in the classical case, the eigenvalues of the momentum operator do not belong to the class L 2 ( R ) . Therefore, they don't describe the physically realizable states of the quantum particle. These eigenfunctions should be regarded as the basic functions, which comprise the complete system of functions.

3

Wave function normalization

To determine the constant normalization A in the expression for the flat wave ( 5 ) , we will take that

(9)

s

This is a particular case of the conventional condition $p,$pdx = 6(p' - p ) for p' E 0. Using the property of the &function, we shall obtain that

Let us specify the peculiarities of such normalization. Firstly, generally speaking, the amplitude is a complex magnitude; secondly, it depends on the eigenvalue of the momentum p . Only when a -+ 1, as the case should be,

A + l / m . Inasmuch as the physical sense applies not to the amplitude itself but to IAI2, the complex nature of A does not contradict unitarity. However, because of the complex nature of the amplitude we may get an impression that we deal with a damping wave; however, actually there is no damping, because A # A ( z , t ) . Besides, the same conclusion results from analysis of


the dispersion expression of the corresponding Hamiltonian. The dependence A = A(p)is not a matter of principle and may be avoided by a suitable choice of the normalization condition. For instance, under condition

dx = 6 ( ~-' K ) ,

$:t$&

(11)

the dependence A(p) is absent. Another important circumstance should be noted as regards the type of the momentum operator. Transition to momentum representation is not a Fourier transformation. Momentum representation should be understood in the sense of f-representation:

4

Translation operator

Lastly, let us derive the formula to express, through the momentum operator

6, the parallel translation operator in space to any finite (not only infinitesimal) distance. From the definition of such an operator it follows:

Fz+$(x) = $(x

-a),

F?-$(x)

= $(x

+ b) ,

(13)

In this case, a and b denote the values of finite displacements but not the coordinate ends of the interval. Expanding the function $(x - a ) in the neighbourhood of the point x into a Taylor series by fractional powers as in (3) and employing the expression for the "right-hand" and "left-hand" parts of the momentum operator,

$7 = - i h i ; - l q ,

6:

= irii;-l@,

(14)

we obtain that

where EE is the generalized exponential function. These are exactly the finite displacement operators we have been searching for. For a -+ 1, for Tz+ we obtain that

The expression T?- could be obtain by substituting in equation (16) a -+ -b and 6 : -+ p?.


5

The shift operator in a superspace of fractional dimension

The modern theories of quantum field specifically unite physical fields of various statistics. Physical fields are unified into super-multiplets which are irreduceable 2,-graded Poincar6 group representation. 8, In the algebra of this group] alongside the translation generators P, in the ususal space, translation generators in the odd-dimensional sector the - i (70) 8, is present. generator Q = In this part of the paper we consider a generalization of the shift generator Q for the case of fractional dimensional of the odd-dimensional sector of the Minkowski superspace. In this case, differently from the shift generator in the even sector, we cannot rely upon physical intuition, because the odd Grassman variables and parameters are unobservable. To obtain the fractional shift operator Q, we shall employ the Balakrishnan formula, konown form functional analysis [191: N

&

By analogy with P,,

Q = lim E-' (T'

-

I ) , where TE= eEQ ,

E'O

(19)

is an element of Grassman's algebra. We take E" as a matrix operator for which, taking into account its Grassmanian nature,

E

( I + E)"

=I

+

a!&

(20)

is an exact equality. Threfore

For translation operators in the even sector, formula (21) leads to the already discussed operator of the fractional Marchaud derivative a;. Therefore, expression (21) may be regarded as the odd Marchaud derivative. Note here that, differently from the usual fractional Marchaud derivative] integration in (21) over E takes place in the whole space of spinor determination] but not within certain area. Formula (21) allows generalization for the c a e of a! > 1.


6

Conclusions

Note that the classical restriction on the smoothness of the wave function $(x) E C2([a,b ] ) does not hold here. The restriction on $(x) follows from the continuity equation; however, in the case of fractional dimension we can show that the condition of continuity is changed, and the limitation on $(x) is reduced to $(x) E d a ] ( [ ab, ] ) . Another note pertains to the structure of the momentum operator. It seems highly significant that the momentum operator consists of two parts the right and left displacements. In classical fractional mechanics, it is quite possible to limit ourselves to one of these two components - $+ or p-. In the quantum case it is impossible, because the full operator of momentum is a Hermitian. The limit transition h + 0 for { a }# 0 means transition to classical fractional mechanics. However, the form of the momentum operator undergoes no qualitative change: fi = $ (p+ fi-), i.e. it consists of two parts, each being proportional to its one-sided derivative. For linear evolutionary equations of classical (not quantum) fractional mechanics this type of structure of the momentum operator may be simplified if p = @+ or p = Ij-. However, here additional considerations are necessary. For the nonlinear fractional evolutionary processes it is impossible in principle, because fi+ = 6- is the condition of smoothness. Note here, too, that all results for the Hermitian operator of momentum are valid in the case of the Riesz quantum derivative: RD,"$(x) K [(-ia)T

+

+

(8) ]: $(x). From the definition of n there follows an interrelation between the momentum and the wave number:

Q: + 1, tc + Ic, and the expression (22) turns into p = hlc. The appearance of the characteristic length scale of I , and the power dependence of the quantum particle momentum on the wave number directly indicate the fractional character of quantum mechanics. Thus we have the Hermitian quantum operator of momentum (7) with the eigenfunctions (5). This allows us to construct the quadratic form of the Hamiltonian H K p 2 instead of the power form H K D,Jp la, and the Hermitian Hamiltonian instead of non-Hermitian proposed in [12,13],and the unitarian Hamiltonian instead of non-unitarian proposed in [14].

for

References [l]M. Kac. Probability and Related Topics in Physical Sciences. Chap. IV., Interscience, New York, 1959.


[2] P. LBvy. The'orie de 17Addition des Variables Ale'atoires GauthierVillaws, Paris, 1937. [3] B.B. Mandelbrot, J.W. van Ness. Fractional Brownian Motionx, Fractional Noises and Applications. SIAM (SOC. Ind. Appl. Math.) Rev. 70(4): 422-437, 1968. [4] J. Klafter, A. Blumen and M.F. Shlesinger. Stochastic pathway to anomalous diffusion. Phys. Rev. A 55(7): 0081-9085, 1987. [5] G.M. Zaslavsky, Fractional kinetic equation for Hamiltonian chaos. Physicu D 76(1-3): 110-372, 1994. [6] G. Zimbardo, P. Veltro, G. Basile and S. Principato. Anomalous d i f i sion and Lvy random walk of magnetic field lines in three dimensional turbulence. Phys. Plasmas 2(7): 2653-2163, 1995. [7] R.N. Mantega and H.E. Stanley. Scaling behaviour in the dynamics of an economic index. Nature 376(6535): 46-48, 1995. [8] B.J. West and W. Deering. Fractal physiology for physicists: Lvy statistics. Phys. Rep. 246(1-2): 1-100, 1994. [9] A. Le Mehautk et al. (Eds.) fiactional differentiotion and its applications. Books on Demand, Norderstedt, 2005. [lo] R. Metzler and J . Klafter. The restaurant at the end of the random walk: rkcent developments in the description of anomalous transport by fractional dynamics. J. Phys. A: Math. Gen. 37: R161-R208, 2004. [ll] G.M. Zaslavsky. Chaos, fractional kinetics, and anotaloub transport. Physics Reports 377: 461-580, 2002. [12] E. Lutz. Fractional Transport Equations for LBvy Stable Processes, Phys. Rev. Lett. 86(12): 2208-2211, 2001. [13] N. Laskin. Fractional quantum mechanics. Phys. Rev. E 62(3): 31353125, 2000. [14] M. Naber. Time fractional Schrodinger equation. J. Math. Phys. 45(8): 3739-3356, 2004. [15] H. Kleinert. Path Integrals in Quantum Mechanics, Statistics and Polymer Physics. World Scientific, Singapore, 1990. [16] V. Marchaud. Sur les dBrivBes et sur les differences des functions de variables rkelles. J. math. pures et uppl. 6(4): 238-235, 1927. [17] S.G. Samko, A.A. Kilbas and 0.1. Marichev. Fractional Integrals and Derivatives. Theory and Applications. Gordon and Breach, Amsterdam, 1993. [18] P. MiSkinis. Nonlinear and nonlocal integrable models. Vilnius, Technika, 2003. [19] E. Hille, R. Phillips. Functional analysis and semigroups. Cambr. Univ. Press. 1961.

Cause of Stock Return Stochastic Volatility: Query by Way of Stochastic Calculus Juho Kanniainen’ Tampere University of Technology Institute of Industrial Management P.O.Box 541, FI-33101 Tampere, Finland (e-mail: juho. kanniainen(0tut. f i) Abstract. This study uses stochastic calculus to investigate the causes of the stock return stochastic volatility. The study aims to advance new explanations of stochastic volatility that hold also if the firm is unleveraged, and if the level of uncertainty about future business conditions does not change. Using the dividend discount model, I show that stock return volatility is admittedly stochastic if future dividends are affected by more than one stochastic state variable. Morever, I study how the mappings of state variables are related to stochastic return volatility. This study also investigates the effects of the discount rate and state variables’ mutual correlation on the level of stock volatility and its fluctuation, finding substantial relationships therein. Keywords: Stochastic calculus, Stochastic volatility, Geometric Brownian motion,

Dividend discount model.

1

Introduction

It is by now a widely accepted observation that the volatilities of individual stocks and aggregate stock markets are not constants, but change stochastically over time. The literature has repeatedly presented sophisticated statistical models t o describe stochastic volatility, but the question of why stock return volatility varies remains open. [Schwert, 19891 presents an extensive analysis of the relation between market volatility and economic activity, confirming Officer’s [Officer, 19731 earlier results that market volatility is higher during economic downturns. This has also been justified by [Hamilton and Lin, 19961, who found that economic recessions are the single most important factor explaining market volatility, accounting for about 60 percent of its variation. Further, changes in volatility are also related t o financial and operating leverage, personal leverage, interest rates, inflation rates, money growth, industrial production growth, trading volume, trading halt, and program trading (see, for example, [Black, 19761, [Christie, 19821, [Mascaro and Meltzer, 19831, [Schwert, 19891, [Schwert, 19901). Stock volatility is also stochastic if a stock has option characteristics; that is, the stock can be viewed as an option on the leveraged firm’s assets (see [Merton, 1974]), and the firm may have numerous growth options. Overall in the literature, the usual explanation of stochastic volatility is either corporate leverage or a change in the level of uncertainty about future macroeconomic conditions. 18

Cause

0.f

Stock Return Stochastic Volatility 19

This study advances new explanations of stochastic volatility by way of stochastic calculus. My explanation does not challenge the earlier explanations but rather complements them. This paper investigates the significance of the above observation by suppressing the other possible causes of stochastic volatility. The assumption of risk-neutrality suppresses leverage effects, the assumption of a constant interest rules out the possibility that the randomness of stock volatility is driven by varying interest rates, and the assumption of constant state variable volatilities eliminates any change in the uncertainty about future macroeconomic conditions. I also examine the effects of state variables’ mutual correlations and the discount rate on the level of volatility and its variation.

2

Model

To suppress the leverage effect, we assume that the risk-neutrality, and hence the discount rate, denoted by p, equals the risk-free interest rate. Moreover, by assuming a constant interest, we rule out the possibility that the randomness of volatility is driven by varying interest rates. The stochasticity of stock volatility arise solely from the dividend process. We denote the dividend stream by {Dt,, D t z , . . . , Dt,}, where dividends occur at (known) times t l , t 2 , . . . ,t,. The stock price is assumed to equal the cumulative present value of its expected dividends, n

v(t)=

Et [Dt,]exP [ - p ( t k

-

t)],

(1)

k=l

Suppose that a discrete dividend stream consists of n discrete dividends. All investors are assumed to monitor the processes of state variables and continuously revising their beliefs regarding to expected dividends. Each dividend is driven by an m stochastic state variable. In addition, the dividends’ state variable vectors need not be equal; that is, the next year’s dividend can be driven by state variables different from those of the dividend paid after five years. Consequently, we specify the dividend process in rather general terms and allow that, for example, each dividend depends on the interest rate and inflation, but that the two first dividends also depend on the oil price, whereas later dividends depend on the price of biodiesel instead of oil. Overall, the matrix X ( t ) E Rmxn, represents the dividend stream information apprearing at time t , and vector X k ( t ) represents the information associated with the stock dividend k (that will be paid at time t k ) . For all k = 1 . . .n, Dtk : Rm ++ R+ is a known mapping, and { X k ( t ) ;t 2 0}, X k E Rm the state variable vector of the dividend Dtk, is a linear diffusion defined on a complete filtered probability space (62, F x k ,P ) . We assume that for all i = 1 . . .m, k = 1 . . . n, { x i k ( t ) ;t 2 0) evolves according to the stochastic differential equation

20


where Wik is a standard Wiener process with the instantaneous correlations dWikdWjl = pik,jldt for all i , j = 1 . . . m, k , 1 = 1 . ..n. Read the above such that Xik is the state variable i of the dividend k (a dividend that will be paid at time t k ) . According to It8’s lemma, the stock price must itself follow the It6 process:

where i ,j = 1 . . .m, k , 1 = 1 . ..n, and where a is the expected price appreciabik E d W i k makes the stock price behave stochastically. tion. The term The rest of the study focuses on just this term.

xi,k

3 3.1

Causes of Stochastic Return Volatility Mappings of State Variable

For simplicity the analysis, let us suppose temporarily that there is only one stochastic state variable driving all the dividends, and that the state variable of the dividends { X ( t ) ;t 2 0 } itself follows the geometric Brownian motion:

d x(t)=ex(t)dt

+ O x ( t ) d w ( t ) , x > 0,

(4)

where 0 E R and 0 E R+ are constants. We will relax assumptions later when we will consider multiple state variables and the alternative processes of state variables. Now the asset price process is in the form

dV = adt

V

av + C-vx -dW ax

= adt

+ OEdW,

where a equals the expected price appreciation and E is the elasticity of the asset price to the state variable. If E is stochastic, so is stock return volatility O&, too. Proposition 1. Suppose that dividends’ state variable x follows geometric Brownian motion. If the elasticity of the asset price t o the state variable is a constant, say (, the asset price V = cxc,where c is a positive constant.

Proof. The solution of the differential equation positive constant. 0

%$ = C is V = cxc,where c is a

Remark 1. The above proposition also says that if the asset price is not in the form of V = cxc,then the elasticity of the asset price to the state variable, and hence also stock return volatility, is not constant.

Cause of Stock Return Stochastic Volatility 21 Example 1 . The above means that the assumption of constant return volatility is invalid, if, for example, the asset price V = clxb + C Z , where b, c 1 , c ~are constants. This is the case if the dividend of time t k ,

If x follows (4), then Now

E

is admittedly stochastic:

av x ax v

--

-

Clxbb C1XbfCZ1

Remark 2. Example 1 has an important economic interpretation. Suppose that the price of output good fluctuates according to (4) and that the dividend of time t k is X ( t k ) - a , where a denotes the production costs of the period. Then return volatility equals cr& and is stochastic. Example 2. Another example of stochastic return volatility could be V = f l ( x ) f z ( x ) . . , where f i , i = 1 , 2 , . . . , is the function of x. This is a case if, for example, the dividend of time tk, Dt, = lnX(tk), when E is stochastic even if discount rate p equals zero and number of dividends n equals one:

+

+.

0

We have now considered transformations that result in return volatility to be stochastic. What kind of dividend transformations do produce constant return volatility? The next proposition clarifies this.

Proposition 2. Suppose that dividends ' state variable x follows geometric Brownian motion. If the dividend of time t has a form Dt, = where a and b are constant numbers, then the elasticity of the asset price to the state variable, E , is constant and equals b. Proof. The proposition is a direct implication of Example 1 with a2 = 0 in (5). 0 3.2

Multiple State Variables

We assume that Dtk

=D(Xk(tk)) =Xlk(tk)

+X2k(tk)

f ' . . f X'TLk(tk)?

(6)

k = 1. . .n, and that the state variables of the dividends { X ( t ) ;t 2 0 } follow the geometric Brownian motion with constant drifts and volatilities :

+

dXilc(t) = QikXilc(t)dt aircXik(t)dWik(t),

(7)

22


where x i k > 0 for all i, k . Now stock price as a cumulative present value of expected dividends can be expressed as

where i = 1. . .m, k = 1. . .n, and where p is the stock’s instantaneous discount rate. The stochastic term in (3) takes the form

i = 1. . . m, k = 1 . . . n. How should we interpret this? What can we say now about stock volatility? Proposition 3. Suppose that there are several (more t h a n one) state variables and that (6), (?’),and (8) hold. T h e n the stock return volatility is a constant if and only zf all the state variables are drived by perferctly correlated W i e n e r processes and volatilities of state variables, cr&, are equal t o each other.

Proof. Without loss of generality, suppose that all dividends depend on the same two state variables] X 1 and X 2 . Technically, for all k = 1.. . n D ( x l ( t k ) ~ X Z ( t k= ) )x l ( t k ) + x Z ( t k ) . These variables follows the geometric Brownian motions driven by the Wiener processes W l ( t )and Wz(t), d W I ( t ) d W z ( t ) = p l z d t . Let W’ be an independent Wiener process with respect to Wl and W2. Then we can write d W z ( t ) = p l z d W l ( t ) d m d W * ( t ) . If aik = cr and plz = 1, then we can then write (9) as follows:

+

=VadW(t),

where d W l ( t ) = d W z ( t ) = d W ( t ) . Thus, the stock price evolves with constant volatility. We must still prove that if return volatility is constant] then all the state variables are drived by perferctly correlated Wiener processes and volatilities of state variables, are equal to each other. Let us do this by showing that if processes are not perefectly correlated or if volatilities are not the equal, return volatility is not constant. Again, suppose that all dividends depend on the same two state vasia,hles, XIand X,. The price of the stock is n

V ( t )=

C { X l ( t )exp((o1

-

P)(tk - t ) )+ ~ 2 ( texp((e2 ) - P)(tk - t ) ) > .

Cause of Stock Return Stochastic Volatility 23 We can then write (9) as follows:

where

i = 1 , 2 . Let &t)dl@(t) = +l(t)dWl(t) + + z ( t ) d W 2 ( t ) , when we find that

$(t)= J$l(t>Z

+ + z ( t > Z + 2PlZ+l(t)+Z(t).

(11)

Therefore, stock price diffusion takes the form

+

dV(t)= aV(t) J(t)V(t)dW(t).

(12)

8

Clearly, the stock volatility is admittedly stochastic even if either 01 = 02 or plz = 0. Note that if return volatility depends positively on the correlation PlZ. 0 I have illustrated the stochastic volatility numerically by generating sample paths for state variablesand supposing that equations (7), (10) and (11) determine state variables, stock price, and volatility. The illustration can be found at http://www.tut.fi/"kanniain/ASMDA/illustration.pdf I assumed two state variables, X , ( t ) and X z ( t ) with a constant correlation and with constant volatilities and drifts. Moreover, I assumed that n dividends will be paid at times (years) 1 , 2 , 3 .. . , and I simulated the time interval ( 0 , l ) (the first year). The illustration shows that stock return volatility may vary considerably over time. Moreover, it also examines the effect of the correlation plz on the price paths concluding that if the correlation increases, the stock price becomes more volatile, and the volatility curve moves upward. If the correlation decreases, also stock volatility decreases, and the volatility curve drops. This is in line with our analytical observation of a positive relation between plz and The result is analogous with portfolio diversification: if st.ate variables do not correlate mut'ually, their fluctuations eliminate each other. Moreover, the numerical illustration argues that the greater (less) the correlation, the greater (less) the stock volatility but with less (greater) fluctuation. T h e result is also quite intuitive. As we can see from the above equations, the mutual proportions of the state variable values clearly affect the level of stock volatility. Suppose that dividends depend on two variables, and that the volatility of the first variable, n1, is less than the volatility of the second variable, 02. If the value of the second state variable, Xz, increases in proportion to the value of the second state variable, X I ,

4.

24


the more volatile state variable takes room, and stock volatility increases. Similarly, if X I increases in proportion to X z , stock volatility decreases. Obviously also the greater (less) the correlation between state variables, the less (greater) their mutual proportion changes over time. Therefore, if state variables evolve with different volatilities and a low, or even negative mutual correlation, stock volatility may fluctuate substantially. We could interpret this result economically in that if a business depends on homogenous (heterogeneous) factors (in the sense of statistical dependency), its volatility is high (low) and does not (does) vary much. I also illustrated the effect of discount rate to volatility and its fluctuation. Here the effect of the discount rate on stock volatility can be either positive or negative. In addition, the discount rate has a great effect on how stock volatility varies over time.

3.3

Alternative Characterizations of State Variables Processes

Finally, we consider alternative processes of the state variables. Suppose that a i k ( t , x i k ) = q i k ( x i , - x&(t)) and b i k ( t , X z k ) = U i k , in which case the state variable { X i k ( t ) ;t 2 0}, z = 1 . . . m, k = 1 . ..n evolves according to the stochastic differential dxik(t)

= qik

( X i k - x i k ( t ) ) dt

+uikdWik(t),

where V i k , X i k , and f f i k are constant numbers for all z, k . This is the so-called mean-reverting process. Now, because for some 7 > t (see, for example, [Dixit and Pindyck, 1994, p. 741)

+ ( X i k ( t ) - X i k ) eXP ( - q i k ( T - t ) ) the stock price with mapping Dt, = X I k ( t k ) f X 2 k ( t k ) + ' . + X m k ( t k ) V ( t )= { x i k + ( x i k ( t ) - X i k ) exp (-'%k(tk t ) ) }exp ( - P ( t k t)). Et

[ x i k ( T ) ] = XEk

1

'

x

-

-

i,k

Stock return volatility is now unquestionably stochastic since the stochastic term in (3) takes the form

Note that stock volatility would be stochastic even if f f i k = cr and W i k ( t ) = W ( t )for all i = 1 . ..m, k = 1 . .. n, t > 0, in which case volatility would be equal to

Remark 3. The reason here is that the variance rate does not grow with x. Therefore, if dividends are driven by such a process, stock volatility remains unquestionably stochastic with linear mappings of state variables.

Cause of Stock Return Stochastic Volatility 25

4

Conclusions

The starting point of this study was that dividends are driven by state variables and that investors monitor the state variables and continuously revise their beliefs regarding to stock price. The paper studied how the mappings of state variables are related stochastic return volatility. Moreover, also multiple state variables were explored. The main result is that the stock return volatility is admittedly stochastic if future dividends are affected by more than one stochastic state variable. Thus, the paper affirms the invalidity of the geometric Brownian motion as models of stock price. We observed that the correlation between state variables has an effect on volatility dynamics according to the greater (less) the correlation, the greater (less) the stock volatility, with less (more) fluctuation over time. In addition, we found that the discount rate affects volatility and its fluctuation positively or negatively.

References [Black, 19761F. Black. Studies of stock price volatility changes. Proceedings of the 1976 meetings of the American Statistical Association, Business and Economic Statistics Section, pages 177-181, 1976. [Christie, 1982lA. A Christie. The stochastic behavior of common stock variances: Value, leverage and interest rate effects. Journal of Financial Economics, 10:407-432, 1982. [Dixit and Pindyck, 1994lA. K. Dixit and R. S. Pindyck. Investment Under Uncertainty. Princeton, New Jersey, Princeton University Press, 1994. [Hamilton and Lin, 19961J . D. Hamilton and G. Lin. Stock market volatility and the business cycle. Journal of Applied Econometrics, 11:573-593, 1996. [Mascaro and Meltzer, 1983lA. Mascaro and A. H. Meltzer. Long- and short-term interest rates in a risky world. Journal of Monetary Economics, 12:485-518, 1983. [Merton, 19741R. C. Merton. On the pricing of corporate debt: The risk structure of interest rates. Journal of Finance, 29:44%470, 1974. [Nielsen, 19991L. T . Nielsen. Pricing and Hedging of Derivative Securities. Oxford University Press, 1999. [Officer, 1973lR. R. Officer. The variability of the market factor of the new york stock exchange. Journal of Business, 46:434-453, 1973. [Schwert, 19891G. W. Schwert. Why does stock market volatility change over time? Journal of Finance, 44:1115-1153, 1989. [Schwert, 19901G. W. Schwert. Stock market volatility. Financial Analyst Journal, 46:23-34, 1990.

On a class of backward stochastic differential equations and applications to the stochastic resonance Romeo Negrea' Department of Mathematics Politehnica University of Timisoara P-ta Victoriei 2, 300006 Timisoara ROMANIA (e-mail: negreabath. uvt .ro)

Abstract. In this paper, we intend to prove a result on the solvabillity of a class of backward stochastic differential equations by the McShane type. We will considered some more general conditions by the coefficient functions and prove a result on the existence and the pathwise uniqueness by the Nagumo type, extended the Athanassov results ([Athanassov, 19901) for the ordinary differential equations. In final, we study the control of some electronic circuits in the presence of the stochastic resonance. Keywords: backward stochastic differential equations, adapted solution, pathwise uniqueness, belated integrals, stochastic resonance, signal-to-noise ratio.

1

Introduction

Noise in dynamical system is usually considered a nuisance. However, in certain nonlinear systems, including electronic circuits and biological sensory systems, the presence of noise can enhance the detection of weak signals. The phenomenon is termed stochastic resonance and is of great interest for electronic instrumentation. The essential ingredient for the stochastic resonance is a nonlinear dynamical system, which typically has a period signal and noise at the input and output that is a function of the input as well as the internal dynamics of the system. The nonlinear component of the dynamical system is sometimes provided by a threshold which must be crossed for the output to be changed or detected. A nonlinear system is essential for stochastic resonance to exists, since in a system that is well characterized by linear response theory, the signal-to-noise ratio at the output must be proportional to the signal-to-noise ratio at the input. Engineers have normally sought to minimize the effect of noise in electronic circuits and communication systems. Today, however, it is acknowledged that noise or random motion is beneficial in breaking up the quantization pattern in a video signal, in the dithering of analog to digital converters, in the area of Brownian ratchets, etc. 26

O n a Class of BSDE 64 Applications to the Stochastic Resonance

27

In the field of control, we usually regard y(.) as an adapted control and

x(.) as the state of the system. We are allowed to choose an adapted control y(.) which drives the state x(.) of the system to the given target X at time t = 1. This is so-called reachability problem. So in fact we are looking for a pair of stochastic processes { x ( t ) y(t), , 0 2 t 5 1) with values in R x R which is Ft-adapted and satisfies the above equation. Such pair is called an adapted solution of the equation and was introduced in [Pardoux and Peng, 19901. In the electronic circuits with stochastic resonance we have some nonlinear system, therefore, must to be considered more general coefficient functions for a backward stochastic differential equations - non-linear and maybe non-lipschitz. On the other hand, in practice, the additional input noise (which is responsible for the resonance) and the external perturbations belong to a large class of noises, then the stochastic calculus by the McShane type (introduced in [McShane, 19741) can be a better approach for the mathematical models of some phenomena. In this paper, we intend to prove a result on the solvabillity of a class of backward stochastic differential equations by the following type:

where { z j ( t ) , 0 5 t 5 l}, j = 1 , 2 , . . . , T is a stochastic process defined on the probability space (R, .F, P ) with the natural filtration {Ft,0 5 t 5 l} and X is a given F1-measurable random variable such that EIX12 < 00. Moreover, f is a mapping from R x [0,1] x R x R to R which is assumed to be P @I B @I B \ B-measurable, where P is the 0-algebra of Ft-progressively measurable subsets of R x [0,1]. Also g is a mapping from 0 x [0,1] x R to R which is assumed to be P x B \ B-measurable. All stochastic integrals are belated or McShane integrals. A version of the equation (1)which has the most important mathematical properties is its canonical form:

2

Preliminary results

In 1974, E.J. McShane, introduced so called belated integrals and stochastic differentials and differential systems which enjoying the following three

28


properties: inclusiveness, consistency and stability. McShane's calculus had proved to very valuable in modeling and is finding applications in physics, engineering and economics ([McShane, 19741, [McShane, 19751). Therefore, in this section we recall some specifically results of the McShane stochastic calculus. Let (Q,.F,P) be a complete probability space and let { F t , 0 5 t 5 u } be a family of complete a-subalgebras of .F such that 0 5 s 5 t 5 a then

Fs c 3 t .

Every process denoted by z with differents affixes will be a real valued second order stochastic process adapted to {Ft,0 5 t 5 u } (i.e. z ( t ) is .?-measurable for every t E [0, u]) and

IE[(z(t)- z(s))"/Fs]l i K ( t - S )

5 a, m = 1,2,4, for a positive constant K having as. continuous sample functions (and we say that the process satisfies a a.s., whenever 0 5 s 5 t

K-condition). It is known (see [McShane, 19741) that if f : [O,u] + L2 is a measurable process adapted to the Ft and if t ---t Ilf(t)lI is Lebesgue integrable on [O, a], then if z1 and z2 satisfy a K-condition, the McShane integrals

exist and the following estimates are true ra

where C = 2Ku4

3

+Ki.

Theoretical results

Some results in weaker conditions than the Lipschitz conditions for the coefficient functions of a backward stochastic differential equation are given in [Mao, 19951, [Lepeltier and San Martin, 19981, [Lepeltier and San Martin, 20021, [Kobylanski, 20001 or [Constantin, 20041, [Negrea, 20061 and many others. A very interesting applications was given, for example, in [Constantin, 20051, [Ladde and Sambandham, 20031, [Srinivasan and Udayabhaskaran, 19821, In [Athanassov, 19901 is proved a uniqueness theorem of Nagumo type for the Cauchy problem generalizing several known uniqueness theorems and sufficient conditions to guarantee the convergence of the Picard successive

O n a Class of BSDE &' Applications to the Stochastic Resonance

29

approximations for the ordinary differential equations. Stochastic generalizations of the results of Athanassov for (forward) stochastic differential equations are given in[Constantin, 1995][Constantin, 19961 [Constantin, 19981, [Constantin and Negrea, 20041 or [Negrea, 20031 and others. But it is known that the uniqueness of the solution of an initial value problem of Cauchy and the convergence of successive approximations are logically independent, i.e. the uniqueness of the solution does not ensure the convergence of successive approximations nor is the converse true. For the equation (4) we consider the following hypotheses: i) gj and its partial derivations are P 8 B 8 B measurable functions; ii) gj(.,O,O) E M 2 ( ( 0 , 1 ) , R ) ,g ( . , O , O ) E M2((0,1),R)and %(.,O,O) E

M 2 ( ( 0 ,I ) , R); iii) there exists u(t) a continuous, positive and derivable function on 0 < t 5 1 with u(0)= 0, having nonnegative derivative u'(t) E L([O,l]),with u'(t) + 00, t + Of such that if p is any one of the functions g j , relation (2)) we have

hjk

(as in

for al121,~2,yl,y2E R, 0 I t 5 1, a n d A = (r+1)C2+(r2+1)C2+(2r2)C2.

iv) with the same functions u ( t )as above,

for all z , y l , y 2 E R,0

< t 5 1.

In [Negrea and Caruntu, 20071 is given the result on the existence and uniqueness of the solution of the equation (4). More exactly, we have the following theorem:

Theorem 3.1. Let be gj and h j k satisfying the above hypotheses for any j , k = 1,.. . , T and X E L2(f2,F1,P, R),then there exists a unique pair (z,y) E M 2 ( ( 0 , 1 ) , R )which satisfy the equation (4) in the canonical form for [S, 11, for any positive constant 6. Now, we will present some results on the stability properties of the solution of the equation (1). We consider families of backward stochastic integral equations

30


whit X E A- a open and bounded set

c R".

In similar way as in [Negrea, 20061 and using the properties (3), (4), (5) and (6) it easy to prove the following two results:

Theorem 3.2.If, f o r any X E A, the coeficient functions fx a n d g x satisfy the hypothesis (i)-(iv), then the family (4) has a unique solution (xx,y x ) E W(0,11). Moreover, if IXx,, - Xxl -+ 0, m -+ 00, then, we have that

Theorem 3.3.h the hypothesis (i)-(iv), XxoI = 0 , then lim Izx

X-Xo

-

we have that if limx+xo IXx

xxo12 = 0 , and lim Iyx X-Xo

-

-

yxOl2= 0 ,

on any compact subset of the domain (0,1], where cp is any functions f or g.

4

Applications to the Stochastic Resonance

In this section we study some applications of a general class of McShane backward stochastic differential equations described above. For a better understanding of the notion of stochastic resonance, we consider the following mechanical experiment - a ball is pushed with a force F on a non-linear path, and let to consider the following three cases: a) Non-perturbed case - if the force F is strong enough and there are not external perturbations, then the ball arrives at the final point. b) Perturbed case - if the there are some external perturbations, the ball do not rise "'the first pick", therefore is increase the force F and the ball pass to the first hill, but if there are more external perturbations the ball can not pass to the second ball and it not return to the initial point for a new increasing of the force F .

O n a Class of BSDE 8 Applications to the Stochastic Resonance

31

c) Perturbed case with a "jumper ball" - the force F pushes not a jumper ball, a ball with a "noise". If there is a "good noise", the ball pass to the all external perturbations and arrive to the final point. A model of one-dimensional nonlinear system that exhibits stochastic resonance is the damped harmonic oscillators with the Langevin equation of motion: rnji.(t)

+ yk(t) =

dU(x) dx

--

+ fi[(t)rnji.(t) + yk(t) = --dU(x) + dE[(t) dx

This equation describes the motion of a particle of mass m moving in the presence of friction . The restoring force is expressed as the gradient of some bistable or multi-stable potential function U(x). In addition, there is an additive stochastic force [(t) with intensity D , and, in generally, it supposed as been a white Gaussian noise. U ( x ) = - a$+b$ In the case of bistable system, in [Gammaitoni et al., 19981, [Harmer et al., 20021 the potential function is a simple symmetric function and, adding a period signal and considering case of time dependent system U(x, t) = U ( z )- Axsin(w,t) = -a-

22

2

+ b-x44

-

Axsin(w,t)

where A and are the amplitude and the frequency of the periodic signal, respectively. More models for the electronic circuits with a stochastic resonance are discrete. In general, they used a state space time series models given by the bellow expressions z n = f (xn-1) un Yn = h(xn) wn

{

+

+

where f and h are non-linear functions and un (input noise) and wn (external noise) are respectively Gaussian or non-Gaussian noises. About these models there is a simple observation: we can not say exactly if the external perturbation is present just at the discrete moments. A continuous model appears as more adequate. This approach is possible just using the theory of stochastic differential equations. Because, the control of the input noise is the key of the benefit of the stochastic resonance, we use the backward stochastic differential equations. The problem, which frequency appear in practice, is the value of initial state 20. This value is "proposed" but in some non-standard external conditions, this make an discontinuity of the simple path of the process {z(t)} After this time (t=O), the adapted control {y(t)} works and we have a continuity of {x(t)} . Therefore, is necessary to consider some general coefficient functions and good stability properties of the solutions. On the other hand, the stochastic resonance make possible a control of the electronic circuits in some external stochastic perturbations by controlling the adapted process {y(t)} (see [Gammaitoni et al., 19981, [Harmer et al., 20021, [Calvo and Chialvo, 2006]), but the control appears as the result of filtering.

32


About above models we make the following remarks: a) in choosing the type of stochastic processes that we shall use a models of the noises we meet a dilemma. On the one hand, there is no physical basis for considering function with properties such as non-differentiability and on the other hand, the involved process must be of a kind that we can manage mathematically. b) in almost approaches, the disturbance is supposed to be Gaussian white noise, but, is known that a electrical system cannot support more than some limited current or voltage difference without destruction. Therefore, the Gaussian white noise which is a special derivate of Brownian motion (in the sense of distributions) must replace or combine with some other noises, more smoothed. For some numerical results, we consider a Chua’s type electronic circuit (see [Harmer et al., 20021, [Beglund and Gentz, 20021)with a discontinuity at time t = 0. More precisely we consider the coefficient functions f ( t , z, y) =

--t$z3+tkz+sin(27rt/T),

g(t,z,y) = y

4-t:, ~ ( 0 )

= 0,

T =X

= 1, z1 =

W.

For the sequences of successive approximations of { ~ ( t and ) } {y(t)} we have the following simple path XI

Y

Fig. 1. Simple path of the signal and the noise. In conclusions we can see the effect of the adapted process in the bistable regime of the state process { ~ ( t ) } .

References [Athanassov, 19901Z. S. Athanassov. Uniqueness and convergence of successive approximations for ordinary differential equations. Math. Japonica, 53:351-467, 1990. [Beglund and Gentz, 2002lN. Beglund and B. Gentz. A sample path approach to noise-induced synchronization: Stochastic resonance in double well potential. Ann. Appl. Probab., 12:1419-1470, 2002. [Calvo and Chialvo, 200610. Calvo and D.R. Chialvo. Ghost stochastic resonance in an electronic circuit. Inter. Journ. of Bifurcation and Chaos, 16:731-735, 2006. [Constantin and Negrea, 20041Gh. Constantin and R. Negrea. An application of schauder’s fixed point theorem in stochastic mcshane modeling. J. Fixed Point Theory, 5:37-52, 2004.

O n a Class of BSDE

tY Applications to the Stochastic Resonance

33

[Constantin, 1995lA. Constantin. Global existence of solutions for perturbed differential equations. Annuli d i Mat. Pura ed Appl., IV:237-299, 1995. [Constantin, 1996lA. Constantin. On the existence and pathwise uniqueness of solutions of stochastic differential equations. Stochastic and Stochastic Reports, 56~227-239, 1996. [Constantin, 1998lA. Constantin. On the existence and uniqueness of solutions of mcshane type stochastic differential equations. Stoch.Ana1. Appl., 16:217-229, 1998. [Constantin, 20041Gh. Constantin. The uniqueness of solutions of perturbed backward stochastic differential equations. J. Math. Anal. Appl., 300:12-16, 2004. [Constantin, 20051Gh. Constantin. An application of the stochastic mcshane’s equations in financial modeling. In J. Janssen and P. Lenca, editors, Proceedings of Applied Stochastic Models and Data Analysis, pages 917-925, 2005. [Gammaitoni et al., 1998lL. Gammaitoni, P. Hanggi, P. Jung, and F. Marchesoni. Stochastic resonance. Rev. of Modern Physics, 70:223-287, 1998. [Harmer et al., 2002lG.P. Harmer, B.R. Davis, and D. Abbot. A review of stochastic resonance: Circuits and measurement. IEEE Trans. on Instr. Measur., 51:299309, 2002. [Kobylanski, 20001M. Kobylanski. Backward stochastic differential equations and partial differential equations with quadratic growth. The Annals of Probability, 28:558-602, 2000. [Ladde and Sambandham, 20031 G.S. Ladde and M. Sambandham. Stochastic versus Deterministic Systems of Differential Equations. Marcel Dekker, Inc., New York, 2003. [Lepeltier and San Martin, 19981J.-P. Lepeltier and J. San Martin. Existence for bsde with sperlinear-quadratic coefficient. Stoch. Stoch. Reports, 63:227-240, 1998. [Lepeltier and San Martin, 2002]J.-P. Lepeltier and J. San Martin. On the existence or non-existence of solutions for certain backward stochastic differential equations. Bernoulli, 8:123-137, 2002. [Mao, 1995lX. Mao. Adapted solutions of backward stochastic differential equations with non lipschitz coefficients. Stoch. Proc. and Their Appl., 58:281-292, 1995. [McShane, 1974lE.J. McShane. Stochastic Calculus and Stochastic Models. Academic Press, New York, 1974. [McShane, 1975lE.J. McShane. Stochastic differential equations. J.Multivariate Analysis, 5:121-177, 1975. [Negrea and Caruntu, 2007lR. Negrea and B. Caruntu. On certain class of backward stochastic differential equations by mcshane type. Analele Univ. Timisoara, Ser. Mat.-Inf, 45:225-234, 2007. [Negrea, 20031R. Negrea. On the uniqueness and convergence of successive approximations for a class of stochastic differential equations. Analele Univ. Bucuresti, Ser. Mat., 52~225-234, 2003. [Negrea, 2006lR. Negrea. On the existence and uniqueness of solutions for certain backward stochastic differential equations. to appear, 2006. [Pardoux and Peng, 19901E. Pardoux and S.G. Peng. Adapted solution of a backward stochastic differential equation. Systems & Control Letters, 14:55-61, 1990. [Srinivasan and Udayabhaskaran, 1982lS.K. Srinivasan and S. Udayabhaskaran. Modeling and analysis of dynamical systems subject to discontinuous noise processes. Journ. Math.Phys.Sci., 16:415-430, 1982.

CHAPTER 2

Distributions

An application of the Extended Waring distribution to model count data variables Jose Rodriguez Avi' , Antonio Conde S h c h e z l , Antonio Jose Shz-Castillo', and Ma Jose Olmo Jinienez' Department of Statistics and Operations Research University of Ja6n 23071 Jakn, Spain (email: jravioujaen.es) Abstract. An extension of the univariate Waring distribution is proposed as an alternative to the U G W D for overdispersed count data in those cases in which the

parameter estimates do not allow the properties of the U G W D to be used, such as the partition of the variance. Specifically, this model is applied to study the set of variables number of hotels per municipality in the Autonomous Region o n Andalusia (Spain) f r o m 1990 t o 2003. Keywords: Waring distribution, Number of hotels, Overdispersed count data.

1

Introduction

To study the impact of tourism in the economy, an interesting variable can be the number of hotels per municipality within a particular geographical framework. Specifically, the aim of this work is to find a probabilistic model to describe, in an appropriate way, the variable X, number of hotels per municipality in the Autonomous Region of Andalusia (Spain) from 1990 to 2003. These data have been compiled from the data bank of the System of Multi territorial Information of Andalusia (SIMA, by its Spanish acronym, [IEA, 20061). The observed frequencies for the 14 variables appear in Table 1.

For all years we obtain discrete count variables that have a minimum in 0 which is also a modal value, whereas the maximum is a very high value

that increases from one year t o another. A summary of the main descriptive characteristics of the 14 variables is shown in Table 2. The aggregation index shows that the 14 variables have strong overdispersion, so the variability is not only due t o randomness. Firstly, an important factor that influences the observed variability is the size of the municipality. However, this is not the only deciding factor. Thus, cities with great tourist importance, such as Mdaga or Granada, have greater number of hotels than Sevilla, the capital of Andalusia and the most populated. The same happens with coastal towns compared with inland towns, etc. Other factors to 35

36

Recent Advances in Stochastic Modeling and Data Analysis - - 91 92 13t i2: i0: 62 62 71 18 24 27 10 12 16 13 13 12 6 8 9 4 7 9 4 2 3 0 1 3 2 2 0 0 3 2 1 0 2 3 1 2

qEi - 90 0 1 2 3 4

5 6 7 8 9 10 11 12 13 14 f15

1 0

7

-

1 1 1 0 9 8 -

-

-

)3 99 14 !7 15 14 7 8 4 4 0 2 2 1 1 1 9 -

9 4 88 34 17 14 14 11 6 2 5 3 1 3 1 1 0 9 -

3 5 81 36 19 14 15 9 8 3 5 3

36 87: 39

- - 37 - 98 - 99 16i 15; l4( 3 1 O i .0; 3 1 32 29 35 14 16 20 18 14 14 10 13 9 9 4 5 8 8 11 9 3 4 5 6 4 4 5 5 5 2 1 1 1 3 4 3 3 1 2 3 1 0 3 2 3 1 2 1 2 1 3 3

- )1 12 - 13 1 2 01 8E 24 24 38 i2 17 5 1 19 13 14 9 11 16 9 4 5 7 LO 6 1 5 6 8 7 6 4 4 4 3 4 4 3 1 1 2 2 4 3 1 1 0 2 3 0 0 1 3 0 0 1 1 0 4 3 2 2 11 13 14 16 LO 18 - LO - 10 - - - 17 - -

-

00 i2: -2: 41 16 13 6 9

Table 1. Observed frequencies for the hotel data from 1990 to 2003

be taken into account are the financial development of the municipality, the business initiative, etc.

Tf=EF

Range Aggregation in( 1991 767 0.8292 13.6227 1992 768 0.9440 16.1492 1993 768 0.9544 16.3899 1994 769 1.0117 19.5980 1995 769 1.0351 16.8479 1996 769 1.1235 19.5309 1997 769 1.1599 20.0537 1998 769 1.1274 18.6001 1999 769 1.1938 20.1484 2000 769 1.3199 22.8808 2001 770 1.3844 23.8886 2002 770 1.4909 27.0162 2003[77011.61561 31.0912

48 54 57 67 56 59 60 59 60 67 68 71 78

16.4287 17.1072 17.1730 19.3714 16.2766 17.3840 17.2892 16.4982 16.8775 17.3353 17.2556 18.1207 19.2444

Table 2. Descriptive summary

An Application of the Extended Wuring Distribution 37

2

Negative binomial model

Since data have overdispersion, a model that could be proposed to describe them is the negative binomial model, N B , that can be seen as the mixture

= Poisson ( A ) AA G a m m a ( a ,k ) This means that the average number of events is not constant but follows a gamma distribution. In this case, the total variance can be split into two components:

+

V ~( X T ) = E A ( V ~( T X l n ) ) V W A( E ( X ~ A .) )

(2)

The second term of (2) represents the variability due to sources of variation that cause overdispersion, whereas the first term reflects the variation due to randomness. The N B ( a , p ) distribution may also be seen as belonging to the Gaussian hypergeometric distribution family, G H D , ([Johnson et al., 20051) because its probability generating function (p.g.f.) is given by:

Table 3 includes the results obtained using the N B model for the hotel data. The estimates have been obtained by the maximum likelihood method.

NB(a,p) x' p-value log-likelihood 1990 (0.0799, 0.0996) 13.1479 0.0686 -626.2 -686.0 1991 (0.0881, 0.0961) 10.2694 0.2466 -760.1 1992 (0.1032, 0.0986) 10.1610 0.3366 -768.3 1993 (0.1057, 0.0997) 10.1809 0.3360 -803.6 1994 (0.1144, 0.1016) 16.7173 0.0533 -827.3 1995 (0.1214, 0.1050) 15.2138 0.0827 -860.0 1996 (0.1242, 0.0995) 15.0755 0.1293 -880.2 1997 (0.1287, 0.0999) 15.7031 0.1085 1998 (0.1418, 0.1117) 33.0476 0.0000 -890.1 1999 (0.1466, 0.1094) 28.6091 0.0014 -919.4 2000 (0.1632, 0.1100) 46.2239 0.0000 -928.6 2001 (0.1786, 0.1143) 46.7014 0.0000 -1026.0 2002 (0.1847, 0.1102) 49.3454 0.0000 -1065.7 2003 (0.1945, 0.1074) 56.3517 0.0000 -1112.1

Year

Table 3. MLE, X2-goodness of fit test and log-likelihood for the hotel data using

the N B model

It should be emphasized that only in the first years - until 1997 - the fits are acceptable whereas in the last years the fits are very bad.

38


3

Fit by a U G W D

The univariate generalized Waring distribution, U G W D , with parameters ( a , k and p ) has also been proposed as a valid model in the case of overdispersion ([Johnson e t al., 20051). It is also a distribution that belongs to the G H D family since its p.g.f. can be expressed as:

This distribution can be obtained from a N B ( a , p ) when the parameter p is not constant but follows a BetaI(k, p) distribution, that is

( p')

P o i s s o n ( A ) A Gamma a , -

Beta1 ( p ,k ) .

(5)

The results obtained fitting the U G W D model and applying the maximum likelihood method ([Rodriguez Avi et al., 20031) are shown in Table 4. It can be seen that the values of the X2-statistic are lower than those provided by the N B fit for all years, and values for the log-likelihood are greater. Year

-

UGWD(a,k,p)

XL 1990 (0.8137, 0.2707, 1.1087) 7.0334 1991 (5.8800, 0.1296, 1.8618) 3.5696 1992 (5.5370, 0.1562, 1.8669) 4.2428 1993 (4.9568, 0.1651, 1.8020) 3.8855 1994 (0.8623, 0.4219, 1.1977) 7.8420 1995 (1.3074, 0.3299, 1.2557) 6.2382 1996 (3.7016, 0.2151, 1.6272) 5.7105 1997 (0.5972, 0.6822, 1.1723) 5.8098 1998 (0.4958, 1.0040, 1.3141) 13.677( 1999 (0.6326, 0.7998, 1.2907) 7.9839 2000 (0.7727, 0.7727, 1.3377) 8.2827 2001 (0.8332, 0.8331, 1.4081) 7.3912 2002 (0.8636, 0.8581, 1.4044) 10.191: 2003 (0.8957. 0.9029. 1.4158) 12.873(

p-value log-likelihood AIC 0.2182 0.7347 0.7515 0.7929 0.2449 0.5122 0.6796 0,6685 0.0572 0.3340 0.4064 0.4951 0.3352 0.1684

-621.71 -681.18 -753.75 -761.52 -794.30 -819.98 -851.34 -872.30 -875.96 -904.80 -963.00 -1003.7 -1042.0 -1084.1

76.66 83.95 84.12 83.81 90.12 91.52 91.52 93.46 96.35 96.35 112.04 106.40 116.67 119.10 ~

Table 4. MLE, X2-goodness of fit test, log-likelihood and AIC for the hotel data using the U G W D model

Comparing Table 3 and Table 4, it is clear that the U G W D model improves the fits that provides the N B model in all cases. Since the U G W D and the NB are not nested models the Vuong test ([Winkelmann, 20031, pg 109) has been carried out as an extension of the likelihood ratio test. The null hypothesis indicates that the two models are

A n Application of the Extended Waring Distribution

39

equivalent versus the alternative hypothesis that the U G W D model is preferable t o the N B one: ffo : E (LUGWD(%GWD) - L B N ( @ B N=) )0

H i :E(LUGWD(~GGW -D L B) N ( @ B N > ))0

(6) (7)

The test statistic is given by

where

is the variance of the individual differences in log-likelihood divided by n. So, under the null hypothesis, the L R statistic converges in distribution to a standard normal distribution. The results of the test for each year appear in Table 5. The p-value is less than 0.1 in all years, except the year 1990, and decreases, specially, from 1998. Therefore, the U G W D model may be considered, in general, to be better than the N B model. Year Statistic 1990 1.0206 1991 1.6696 1992 1.7954 1993 1.7986 1994 1.4743 1995 1.3395 1996 1.8765 1997 1.2916 1998 2.2484 1999 2.2799 2000 2.8479 2001 3.1587 2002 3.2674 2003 3.6413

7-valuc 0.1537 0.0475 0.0363 0.0360 0.0702 0.0902 0.0303 0.0982 0.0123 0.0113 0.0022 0.0007 0.0005 0.0001

Table 5. Vuong test between the U G W D and the N B

Many studies have demonstrated the main properties of the U G W D ([Irwin, 19681, [Irwin, 19751, [Xekalaki, 1983133, [Xekalaki, 1983a], among others). Particularly, in order to interpret the data variability, the decomposition

40


a s a mixture of three distributions provides a partition of the variance into three components ([Irwin, 19683):

when p > 2. The first of these is related to random factors, the second to the variability due to external factors that affect the population (liability), and the third to the differences in the internal conditions of the individuals (proneness) ([Irwin, 19681). However, p^ < 2 in the 14 years studied, so the fitted distributions have infinite variance.

4

Fit by an Extended Waring distribution

Trying to solve the aforementioned problem, the hotel data will be modelled by a tetraparametric G H D of type I ([Rodriguez Avi et al., 20031) whose p.g.f. has the expression

with a , P , y > 0, 0 < A 5 1 and 0 < y. This distribution is called Extended Waring distribution, E W D , ([Rodriguez Avi et al., 20061) and includes, as limit cases, the U G W D ( u ,k , p ) when a = a , P = k , y = a k p and X = 1 and the N B when y is equal to a or /3 and X = 1 - p . Moreover, like the U G W D , the E W D can be expressed as a mixture. Specifically:

+ +

(

Poisson ( A ) A G a m m a a , A

- )' 1-X(1-P)

)

A P

G B e t a (y - a - PIP,a , A ) ,

(12) where G B e t a (y - a - p, p, a , A) denotes a generalization of the Beta distribution whose density function is

with y > /3 > 0, a > 0 and 0 < X 5 1 ([Rodriguez Avi et al., 2007al). A similar continuous distribution appears when [Rodriguez Avi et al., 2007bl develop a generalized beta binomial distribution, GBB. Once again the MLE are computed. A summary of all the fits is shown in Table 6 where it can be seen that the results are very similar to those obtained by the U G W D . Taking (12) into account, the variance can also be split into three compcnents: randomness, liability and proneness. X < 1 in all cases, so the fitted model has finite variance and the three components can be computed as it is shown in Table 7. h

A n Application of the Extended Waring Distribution 41 AIC Year EWD(%P;7;4 xs p - d u e log-likelihood -620.22 77.46 1990 (0.3682,0.3672,1.4041,0.9736) 8.0857 0.1516 84.41 -680.94 1991 (2.6136,0.1464,3.8107,0.9821) 3.8337 0.5726 -753.58 86.50 1992 (3.1237,0.1707,4.5378,0.9870) 4.6440 0.5902 -761.28 85.16 1993 (2.2502,0.1919,3.5235,0.9840) 4.0410 0.6711 -793.15 89.73 1994 (1.1743,0.2906,2.4461,0.9852) 5.3354 0.5016 -818.62 91.55 1995 (0.5193,0.4994,1.7711,0.9742) 4.1903 0.5554 91.07 1996 (0.9950,0.3260,2.1547,0.9791) 4.9794 0.6625 -850.67 -870.40 92.96 1997 (0.9950,0.3415,2.1749,0.9792) 3.2506 0.8609 96.30 -875.10 1998 (0.6287,0.6284,2.2798,0.9859) 12.5357 0.0510 -903.93 96.30 1999 (0.6481,0.6469,2.3196,0.9863) 6.7368 0.4568 112.95 -962.42 2000 (0.7239,0.7239,2.5868,0.9905)12.7388 0.0787 -1003.1 107.82 2001 (0.7964,0.7955,2.8477,0.9930) 7.7258 0.3574 118.41 -1041.7 2002 (0.8238,0.8235,2.9047,0.9936)12.3208 0.1375 120.73 -1083.9 2003 (0.8692,0.8685,3.0368,0.9952) 12.8017 0.1189

Table 6. MLE, X2-goodnessof fit test, log-likelihood and AIC for the hotel data using the E W D model

G

Year Total variance b Randomnes 3.12% 3.69% 2002 40.4804 3.94% 2001 35.2344 2000 31.4221 4.19% 4.76% 1999 25.0598 4.88% 1998 23.1130 22.1181 5.25% 1997 5.25% 1996 21.3852 5.87% 1995 17.5759 1994 21.4973 4.70% 1993 18.4643 5.17% 1992 5.02% 18.7985 1991 5.30% 15.6626 1990 5.55% 12.9923

6 Liabilitj '0 Pronenesz 50.96% 55.83% 56.55% 58.80% 61.34% 61.81% 50.55% 50.45% 65.94% 49.28% 64.14% 70.71% 67.28% 71.96%

45.91% 40.48% 39.31% 37.01% 33.90% 33.31% 44.20% 44.30% 28.19% 46.02% 30.69% 24.28% 27.42% 22.49%

Table 7. Partition of the variance for the hotel data

5

Conclusions

The E W D distribution is more flexible than the U G W D distribution to model overdispersed count data sets where the excess dispersion is assumed to be due to external factors (modelled by the gamma ditribution) and internal factors (modelled by the beta distribution). So, there are data sets that are adequately modelled by the E W D but not by the U G W D ([Rodriguez Avi et al., 2007al). When the fits provided by both models are acceptable and similar, the best model is the U G W D since it has less number


of parameters, according to t h e principle of parsimony. However, t h e estimation results sometimes do not allow t h e inherent properties of t h e U G W D to b e exploited, in particular, t h e partition of t h e variance. I n these cases, t h e E W D model is proposed as an excellent alternative to t h e U G W D model. This is the example included in this work.

References [IEA, ZOO6]Institute of Statistics of Andalusia IEA. System of Multi territorial Information of Andalusia (SIMA). Web access: http://www.juntadeandalucia.es/institutodeestadistica/sima/indexZ.htm, 2006. [Irwin, 196815. 0. Irwin. The generalized waring distribution applied to accident theory. Journal of the Royal Statistical Society A , 131:205-207, 1968. [Irwin, 197515. 0. Irwin. The Generalized Waring Distribution. Part I. Journal of the Royal Statistical Society A , 138:18-31, 1975. [Johnson et al., 2005lN. L. Johnson, A.W. Kemp, and S. Kotz. Univariate Discrete Distributions. Wiley, New Jersey, 2005. [Rodriguez Avi et al., 2003]J. Rodriguez Avi, A. Conde Shchez, A.J. S&ez Castillo, and M.J. Olmo Jimknez. Estimation of parameters in gaussian hypergeometric distributions. Communication in Statistics, Theory and Methods, 32:11011118, 2003. [Rodriguez Avi et al., 2007alJ. Rodriguez Avi, A. Conde Sanchez, A.J. S&ez Castillo, and M.J. Olmo Jimknez. A new extension of the waring distribution. Computational Statistics and Data Analysis, in press, 2007. [Rodriguez Avi et al., 2007blJ. Rodriguez Avi, A. Conde Sbchez, A.J. S&ez Castillo, and M.J. Olmo Jimknez. A generalization of the betabinomial distribution. Journal of the Royal Statistical Society, Series C: Applied Statistics, 56:in press, 2007. [Rodriguez Avi et al., 2006lJ Rodriguez Avi, A. Conde Shchez, A.J. S&ez Castillo, and M.J. Olmo Jimhez. Extended waring bivariate distribution. In Callej6n J . Herrerias J. Herrerias, R., editor, Distribution Models Theory, pages 221-232, 2006. Econometric Analysis of Count Data. [Winkelmann, 2003lR. Winkelmann. Springer, Heidelberg, 2003. [Xekalaki, 1983alE. Xekalaki. Infinite divisibility, completeness and regression properties of the univariate generalized waring distribution. Annals of the Institute of Statistical Mathematics, 35279-289, 1983. [Xekalaki, 1983blE. Xekalaki. The univariate generalized waring distribution in relation to accident theory: Proneness, spells or contagion? Biornetrics, 39:887895, 1983.

Estimation of simple characteristics of samples from skewed and heavy-tailed distributions Zdenek Fabi&n Institute of Computer Science Academy of Sciences of the Czech Republic Pod vodarenskou veii 2 Prague 18200 Czech Republic (e-mail: zdenekQcs.cas .cz) Abstract. Abstract. We present new characteristics of the central tendency and dispersion of data samples. They are constructed from estimates of parameters of underlying distributions and make possible an easy comparison of results obtained under different assumptions. Keywords: scalar inference function, generalized moment method.

1

Introduction

Johnson score - a scalar inference function - was introduced by [1,2] for a large class of continuous probability distributions. It was shown that the Johnson score moments exist under mild regularity conditions even in cases of distributions without mean and variance. The first moment describes the central tendency of the distribution and the reciprocal value of the second moment the dispersion of the values around the central point. It seems that whereas the mean m = J z d F ( z ) and variance g2 = z 2 d F ( z )-m2 compare the properties of distribution F with the standard (with the the normal distribution), the new characteristics have an ability t o compare distributions within parametric families even when distributions are skewed and/or heavytailed. Usually, having a n idea about the type of the underlying distribution, it is to estimate the parameters. We argued for a slight change of view: there are the sample Johnson mean and sample Johnson variance, which are to be estimated as characteristics of data samples taken from the distribution under consideration. They make possible t o compare results of estimation for various assumed distribution families parametrized by arbitrary ways.

2

Johnson score

Let us define the basic concept. Definition 1. Let F be distribution with support X = ( a ,b) C_ R and density f continuously differentiable according t o the variable. Let mapping 43

44


77 : X

+

R be defined by f

77b)=

x log(x - a ) (x - a ) log iogjb - .j

if ( a , b ) = R if-ca 6,then the process Qn converges in distribution t o a n improper limiting sequence.

Proof. We have from Wald's equation and the memoryless property of the Poisson process EEn = A+ZErF - p'bEr1 - 6. (3) Since r; is a stationary and ergodic sequence then Sn, which is generated by T; (and other i.i.d sequences), is also a stationary ergodic sequence (see


62

[Kernane and Aissani, 20061). If condition (A% - pL+6)E71 < 6 is satisfied then EEn < 0. From this and Example 11.1 of [Borovkov, 19981, there exists a stationary sequence of renovating events with positive probability for Qn. Using Theorem 11.4 of [Borovkov, 19981,-the SRS Qn is strong coupling convergent to a unique stationary sequence Q, = UnQ0, where is measurable with respect to the a-algebra generated by 7; and U is the shift transformation of random variables generated by 7; (i.e., 7i+1 = V7;). Since T; is ergodic then is also ergodic from the fact that it is an U shifted sequence. From Loynes’ scheme [Loynes, 19621 (for more details see [Gyorfi and Morvai, 20021) - it follows that the SRS is coupling convergent to the sequence Qn where QO= sup,>o V,. Since the strong coupling convergence implies the coupling then the result follows. For the instability condition, if it holds that (A% - p+6)E~; > 6 then E& > 0. It is well known that for SRS of the form Qn+l = ( Q n En)+, the condition E& > 0 implies that the SRS converges in distribution to an improper limiting sequence (see Theorem 1.7 in [Borovkov, 19761).

00

an

-

+

Remark 1. In the case of an i.i.d service times (general distribution B ) and

N ( t ) the counting process of a non-delayed renewal process having cycle lengths i.i.d.-B, then we obtain a more general result, that is for the condition of stability X%E7, - ~ N ( T ;oV,. If A%(l - p*(A-)) > A-6, t h e n the process Qn converges in distribution to a n improper limiting sequence. I

Remark 2. By assuming an i.i.d sequence for inter-arrival times of negative customers with exponential distribution with rate A- = ( l / E ~ c and ) single arrivals and services, hence ii = 1 and 6 = 1, we obtain in Proposition 1 the well known stability condition for M/M/l queue with negative customers (see [Harrison and Pitel, 19931): (A+

-

A-)(l/p+)

< 1.

(6)

In Proposition 2, if we assume an i.i.d sequence of service times with exponenA-) tial distribution with rate p+ = (l/E&), we have p*(A-) = p+/(p+ and condition 6 holds true also for this case.

+

3 Removal of Customers at the Tail of the Queue 3.1

Stationary ergodic arrivals of negative customers

The sequence of inter-arrivals of negative customers {T;} is assumed stationary ergodic and customers are removed from the tail of the queue at the epoch of arrivals t , in batches of random sizes d,, with {d,} an i.i.d sequence with mean 2.We assume that the input flows of customers, size of batches and service times are mutually independent. Let Q , embedded just before the arrival of a negative customer. The representation of Qn as a stochastic recursive sequence (SRS) is given by:

i=l We have the following result:

Proposition 3. If (A% - p+b)E~; < 2,t h e n the process - Qn i s strong-coupling convergent t o a unique stationary ergodic regime Qn such that QO = SuPn>o V n . If (A% - p+6)E~; > 2,t h e n the process Q , converges in distribution t o a n

improper limiting sequence. 3.2

Stationary ergodic service times

We assume now that the service times S, are stationary and ergodic and the inter-arrivals of negative customers are i.i.d exponentially distributed with

64


rate A-. The process Qn is embedded immediately after the end of the ( n - 1 ) s t service time. The process Qn satisfies the following recursion:

We have the following proposition:

Proposition 4. If (X+si - X-1)ESl < b, t h e n the process Qn i s strong-coupling convergent t o a unique stationary ergodic regime Qn such that QO = SUPn>o Vn. If - X-E)ES1 > b, t h e n the process Qn converges in distribution t o a n improper limiting sequence.

-

Negative customers breaking the server An interesting generalization in practice is to consider that the negative customer not only eliminates regular customers but also causes a breakdown to the server, which must immediately be repaired. As in computer systems where the entry of a virus causes the elimination of programs and thus requires its treatment by an antivirus, and this time of repair may be random. Assume that the server fails at times. Upon the arrival of a negative customer the server fails and takes immediately a repair time Ri, i = 1 , 2 , .... The sequence of repair times {Ri}is assumed to be stationary and ergodic. We assume that after a repair time the server is as good as new and the service of a customer is cumulative. The SRS modeling the system has now the form: NA+(Sn)

Qn+l=(Qn+

1 i=l

ai-

NA-(sn) NA-(Sn)NA+(Rz) ci+ aj - b n ) i=l j=1 i=l

C

C 1

+

.

(9)

We have the following proposition:

-

Proposition 5. If ( X + t i ( l + X-ER1) - X - E ) E S l < 6, t h e n the process Qn i s strong coupling convergent t o a unique stationary ergodic regime Qn such that QO = V,. If ( A + t i ( l + X-ERl) - A-i?)ESl > 5, t h e n the process Qn converges in distribution t o a n improper limiting sequence. Retrial queueing systems with negative customers We can obtain in a similar manner stability and instability conditions for a special class of queueing systems called retrial queues. We consider arrivals in batches for regular customers and if a group of an arrival finds the server busy then the whole batch joins a group of blocked customers called "orbit". Otherwise, if the batch arrival finds the server idle, one of them takes his service and the others join the orbit. Customers in orbit reapplies at random times to get

O n the Stability of Queues with Negative Arrivals 65 served according to the following versatile policy. The probability of having a retrial during the time interval ( t ,t At],given that j customers were in orbit at time t, is (6(l - 60j) j v ) A t o ( A t ) . This versatile retrial policy, introduced by [Artalejo and Gomez-Corral, 19971, incorporates the classical linear policy and the constant one. If v = 0 we obtain the constant retrial policy studied by [Fayolle, 19951. Denote by u, a sequence of independent random variables uniformly distributed on [0,1] and generating the type of customer that gets the service (external or from the orbit) at the end of the successive service periods. The stochastic recursive sequence modeling the dynamics of the system has the following representation: NA+(Sn) Nx- ( S n ) ci Ci=l Qn+l = (Qn Ci=l ai (a1 - l ) I ( U n I x + + ~ + Q1 ~ ~

+

+

0fQnv A++o+Q,,,

I)+.

+ +

+

bnI(un 5 a1 is the size of the first batch arrival if the external arrival occurs before a retrial from the orbit. We have the following proposition: Proposition 6. 1) If v = 0, 6 > 0 and ( X f E - X-E)ESl < ( A + ~ ~ f ~ e ' ) then th,e proce_ss Qn is stronx coupling convergent to a unique stationary er- V,. godic regime Q, such that QO = sup,>o

If ( X b - X-E)ES1 > then the process Q, converges in distribution to an improper limiting sequence. 2) If v > 0, 0 2 0 and (A% - X-E)ESl < b, then the process- Q, is strong coupling convergent to a unique stationary ergodic regime Q, such that Qo = SUP,>O - Vn. If (A% - X-E)ESl > 6, then the process Q, converges in distribution to a n improper limiting sequence. Remark 3. By considering a system with retrials without negative customers and service is provided in single then we obtain the result of Theorem 7.1 in [Kernane and Kissani, 20061 for a retrial queue with versatile retrial policy and batch arrivals.

4

Work Removals

Consider a single server queue with work removals. The inter-arrival times of regular customers {T:} and service times {S,} are general stationary and ergodic sequences. Let t, the arrival epoch of the nth regular batch of customers. During T:, NA(T;) negative arrivals occur removing t,he with mean E . We assume that i.i.d amounts of work c a , i = 1,...,NA-(T$) the input flows of customers, size of batches and service times are mutually independent. Denote by W ( t )the workload at time t. The following recursion holds for the workload process Wn = W(t,-):

i t1

,

66


Denote by En = S , - T ; - ~ Z < ' ~ ' ) cy. We have Etn = ES~-ETF-X-~ET:. Using the same approach as for the precedent propositions we have the following result:

+

< (1 X-E)ET,~,then the process W, is strong coupling convergent to a unique stationary ergodic regime W, such that WO= suPn>o V n . If ES1 > (1 X-E)ET$, then the process W, converges in distribution to an improper limiting sequence. Proposition 7. If ES1

-

-

+

Remark 4. If we assume and i.i.d. sequence of service times with mean 0 a Poisson process with rate A+ = ~/ET; then we obtain the result of stability given in [Boucherie and Boxma, 19951 by X+p < 1 A-2.

+

References [Altman and Borovkov, 1997lE. Altman and A.A. Borovkov. On the stability of retrial queues. Queueing Systems, volume 26, pages 343-363, 1997. [Artalejo and Gomez-Corral, 19971J.R. Artalejo and A. Gomez-Corral. Steady state solution of a single-server queue with linear repeated requests. J. Appl. Prob., volume 34, pages 223-233, 1997. [Borovkov, 1976lA.A. Borovkov. Stochastic Processes in Queueing Theory. John Wiley and Sons, 1976. [Borovkov, 1998lA.A. Borovkov. Ergodicity and Stability of Stochastic Processes. John Wiley and Sons, 1998. [Boucherie and Boxma, 1995lR.J. Boucherie and O.J. Boxma. The workload in the M/G/l queue with work removal. Probab. Engineering and Informational Sci., volume 10, pages 261-277, 1995. [Boucherie and Van Dijk, 1994lR.J. Boucherie and N.M. Van Dijk. Local balance in queueing networks with positive and negative customers. Ann. Operat. Res., volume 48, pages 463-492, 1994. [Chao, 1995lX. Chm. A queueing network model with catastrophes and product form solution. Operations Research Letters, volume 18, pages 75-79, 1995. [Chao and Pinedo, 1995lX. Chao and M. Pinedo. Networks of queues with batch services, signals and product form solutions. Oper. Res. Letters, volume 17, pages 237-242, 1995. [Fayolle, 1995lG. Fayolle. A simple telephone exchange with delayed feedbacks. Teletrafic Analysis and Computer Performance Evaluation (0.J. Boxma, J. W. Cohen, and H. C. Tijms, eds.), Elsevier Science, Amsterdam, pages 75-79, 1986. [Gelenbe et al., 19911E. Gelenbe, P. Glynn, and K. Sigman, Queues with negative arrivals. J. Appl. Prob., volume 28, pages 245-250, 1991. [Gelenbe, 19911E. Gelenbe, Queueing networks with negative and positive customers and product form solution. J. Appl. Prob., volume 28, pages 656463, 1991.

On the Stability of Queues with Negative Arrivals 67 [Gyorfi and Morvai, 2002lL. Gyorfi and G. Morvai. Queueing for ergodic arrivals and services. In I. Berkes,(ed.) et al., Limit theorems in probability and statistics. Fourth Hungarian colloquium on limit theorems in proba. and Stat, Balatonlelle, Hungary, 1999. volume 11. Budapest: Janos Bolyai Mathematical Society pages 127-141, 2002. [Harrison and Pitel, 1993lP.G. Harrison and E. Pitel. Sojourn times in single-server queues with negative customers. J. Appl. Prob., volume 30, pages 943-963, 1993. [Harrison and Pitel, 1996lP.G. Harrison and E. Pitel. The M/G/l queue with negative customers. Adv. Appl. Prob., volume 28, pages 540-566, 1996. [Henderson, 1993lW. Henderson. Queueing networks with negative customers and negative queue length. J. Appl. Prob., volume 30, pages 931-942, 1993. [Henderson et al., 1996lW. Henderson, B.S. Northcote and P.G. Taylor. Geometric equilibrium distributions for queues with interactive batch departures. Ann. Operat. Res., volume 48, pages 493-511, 1994. [Jain and Sigman, 1996lG. Jain and K. Sigman. A Pollaczek-Khintchine formula for M/G/1 queues with disasters. J. Appl. Prob., volume 33, pages 1191-1200, 1996. [Kernane and Aissani, 2006lT. Kernane and A. Aissani. Stability of Retrial Queues with Versatile Retrial Policy. Journal of Applied Mathematics and Stochastic Analysis, 2006:pages(16), 2006. [Loynes, 1962lR.M. Loynes. The Stability of a Queue with non-independent interarrival and Service Times. Proc. Cambridge Philos. SOC.,58:497-520, 1962. [Zhu and Zhang, 2004lY. Zhu and Z.G. Zhang. M/GI/l queues with services of both positive and negative customers. J. Appl. Prob., volume 33, pages 11911200, 1996.

Random Multivariate Mult imodal Distributions George Kouvaras' and George Kokolakis2 National Technical Uiiiversity of Athens Department of Mathematics Zografou Campus 15780 Athens, Greece (email: gkouvhath. ntua. gr) National Technical University of Athens Department of Mathematics Zografou Campus 15780 Athens, Greece (email: Kokolakishath. ntua. gr)

Abstract. Bayesian nonparametric inference for unimodal and multimodal random probability measures on a finite dimensional Euclidean space is examined. After a short discussion on several concepts of multivariate unimodality, we introduce and study a new class of nonparametric prior distributions on the subspace of random multivariate multimodal distributions. This class in a way generalizes the very restrictive class of random unimodal distributions. A flexible constructional approach is developed using a variant of Khinchin's representation theorem for unimodal distributions. Results using our approach in a bivariate setting with a random draw from a Dirichlet process are presented. Keywords: Convexity, Dirichlet process, Unimodality-Multimodality, Polya trees, Random probability measures.

1 Introduction Much of nonparametric Bayesian inference has proceeded by modelling the unknown cumulative distribution function (c.d.f.) as a stochastic process. In a fundamental paper, [Ferguson, 19731, a random process, called the Dirichlet process, was defined as a distibution on ( P , S ) ,where P is the collection of all probability measures on a measurable space (X,.F), endowed with a calgebra S . The major drawback of a Dirichlet process is that it selects discrete distributions with probability one [Ferguson, 19731 and [Blackwell, 19731. Several different classes of nonparametric priors, which all contain the Dirichlet process as a particular case, have been proposed. It seems worth mentioning, among others, the mixture of Dirichlet processes [Antoniak, 19741, which is a Dirichlet process where the base measure is itself random and the mixture of Dirichlet process prior [Lo, 19841, which is a convolution of a Dirichlet process with an appropriate kernel. After the work of previous authors, the study of absolutely continuous random probability measures has 68

Random Multivariate Multimodal Distributions

69

become a very active area of research, touching on both analytical and simulation based approaches, cf. Polya Trees [Lavine, 19921, [Lavine, 19941 and [Kokolakis and Dellaportas, 19961, Dirichlet diffussion trees [Neal, 20011 and Levy-driven Processes [Nieto-Barajas and Walker, 20051. For reviews of nonparametric priors the reader is referred to [Walker et al., 19991, [Hjort, 20031, [Dey, 19981, [Ghosh, 20031 and the references therein. In this paper, we present a Bayesian nonparametric inference for unimodal and multimodal random probability measures on a finite dimensional Euclidean space that have finite expected number of modes. As a consequence, we get a random probability measure that admits a derivative almost everywhere in IRd. The paper is organized as follows. Section 2 has the essential theoretical background on multivariate unimodality to implement our methodology. In section 3, a detailed description of partial convexification procedure is provided. Random bivariate multimodal probability measures are constructed and possible modifications and extensions are discussed in section 4.

2

Univariate and multivariate unimodality

An important property of a distribution is unimodality. A univariate c.d.f. F is said to be unimodal with mode (or vertex) at m, if F is convex on (-m, m) and concave on ( m ,GO). We make use of unimodality to get absolutely continuous distribution functions. 2.1

Unimodality on R

For univariate distributions there is a well known representation theorem due to Khinchin (see [Feller, 19711, p.158) that refers to the classical univariate unimodality.

Theorem 1. A real valued random variable X has a unimodal density at 0 i f and only i f it is a product of two independent random variables U and Y , with U unaformly distributed on ( 0 , l ) and Y having an arbitrary distribution. This can be expressed in the following equivalent form cf. [Shepp, 19621, [Brunner, 19921 and (Kokolakis and Kouvaras, 20071.

Theorem 2. The c.d.f. F is convex on the negative real line and concave on the positive, if and only i f there exists a distribution function G on R such that F admits the representation:

for all x points of continuity of G .

70

2.2

Recent Adva.nces in Stochastic Modeling and Data Analysis

Unimodality on Rd

For multivariate distributions, however, there are several different ways that unimodality is defined. Among the main types of multivariate unimodality there are the following: the beta unimodality, which is generated by the Beta distribution, Beta(m,u ) , instead of Uniform distribution on the interval (0, l), and contains the classical univariate unimodality and some of the existing multivariate notions of unimodality as special cases (star unimodality, u-unimodality), the linear unimodality, which is characterized by the unimodality of the distribution of any linear combination of the components of a random vector and the strong unimodality, which is defined as a convolution of unimodal distributions. An extended study of different types of unimodality and their useful consequences can be found in [Dharmadhikari and Kumar, 19881 and [Bertin et al., 19971. In what follows we focus our attention on the Khinchin’s classical unimodality extended to the multivariate case. For the sake of simplicity we restrict ourselves to the bivariate case. Results for higher dimensions can be easily derived. According to the classical unimodality [Shepp, 19621 we have: Theorem 3. The c.d.f. F is unimodal at 0 i f and only i f there is a random vector ( X l , X 2 ) with c.d.f. F , such that ( X l ,X 2 ) = ( K U 1 ,Y2U2)l

(2)

where (Y1,y2) and ( U l ,U2) are independent random vectors, (U1,U2) is uniformly distributed on the unit square and (Y1,Yz) having an arbitrary c.d.5 ti.

Equivalent to Theorem 3 is the following.

Theorem 4. The c.d.f. F is unimodal at 0 i f and only i f for all points of continuity of G ,

( 2 1 , ~ ~ )

F ( z i , 2 2 ) = ~ 1 F ’ ~ ( 2 1 , 2 2+ ) z2FZ2(51,z2)- z i ~ f ( z i , z 2+)G(21,22), (3) where G is an arbitrary c.d.f., with subscripts in (3) denoting partial derivatives. According to the above procedure, i.e. by the component wise multiplication of two independent random vectors (Y1,Y2) and (U1,U Z ) ,where the latter is uniformly distributed on the unit square, we always get a c.d.f. F with a single mode at zero, no matter what the distribution G, we start with, is. To overcome the limitation of getting always a single mode at zero we propose the following “partial convexification” procedure.

3 Univariat e and multivariate partial convexification Partial convexification procedure of a c.d.f. G relies on using U ( a ,1) distributions, with 0 < a < 1, instead of U ( 0 , l ) [Kokolakis and Kouvaras, 20071.

Random Multivariate Multimodal Distributions

71

The parameter (Y can be fixed, or random with a prior distribution p ( ~ ) , on the interval (0, 1). According to this, we obtain a prior distribution on the subspace of multimodal c.d.f.’s. The expected number of modes of F increases from one, when (Y = 0, to infinity, when a = 1, having a finite number of modes when 0 < LY < 1. This means that when 0 < LY < 1, the c.d.f. F ( x ) alternates between local concavities and local convexities, i.e a “partial convexification” of F is produced. Definition 1. The d-variate c.d.f. F is called partially convexified if there exists a random vector X = (XI,.. . ,Xd) with c.d.f. F , such that

(XI,. . . ,Xd) = (YIUl,. . . YdUd),

(4)

where Y = (Yl,.. . ,Y d ) and U = (Ul,. . . ,U d ) are independent vectors, U is uniformly distributed on the rectangle ( ( ~ 11) , x . . . x ( a d , 1) and Y having an arbitrary d-variate c.d.f. G. 3.1

Partial convexification on

IR

Using the partial convexification procedure, Theorem 2 can be expressed [Kokolakis and Kouvaras, 20071 in the following form. Theorem 5. The c.d.f. F is partially convexified i f there exists a distribution function G on R such that F admits the representation:

for all x, z / u , points of continuity of G. 3.2

Partial convexification on

R2

A generalization to the bivariate case using Uniform distributions U ( ( Y l), ~, i = 1,2, with parameters ai fixed in the interval (0, l), is as follows. Theorem 6. The c.d.f. F is partially convexified i f there exists a distribution function G on IR2 such that F admits the representation:

F(xI,~

2= ) X

+

I F ~( ~, 1 , ~ 2 x2FZz ) ( 2 1 , ~-) x 1 ~ 2 f ( x i~,

+ &(xi,~

2 )

2 )(6)

where

for all (x1,x2), (x1/a1,22), (zlrx2/a2) and (xi/ai,x2/a2) points of continuity of G.

72


Proof. We have: F(Xl,X2) =

P[X1 I

X l , X 2

i x 2 1 = P[UlYl 5 X l , U Z Y 2 5 5 2 1

For xi 2 0,

for x;

1+ d 4 (

x 2

>I+..*+p[& ( x k

1

0-

Figure 1. Approximation of transformed rate of returns of American Express Co by lognormal and Wald distribution. We have worked out ourselves the mentioned simulation algorithm for Survival copula and for two parameter Copula 4.24 from Nelsen (1999), p. 124. Algorithm for Survival copula is the following: 0 Simulate 3 independent random variables u l , u 2 , U3 from

W O , 1) 0 0

Set XI = e-I(u1) Calculate C1= ln(1- 6 In u l ) Find 1-(1-61n(F2 (x2)))eC'

1-eC1 ~

x2 t root[(1 - 6 In( F2(x2)))e

6

-242-e

,x2]

Stochastic Risk Capital Model f o r Insurance Company 119

Calculate C2 = ln[(l - S In ul)(l - S In u2)] I-(I-JI~(F,(x~)))~'~ 0

Find x3 t root[(l- 8 h(F3(x3)))e

6

The algorithm for Copula 4.24 is: 0 Simulate 3 independent random variables ul, u2, U3 from

W O , 1)

'

0

set xl = e - l ( u 1 ) .

0

Calculate C, = (uI-" -1)'.

0

Calculate C, = (ul-"

- 1)' + (u2-" - I)'.

x3 t root[((c2+ ((F,(x3))-"- 1)"F

1

-1-a

+ 1)"

1

x(c2 +((F,(x3))-"-l)p>-2+p

We have used also the standard algorithm to simulate Gaussian copula (see Cherubini, Luciano, Vecchiato (2004)). We investigated all mentioned copulas (Clayton copula, Frank copula, Survival copula, Copula 4.24 (Nelsen (1999) example 4.24)) with lognormal and Wald distributed marginals. Copula parameter and the copula itself for hrther applying was found by using the fitting measure

cc H

sEE=

H

k=l J = 1

H

(Oi, j , k -,i' i=l

H3

j,k

1'


where maximum lengths of each axis are divided into H segments and ff 3 rectangles created, Oi,j,kis observed frequency in each rectangle and Si,j,k is the simulated frequency in each rectangle under the condition that size of the simulated data coincides with data size. Average values of the fitting measure and its standard deviation with the best copula parameter are shown in Table 4. Marginals

Clayton copula

Frank copula

S = 1.4 Lognormal, lognormal, lognormal Wald, Wald' Wald

Survival copula

6 = 7.5

p = 12.256 0 = 0.782 p = 7.486 0 = 0.578

p = 8.470 0 = 0.489 p = 8.903 0 = 0.563

Copula 4.24

S=-O.1

a=5, ~=1.4

p =20.156 0 = 1.109 p = 19.023 0 = 0.989

p = 13.024 0 = 0.566 p = 14.356 0 = 0.675

Gaussian copula

p = 16.34 0 = 0.967

As it is possible to see from Table 3 the best approximation is reached with Clayton copula with Wald marginals. Finally we have simulated 50 000 data 50 times with that copula, have calculated portfolio with different proportions of securities in it and have found average of empirical VAR(99.5%) of portfolio. Results with several proportions of securities are shown in the Table 5. As it is possible to see from mentioned table, it is possible to find optimal structure of portfolio that gives better 99.5% VAR to compare with 32% or 45% decrease. Table 5. Portfolio VAR and corresponding rate of return in comparison with

0,4 0.2

I I

0,2 0.2

I I

0,4 0.4

I I

1.221 1.395

I

I

-0.221 -0.395

I

I

0 0.6

I

I

0.4

0.4

I

I

0.6 0

I

I

1.106

1.201

I

I

-0.106

-0.201

4 CONCLUSIONS Solvency I1 papers claim that no diversification can be taken into account but in the same papers is advised to add all risks with correlation matrix. Like it is possible to see from Table 4 diversification plays large role in solvency capital determination. The negative point is that it is difficult to

Stochastic Risk Capital Model for Insurance Company

121

find appropriate copula because most widely used copulas have only one parameter and it is difficult to fit the distribution to data. Even two-parameter Copula 4.24 did not give better approximation..

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

Bagany M. (2006) Economic capital: a plea for the Student copula. 28"' International Congress of Actuaries, http://papers.ica2O06.com/30 14.html. Balakrishan, N., Nevzorov, V. (2003) B. A Primer on Statistical Dism'butions.Wiley, New York. Breymann W., Dias A,, Embrechts P. (2003) Dependence Stsuctures for Multivariate High-Frequency Data I Finance, Quantitative Finance 3(1) 116, m.math.ethz.ch/-baltes/ftp/papers.html. CEIOPS'advice to European Commission in the light of Consultation Paper CP15-20, m.gcactuaries.org, Solvency I1 page. ceiops_qis2_final,www.pcactuaries.org, Solvency I1 page. Cherubini, U., Luciano, E. (2001) Value at risk trade-off and capital allocation with copulas. Econ. Notes, 2001, v01.30,235-256. Cherubini U., Luciano E., Vecchiato W. (2004) Copula Methods in Finance, Wiley, New York. Embrechts P., Hoing A., Juri A.(2003) Using Copulae to bound the valueat-Risk for functions of dependent risks, Finance dt Stochastics 7(2) 145167, www .math.ethz.ch/-baltes/ftp/papers .html Embrechts P., Hoing A. (2006) Extreme VAR scenarios in higher dimensions, www.math.ethz.ch/-baltes/ftp/DaDers.html. Fabien F.(2003) Copula: A New Vision forEconomic Capital and Application to a Four Line of Business Company, www.actuaries.org/ASTIN/Co~lo~uia/Berliil/F Frees, E. W., Valdez, E. A. (1998) Understanding relationships using copulas. North American Actuarial Journal, vol. 2, 1-25. Hurlimann W. (2002) An alternative approach to portfolio selection, in Proceedings of the 12"' international AFIR Colloquium, Cancun, Mexico. Markowitz, H. (1959) Porvolio Selection. Wiley, New York. Micocci M., Masala G. Basktesting value-at-risk estimation with non Gaussian marginals. Nelsen, R B.(1999) An Introduction to Copulas. Springer-Verlag, New York. Panjer H H. (2003) Risk, Solvency and Allocations of Capital in Financial Institutions. Financial Engineering News, 2003, vol. 30, MarcldApril, www.fenews.codfen30.

Measuring demographic uncertainty via actuarial indexes Mariarosaria Coppola' , Eniilia Di Lorenzo', Albiiia Orlando3, and Marileiia Sibillo4

*

University of Naples Federico I1 via Rodin6 22 80134 Napoli, Italy (e-mail: m. coppola0unina. it) University of Naples Federiro I1 via Cintia, Complesso Monte S. Angelo 80126 Naples, Italy (e-mail: diloremiounina. it) CNR, Istituto per le Applicazioni del Calcolo Mauro Picone via P. Castellino 111 801213 Naples, Italy (e-mail: a. orlando0iac. cnr . it) University of Salerno via Ponte Don Melillo 84084 Fisciano (SA), Italy (e-mail: msibilloounisa. it)

Abstract. Aim of the paper is the analysis of the behaviour of risk filters connected to the demographic risk drivers for a portfolio of life annuities. The model, easily suitable to the rase of pension annuities, involves the evolution in time of the mortality rates, taking into account the randomness of the financial scenario. Within this context, the uncertainty in the choice of the deiriograpllic scenario is measured and the analysis is also supported by the VaR sensitivity to this risk source. Keywords: Longevit,y risk, Value at Risk, Risk index.

1 Solvency assessment principles: the importance of the longevity risk 1.1

Basic guidelines on solvency

In a solvency assessnient framework the keystone is represented by the capability of assets to cover obligations. Of course such valuations can be pursued on a run-off basis, i.e. considering only written business, or as an on-going concern, i.e. considering also future new business. As explained by [Cocozzaet al., 2006a], several a.pproaches have been considered in literature and in practice, for instance consistently with a surplus analysis, according t o an actuarial perspective, or with an income valuation. according to an accountant perspective, or with a profit analysis, according t o an ecoiioinic point of

122

Measuring Demographic Uncertainty via Actuarial Indexes

123

view. In any case, whatsever valuation perspect.ive one chooses (that can be aimed, for instance, t o the financial policy of the company or whatever company's policy), it is quite iinportant what follows (cf. [Cocozzaet d., ZOOCia] for a deeper understanding): 0 0

0

identifying the risk drivers affecting solvency, iiieasuring these risk sources, defining the capital requirements apt to cover liabilities.

Categorizing any possible risk must be opportunely addressed t o a specific purpose, ta.king into account several points of view: the shareholder's, the policyholder's, the supervisory a,uthority's ones. Focusing on a run-off basis; according to a. supervisory authority perspective, two risk components basically affect a life insurance system, that is the demographic and the financial risk drivers. Demographic risks are due to the differences between the anticipated mortality rates and the actual death frequencies; analogously financial risks are due to the differences between the actual return on assets (obtained investing premiums or contributions) and the interest rates adopted in the technical bases. Capital requirements can he based on fixed ratio systems or risk-based approaches. As well known, the first procedure, structured on a fixed percentage of a quantity representative of the risk exposure, which determines the solvency margin, does not capture the company's pecularities in terms of risk profile. The risk-ba.sed a.pproa.ch is cha.ra,cterized by connecting ca.pita.1 reqiiirement's to the specific risk profile of t,he insiirance business, that) is (cf. [Cocozzaet al., 2006bl) asset risk, insurance risk, interest ra.te risk, business risk (see RBC requirements developed by NAIC). On the other ha.nd, according to the Ra.sle indica.tions., iiiterna.1 models are realized connecting the siirpliis level t o the company's results and linking capital requirements to ruin probabilities. This approach is framed into a stochastic model involving an overall valuation perspectivc, apt, t,o assess solvency in connection t o the company's activities. In general solvency can be assessed on the basis of the ineqiialit,y:

q . 4 , - Lt 2 0 ) = E ,

E > O

where At and Lt. are; respectively, assets and liabilities at time t . Within a risk mapping procedure, in the following we analyze the impact on the liabilities of the systema.tic deniogra.phic risk, ta,kiiig into a.ccount its intera.ction with the financial risk soi.irces. In this order of ideas the int,ernal risk profile is deeply investigated by means of suitable risk indexes, apt to describe the impact of the selected risk source, as well as additional costs can be estimated with a fixed reliability degree.

124


1.2

An overview on longevity risk: the dynamic aspect of mortality and the projected survival models

The traditional survival models describe the age behaviour of mortality in a wide period, from age 0 t,o the maximum att,ainable age, attributing to the force of mortality, thereby to the probabilities of death, values deriving from period iiiortality observations. Such models assume the mortality trend static in time and manifest itself t o be inadequate in capturing the betterment in the mortality trend, known as the longevity phenomenon, particularly observed beginning from the past century in all industrialized countries (cf. [Pita.cco 20041). The curves of dea,ths reveal to be strongly clia.ra.cterizedby the increasing concent.rat#ionin the inode (recta,ngi~larization)and at the same time by the mode random approaching toward very old ages (expansion), effecting decreasing mortality profiles. In substance, the evolut,ion in time of the number of deaths presents, besides the xcidental deviations from the expected values, systematic deviations too. In actuarial calculations, period-based assuinptioiis for siirvival probabilities mean to represent the future as the past (even if recent past), without taking into account that future death probabilities will he most likely less than the corresponding period-ba,sed ones. In the fair value framework, the question of the correct choice of tools modelling the mortality pattern becomes noda.1 observing that no indications about mortality systematic deviations from the expected values come out from the market, so that values are obtained as marked to model values. Otherwise the longevity bond market is not perfectly developed yet for the incompleteness of the a.ges a.t issue represented in the contra.cts a.nd for the aspect of the incomplete liquidity. As suggested in [Ballottaet al., 2006], in the current valuation approach the deinogmphic phenomenon can be represented a.s the expecta.tion of its best estimate, considering the market neutral with respect to the two aspects, systematic and unsystematic, of the demographic risk. In this basic assumption, the. projec morta.lity tables., founded on models capt'iiring t,he dynamic aspects of the human life, in particular with respect to longevity, coiistitute the tool for fronting the problem. The consequence in the risk mapping is tha.t the demographic risk component results from the combined action of the mortality risk, deriving from the accideiital deviations of the number of deaths from the expected value, and the risk arising froin the use of a. ta.ble not a.dequa.te in representing the future mortality, the demographic m,odel risk. The demographic model risk is studied in the death benefit context in [Olivieri , 20011, in which term a,ssura,nce portfolios are considered paying att,eiition to new deceases. In [Coppola et al., 2006b], the impact, of mortality ta,bles with different degree of projection is analysed with respect to the fair valuation of the mathematical provision in t,he part,icular case of a term assurance with decreasing capital. In that paper the influence of the de-


125

mographic model risk on the periodic premium duration is practically shown and measured. On the other side, due to their long dura.tion a,nd ma.nifold payments, the actuarial appraisal involving life annuities is indeed a business section heavily affected by the demographic model risk. The choice of a mortality table with an iiiifair degree. of projection can cause remarkable iinderestiination of future costs, including insurers’ or pension fund liabilities or, otherwise, too high amounts constrained for reserves. In light of these considerations, in the life benefit contract framework, the longevity component holds a primary role in the insura.nce business risk ina.na.geiiient a.nd in the solvency a.ssessment. In papers by [Olivieri , 20011 and [Coppolaet al., 20021, the longevity risk is shown to be independent on the portfolio size and the model risk is introduced and measured outlying different inort,ality scenarios. Di Lorenzo and Sibillo (cf. [Di Lorenzoet al., 20021 draw the behaviour of the this risk for the mathematical provisions of a life annuity portfolio as fiiiict,ion of age at issue and portfolio size, picking out a critical age in correspondence of which the projection risk changes its trend. In the same paper some considerations arising from the actuarial functions’ pattern are developed. The relative importance of the two demogra.phic risk coiiipoiients, na.mely the insura.nce risk and the model risk, are coinpared in [Coppolaet d., 2002], together with the contribute of the stochastic interest rate to the global portfolio risk.

2

The mathematical provision fair value in living benefit products

The risk analysis we propose is based on stochastic assumptions both for the financial coinponeiit, due t o the random movements of the interest rates, and for the demographic one. The stochastic framework we will work in is outlined in its evolution in time by the information flow containing the financial and the demographic information formalized in the two probability F’, P’), (0, F”, P”), where F’ and F” a.re the a-a.1gebra.sreferred spa.ces (0, t o the financial and survival information flow respectively; we suppose the independence of the randomness in interest rates on the randomness in mortality rates, as commonly assumed. The two spaces generate the probability space (Q, F, P ) , represented by the filtration { F k } C F , with Fk = FL U 8‘: and {FL} C F’, { F ’ k } C F . The financial market in which the valuation is framed is assumed to be frictionless, with continuous trading, no restrictions on borrowing or short sales, the zero bonds and the stocks being iiifinitely divisible. We consider the case of payments due in case of life at the end of each period to persons belonging t o a.n initia.1 group of c coeval individuals aged 2. If w ( t , j ) is the market value at t,iine t of one monetary unit! due at time j and X j is the stochastic cash flow at time j , we can write the fair value at, t,iine t of the stochastic stream of loss from t to the coiitract,ual ending time:


In formula (1) Nj indicates the number of claims at time j in a portfolio of hoinogeneous policies, coinciding with the iiuinber of survivors at that time, in the case of life annuities. In (1) the generic element of the trading strategy replicating the portfolio flow at time j is constituted by N J X , units of unitary zero coupon bond issued in t and maturing in J with the following price (see [Coppola. et al., 2006bl):

The operator E in formula (1) denotes t,he expected value under a risk neutral probability measure hypotheses, deriving by the completeness of the market. This assumption, acceptable in relation to the financial aspect of the operation, is less realistic for its demographic component. As suggested in [Ba.llotta.et al., 20061, the expected va.lue under a. risk neutra.1 proba.bility measure can be calculated using the most opportune probability measure for the demographic component, taking into account both the systematic and the unsystematic mortality risks. In the case of an immediate life annuity portfolio, following [Coppola et al., 2006a], having iiidicated by t p , the probability that the individual aged z survives at the age 2 t , we write:

+

3

Risk indexes in living benefit contract fair valuation

Measuring the impact of the demographic model risk on a fiiiaiicial undertaking involving life benefit payments means quantifying its variability due to t,he raidomness in the. choice of the mortality table, the effects of the other two risk components (stochastic interest rates and random deviations of mortality) having been averaged out (cf . [Coppolaet ul., 20021). Referring to the case considered in section 2, we measure the impact of the demographic model risk ( D M R M ) on the value V, calculated in formula (3), proposing the. following expression (see [Coppola et al., 2006aI)

DMRM

= Va~[IE[v,lKtl] = c ’ V a r [ E [ x Xjjp,jp,.+ttJ(t,

j)lKt]].

(4)

j>t

having estimated by the conditioning on h’t the randomness in the choice of the survival function used to determine the survival probabilities.

127


Numerical applications

4

First of all we estimate the risk index DMRM in forinula (4),in the case of a portfolio of c = 1000 immediate unitary life annuities, a,ccording to a demographic scenario represented by the mortality rates involved by 0

0 0

the Italian inale life tables S I M 2002, the Weibul model with parameters cy = 85.2 and y = 9.15, the Lee-Ca,rter model a.pplied to the 1ta.lia.nda.ta. (referred to the Ita.lian popillation from 1947 to 1999 and from age 0 to 109, where 109 is the limiting age).

with reliability degrees, respectively, 0.2, 0.3, 0.5. The first table is the less projected one a.nd the third ta.ble the most projected. The fiiiancia.1sceimrio is represented by a CIK process for the interest rates, adjusted on daily average yields on fixed interest securities over the period January 1999- July 2006. t=5

t-10

t=l5

t=20

t=25

t=40

t=50

z = 40 984,8022 1,078,248 1,542,4113 1,984,756 2,480,825 3,156,176 1,278,737 0 x = 65 1,198,026 1,429,125 1,285,527 1,200,000 410,178.8 0.01886

Table 1. Projection Risk -(Model risk) estimation, reliability degrees: 0.2, 0.3, 0.5

Consistently to the results obtained in [Di Loreiizod al., 20021, t.he values in Table 1 show that the projection risk, for every fixed age at issue, generally increases with the time of the reserve valuat.ion, until the age reached at that time assumes a certain (high) value (depending on the pecularities of the selected tables). This valuation time is higher the younger the insured is. The results obtained in Table 1 reflect the reliability degree assigned t o each survival model, as the results in Table 2 confirm, where the probabilities given t o each survival table are, respectively, 0.2, 0.6, 0.2. t=5

t-10

t=15

t=20

t-25

t=40

t=50

z = 40 616,076.6 745,656.3 952,733,3 1,217,554 1,516,759 1,951,206 800,480.4

z = 65 733,151.5 874,301.6 790,432.4

741,963 253,750.6

0.278133

0

Table 2. Projection Risk -(Model risk) estimation, reliability degrees: 0.2, O.G, 0.2

It is interesting t o observe tha.t for the insured a.ged LL: = 40, the risk index, capturing the projection uncert,aint,y, increases until a valuation t,iine greater than the time in which 5 = 65 reaches his hit, since the results incorporate the behaviour of t,he Lee-Carter survival probabilities, where t,he survival dynamic in time reflects the starting age. In confirmation of this, we observe in

128 Recent Advances in Stochastic Modeling and D ata Analysis

Table 3 the behavioiir of t,he va,lues, corresponding t,o the ages z = 40, z = 60 and z = 65, using the Weibull model with parameters ( a = 83.5;g = 8), ( a = 85.32; 8 = 9.15), ( a = 87.0; ,/? = 10.45) and reliability degree, respectively, 0.2, 0.6, 0.2. In fact in this case the three reserves corresponding t o the ages z = 40, z = 60 and z = 65 begin t o decrease when the corresponding insured is between the ages 65 and 70. The behaviour for older ages is due t o the intera.ctions between z a,nd t rela.ted to the projected probability t p , and the residual duration of the contract (cf. also [Di Lorenzoet al., 20021).

t=5 t-10 t=15 t=20 t=25 t=30 t=40 t=50 z = 40 303,495.2 478,901.2 493,899.7 547,731.1 552,953, 1 480376.3 159,494.5 4,576.932 z = 60 421,733.1 406,077.3 279,729 138,056.7 37,788.14 18.73209814 12.1783 0 z = 65 188.557.7 139,OG7.2 118, GG7.9 32.894.78 3.1G0.081 0.223 0.048717 0

Table 3. Projection Risk -(Model risk) estimation, Weibul model

The following tables reporte the quantile reserve values obtained by means of a siiriulatioii procedure developed in [Orlandoet al.. 20061. Here, in order t o have a better approximation of each simulated path for the interest rate, we consider a weekly sample interval. t=5 t-10 t=15 t=20 t=25 t=40 t=50 SIM2002 21,278.48 18,516.65 15,891.98 13,495.11 10,737.45 4,649.763 1,759.868 Weibul 22.218.74 19.370.08 16.637.88 14.086.61 11.225.09 5.021.374 1.818.434

Table 4. Quantile Reserve. Confidence level=95%, x=40 c= 1000

t=40 t=50 t=5 t=lO t=15 t=20 t=25 SIM2002 21,285.61 18,522.31 15,896.32 13,498.28 10,739.62 4,650.199 1,759.966 Weibul 22,226.47 19,376.21 16,642.60 14,090.07 11,227.46 5,021.893 1,818.523 LeeCarter 23.816.98 21,210.78 18,839.69 16,726.03 14,199.61 10,524.92 4,698,106 Table 5. Quantile Reserve. Confidence level=99%, x=40 c=1000

We see that, if the projection degree increases, the quantile reserve for both the considered confidence levels increases. This trend is stronger in the case of the Lee-Carter table and when the t,iine of valuation is high. Finally, all the numerical results shows that, in a solvency perspective, it is


129

opport,iiiie to take into account context,iially different kinds of risk indexes. in particular referred t o the uncertainty in t h e choice of survival tables a n d to t h e quantification of t h e fiuancial position for different tables. In fact, t h e risk index i n (4)provides a measure of t h e overall effect of t h e uncertainty i n the choice of t h e deiriographic scenario. whilst t h e evolution i n time of t h e quantile reserves is a concrete indicator, in terms of fiiiancial results, of the a.bove choice.

References [Ballottaet al., 2OOG]L. Ballotta and S. Haberinan The fair valuation problem of guaranteed annuit,y options: The stochastic niortality environment case. Insurance: Mathematics and Economics, 38, 2006. [Cocozzaet al., 200Ga,]R. Cocozza and E. Di Lorenzo Solvency of life insura.nce companies: methodological issues. Journal of ilctuarial Practice, 13:81-101, 2006. [Cocozzaet al., 200GblR. Cocozza, E. Di Lorenzo A . Orlando and M. Sibillo In ProThe VaR of the Mathematical Provisions: critical issues. ceedin.gs of the In.tem,ation,a.l Conjeren.ceof Actuni-ies, 2:95-102, 2004. 1ittp://www.papers.ica200G.coiii/Papiers/2002/2002.pdf, 2OOG. [Coppolaet ul., 2002lM. Coppola, E. Di Lorenzo and h l . Sibillo Further Remarks on Risk Sources Measuring in the case of a Life Annuity Portfolio. Journal of Actuarial Practice, 10:229-242, 2002. [Coppolaet al., 2003]M. Coppola, E. Di Lorenzo and M. Sibillo Stochastic a.nalysis in life office managenlent: application to large annuity portfolios. .4pplied Stochustic Models in Business u71d In&ustry, 19:31-42, 2003. [Coppolaet al., 2OOGa]M. Coppola, V. DAniato, E. Di Lorenzo, M. Sibillo Risk measurement and fair valuation in the life insurance field. InEcople: from tradition t,o complexinty, Ca.pri, 2006. [Coppolaet al., 200Fb]M. Coppola, V. DAniato, M. Sibillo Fair value and demograpliic aspects of the insured loan. InProcee&inys of 1 Uth Intel,national Congress on Insurance: Mathematics and Economics Leuven, http://www. kuleuven. be/ime2U06/abstract.php ?zd=80. [Di Lorenzoet al., 2002lE. Di Lorenzo and M. Sihillo Longevit,y risk: measurement and application perspectives. InProceedings of the I 1 Conference in Actuarial Scie71ce, Sarnos, 2002. [Olivieri , 2001lA.M. Olivieri Uncertainty in niortality projections: an actuarial perspective. Insurance: Mathematics and Economics, 29:231-245, 2001. [Olivieriet ak., 2003lA.M. Olivieri and E. Pitacco Solvency requirements for pension annuities. Journal of Pension Economics and Finance, 2: 127-154, 2003. [Orlandoet ul., 20061A. Orlando and M. Politano Further remarks on risk profiles for insurance participating policies. InProceedings of the Ma4F Conference, Salerno, 2006. [Pitacco , 2004lE. Pitacco Sun~icralm.odels in a dgnnmic context: a surtieg. Insurance: Mathematics and Economics, 35(2):279-298,2004,

Location as risk factor Spatial analysis of an insurance data-set Ildikb VitBz' Department of Probability Theory and Statistics Eotvos L o r h d University P b m h y P. stny. 1/C Budapest, Hungary (e-mail: vildikoOcs.elte.hu) Abstract. Our aim was to examine the territorial dependence of risk for household insurances. Besides the classical risk factors such as type of wall, type of building, etc., we consider the location associated to each contract. A Markov random field model seems to be appropriate to describe the spatial effect. Basically there are two ways of fitting the model; we fit a GLM to the counts of claims with the classical risk factors and regarding their effects as fixed we fit the spatial model. Alternatively we can estimate the effects of all covariates (including location) jointly. Although this latter approach may seem to be more accurate, its high complexity and computational demands makes it unfeasible in our case. To overcome the disadvantages of the distinct estimation of the classical and the spatial risk factors proceed as follows: use fist a GLM for the non-spatial covariates, and then fit the spatial model by MCMC. Refit next the GLM with keeping the obtained spatial effect fixed and afterwards refit the spatial model, too. Iterate this procedure several times. We achieve much better fit by performing eight iterations. Keywords: GLM, insurance, Markov Chain Monte Car10 (MCMC), Markov

random field, Spatial statistics.

1

Introduction - Risk models with spatial components

Numerous models are known in the literature for estimating the spatial effect in various problems such as disease mapping, deliquency, or number of accidents [Aratb et al. (2004)], [Gilks, W. R. et al.]. Alongside the spatial effect, however, other influential variables may exist, and they can be quite different depending on the subject of study. For example age and sex are important factors in the case of a motor TPL insurance, while they are insignificant for household insurances. In the latter case type of wall and type of building seem to be important parameters of the contracting partner affecting the number of claims. Besides these more usual (hereinafter: classical) variables location can be an important risk factor worth including into the model. Basically, there are two ways of dealing with the classical and the spatial risk factors; we can estimate the classical ones first, and then fixing their values as if they were known, estimate the spatial effect, or we can estimate all effects simultaneously. Both methods work better than the ones ignoring territorial dependence when the considered phenomenon does contain a 130

Location as Risk Factor

131

spatial effect, but as the usual critic of the first method goes it is not reasonable to cope with the different variables separately, simultaneous estimation is more accurate. Though this latter concept seems to be more desireable but the realisation can be hampered. These models are usually several- level hierarchical Bayes models as spatial effects are generally choosen to be random variables with some prior distribution and using MCMC algorithm for estimating the parameters is often necessary. In this case too many parameters result in cumbersome full conditionals and enormous running time. To avoid this difficulty we suggest the following. Fit first a generalized linear model to the classical risk factors then keeping the received parameters fixed estimate the spatial effect. Return then to the classical risk factors and refit the GLM keeping now the spatial effect fixed, and then move on to the spatial estimation with the newly fixed GLM parameters. By iterating these steps much better predictions can be obtained and the disadvantages of the distinct estimation of the classical and the spatial risk factors can largely be eliminated. The paper is organized as follows. Section 2 describes our model. Section 3 presents the MCMC implementation. The results of the parameter-estimation are presented in Section 4. Section 5 tests the goodness of model choice by comparing the results from our model with ones obtained from some alternatives.

2

Model construction

Selecting the explanatory variables in the generalized linear model we found that: "time spent in risk", "type of building" (4 types), "type of wall" (4 types), "type of roof' (6 types), "type of tarif" (4categories), and "population size of the locality" (10 groups), (all of them are factor-type variables in the GLM) are the significant classical risk factors. As our aim is to analyse the effect of location we include the region of the contract into our model. We build up a hierarchical Bayes model, because this way we can incorporate our prior belief about the structure of the regional effects. At the lowest level of the hierarchy we suppose the counts of claims to comply with the Poisson distribution: yi N P d s ~ m ( t.iEi . A,= 1 . 1 5 t G

enp( - (logt - 1.39)’ 2.66

It can be seen from Figure 2 that this distribution adequately captures the shape of the data, even though it doesn’t manage to depict the large peak at the beginning, owing to the large frequency of zero waiting. This initial lognormal model allows us to reasonably describe the distribution of the trolley waiting times. To build an improved model for trolley waiting times it will be useful to incorporate patient covariates which are shown to be associated with trolley waiting times. For example results of the dataset indicate a strong association between patient age group and waiting time (p value < 0.0001). As patient age group increases their waiting time significantly increases. Development of further models which account for age and other influencing variables will improve the modelling ability.


Figure 2. Fitting a lognormal model to patient trolley waiting times

0.25

I

0.2

0.1 5 h

4-

v c

0.1

0.05

0

Conclusion This paper presents the first steps in establishing a clear model for A&E trolley waiting times. Firstly a nai've Bayes classifier was developed to identify the two streams of patient entering A&E. The optimal classifier was constructed using patient priority code and arrival method as defining features. For those patients with a DTA it was found that a lognormal distribution reasonably fitted the associated trolley waiting time. Future work will be concerned with further modelling of the trolley waiting time. Influencing patient attributes will be considered and introduced, to produce a more accurate model, for example, the age of the patient. Also, the use of phase-type distributions will be explored to see if these specialised distributions are a suitable tool for modelling trolley waiting times. By modelling the waiting process, it should lead to a greater understanding of the problem trolley waits and the factors which are contributing to it.

Identifyzng Heterogeneity of Patients using a Bayesian Model

171

References [BBC News, March 20051 available at htttx//news.bbc.co.uMl/hi/northerni r e l a d 4 3 10 133.stm Accessed on 11/12/2006. [BBC News, September 20031 available at httr,://news.bbc.co.t1Ml~ii/health/311SS6O.stm Accessed on 14/12/2006. [Community Care Bill, 20041 available at htt~://~~~.~arliament.u~commons/lib/researchlr~2002/r~O2-066.pdf Accessed 12112/2006 [Cox 19551 D.R. Cox. A use of complex probabilities in the theory of stochastic processes. Proceedings of the Cambridge Philosophical. Society, 51:.313-319, 1955 [DHSSPS, 20051 available at htt~://www.dhss~sni.aov.uWkev-facts-99-05.~df Accessed on 10/12/2006 [Domingos and Pazzani, 19971 P.Domingos and M. Pazzani. On the optimality of the simple Bayesian classifier under zero-one loss. Machine Learning, 29:103-130, 1997. [Faddy and McClean, 19991 M. Faddy and S.I. McClean. Analysing data on length of stay of hospital patients using phase-type distributions, Applied Stochastic Models in Business and Industry, 15:311-317, 1999. [Hellerstein et al, 20001 J. Hellerstein, T. Jayram and I. Rish. Recognising end-user transactions in performance management. In Proceedings 0fAAAI-2000, pages 596602,2000. [Marshall and McClean, 20031 A.H. Marshall and S.I. McClean, Conditional phasetype distributions for modelling patient length of stay in hospital, International Transactions in Operational Research, 10:565-576 , 2003. [Neuts, 19811 M.F. Neuts. Matrix Geometric Solutions in Stochastic Models, John Hopkins University Press, Baltinmore, Maryland, 1981. [NHS Plan, 20001 available on www.dh.eov.uk Accessed on 12/12/2006 [NHS Information Authority, 20041 available at ?searchterm=troII ey+ wai t Accessed on 14/12/2006 [Rish 20011 I. Rish. An empirical study of the nai've Bayes classifier, Proceedings of IJCAI 2001 Workshop Empirical Methods in Artificial Intelligence. 2001 [Rish et al, 20011 I. Rish, J. Hellerstein and T. Jayram. An analysis of data characteristics that affect nai've Bayes performance. Technical Report RC21993, IBM T.J. Watson Research Centre. 2001

Modelling the total time spent in an Accident and Emergency department and the associated costs Barry Shaw and Adele H. Marshall Centre for Statistical Sciences and Operational Research Queen’s University of Belfast Belfast, Northern Ireland, BT7 1 N N (e-mail: barry .shawQqub. ac .uk) Abstract. The aim of this paper is to analyse the total length of time spent by a group of patients in an Accident and Emergency (A&E) department using a multistage Markov model. A patient’s pathway through A&E consists of a sequence of stages, such as triage, examination, and a decision of whether to admit to hospital or not. Using Coxian phase-type distributions, this paper models these stages and illustrates the difference in the distribution of t,he time spent in A&E for those patients who are admitted to hospital, and those patients who are not. A theoretical approach to modelling the costs accumulated by a group of patients in A&E is also presented. The data analysed refers to the time spent by 53,213 patients in the A&E department of a hospital in Northern Ireland over a one year period. Keywords: A&E department, costs, Markov model, patient flow, waiting time.

1 Introduction Accident and Emergency (A&E) departments provide the main passageway t o a hospital for emergency inpatients and are one of the principle contacts with hospital services for many members of the public. Over 14 million patients use A&E services in the United Kingdom (UK) each year [Department of Health, 20031. In Northern Ireland this number is approximately 700,000 [DHSSPSNI, 20051. The total time spent in A&E is used as the key measurement of a department’s performance because it is this that is of concern to those attending the department, and has clinical implications. Analysing the total time spent in A&E gives a more robust indicator of performance of the whole system, rather than measuring performance based on just one part. The importance of the total time spent in A&E, from arrival t o admission into hospital or discharge, is emphasised by the U.K. government’s target that 98% of patients should spend no more than 4 hours in any A&E department [Department of Health, 20001. In Northern Ireland a similar target of 95% is being implemented by March 2008. The aim of this paper is to analyse the total length of time spent in an A&E department using a multi-stage Markov model. This is an appropriate approach since the patients in an A&E department proceed through a series of

172

Modelling Total Tame Spent an A&E and Associated Costs

173

different stages relating to arrival, assessnzent: examination, decision to admit ( D T A )to hospital, and admittance or discharge. It is increasingly recognised that by more accurately inodelling the flow of patients in hospital, policy rnakers may understand better the case-mix of patients being admitted. Identifying this heterogeneity and the associated length of stay is important in allowing health care managers to understand the syst,ein act,ivit,y. Different, stat’istical techniques have been used to model the flow of patients through various other hospital departments, see for example [Harrison and Millard, 19911, [Gorunescu e t al., 20021 and [Faddy and McClean, 19991. The latter of these used a Coxian phase-type distribution to analyse the flow of’ patients through a geriatric ward, where the phases of the Markov model could be thought of as the stages of care experienced by a patient. Further work expanded this idea by incorporating costs into a three-stage model [McClean and Millard, 19981 and a generalised 2-stage model (Shaw and Marshall, 20061, enabling the costs to be estimated for a group of patients’ overall duration of stay. These techniques have previously been applied to a geriatric ward, however from analysing the distribution of the total length of time spent in an A&E department, a similar approach could be adopted here. The patients admitted into A&E may be considered as two sets - those who are admitted to hospital ( D T A = y) and those who are not admitted t o hospital ( D T A = n) and are discharged home or referred to their GP. This paper aims t o illustrates two things: (1) the use of the Coxian phase-type distribution as a modelling technique for the flow of patients through an A&E department, and (2) a theoretical approach t o model the expected cost for a group of patients’ accumulated duration of time in an A&E department, given appropriate costing data. The remainder of the paper is constructed as follows. Section 2 provides the background of Coxian phase-type distributions, section 3 illustrates an application using data from the A&E department of a hospital in Northern Ireland, while section 4 discusses the methodology for analysing the costs of a group of patient’s duration of time in A&E. Section 5 concludes the paper.

2

The Coxian phase-type distribution

A Coxian phase-type distribution [Cox, 19551 describes the time to absorption of a finite Markov chain ill continuous time, where there is one absorbing state (or phase) and the process starts in the first of k transient states, see Figure 1. The process sequentially moves through the ordered transient states, with the choice of departing into the absorbing state at any time. The parameters of the Coxian phase-type distribution, the X,’s and pz’s,estimated from the data, describe the transition rates through the ordered transient states and the transition rates from the transient states t o the absorbing state, respectively. Due to the highly skewed nature of the total length of time spent in an A&E department, the Coxian phase-type distribution provides a robust

174 Recent Advances in Stochastic Modeling and Data Analysis A1

1

-

A2

2

-

3

----> k

method for modelling the variable of time to absorption. The probability density function (pdf) of the random time variablc T, representing the time until absorption, is given by

where Q, a sub-matrix of transition rates rcstricted t o the transient phases, is of the form

0 0

0 0

Q=

p is a vector of probabilities defining the initial transient phases, given by p= (l00...00)

(3)

and q is the transpose of a 1 x k vector of transition rates from transient phases to the absorbing phase, given by

As well a s having been previoiisly iised to model patient, flow, t,he Coxiaii phase-type distribution has also been used to model the length of treatment times of patients in a suicide study [Faddy, 19941, the telephonc service times for customers of an Israeli bank [Ishlay, 20021 and the incubation time of AIDS [Aalen, 19951. Such applications requirc the estimation of the parameters of the Coxian phase-type distribution from some observational data.

Modelling Total Time Spent in A&E and Associated Costs

175

One technique for doing this is maximum likelihood estimation. This paper uses MATLAB t o implement an optimization function that maximises the log-likelihood for a I-phase distribution, then a 2-phase distribution and so on. By ta.king into coiisidera.tion both the fit of the model to the da.ta. and the complexity of the model, the minimum Bayesian Information Criterion is used to decide the most appropriate number of phases. The corresponding parameter estimates of the distribution are also obtained.

3

Application

The data refers t o the total length of time spent by 53,213 patients in the A&E department of a hospital in Northern Ireland between April 2005 and March 2006. Depending on the severity of the admittee’s injury or illness, they will pass through a sequence of stages in the department, before either being admitted into hospital or discharged, see Figure 2. On arrival t o A&E,

Arrival

Examination

Fig. 2. Patient pathway through an A&E department

a patient may wait for a period of time before seeing a triage nurse for initial assessment. After this they may ‘queue’ for a further time period before they are given a further examination by a doctor. The next stage is the DTA. If a patient is admitted t o hospital, they can wait in the A&E department for a further period of time before proceeding to the ward. Of the 53,213 patients analysed in our data set, 12,917 (24%) are admitted into a hospital ward. These patients have a mean (median) time in A&E of 372 (237) minutes. If a patient is not admitted to hospital they are discharged home or referred t o


their GP. These patients have a mean (median) time in A&E of 129 (100) minutes. A Mann-Whitney test carried out to aiialyse the iiiflueiice of D T A on the total time spent in A&E produces a p-valuc < 0.001. The results of fitting DTA

No. of phases

BIC score

Y

4

175,680

n

3

464,267

Fitted parameters @1=0.00004, @~=0.00001,@3=0.01569, @4=0.00207, x^1=0.01152, &=0.01484, &=0.00893. @1=0.00011, @z=O.01080, @3=0.23641, X^I =0.01013. x^2=0.02722.

Table 1. Results of fitting the Coxian phase-type distribution to those patients categorised as DTA = y and DTA = n.

separate Coxian phase-type distributions t o the total time for D T A = y and D T A = n patients are shown in Table 1. The Bayesian Information Criterion, BIG' = -2 x LL nln(N), was used t o determine the most suitable number of phases for ea.ch fit. L L represents the log-likelihood va.lue for ea.ch fit, n represents the number of parameters of the Coxian phase-type distribution and N represents the number of data cases. The most suitable number of phases for the time spent in A&E of those patients who were admitted t o hospital is four. For those patients who were not admitted to hospital, a four phase fit, produced a better BIC score t,liaii that for three phases, however, after several fits one or more of the parameters were zero. This changes the distribution being fitted, and so it was decided to use three phases. The visual fit between the three phases and four phases was near negligible. Figures 3 and 4 show the graphical fit of the four phase arid three phase distributioiis to the D T A = y and D T A = n data, respectively, truncated at 1000 minutes. Although no attempt is made here to physical interpret the fitted phases of the Coxian phase-type distributions, it is note worthy that the number of fitted phases found from the data correspond to the number of physical stages illustrated in Figure 2. For those patients admitted t o hospital ( D T A = y), their pathway consists of four components. For those patients discharged or referred ( D T A = n ) , their pathway consists of three components. Previously, using data sets describiiig the flow of patients through other hospital wards, a physical int,erpret,atioii has been attached to t,he fitted phases of the model [McClean et al., 20051. For example, in geriatric medicine, collaboration with geriatricians has helped label the phases of a 3-phase distribution as acute-, medium- and long-term care. By attaching financial costs to these phases, or stages of care, a model was developed enabling the estimation of the costs for the duration of time spent in care by a group of patients [McClean and Millard, 19981. Further collaborative

+

Modelling Total Tame Spent in A & E and Associated Costs

177

0.0045

0.004

1

1 -data

-f(t)

[4 phases]

time (minutes)

Fig. 3. Fitted 4-phase Coxian phase-type distribution for the time spent in A&E of those patients admitted to hospital.

between modellers, A&E clinicians, and financial administrators would offer a similar model for the application discussed in this paper. The following section briefly discusses the theoretical approach that could be adopted for modelling the expected cost for a group of patients' accumulated duration of time in A&E.

4

Modelling costs in an A&E department

By considering the number of patients in each phase i, n, , the cost per patient per time unit in phase i, c,, and the continuous random variable T, representing the time a patient spends in phase i, the aim is to derive the moment generating function (MGF) of the total costs for all patients in A&E, T N . The costs, c, are phase dependent, but time homogeneous. If D,, represents the total cost per patient that leaves phase j , given they started in phase z , then the MGF of D,, is given by

178 Recent Advances in Stochastic Modeling and Data Analysis 0.008

0.007

0.006

0.005

r ’?

0.004

Q

0.003

0.002

0.001

0

1

46

91

136 181 226 271 316 361 406 451 496 541 586 631 676 721 766 811 856 901 946 991

time (minutes)

Fig. 4. Fitted 3-phase Coxian phase-type distribution for the time spent in A&E of those patients discharged or referred.

where X i and pi are the parameters of the Coxian phase-type distribution. If the discrete random variable Z,iJ,where 1 5 i 5 k and i 5 j 5 k , represents the number of subjects who leave A&E from phase j , given that they started in phase i, then the following MGF can be derived

where p i j is the probability of leaving phase j , given the patient started in phase i. Assuming that TN is a random variable representing the total cost for all patients while in A&E, then using (5) and (6) the MGF of T N ,M T ~ , is given by

The expected future total cost, C=E(TN), for a group of patients’ accumulated duration of time in an A&E department is then derived by taking the

Modelling Total T i m e Spent an A&E and Associated Costs

179

first moment of (7) and substituting z=O [Shaw, ZOOS]. The approach outlined here offers a possible technique that may be used for an A&E department. Considering, for example, those patients in section 3 who, having spent a period of time in A&E, are admitted into hospital. Assuming, through manipulation of available financial data, that the cost per patient per minute in each of the four stages of care was Eu, Eb, Ec and L d , respectively. These costs would be representative of the different stages of care a patient may undergo while in A&E. Then using ( 7 ) ,C is given by

By altering the parameters of the model, policy changes, such as opening an additional examination room or changing the number of staff in certain parts of the A&E department, could be investigated as potential benefits of such a model t o hospital managers.

5

Conclusion and further work

The importance of analysing and modelling the total length of time patients spend in an A&E department can only be highlighted by the recent targets introduced by the U.K. government. Modelling the flow of pat.ients t,hroiigli the system is important, allowing policy makers to test possible changes before implementing them on a real department. This paper has illustrated how the Coxian phase-type distribution may be utilised t o model the total length of time spent by a group of patients in an A&E department. Although no physical interpretation was attached to the phases of both distributions, it was signifimnt to note tha.t the number of fitted phases corresponds to the number of stages a patient proceeds through while attending A&E. Both distributions provide good fits to the empirical data. The last section of this paper illustrates a possible approach that could be taken for modelling the costs for a group of patients in A&E if appropriate financial dat'a was obtained. This is one avenue that could be considered as future work. A costing model would give hospital administrators and clinicians the ability to foresee the economic and non-economic consequences of any theoretical changes made to a health care system before implementing the changes on a real A&E department.

180

Recent Advances in Stochastic Modeling and D a t a Analysis

References [Aalen, 199510. Aalen. Phase type distributions in survival analysis. Scandavian Journal of Statistics, 22:447-463, 1995. [Cox,1955lD.R. Cox. A use of complex probabilities in the theory of stochastic processes. PTOC.Camb. Phil. Soc., 51:313-319, 1955. [Department of Health, 2000]Department of Health. The nhs plan. 2000. Hospital activity statis[Department of Health, 2003JDepartment of Health. tics. Available: www.performance.doh.gov.uk/hospitalactivity/data-requests, 2003. [DHSSPSNI, 2005lDHSSPSNI. Northern ireland hospital statistics 1999/2000 to 2004/2005 key facts. Available: http://www.dhsspsni.gov.uk/key-facts-9905.pdf, 2005. [Faddy and McClean, 19991M. Faddy and S.I. McClean. Analysing data on length of stay of hospital patients using phase-type distributions. Applied Stochastic Models in Business and Industry, 15:311-317, 1999. [Faddy, 1994lM. Faddy. Examples of fitting strii.rt,iired phase-type distributions. Applied stochastic models and data analysis, 10:247-255, 1994. [Gorunescu et al., 2002]Gorunescu, -, and et al. A queueing model for bedoccupancy management and planning of hospitals. Journal of the Operational Research Society, 53:19-24, 2002. [Harrison and Millard, 19911G. Harrison and P.H. Millard. Balancing acute and long term care: the mathematics of throughput in departments of geriatric medicine. Meth Inform Med, 30:221-228, 1991. [Ishlay, 2002]E. Ishlay. Fitting phase-type distributions to data from a telephone call center. Research Thesis: Israel Institute of Technology, 2002. [McClean and Millard, 1998]S.I. McClean and P.H. Millard. A three compartment model of the patient flows in a geriatric department: a decision support approach. Health Care Management Science, 1 (2):159-163, 1998. [McClean et al., 2005]S.I. McClean, -, and et al. Markov model-based clustering for efficient patient c x e . 18th IEEE Symposium on Computer-Based Medical Systems, pages 467-472, 2005. [Shaw and Marshall, 2006lB. Shaw and A.H. Marshall. Modeling the health care costs of geriatric inpatients. IEEE Transactions on Information Technology in Biomedicine, 10 (3):526-532, 2006. [Shaw, 2006]B. Shaw. An extended bayesian network approach to model the health care costs of patient spells in hospital. PhD Thesis: Queen’s University Belfast, 2006.

CHAPTER 5 Markov and Semi Markov Models

Periodicity of the Perturbed Non-Homogeneous Markov System M. A. Symeonaki' and P.-C. G. Vassiliou2 Panteion Univeristy Department of Social Politics, 136 Syggrou Av., 17671 Athens, Greece (e-mail: msymeon(0unipi.gr ,) Aristoteleion University of Thessaloniki, Department of Mathematics 54006, Thessaloniki, Greece (e-mail: v a s i l i o u h a t h .auth.gr) Abstract. In this paper the periodicity of a perturbed non homogeneous Markov system (P-NHMS) is studied. More specifically, the concept of a periodic P-NHMS is introduced, when the sequence of the total transition matrices {Q(t)}Eodoes not converge, but oscillates among several different matrices, which are rather close to a periodic matrix Q, with period equal to d. It is proved that under this more realistic assumption, the sequence of the relative population structures {q(t)}zo splits into d subsequences that converge in norm, as t -+ 00. Moreover, the asymptotic variability of the system is examined using the vector of means, variances and covariances of the state sizes, p ( t ) . More specifically, we prove that the vector p ( t ) also splits into d subsequences that converge, as t + 00, and we give the limits in an elegant closed analytic form. Keywords: Markov processes, Markov systems, Perturbation theory.

1

Introduction

Consider a NHMS as introduced in [Vassiliou, 19821. The main purpose of this paper is to provide the concept of a periodic P-NHMS and to study the asymptotic behavior and variability of the system. This is an effort to provide a general framework for a number of periodic systems in manpower planning, where the sequence {Q(t)},00=, does not converge as t -+ co, but lies rather close to a stochastic matrix Q, whose period is equal to d, for each t. Applications for periodic systems and a fair number of conclusions concerning a NHMS when the embedded Markov chain is periodic are given in [Georgiou and Vassiliou, 19921, [Tsaklides and Vassiliou, 19881 and [Tsaklides and Vassiliou, 19921. Perturbed Markov chains are studied in [Meyer and Shoaf, 19801, [Meyer, 19801 and [Meyer, 19941, where the sensitivity of Markov chains in changes in the transition probabilities is studied. In [Vassiliou and Symeonaki, 19971 and [Vassiliou and Symeonaki, 19991 the concept of a P-NHMS both in discrete and continuous time is presented, 182

Periodicity of Perturbed Non-Homogeneous Markov System

183

in order to examine the sensitivity of a NHMS. In this paper it is assumed that the embedded non homogeneous Markov chain {Q(t)}& is of the form: Q(t) = Q - E q ( i ) , W = 0,1,2, ..., where &,(i) is randomly selected from a finite set of matrices Eq. More specifically, we assume that for each t , the matrix P(t) is of the form:

P(t) = P - E P ( i ) , t = 0,1,2, ...

(1)

with:

P(t)l' I 1', P1' I 1', P 2 0, P(t) 2 0 where 1' = [l,1,...,11' and the matrix EP(i) is randomly selected from the &p(2),...,EP(w)} according to the probabilities: finite set &I = {L$,(l),

prob{&p(t)= & p ( i ) } = ci

> 0, V i = 1,2, ...,V.

(2)

Assume moreover that for each t , the vector p,(t) is of the form: p,(t) = po - ~ ~ ( where i ) , p,(t)l' = 1

(3)

where the vector ~ , ( i is ) a perturbation vector for the vector p,(t) and is ~ E0(m)} ( 2 ) , according randomly selected from the finite set &, = { ~ ~ ( 1 ) , ~..., to the probabilities:

p r o b { ~ , ( t )= ~ , ( i )= } c,i > 0,Vi = 1,2, ..., m.

(4)

For the sequence of loss probabilities we have that:

prob{Ek+l(t) = ~ k + l ( i ) } = ci

> 0, f o r

i = 1 , 2 , ...,w

(6)

Consider now the embedded non homogeneous Markov chain {Q(t)}Z",o. Then, 'dt, Q(t) is of the form:

Eq(4

= &P(S)

+ P;+,&o(j) + &+l(S)Po- .;+l(+o(.d

(8)

for t = 0,1,2, ... and let that prob{&,(t) = E q ( i ) } = cqi > 0, for i = 1,2, ...,m w and Q - E q ( i ) 2 0. We suppose moreover, that the matrix Q is d-periodic, i.e. d is the least positive integer, such that Qdfl = Q. Assuming that c1,..., Cd-1 are the cyclic subclasses of Q , let Qi denote the submatrix of Q corresponding to the transition probabilities from Ci-1 to Ci. We also

c,,

184


assume that for h = 1,2, .... m w , the matrix &,(h) do not have nonzero elements, where the matrix Q has zero elerrierits. More specifically the following condition is required: if ~

# 0, then q i j # 0, 'di,j E S

i j

Then:

E [ Q ( t )= ] Q h

-

gq 2 0

xy:l

where &, = &,(h)prob{Q(t) = Q - &,(h)}. The system we have just described is called a periodic perturbed non homogeneous Marlcov system. The following Lemma is now proved:

Lemma 1. The matrix E[Q(t)]is a stochastic, d - periodic matrix. Proof. Following the same steps as in Theorem 3.1 in [Vassiliou and Symeonaki, 19991 we have that E [ Q ( t ) is ] a stochastic matrix. Due to the fact that the matrix &,(i) has zero elements exactly where the matrix Q has zero elements, we have that the matrix &,(i) has the following form, for each i = 1 , 2 , .... mu:

0 0

Eq0(i)

0

&,(i) =

0 0 Eqd&-l(i) 0 [ .

1.

0 ... &ql(i) 0 . . . . 0 . . ' &qd&2(i) 0 ". 0 0 " '

(9)

We do not exclude the fact that E q j ( i ) = 0 for some j. Thus, the ma-

2,

= Q - E q j ( h ) }= will also have the above trix C;I"="lEqj(h)prob{Qj(t) form, since p r o b { Q j ( t ) = Q - E q j ( h ) }> 0. Considering the fact that Q is h

d-periodic we conclude that E [ Q ( t ) = ] Q - &, is also a d-periodic matrix. and E [ Q ( t ) ]are respectively of the following Consequently, the matrices form: I 0 r,, ..' 0 0 0 E,, . . . 0 . . . . . . E, =

2,

h

,o

h

h

0 t

and

i-P d - 1

0 0

0 0

Eqd-2 0

_ ' '

_ ' '

Periodicity of Perturbed Non-Homogeneous Markou System 185 A

0 0

Qo - &qo

0

0 0.

= [Qiocd-I

... ... ...

0 Q i - 6 1

0 0

Qi - 6 d - 2

...

'

where:

h=l

In the following lemma the limit of the matrix E [ Q ( t ) ]td as t estimated.

+

00

is

Lemma 2. Let a periodic - P - N H M S and let Q be a d- periodic stochastic matrix. T h e n matrix B = limt+m{EIQ(t)]}td i s the diagonal matrix given by:

where kxi

7=2

7=2 s=o

i--T+2

m=d-r+2

s=i

k=O

and A! is the generalized group inverse ([Meyer, 19801) of matrix Ai = I Qi, for every i = 0,1,2, ...,d - 1. -d

Proof. According to Lemma 1, the matrices Qd and {1!3[Q(t)]}~ = Q (Q - c $ ) ~are given respectively by the following relations:

and

-

--

where Xi = QiQi+l . . . Qd-lQo. . . Qi-l and Xi = Q i Q i + l . are regular stochastic matrices. Now it is true that:

--

A

QiQi+i = (Qi - &q;)(Q,+i

A

- &,,+I

I = QiQi+l

-

-

. .Q d - l Q , . . h

-G I

-

. Qi-1

=

186 Recent Advances in Stochastic Modeling and Data Analysis A

A

+

A

-

where Ei, = QiEqi+l EqiQi+l. Repeating this process will yield:

-

QiQi+l.

-

. .Q d - i Q ,

&ij

-

&id-l

=

{

= QiQi+l. . . Qd-l&qo

. . .Q d - i Q o

-

h

+

+

A

=

d-1

A

-

if i A

+j

-

A

A

>j

Id

&ij+lQj-d+i

i

we have that:

if i + j > d

-

and E,, = Eqo, Q d =

&qiQi+l

i+l

+

-L

Now if i

&id-l-iQo.

Qi+j Q i Q . . . . Qi+j-lgqi+j Q i Q i + l . . . Q , . . . Qj-d-l+i&qj-d+i %+l

1 then Ei1 = QiEqi+l s < 0. Then:

If j

+

A

A

where

A

= QiQi+i

i--7

- -

Q,, Q ,

= I if

i-1

According to Lemma 1, the matrses %i are regular, stochastic matrices, 'dz = 0 , 1 , ..., d - 1.. The matrices E i d - 1 are therefore perturbation matrices b'i = 0 , 1 , ...,d - 1. Thus, from Theorem 4.1 in [Meyer, 19801, we have that limt,, = ni(I & - l A % ) p land consequently 6 is given by Relation (12).

Ei

+

We now give the following theorem:

Theorem 1. Let a P-NHMS like the one we have j u s t described and let that Q be a d-periodic stochastic matrix. Let also that the system i s expanding, i.e. T ( t )2 T ( t - 1 ) and that limt,, = 0. T h e n the sequence of the relative structures { q ( t ) } E 0 splits into d subsequences, with limits:

$#

m=O

where: ,s

= lim 12103

c

n-l

k=O

+

A T ( k d m) T(nd+m)

and B is given by relation (12). Proof. It is true that in a P-NHMS the following equation holds:

E[Q(t)l= E[P(t)1+ E [ P k + i ( t ) l E [ P ~ ( t ) ] .

(14)


187

Moreover it is true that:

+

Without loss of generality we assume that t = n d Relation (15) can be written in the following form:

r-l

C

u=o

+

AT(nd u) T(nd+u) (Q -

T

(0 5 T 5 d

-

1). Then

zq)r-u-l.

We now prove that the second term of (16) converges, when t -+co. Let:

From Proposition 3 in [Tsaklides and Vassiliou, 19881 we conclude that: lim

t-m

lim

t-w

II(Q

,.

- €4) (n-k)d+(r-m-1) - B(Q -

gq)T-m-lI I = O 11

II(Q-gq)("-k))df(r-m -B - l()Q - g q ) d + r - m - l = O

i f r > m + l , and if

T

< m + l . (18)

IS . convergent 'dm = 0,1, ..., d - 1 since Now, the sequence it is limited from above and is a monotonically increasing sequence of time. Let the limits of the sequence be:

sm = lim n-w

c

n-l

k=O

AT(kd T(nd

+ m)

+ ?-) .


It is true, using also Relations (17) and (18) that: n-1

AT(kd + m, (Q - gq)(n-k)d+(T-m-l) T(nd+r)

lim

n-00

k=O

if r > m + l if r < m + l

'

From Relations (17) and (19) we have that:

m=r

m=O

We now consider the third part of Equation (16), for which it is: r-l

AT(nd +u)

IIC u=o T ( n d + u )

(Q - ~ q ) r - u - l l l

'-' A T ( n d + u )

I

u=o

T(nd

+r )

which converges to 0 due to the hypothesis. Moreover, since T ( t )+ M, the first part of (16) converges to 0. Hence: d-1

C

q,(m) = ;lEq(t)= {p, - E [ E , ( ~ ) ] } B (Q - I^q)dfT-mfl. m=O

We now study the asymptotic variability of the system using the vector of means, variances and covariances p ( t ) . More specifically, let:

P ( t ) = (E"l(t)I, E"2(t)l,

...I

E"k(t)ll

COV(N1( t ) Nl , ( t ) )C, O V ( N l ( t ) , N2 ( t ) I)'", COV(N1( t )I Nk ( t ) I)COV(N2( t ) Nl , (t)), COV(N2(t),N 2 ( t ) ) ,

"')

cov(Nz(t),N k ( t ) ) ,..., COV(Nk(t),N k ( t ) ) ) .

(22)

Then the following theorem holds:

Theorem 2. Let a P-NHMS like the one we have just described and let that Q be a d-periodic stochastic matrix. It is also assumed that the system is expanding, i.e. T ( t ) T ( t - 1 ) and that limt,, T ( t )= T . Then the sequence of the vectors { p ( t ) } E osplits into d subsequences, with limits:

>

Pr+I(M) = t-oo lim P ( t d r+ 1

+ + 1) = P(O)E[ ?-

VQ0,?I

d-1

C w r n ~ [ ~ o ( m ) ~+~ [C~ q w~ m, ~~ I[ ~ o ( m ) l ~ [ ~ q ~ ,

+{T - T(O)I{

m=O

m=r+2


189

b'r = 0,1,2, ..., d - 1, where:

and limn+m E [V,(n,d

+ v, nd + r)] =

if r > v - 1

E [vqT,u] if T < 'u - 1 .

References [Georgiou and Vassiliou, 1992lA. Georgiou and P.-C. G. Vassiliou. Periodicity of asymptotically attainable structures in non-homogeneous markov systems. Lin. Alg. Appl., pages 137-174, 1992. [Meyer and Shoaf, 198OlC. D. Meyer and J. M. Shoaf. Updating finite markov chains by using techniques of group matrix inversions. J. Statis. Comp. Sam., pages 163-181, 1980. [Meyer, 198OlC. D. Meyer. The condition of a finite markov chain and perturbation bounds fot the limiting probabilities. Siam J. Alg. Disc. Meth., pages 273-283, 1980. [Meyer, 1994lC. D. Meyer. Senstivity of the stationary distribution of a markov chain. Siam J. Mat. Anl. Appl., pages 715-728, 1994. [Tsaklides and Vassiliou, 1988lG. Tsaklides and P.-C. G. Vassiliou. Asymptotic periodicity of the variances and covariances in non-homogeneous markov systems. J. Appl. Prob., pages 21-33, 1988. [Tsaklides and Vassiliou, 1992lG. Tsaklides and P.-C. G. Vassiliou. Periodicity of infinite products of matrices with some negative elements and row sums equal to one. Lin. Alg. Appl., pages 175-196, 1992. [Vassiliou and Symeonaki, 1997lP.-C. G. Vassiliou and M. A. Symeonaki. The perturbed nonhomogeneous markov system in continuous time. Appl. Stoc. Mod. and Data Analysis, pages 207-216, 1997. [Vassiliou and Symeonaki, 1999lP.-C. G. Vassiliou and M. A. Symeonaki. The perturbed nonhomogeneous markov system. Linear Algebra and Applications, pages 319-332, 1999. [Vassiliou, 1982lP.4. G. Vassiliou. Asymptotic behavior of markov systems. J. Appl. Prob., pages 851-857, 1982.

On the moments of the state sizes of the discrete time homogeneous Markov system with a finite state capacity G. Vasiliadis and G. Tsaklidis Department of Mathematics Aristotle University of Thessaloniki 54124, Thessaloniki, Greece (e-mail: gvasilhath.auth.gr, tsaklidiamath.auth.gr)

Abstract. In the present paper we study the evolution of a discrete-time homogeneous Markov system (HMS) with a finite state capacity. In order to examine the variability of the state sizes, their moments are evaluated for any time point, and recursive formulae for their computation are derived. Also a recursive formula is provided for the moments of the overflow size due to the finite state's capacity. The p.d.f. of the overflow size follows directly by means of the moments. Keywords: Stochastic population systems, discrete-time homogeneous hlarkov models, Markov systems.

1

Introduction

Consider a discrete-time Homogeneous Markov System (HMS) with state space S = 1,2,. . . ,k. Symbolize by t , t = 0,1,. . . , the time variable. Every member of the HMS may be in one and only one of the states 1 , 2 , .. . , k at some time point t according to some characteristic, i.e., the states of the HMS are exclusive and exhaustive. The members of the system can be human populations, biological organisms, etc. Let p i j denote the one-step conditional (time independent) transition probability for a member of the HMS of moving from state i to state j, and P = ( p i j ) the respective k x k transition matrix. The population structure of the HMS at time t is given by the (column) state vector n(t) = (n1(t),nz(t),' . . , n k ( t ) Y , where ni(t)stands for the number of the members at state i, i = 1 , 2 , . . . , k , and the superscript denotes transposition of the respective vector (or matrix). Moreover, we will denote by nij(t),i , j = 1 , 2 , . . . , k , the number of the members of the HMS who are moving from state i to j in the time interval [t,t l ) , and by ni(t) the vector of the numbers of transitions from state i to the states 1 , 2 , . . . ,k , in the time interval [t,t l ) , i.e.,

+

+

190

Moments of State Sizes of Discrete Time Homogeneous Markov System

191

for i = 1 , 2 , . . . , k , t = 0 , 1 , . . .. Then ni(t)/ni(t)

-

Multinomial ( n i ( t ) ; p i l , p i z , .. . ,pile),

for i = 1,2, . . . , k , t = 0,1, . . . ([Bartholomew, 19821). In the literature, such Markov systems, are called homogeneous (HMS) if P = ( p i j ) , or non-homogeneous (NHMS) if P = ( p i j ( t ) ) . They appear in manpower planning ([Bartholomew, 19821, [Gani, 19631, [Tsaklidis, 19941, [Vassiliou, 19821, [Vassiliou, 19971), in demography ([Bartholomew, 19823), biology, etc. For example, a HMS can be used in order to describe the evolution of an university system ([Gani, 1963]), the patients’ flows and costs in a hospital ([McClean et al., 19981, [Taylor et d.,2000]), the pollution of a biological system ([Patoucheas and Stamou, 1993]), etc. The evolution of a Markov system is usually examined by studying the evolution of the state vectors n(t), t = 0 , 1 , . . . , and their expectations, variances and covariances. In particular their asymptotic behaviour is of central importance in the study of the system. Results concerning the convergence of the expectations of the state vectors n(t), t = 0 , 1 , . . ., can be found for instance in [Kipouridis and Tsaklidis, 20011 and [Taylor et al., 20001, while in [Tsaklidis and Soldatos, 20031 the trajectories of the expectations of the r.v.’s n(t),t = 0,1,. . ., are examined in order to interpret a HhlS as an elastic solid. Let the size of the HMS be equal to N E N+. Note that nj(t 1) = k CiZ1 n i j ( t ) ,and consequently the probability distribution function (p.d.f.) of nj(t+l) is the convolution of the p.d.f.’s of n i j ( t ) , i = 1 , 2 , . . . , k . Since it is not convenient to deal with the convolution, only the means, variances and covariances of the r.v.’s nj(t)as well as their asymptotic distributions have been evaluated so far ([Bartholomew, 19821, [Vassiliou, 19821, [Vassiliou, 19971) etc). For this reason it is very useful to investigate the distributions of the r.v.’s n j ( t )by means of their moments.

+

2

Moments of the r.v.’s ni(t)

We consider the case of a homogeneous Markov system the s - th, s E { 1,2,. . . , k } , state of which has finite capacity c. Conventionally we assume that c E N. We will denote this Markov system by HMSIc,. We assume that the overflowing members leave the system. Let e ( t ) symbolize the number of the members who are overflowing at time t and Nt the size of HMS/c, at time t. Moreover, we will denote by m i @ ) the number of the members who decide k to move to state i in the time interval [t-1, t ) ,that is m i ( t ) = Cj=l nji(t--1). In what follows, we will use a new vector product denoted by ” x ’I: If x is a (column) vector then xT x xT is a row vector, that can be derived from the Kronecker product xT 8 xTIby replacing the products appearing in xT @ xT with the respective factorials. For example, if x = ( 2 1 , Q ) ~ since , XT 8 x T = ( 2 : , 2 1 2 2 , 2 2 2 1 , 2 2 2) ,


192

then XT

x

XT

= (xl(2) , X (1) I z2 (1),x2 (1)21 (1), x W 2 ),

that is XT x XT =

(Zl(X1

- 1),2122,X2X1,22(22 -

1)).

Moreover, we will use the following proposition ([Vasiliadis and Tsaklidis, 2007bl).

Proposition 1. Let a HMS/c, with transition matrix P = (pij) . Then: I . The r - t h factorial moment E[n?)(t)]of the r.v. n,(t), s E { 1 , 2 , . . . , k } , is given by

2. The mixed factorial moments of the state sizes ni(t) are given by

where ri E

N,i = 1 , 2 , . . . ,k .

By means of Proposition 1 the factorial and mixed factorial moments of the r.v.'s ni(t),i = 1 , 2 , . . . ,k , can be computed using the following theorem:

Theorem 1. Let a discrete-time HMS/c, with transition matrix P . Then

I. E [ n T ( t )x . . . x n T ( t ) ]= E [ m T ( t )x . . . x mT(t)] - eT(t)= P r

-

= E [ n T ( t- 1) x

/

Y

r

. . . x nT(t- I)]? 8 . ~8 .$1 "

\

\

-

eT(t)

4

r

r

2. E [ n T ( t )x . . . x nT(t)] = (nT(o) x . . . x n T ( 0 ) ) ( P t @. 8 . ~ '- ) -

cg:,

€+ !

- j)(Pj8

T

\

Y

T

8 Pj),

/-

r

Moments of State Sizes of Discrete T i m e Homogeneous Markov System

193

where mT(t)= ( m l ( t )ma(t), , . . . , m k ( t ) ) ,and e T ( t )as a 1x kT vector, whose element at the same place where in the vector E[rnT(t)x . . . x mT(t)] appears

equal to 0. Proof. 1. Taking into account that ([Vasiliadis and Tsaklidis, 2007al)

and Proposition 1, we derive the desired result. 2. By means of the first part of the theorem we have that

E [ n T ( t )x . . . x nT(t)] = E[nT(t- 1 ) x . . . x nT(t- 1 ) ] ( PC3 . . . C3 P ) - e T ( t ) P

\

-4

r

3

T

J-

r

The factorial moments of the overflow size

In order to evaluate the factorial moments of the state sizes, given in Theorem 1,the computation of the factorial moments of the overflow size and the mixed factorial moments E [ e ( r * ) ( t ) n('"(t)]are needed. For this purpose we denote by et the total overflow till time t . Then the following theorems hold.

n!=l %#S

Theorem 2. Let a HMS/c, with transition matrix P = ( p i j ) , i ,j = I, 2,. . . , k . The r - t h factorial moment of the r.v. e ( t ) , at any time t , i s given by


Proof. The r - th factorial moment of the r.v. e ( t ) / e t - l , is given by No-et-l-c

C

~ [ e ( ' ) ( t ) / e , - l= ]

x ( ' ) P ( e ( t )= x / e t - l ) .

(1)

x=o

The probability density functions (p.d.f.'s) of m i ( t ) / e t - l , i = 1,2,. . . ,k, satisfy the relation ([Vasiliadis and Tsaklidis, 2007al)

1 P [mi@) = n/et-l] = n!

C

j=O

(-l)j

T E[m,!"fj'(t)].

(2)

From (2) we get

P ( e ( t )= x / e t - l )

= P(m,(t) = c

+ x/et-l)

Then, from ( 1 ) and (3), we deduce

In order to evaluate the mixed factorial moments of the r.v.'s e ( t ) ,ni(t), i = 1,2,. . . , k , i # s, the next proposition is needed. Proposition 2. Let a discrete r.w. x = ( X I ,x2,.. . ,xk),where X I + x2 -I. . . + XI, = N , N E N. Then, the joint distribution of X satisfies the relation

+ + +x k =

Proof. For a discrete r.v. x = ( X I ,x2,.. . ,x k ) , with x1 x2 . . . N , N E N, the mixed factorial moments E [ n t = l X i T i ) ] where , ri E r1 7-2 . . . rk = N , are given by

+ + +

n

N and

k

=

Z!'")P(x1

XI+

= 2 1 , x2 = Z2,.

. . ,XI, = 2 k ) .

...+X & = N Z=l

+ + +

Since r1 7-2 . . . rk = N , all but one of the summands appearing in the latter sum are equal to 0; only the term arising for 2 1 = T I , x2 = 7-2, . . . ,xk = rk is not equal to 0. Thus, we have = T I ,Xz = T z , . ..,Xk = T k ) .

Moments of State Sizes of Discrete Time Homogeneous Markov System 195 Hence

Now we provide the following theorem. Theorem 3. Let a HMS/c, with transition matrix P = ( p i j ) , i , j = 1 , 2 , . . . ,k . T h e n the mixed factorial moments E [ e ( T s ) ( t ) n ! =niTi)(t)], l ri E M, i = a#s 1 , 2 , . . . ,k , at any time t , are given by No -c No - I -C

k

k

where the summation CZiis extended over all xi, i = 1 , 2 , . . . , k , i # s, which satisfy the condition X I . . . x k = NO- j - c - 1.

+ +

Proof. We have that

or

i=l i#s

. P ( m l ( t )= X l , m 2 ( t ) = 2 2 , . . . ,m,(t) = c

+ 1,. . . ,mk(t)= x k / e t - I ) ,

where the summation CZiis extended over all xi, i = 1 , 2 , . . . , k , i which satisfy the condition X I . . . x k = NO - et-l - c - 1. Then, from Proposition 2 we get

+ +

(4)

# s,


Hence k

k

i=l

i=1 a#*

i#s

Nn-c

k

j=O

i=l i#a

and by ( 5 ) , we get the desired result. The p.d.f. of the total overflow et can be evaluated using the following theorem.

Theorem 4. Let a HMS/c,. The p.d.5

of

et satisfies the recursive relation

for 1 5 x 5 NO- c, and Nn-c x=1

Proof. For 1 5 x 5 No - c we have

+

P(et = x/et-l = 1) = P ( e ( t ) et-1 = x/et-l = I ) = P ( e ( t )= II: - Z/et-l = 1 ) = P ( m , ( t ) = c z - I/et-l = I ) .

+

We get from (2) that

1 ( c + x - Z)!

No-Z-c-x+Z

j=o

For z = 0, we derive No-c

P(et = 0) = 1 -

C P(et

j=1

=j ) .

Moments of State Sizes of Discrete Time Homogeneous Markov System

4

197

Concluding remarks

T h e evaluation of t h e moments of t h e state sizes becomes very useful in real applications, since they determine completely the state sizes’ distributions. The problems treated in sections 2-3 can be examined if capacities are considered for all the states of t h e system. In this case t h e overflowing members can be assumed to enter a new (nominal) state, say k 1. Then, new transition probabilities, and t h e respective transition matrix, shall be computed. It is very interesting t o notice that the transitions between t h e states of this new system are not independent. Thus t h e results given in t h e literature for the classical HMSs cannot be generalized directly for the HMS with capacities. Nevertheless, the evaluation of t h e moments of t h e state sizes is still a very useful tool for t h e determination of the distributions of t h e state sizes and the examination of t h e evolution of t h e system.

+

References [Bartholomew, 1982lD.J. Bartholomew. Stochastic Models for Social Processes, volume 3rd edn. J. Wiley, New York, 1982. [Gani, 196315. Gani. Formulae for projecting enrolments and degrees awarded in universities. J. R. Statist. SOC.A., 126:400-409, 1963. [Kipouridis and Tsaklidis, 200111. Kipouridis and G. Tsaklidis. The size order of the state vector of discrete-time homogeneous markov system. J. Appl. Prob., 38~357-368,2001. [McClean et al., 1998lS.I. McClean, -, and et al. Using a markov reward model to estimate spend-down costs for a geriatric department. J. Operat. Res. SOC., 10:1021-1025, 1998. [Patoucheas and Stamou, 1993lP.D. Patoucheas and G. Stamou. Non-homogeneous markovian models in ecological modelling: a study of the zoobenthos dynamics in thermaikos gulf, greece. Ecological Modelling, 66:197-215, 1993. [Taylor et al., 2000lG.J. Taylor, -, and et al. Stochastic model of geriatric patient bed occupancy behaviour. J. R. Statist. SOC.,163 (1):39-48, 2000. [Tsaklidis and Soldatos, 2003lG. Tsaklidis and K.P. Soldatos. Modelling of continuous time homogeneous markov system with fixed size as elastic solid. Appl. Math. Modell., 27:877-887, 2003. [Tsaklidis, 1994lG. Tsaklidis. The evolution of the attainable structures of a homcgeneous markov system with fixed size. J. Appl. Prob., 31:348-361, 1994. [Vasiliadis and Tsaklidis, 2007alG. Vasiliadis and G. Tsaklidis. On the distributions of the state sizes of discrete time homogeneous markov systems. Methodology and Computing in Applied Probability (Under publication), 2007a. [Vasiliadis and Tsaklidis, 2007bl G. Vasiliadis and G. Tsaklidis. The discrete-time homogeneous markov system with a finite state capacity. Proceedings of the 20-th Pan. Statist. Conference, Nicosia, 2007b. [Vassiliou, 1982lP.-C.G. Vassiliou. Asymptotic behaviour of markov systems. J. Appl. Prob., 19:851-857, 1982. [Vassiliou, 1997]P.-C.G. Vassiliou. The evolution of the theory of non-homogeneous markov systems. Appl. Stoch. Models Data Anal., 13, no. 3-4:159-176, 1997.

Copulas and goodness of fit tests PA1 Rakonczai’ and AndrAs Zemplkni’ Eotvos Lor6nd University Department of Probability Theory and Statistics Budapest, Hungary (e-mail: pauloQmath. elte .hu) Eotvos LorAnd University Department of Probability Theory and Statistics Budapest, Hungary (e-mail: zempleni@math. elte.hu) Abstract. There are more and more recent copula models aimed at describing the behavior of multivariate data sets. However, no effective methods are known for checking the validity of these models, especially for the case of higher dimensions. Our approach is based on the multivariate probability integral transformation of the joint distribution, which reduces the multivariate problem to one dimension. We compare the above goodness of fit tests to those, which are based on the copula density function. We present the background of the methods as well as simulations for their power. Keywords: copulas, goodness of fit test, probability integral transformation.

1

Introduction

In the last decades the question of multivariate modeling became also tractable, by the vast number of recorded data and the powerful computing equipment readily available. However, the methodology has not always been kept pace with the available resources: one can easily fit multivariate models by one software or another, but there is not always a suitable method at hand for checking the goodness of the fit. Copulas, simple yet powerful tools for modeling, ensuring the separation of marginal modeling and dependence questions, have been re-invented in the 1990s and their use has been expanded rapidly since then. One natural area of their applications is in the financial mathematics, where they are often used t o model the dependence structure between assets or losses, stock indices and so on. The paper is organized as follows. In Section 2 we recall the definition of the Archimedian copula and outline its most relevant properties. In section 3 we demonstrate two completely different approach for investigating the goodness of fit. In Section 4 we apply the presented methods for real financial data sets and compare their power. We show possible extensions of the wellknown bivariate methods t o higher dimensions. In this aspect the open-source R package played a leading role, see http://www.rproject.org/ for its description and the available packages 198

Copulas and Goodness of Fit Tests

2

199

d-dimensional Archimedian copulas

In the following we present the basic elements of copula theory for the class of the d-variate Archimedian copulas, which is the extension of the very popular notion of bivariate Archimedian copulas, so it possesses similarly favorable characteristics. Let us consider a copula generator function: $Q(u): [0,1] + [O,oo],which is continuous and strictly decreasing with $(1) = 0. Then a d-variate Archimedian copula function is

The d-copula inherits the beneficial properties of its bivariate ancestor, however it has a limitation that for a fixed family $6, there are only a few parameters to capture the full dependence structure. Since all the d - 1 dimen(1,u2, ...,U d ) = sional margins of an Archimedian copula are identical: ... = c,@(u~, ...,~ d - 1 ~ 1=) qj,’(~f:; +o(ui)),it assumes a certain symmetry among the coordinates. Anyway, since the main aim of this paper is to introduce some appropriate methods for checking a given model’s validity and not to develop involved copula models, the Archimedian families are pretty eligible. Indeed in the course of the next sections we deal mostly with the Clayton copula family, but we emphasize that the presented methods can be adapted for any Archimedian models in the same way. The generator function of the Clayton copula is given by $ Q ( u= ) u-’ - 1, hence +il(t) = ( t + l ) - i . The Clayton d-copula function, also known as Cook and Johnson’s family, is given by:

c,,

with 9 > 0. Simulations can be performed by general methods, such as the conditional sampling, which can be computed quite easily with the help of the derivatives of the function $ i l ( t ) , for details see [l]. Beyond the simulation, another relevant question is the parameter estimation. An easy method for the bivariate case is based on Kendall’s T [cite:] In the general, d-dimensional case one may use the by the form of 8 = so-called maximum pseudo-likelihood method (see [5]), based on the copula density function:

&.

200

3


Goodness of fit t e s t s

In this section we discuss the goodness of fit statistics in two subsections, first the tests related to the cumulative distribution function and then those based on the probability density function. 3.1

GOF statistics based on PIT

Let a random vector X = ( X I ,...,X,) possess a continuous d-variate copula model c = (CQ) with unknown margins F‘1, ..., Fd and (X11,...,& I ) , ..., ( X I n r...,X d n ) , n 2 2 a random sample from X. Let the distribution function of the probability integral transformation V = H ( X ) be denoted by

K ( e , t )= P(H(X) 5 t ) = p(CB(Fl(xl), ...>F l ( X d ) )5 t ) .

(4)

In the case of the Archimedian copula family (4) can be computed as follows

( t ~Clayton . copulas this can be given as where f i ( 0 , t ) = $ q 5 ~ 1 ( z ) / z = ~ oFor

nii’,(m+

where q(e, i, rn) = $9). Define the empirical version of K as

where Ein = x ; = , l ( X 1 k 5 xli,...,Xdk 5 Xdi). The test statistics we propose for checking the goodness of fit is based on the comparison of the parametric estimate K ( & , t ) of K ( Q , t )with its empirical counterpart K n ( t ) (for further details see [4]). Known tests for the bivariate case use some continuous functional of the Kendall’s process n,(t) = fi(K(&,t) - K n ( t ) )such as Sn = J i ( r ~ . ~ ( t ) )and ~ d tTn = supost 0 U is a matrix with all its elements equal to 1. For the homogeneous case equation (1)becomes

n

Q ( n ) => w ( n )

+ 1C ( m ) Q ( n

-

m),

(2)

m=l

where

C ( m ) is the Hadamard product of the matrices P, H(m), i.e., C ( m ) = PoH(m) 'W(n) = IO{c:=,+1 C(m)U>. Also from F'rom Papadopoulou and Vassiliou (1994) we get that the limit of Q(n,s ) } equals 0

where 0 P' = limn+mP7L, P = lzms-,mP(s), assuming that P is an irreducible and regular stochastic matrix 0 W = Cz=, nW(n) and W is a diagonal matrix whose i-th element is the mean waiting time in i when s -+ co and W(n) = lim,,,W(s,n).

3

The homogeneous semi-Markov model with fuzzy states

Let SF = { F I ,F 2 , .. . , F N } be the fuzzy state space. In most cases the number N of fuzzy states is much smaller than the number M of states of the initial semi-Markov process. Then the probability that the process is in the fuzzy state F,. (r = 1 , 2...,N ) equals M

Prob(F,) = -ypFT(i)Pl.ob(z),

(4)

i= 1

where 0

p ~ ? , (is. )the membership function of the fuzzy event F,., 0

5 p ~ , ( i )5 1

Discrete Time Semi-Markov Models with Fuzzy State Space 209

cr=l

for every i and N p F , ( i ) = 1. The membership function p ~ , . ( .describes ) the grade of membership of the fuzzy state F, to the crispy state i. 0 Prob(i) is the probability that the process is in state i of the initial state space. Now as described above and taking into account Bhattacharyya (1998), the fuzzy transition probabzlztaes of the process can he defined as: P F ~ F=, prob{the process moves to fuzzy state Fj at its next transition / the process entered fuzzy state Fi at its previous transition} = prob{X,, = Fj/Xp = Fi}. Then from (4)and the latter definition we have

In what follows we will define the corc matrix of the fuzzy semi-Markov process. Let us define as: Cpi,(m) = prob{the process holds in Fi m time units and makes its next transition to Fj / the process entered Fi at its previous transition}.

Lemma 1 It is true that

Proof We have that C F , F , ( ~=)prob{the process holds in Fi m time units and makes its next transition to Fj , the process entered Fi at its previous transition}/prob{ the process entered Fi at its previous transition} = ELl prob{the process holds m time units in state r, makes its next transition to s, entered state r at its previous transition}.

c,"=,

M p F i ( T ) p F j ( s ) / c r = lP o r p F i (T)

-

c::,

P"+Fi

(r)

Using similar reasoning as in Bhattacharyya (1998), the transition proba~ , the probabilities of the holding times C F , F , ( ~for ) the bilities P F ~ and fuzzy process can he interpreted in matrix notation by means of the corresponding matrices of the initial process. We denote: PF = { p F i F j }

210


MF = { p : } , where p$ = p F j ( i ) , i = 1 , 2...M , j = 1,2, ...N Pois an NxN diagonal matrix with its (2, i)-th element equal to porpp, (r)]-' 0 D is an MxM diagonal matrix with its (i,i)-th element equal to poi Then it c m be easily seen that

[EL,

0

pF = D M ~ P , P M ~ ,

(7)

and then the k-step transition matrix (PF)' is given by

(PF)' = D M k P o ( P M ~ D M k P o.)..( P M F D M ~ P , ) P M F ,

(8)

where the number of the parenthesis is k-1 terms. In order to simplify notation we adopt Einstein's notation to get

(PMFDMk Wij = P o j P i , ~ s ~ , s ~ j s

(9)

Under mild conditions concerning the initial probabilities poj and the matrix MF, the product PMpDMkP, turns out to be a primitive matrix. For example, this is the case if puspjs and poi are non-null. Then (PF)', k = 1,2, ..., becomes a fully regular (stochastic) matrix. Similarly, the fuzzy core matrix is of the form

C ~ ( r n .=) DMkP,C(m)MF.

(10)

In the following lemmas we will prove that PF is a stochastic matrix and CF,F, are probability functions, as in the classical process. (a)

Lemma 2 The transition matrix of the fuzzy model is stochastic, i.e.,

Discrete Time Semi-Markov Models with Fuzzy State Space

211

For the probability functions CF,F,(.) yields: Lemma 3

It is true that

Proof We have that

Now, since

C:=, h,.,(m)

= 1 for every s, then

Last, following probabilistic arguments it can be proved that the interval transition probabilities for the fuzzy process are given by ri

Q F ( ~=') W F ( ~+)

C C F ( ~ ) Q F- (m), ~

(13)

ni= 1

where >WF(n) =

Io{cz=TL+l DMkPoC(m)MFU).

Similarly, the above equation for the non homogeneous model can be proved to be of the form

References l.Bhattacharyya, M (1998): Fuzzy Markovian decision process. Fuzzy scts and sustems, 99, 273-282. P.Cinlar, E (1969): Markov renewal theory. Adv. Appl. Prob., 1, 123-187. d 21, 7273.Cinlar, E (1975): Markov renewal theory: a survey. M u ~ ~ u y e , n s tSLL., 752. 4.Cinlar, E (1975): Introductzon t o stochastzc processes., Prentice-Hall, Englewood Cliffs, NJ. B.Howard, R.A. (1971)' Dynanzzc Probabzlzstzc systems. Wiley, Chichester.

212


6.Iosifescu - Manu, A . (1972): Non homogeneous semi-Markov processes. StudLasz Cercetuan A4atematice, 24, 529-533. 7.Janssen, J . (1986): Senti-Markov models: Theory and Applications. ed. J. Janssen, Plenum Press, New York. 8.Janssen, J. and R. De Dominics (1984): Finite non homogeneous semi-Markov processes: Theoretical and computational aspects. Iiwur.utLcc: M u ~ ~ ~ ~ J ~ u ~ L and Economics, 3, 157-165. 9.Janssen, J . and N. Limnios (1999): Semi-Marko~~ models and Applications. J. Janssen and N. Limnios Eds, Kluwer Academic Publishers, Dordrecht. lO.Keilson, J (1969): On the matrix renewal function for Markov renewal processes. Aim. Malh. Slulrsl., 40, 1901-1907. ll.Keilson, J (1971): A process with chain dependent growth rate. Markov Part 11: the ruin and ergodic problems. Adv. Appl. Prob., 3, 315-338. la.McClean, S.I. (1980): A semi-Markovian model for a multigrade population. .I. , 846-852. Appl. P r ~ b . 17, lS.McClean, S.I. (1986): Semi-Markov models for Manpower planning. In SemiMarkov models: Theory and Applications, 283-300. Plenum Press, New York. 14.Mclean, R. A. and M. F. Neuts (1967): The integral of a step function defined on a semi-Markov process. Sia7n. .I. A p i ~ l .Math.., 15, 726-737. 15.Papadopoulou, A.A. & P.-C.G. Vassiliou (1994): Asymptotic behavior of non homogeneous semi-Markov systems. Lznear Algebra and Its Applications, 21 0, 153-198. 16.Pyke, R. and R. A. Schaufele (1964): Limit theorem for Markov renewal process. Ann. Math. Statist., 55, 1746-1764. 17.Teugels J.L. (1976): A bibliography on semi-Markov processes. J . CoTrip. A p p l . Math., 2, 125-144. 18.Vassiliou, P.-C.G. and A.A. Papadopoulou (1992): Non homogeneous semiMarkov systems and maintainability of the state sizes. .I. Appl. Proh., 2.9, 5 1S534. 19,Vassiliou, P.-C.G. , A. Georgiou and N. Tsantas (1990): Control of asymptotic variability in non homogeneous Markov systems. J . Appl. Prob., 27, 756-766. 20.Zadeh, L.A. (1965): Fuzzy sets. Information and Control, 8, 338-353. 21.Zadeh, L.A. (1968): Probability measures of fuzzy events. .J. Math. An.al. A p p l . , 23, 421-427.

An application of the theory of semi-Markov processes in simulation Sonia Malefaki and George Iliopoulos Department of Statistics and Insurance Science University of Piraeus 80 Karaoli & Dimitriou str., 18534 Piraeus, Greece (e-mail: { smalef ak ,geh}Qunipi .gr) Abstract. Importance Sampling (IS) is a well-known Monte Carlo method which is used in order to estimate expectations with respect to a target distribution 7r, using a sample from another distribution g and weighting properly the output. Here, we consider IS from a different point of view. By considering the weights as sojourn times until the next jump, we associate a jump process with the weighted sample. Under certain conditions, the associated jump process is an ergodic semiMarkov process with stationary distribution 7 i . Besides its theoretical interest, the proposed point of view has also interesting applications. Working along the lines of the above approach, we are allowed to run more convenient Markov Chain Monte Carlo algorithms. This can prove to be very useful when applied in conjunction with a discretization of the state space. Keywords: Importance sampling, properly weighted samples, Markov chain Monte Carlo, semi-Markov process, limit distribution, discretization.

1 Introduction One of the most common and difficult t o handle problems in computational stastistics and especially in Bayesian analysis is the estimation of integrals of the form

.i, (X, B(X)), E,(h) :=

h(z)T(dz),

a probability distribution T and a function for a measurable space h E C1(n). For this purpose, many Monte Carlo (MC) and Markov chain Monte Carlo (MCMC) methods have been developed. A well-documented introduction t o this topic is presented by Robert and Casella [Robert and Casella, 19991 and Gilks et al. [Gilks et al., 19961. Importance sampling (IS) [Marshall, 19561 is one of' the most popular and well-known MC methods for handling succesfully such problems. The basic idea behind IS is that instead of sampling directly from the target distribution T , a sample ( 5 1 , . . . , 2), is generated from another distribution g which is easy t o sample from and with support at least the same as T . Then, E,(h) is estimated by

213

214


where w(zi) := r ( z i ) / g ( z i )which , is called importance weight. The most frequently used estimator is LAs since it can be used in more general settings, such as cases where T is known up to a multiplicative constant. Moreover, it can be mentioned here that the assumption of independent samples is not so crucial. IS estimators converge to E,(h) even if the sequence of z's forms a Harris ergodic MC, due to the Ergodic theorem. On the other hand, since all draws are from the proposal distribution g, IS seems at a first glance to fail obtaining samples from T . This makes IS not to be a proper method for the cases when the aim is to estimate features of the distribution that cannot be expressed as expectations, such as quantiles. Recently, it has been proven in [Malefaki and Iliopoulos, 20071 that under certain conditions the g-sample converges in a sense to the target distribution T . By considering the importance weights as sojourn times until the next jump, a jump process is associated with the weighted sample. In the case that the original sample sequence forms an ergodic Markov chain, the associated jump process is an ergodic semi-Markov process with stationary distribution T . From this point of view IS does not differ much from MCMC schemes, in that it exhibits convergence to the target distribution as well. Some of the well-known MCMC procedures are special cases of the above mentioned jump processes (e.g. the Metropolis-Hastings algorithm, see Subsection 3.1). This is also true for general properly weighted samples with respect to a target distribution r. Behind the theoretical interest of the above approach, it can be used in order to facilitate MCMC approximates. Combining the above approach with the discretization of the state space, one is allowed to run a more convenient MCMC algorithm with a diEerent target distribution and then weight properly the obtained output. This paper is organised as follows: In Section 2, we give some basic definitions and the main results of the paper. In Section 3, Metropolis-Hastings is connected with the context of jump processes. Moreover, we give a toy example to illustrate the method and then this method is applied to a benchmark example in Bayesian analysis. Finally, we conclude this paper by providing a short discussion.

2

Main results

The concept of a properly weighted sample has been introduced by Liu and Chen [Liu and Chen, 19981 as a generalisation of the standard IS method. An equivalent and more convenient definition is

Definition 1. [Liu, 20011 A set of weighted random samples called proper with respect to T if

for some positive constant

K,

where

Xi

-

g.

(Xi, [i)lcicn is

An Application of Theory of Semi-Markov Processes in Simulation

215

In the sequel, Malefaki and Iliopoulos [Malefaki and Iliopoulos, 20071 associated a jump process with any infiiiite weighted random sequence in the following sense.

Definition 2. Consider a weighted sequence (X,, strictly positive weights. Define So = 0, S, =

xrzi

Nt

:= sup{n :

s, < t } ,

where the ['s are ti,n 3 1, and let

[n)nE~+,

t 3 0.

(x)tao

Then, the stochastic process Y = defined by yt := X N ~t ,3 0 , will be called the jump process associated with the weighted sequence ( X n ,En),Ez+. T h e definition ensures that the process Y has right continuous sample paths which also have left hand limits. However, if the support of En's is a subset of N = {1,2,. . .}, we will consider the process Y only for t E Z + , i.e. we set Y = (Yo,Y1,Yz,.. .). If this is the case, limits of quantities related t o Yt should be suitably interpreted.

Proposition 1. Assume that the sequence X = (X,),€Z+ is a homogeneous Harris ergodic Marlcow chain with state space ( X ,B ( X ) ) having an invariant depends solely on X , with probability distribution g and the distribution of E{EnIXn = x} = K W ( Z ) = m-(Ic)/g(x)for some K > 0 . Then, for the jump process ( y t ) t ) o associated with the weighted sequence (X,, E n ) n E ~ + , it holds that lim P{yt E A } = 7r(A), V A E B ( X ) .

cn

tTm

Proof. T h e result follows from the standard theory of semi-Markov processes [Limnios and OpriSan, 20011. Under the above assumptions, Y is a n ergodic semi-Markov process with embedded Markov chain X and respective sojourn times ( E n ) n E Z + . Thus,

as is claimed. Setting deterministically

En = w(X,),

we have the following:

Corollary 1. If (Xn),€z+ forms a Harris ergodic Marlcov chain with stationary distribution g , then the jump process associated with the weighted sequence (X,, W ( X , ) ) , ~ Z +has 7r as limit distribution. Any sequence of independent g-distributed random variables trivially forms a n ergodic Markov chain with stationary distribution g. Thus, Corollary 1 covers also the original importance weighted sequence. The requirement that the distribution of (, depends only on x, seems rather restrictive. However, zn could be a block of specific size allowing En

216


to depend on more than one term of the original sequence. (Note that the standard definition of a semi-Markov process allows the sojourn time of X , depending on both X , and X,+1.) As already mentioned in the Introduction, some of the well-known sampling schemes are special cases of this context. Moreover, another potential application of Proposition 1 is the following: Let 7r be a target distribution of which some full conditional distributions are difficiilt, to sample from. Instead of hybridizing Gibbs sampler using suitable Metropolis steps, one can replace 7r by another target distribution g of which all full conditional distributions are easily handled, run the Gibbs sampler and fiimlly weight the output. The weighted sample will be the realization of a converging .jump process (see Section 3).

3

Examples

In this section we connect Metropolis-Hastings (MH) alorithm with the context of jump processes. Moreover, we give a simple example of how one can facilitate MCMC algorithms by discritization of the state space followed by proper weighting of the sample, so that the associated jump process converges to the target distribution. 3.1

The Metropolis-Hastings algorithm

Consider an arbitrary MH algorithm [Metropolis et al., 1953; Hastings, 19701 with target distribution 7r and proposal q ( , l . ) , that is, at time t 1 given yt = y, draw 2 q ( z l y ) and set yt+l = z with probability

+

-

or X+l = y with probability 1 - a ( y , 2). Although it is well-known that the algorithm defines a reversible Markov chain with stationary distribution 7 r , let us consider it from a different point of view. + a Markov chain with transition density Let X = ( X n ) n E ~be

(Notice that this is exactly the density of the accepted states of the above MH algorithm.) It can be easily verified that g ( z & - 1 ) sat)isfies the detailed balance condition g ( z i - 1 ) g ( x i l z i - l ) = g ( z i ) g ( z i - 1I z i ) , where g ( z ) x J min(7r(x)q(zlx),7 r ( z ) q ( z / z ) } p ( d z ) .This function, when normalized, results in a probability density function, hence it is the stationary distribution of the Markov chain X . Weight now xi by ti drawn from the geometric distribution with success probability J a(zi,z)q(zlzi)p(dz). Since

E{tlzi} =

{.f ~ ( z iz ),& I Z i ) p ( d z ) } - '

x T(zi)/g(zi),

(1)

A n Application of Theory of Semi-Markov Processes in Simulation 217 the sequence ( X n r ( n ) n E is ~ +properly weighted with respect to 7r. It is immediately seen that the associated jump process is the original MH output (Y,)tGz+which is known t o be a pure Markov chain (rather than a general discrete time semi-Markov process). The above analysis suggests that we are allowed t o use any distribution (beyond the geometric) for the weights provided that (1) is sat,isfied. In particular, if & m ~ ( z i=) 7r(zt)/.q(zi)is chosen, the variance of estimators of certain expectations of interest will be minimized. However, direct calculation of this importance weights is in general computationally demanding or even infeasible making such a task hard to accomplish. Moreover, the geometric distribution comes out naturally since each simulation from g(.l.) automatically generates the corresponding geometric weight. 3.2

A toy example

-

Consider a random variable X Beta(2, 2), 0 5 z 5 1. The density function of X, up t o the normalizing constant is: 7r(z) 0;

z(1- z).

We discretize the interval [0, 11 into m = 10 equal length bins and choose as target distribution the g(x) 0: q m l ( 1 - " [ m ] ) .

+

where x [ m ~= ( [ m z ] 0.5) /m. In order t o sample from g , first draw a bin from the discrete distribution

-

+

and then simulate U U ( 0 , l ) and set z = ( j u l ) / m . In the above scheme the importance weights are w ( z ) = 7 r ( L c ) / g ( z )and according t o the Proposition 1the jump process associated with the weighted output converges t o 7r. Fig.l shows the histogram and the convergence of the weighted mean 2;' = C~="=,(zi).i/C~="=,(zz) with m = 10, computed from a n output of 10000 updates after a burn-in period of 1000 iterations t o the mean E ( X ) . In this connection, it may be remarked that this method does not require a large m in order t o converge. Moreover, we note the fast convergence of 2;'. Finally, we can point out that the convergence of 2;' is not worse than the corresponding one of the sample mean of an iid sample from the target distribution, as we can see in Fig.1.

3.3

~

Dugongs dataset

The proposed method has also been applied t o the well known dugongs dataset which was originally analyzed by Ratkowsky [Ratkowsky, 19831. This

218


m

0.65~ 0.6 0.55

Fig. 1. Histogram of the weighted sample ( x i ,w ( ~ ~ ) ) ~ ~ and ~ ~ convergence ~ o o o o of line) and of the sample mean of an iid sample from the target distribution (dashed line).

?ks with m = 10 (solid

particular dataset is among the standard examples which is used by many authors in order to illustrate and compare several sampling techniques. The data consist of length (y) and age (z) measurements for n = 27 dugongs captured near Townsville, Queensland. Carlin and Gelfand [Carlin and Gelfand, 19911 modelled the data using a nonlinear growth curve with no inflection point and an asymptote as z tends to infinity. Specifically, they assumed that

yi-N(wwhere N , , ~ , I -> 0 and 0 < y prior for the parameters:

-

" 0 , 7,1)1(. > O ) , P

with ra = (a3Bl

Yl.)

70

=

O p , 7 - 1 ) ,

i = l,...,n,

< 1. We consider the following relatively vague

-

N(0,T,l)1(3 > O), y

- U ( 0 ,1),

7

N

G(kl k ) ,

and k = lop3. The posterior distribution of 6, =

is

Sampling from the full conditional (posterior) distributions of a , 0 (truncated normal) and 7 (gamma) is a straightforward task but this is not the case for y. Instead of using a Metropolis step, we can adopt the following strategy. We choose a different target distribution, namely, g(0ldata) by discretizing the sample space of y into m equal length bins, with y being uniformly distributed within each bin. The form of the new target distribution is

A n Application of Theory of Semi-Markou Processes in Simulation 219

where 7jm]is the point that the maximum of r(0Idata) with respect to y,is achieved in each bin. In the above scheme the importance weights are

w(e)

c(

n(0 I data) g(0ldata) ’

These points are selected in order for the weighted sample mean t o have finitme variance. Sampling from the full conditional distribution of y is now a n easy task: one can first draw a bin from the discrete distribution

-

and then simulate U U ( 0 , l )and set y = ( j + u - l ) / m . According to Proposition 1, the jump process associated with the weighted output converges t o rr. Fig.2 shows the histogram and the convergence of the weighted mean = CZ, w(&)yi/ Cy=lw(&),with m = 20, computed from the output of 10000 updates after a burn-in period of 1000 iterations to the posterior mean E{yldata}. The graphs for the rest of the parameters are similar. An interesting feature arising from the above weighted scheme is that the autocorrelations almost vanish (see Fig.2). Hence, the standard errors of the estimates of the parameters can be calculated easily. The decrement in autocorrelations is also similar t o all the parameters of the model.

+A”

0.6

0.7

0.8

1.

L

0.7

0.65

0.3

1.

0.75

0.75

0.5

0.5

0.25.

0.25 11111,

Fig. 2. Histogram of the weighted sample (yi, z u ( O i ) ) l ~ i ~ l o o oconvergence ~, of 9;” and the autocorrelatons of y and weighted y with m = 20 for the dugongs dataset.

220

4


Discussion

T h e a i m of this paper is to stress o u t that t h e proper weighting of a Markov chain’s o u t p u t can be used i n order to obtain samples from t h e target distribution. T h i s is accomplished by considering i t from a different point of view, namely, associating an appropriate weakly convergent j u m p process to the weighted sample. Man y well known simulation schemes, including the MH algorithm, fall in this context. Moreover, contrary to w ha t is thought, this is t h e case for t h e s t an d ar d IS o u t p u t . Hence, IS c a n also be used in order to obtain (approximate) samples from t h e t ar g et distribution. Besides its theoretical interest, t h e benefit of the proposed point of view is significant if i t is applied i n conjunction with the discretization of t h e state space i n order to facilitate M C M C algorithms (as in Subsection 3.3).

References [Carlin and Gelfand, 1991lB.P. Carlin and A.E. Gelfand. An iterative Monte Carlo method for nonconjugate Bayesian analysis. Statistics & Computing, 1:119128, 1991. [Gilks et al., 1996lW.R. Gilks, -, and et al. Markov Chain Monte Carlo in practice. Chapman & Hall, New York, 1996. [Hastings, 197OlW. K. Hastings. Monte Carlo sampling methods using Markov chains and their applications. Biometrika, 57:97-109, 1970. [Liu, 2001 1J.S. Liu. Monte Carlo Strategies in Scientific Computing. SpringerVerlag, New York, 2001. [Liu and Chen, 1998lJ.S. Liu and R. Chen. Sequential Monte Carlo methods for dynamic systems. Journal of the American Statistical Association, 93: 10321044, 1998. [Limnios and Oprigan, 20011N. Limuios and G. OpriSan. Semi-Markov processes and reliability. Statistics for industry and technology, Birkhauser, 2001. [Malefaki and Iliopoulos, 2007]S. Malefaki and G. Iliopoulos. On convergence of properly weighted samples t o the target distribution. Journal of Statistical Planning and Inference, (to appear). [Marshall, 1956lA.W. Marshall. The use of multi-stage sampling schemes in Monte Carlo computations. In M.A. Meyer, editor, Symposium on Monte Carlo methods, pages 123-140, Wiley, New York, 1956. [Metropolis et al., 1953lN. Metropolis, -, and et al. Equations of state calculations by fast computing machines. The Journal of Chemical Physics, 21:1087-1091, 1953. [Ratkowsky, 1983lD. Ratkowsky. Nonlinear regression modeling. Marcel Dekker, New York, 1983. [Robert and Casella, 1999lC.P. Robert and G. Casella. Monte Carlo statistical methods. Springer-Verlag, New York, 1999.

On a numerical approximation method of evaluating the interval transition probabilities of semi-Markov models Dimitrios Bitziadis', George Tsaklidis2and Aleka Papadopoulou2

' Technological Institute of Thessaloniki P.O. Box 14561 54 10 1 Thessaloniki, Greece (e-mail: BitziadisDOgiraeusbank.gr) Department of Mathematics Faculty of Science Aristotle University of Thessaloniki 54124 Thessaloniki, Greece (e-mail: [email protected]) Abstract. For the classical semi-Markov model, either time homogeneous or nonhomogeneous, an examination of the convergence of the interval transition probabilities Pi,(s,t) as t-m is presented using an approximation method provided by [R. De Dominics and R. Manca 19841. Especially, we examine the dependence of the accuracy of the respective numerical method on the various values of the step h, in finding the transition interval probabilities, and we investigate the complexity of this algorithm.

Keywords: Semi-Markov process, Non homogeneity, Numerical methods.

1. Introduction

In what follows denote by E={ 1,2,...,n} the state space of the Markov model and by t the (continuous-) time parameter. The states 1,2,...,n are exclusive and exhaustive, i.e., every member of the system may be in one and only one state at some time point t. Also denote by P,, i,j =1,2, ...,n, the probability that a member who entered state i on its last transition will enter state j on its next transition. The transitions between the states occur according to either a homogeneous or a non-homogeneous semi-Markov chain (the embedded semi-Markov chain). Let P={P,} be the matrix of transition probabilities and G(s,t)={G,(s,t)} the matrix of the distributions of the holding times, i.e., G,(s,t)=prob{the holding time in state i is less than or equal to t-s, given that the entrance in i occurred at time s and next transition will take place in state j}. 221

222


The core matrix of the semi-Markov process is Q(s,t) = { Q,(s,t)}, with Q(U) = POG(s,t) = {P,GkJ(s,t)}, where the symbol “0” stands for the Hadamard product. Finally, we denote S(s,t)=diag{S,(s,t)}, where S,(s,t) = CQJ(s,t)= prob{ the holding time in state i is less than or equal J

to t-s, given that the entrance in i occurred at time s } The aspects that have been studied in discrete and continuous time semi-Markov models, as outlined above, are the asymptotic behaviour, stability, variability, attainability etc. Semi Markov models arise in physics, actuarial work, biometry and manpower planning. Basic results can be found in [McClean, 1978,1980,1986], [Mehlman, 19791, [Bartholomew, 19861, [Howard, 19711, [Janssen and Limnios, 19991, [R. De Dominics and R. Manca, 19841, [Papadopoulou and Vassiliou, 19941 and [Vassiliou and Papadopoulou, 19921. The most characteristic features of a semi-Markov model are related to the interval transition probabilities of the model. A recursive formula for the interval transition probabilities is 1

P,(s,t>=4J(l-s,(4)

+

Cjp,(u,W(ak(S,4)

f

(1)

k s

Relation (1) can be approximated numerically by the recursive relation 1

c,s,j,f= q,s,j,t+

Cc&,k,he,h,j,t k h=s

9

(R. De Dominics and R. Manca (1984)), where:

The step of the approximation is equal to 1, i.e., the calculations in (2) are carried out only for h = s, s+l, s+2,. ..t Q,s,J,, =q,J(l-,S,,s,,) is the approximation of the probability that the process holds in i without any transition in the time interval (s, t ) S,,s,,=cQ,,,,,, is the approximation of the probability that the J

chain moves fiom state i into any other state before time t, given that the entrance in i took place at time s =q,s,J,tq,J is the approximation of the i,j element of the core

a,,,,,,

matrix of the semi-Markov process y,s,k,h is the difference Q,s,k,h -Q,s,k,h-,for the time interval (h-1,h) if t>s+l,which interprets an approximation of the p.d.f. c,&)

Numerical Approximation of Semi-Markou Models

223

if t=s+l, and y,b,j,b =O. The recursive formula (2) in matrix notation becomes

y,s,k,b =Q,s,k,b

I

K,t = D,,l

+ cy,bpb,t 3

(3)

b=S

with initial conditions P,,, = D,,,= I and Vb,b=0, where the i,j-th element of P,,, equals & J , t , the i,j-th element of D,,, equals L?s,J,l, and the i,j-th element of V,,hequals y,s,J,b. ARer some manipulations, formula ( 3 ) can be written in the form UP=D, where U is an upper block triangular matrix with the (i,i)-th block equal to the unity matrix and the (i,j)-th block (ifj) equal to -y,J, P is a block upper triangular matrix with the (i,i)-th block equal to the unity matrix and the (ij)-th block (ifj) equal to p,,, , D is a block triangular matrix with the (i,i)-th block equal to the unity matrix and the (i,j)-th block (ifj) equal to D,,, . It is interesting to investigate the convergence of ( 3 ) numerically. Using a simulation program created in PowerBilder we examine, for various values of the step h, the dependence of the accuracy of the numerical method based on ( 3 ) in order to find the transition interval probabilities. We treat both classical cases, i.e., when the holding times depend on the time of entrance into some state or not (non-homogeneous and homogeneous case respectively). Finally, we investigate the complexity of the algorithm provided by ( 3 ) . 2. Homogeneous case

We applied the approximation method provided by (3), for the example of the homogeneous semi-Markov system given in Papadopoulou and Vassiliou (1999), where

9/15 5/15

1/15

1/10 8/10 1/10 1/20 4/20 15/20

2e-2t 3e-3~ 4e4' 2e-2' e-' 3e-3' 3e-3' 2e-2t e-'

224


Then the core matrix Q(s,t)= POG(s,t) and the matrix D(s,t) can be evaluated easily for every s and t. We carried out the simulation using relation (3), with iterations 1 to 50 for s and t, and step-size h equal to 1, 0.5, 0.3, 0.2 and 0.1. We used double precision floating-point variables, with 15 digits of accuracy and range fiom 2.2E-308 to 1.7E+308. The results concerning the computation of the transition probabilities, via (3), are given below. Result analysis for the evaluation of the transition probabilities step h=l, time interval [30,50], execution time 2 min [1,1] = 0,13621 [1,2] = 0,58660 [1,3] = 0,27719 [2,1] = 0,13617 [2,2] = 0,58662 [2,3] = 0,27721 [3,1] = 0,13503 [3,2] = 0,58026 [3,3] = 0,28471 step h=0.5, time interval [30,50], execution time 6 rnin [1,1] =0,11662 [1,2]=0,59928 [1,3] =0,28410 [2,1] = 0,11663 [2,2] = 0,59941 [2,3] = 0,28396 [3,1] = 0,11629 [3,2] = 0,59664 [3,3] = 0,28707 step h=0.3, time interval [29.8,49.6], execution time 20 rnin [1,1] = 0,10738 [1,2] = 0,60561 [1,3] = 0,28698 [2,1] = 0,10739 [2,2] = 0,60574 [2,3] = 0,28683 [3,1] = 0,10718 [3,2] = 0,60371 [3,3] = 0,28907 step h=0.2, time interval [30,50], execution time 60 rnin [1,1] = 0,10254 [1,2] = 0,60902 [1,3] = 0,28851 [2,1] = 0,10255 [2,2] = 0,60913 [2,3] = 0,28839 [3,1] = 0,10240 [3,2] = 0,60756 [3,3] = 0,29011 step h=0.1, time interval [30,50], execution time 180 rnin [1,1] = 0,09755 [1,2] = 0,61227 [1,3] = 0,28989 [2,1] = 0,09756 [2,2] = 0,61238 [2,3] = 0,28977 [3,1] = 0,09745 [3,2] = 0,61109 [3,3] = 0,29117 Theoretical values of the transition probabilities [ 1,3] = 0,2968 [ 1,1] = 0,0942 [ 1,2] = 0,609 [2,1] = 0,0942 [2,2] = 0,609 [2,3] = 0,2968 [3,1] = 0,0942 [3,2] = 0,609 [3,3] = 0,2968

Numerical Approximation of Semi-Markov Models

225

As an example, we present for h=0.1 the trajectories of the interval transition probabilities P,2(30,30+t), P22(30,30+t) and P32(3O730+t), t=1,2,..,20. Convergence-graph of the three second column elements of the interval transition probabilitv matrix with step h=O. 1

Convergence-graph of the three third column elements of the interval transition probability matrix with step h=O. 1

I

I

C2DD' 0,100~ 4

2 4 6 8 10 11 14 I 6 18 20 22 24 26 B 30

Time

I 4 $ 8 10 12 I k 1% C I 20 2 14 2d 18

Time

Conclusion: There are significant estimation errors if h=l, 0.5, 0.3 and 0.2. Even for h=0.1 there are estimation errors appearing at the third, second and third decimal place for the first, second and third columns of the transition matrix respectively. Thus, the step-size h must be small enough - in connection with the estimation accuracy needed. On the other hand the execution time becomes considerable large; in our case, for h=O.1 it becomes three times more than for h=0.2. 3. Non-homogeneous case

Let the transition matrix be the one used in the homogeneous case, and the non-homogeneous matrix of the holding times be as follows:

226


The results concerning the computation of the transition probabilities by means of (3) are given below. step h=l, time interval [30,50], execution time 2 min and 30 sec [1,1] = 0.13553 [1,2] = 0.58724 [1,3] = 0.27724 [2,1] = 0.13548 [2,2] = 0.58729 [2,3] = 0.27723 [3,1] = 0.13427 [3,2] = 0.58036 [3,3] = 0.28537 step h=0.5, time interval [30,50], execution time 7 min and 10 sec [1,1] =0.11616 [1,2]=0.59970 [1,3] =0.28415 [2,1] = 0.1 1617 [2,2] = 0.59985 [2,3] = 0.28398 [3,1] = 0.1 1579 [3,2] = 0.59672 [3,3] = 0.28749 step h=0.3, time interval [30,50], execution time 23 min and 30 sec [1,1] = 0.11707 [1,2] = 0.60590 [1,3] = 0.28699 [2,1] = 0.10708 [2,2] = 0.60605 [2,3] = 0.28683 [3,1] = 0.10684 [3,2] = 0,60372 [3,3] = 0.28939 step h=0.2, time interval [30,50], execution time 68 min [1,1] = 0.10233 [1,2] = 0.60923 [1,3] = 0.28852 [2,1] = 0.10234 [2,2] = 0.60936 [2,3] = 0.28837 [3,1] = 0.10217 [3,2] = 0.60754 [3,3] = 0.29037 simulation step h=0.1, time interval [30,50], execution time 190 min [1,1] = 0.09744 [1,2] = 0.61241 [1,3] =0.28987 [2,1] = 0.09745 [2,2] = 0.61253 [2,3] = 0.28974

Numerical Approximation of Semi-Markov Models

227

[3,1] = 0.09732 [3,2] = 0.61 103 [3,3] = 0.29138 theoretical values of the transition probabilities [ 1,3] = 0,2968 [ 1,1] = 0,0942 [ 1,2] = 0,609 [2,1] = 0,0942 [2,2] = 0,609 [2,3] = 0,2968 [3,1] = 0,0942 [3,2] = 0,609 [3,3] = 0,2968 Conclusion: In the same manner as in the homogeneous case, a small h is needed in order to achieve accurate values of the transition probabilities. Thus, considerable errors are observed for h=l, 0.5, 0.3 and 0.2. For h=O. 1, the errors appear at the third, second and third decimal digit for the first, second and third column of the transition matrix respectively; in that sense the transition probabilities are close to the theoretical values if h10.1. The execution time (190 min) for h=O. 1 is almost three times more than for h=0.2.

4. On the algorithmic complexity 1. Discrete case. Evaluation of Pst, by means of (3), needs N3 multiplications and (N-1)N2 additions for the computation of Vs,hPh,t. Thus the time complexity of computing Pst is OW3); it can be reduced, for instance using Strassen's algorithm to become O(n22807) (but the error bound is weaker than that of the traditional algorithm). We notice that the same conclusion holds if the method provided by Vassiliou and Papadopoulou (1992) is used, since the respective equation applied, is of the same structure as (3).

2. Continuous time. For the algorithm given in De Dominics and Manca (1984) a small-enough step-size h shall be applied in order to achieve the desired accuracy for the computation of the transition probabilities. We note that especially for the evaluation of the asymptotic probabilities a formula in closed analytical form is given by Vassiliou and Papadopoulou (1992), where matrix inversion is needed. Using Gaussian elimination the time complexity of matrix inversion is O(N3). The time needed for the computation of the asymptotic transition probabilities by means of (3) in that case, is larger than the time needed using the direct evaluation of the asymptotic probabilities by means of the formula using matrix inversion, as carried out -for instance- by MATLAB or Mathematics etc.

228


References

[Bartholomew, 19861 D. J. Bartholomew. Social applications of semiIn Semi Markov models: Theory and Markov processes. Applications. New York:Plenum Press, 463474,1986. [De Dominics and Manca, 1984lDe Dominics R. and R. Manca. An algorithmic approach to non homogeneous Semi Markov processes Commun. Statist. - Simula. Computa., 13(6), 823-838, 1984. [Howard, 19711A. R. Howard. Dynamic probabilistic systems Chichester: Wiley, 1971. [Jansen and De Dominics, 19841 J. Jansen 'and R. De Dominics. An algorithmic approach to non homogeneous Semi Markov processes. Insurance:Math. Econ. 3: 157-165, 1984. [Janssen and Limnios, 1999lJ. N. Janssen and N. Limnios eds. Semi Markov Models and Applications. Vol.15, 23-32. Dordrecht Kluwer Academic Publishers, 1999. [McClean, 19781s. I. McClean. Continuous time stochastic models for a multigrade population. J. Appl. Prob. 15: 26-32, 1978. [McClean, 19801s. I. McClean. A semi Markovian model for a multigrade population. J.Appl. Prob. 17: 846-852, 1980. [McClean, 19861s. I. McClean. Semi Markov models for manpower planning. In Semi Markov models: Theory and Applications. New York:Plenum Press, 283-300, 1986. [Mehlman, 19791 A. Mehlman. Semi Markovian manpower models in continuous time. J. Appl. Prob. 16: 416-422, 1979. [Papadopoulou and Vassiliou, 1994lA. Papadopoulou and P.-C. G. Vassiliou. Asymptotic behavior of non homogeneous semi -Markov processes. Linear Algebra Appl. 210: 153-198, 1994. [Vassiliou and Papadopoulou, 19921 P.-C. G. Vassiliou and A. Papadopoulou. Non homogeneous semi -Markov systems and maintainability ofthe state sizes J. Appl. Prob. 29: 519-534, 1992.

Markov property of the solution of the generalized stochastic equations KHALDI Khaled Univdrsitd dc Boumcrdcs FacullE des Sciences DBpartement de MathCmatiques BOUMERDES 35000

ALGERIA (email: k k h a l d i h b b . dz) Abstract. Some models of probabilities are described by generalised stochastic equations. These models lead to the resolution of boundary problems for random distributions (generalized equations). We are interested in the equation Lz = f in S C IRd where L is a linear operator, f is a random distribution and to the class of boundary conditions on the frontier I’ = a s in ordcr t o dcfinc for thc corresponding boundary conditions. The resolution of boundary problems for random distributions lead to the Markov property for the solution of these equations. AMS Classif : Primary 60G12; Secondary 47A15 Keywords: Linear operator, stochastic equation, stochastic distribution, Markov property.

Some models of probabilities are descripted by generalised stochatic equations. These models (like that prediction) lead to the resolution of boundary problems for random distributions (generalizedequations). We are interested in the equation Lx = f in S c IRd where L is an linear operator, f is a random distribution and to the class of boundary conditions on the frontier r = dS in order to define for the corresponding boundary conditions. The resolution of boundary problems for random distributions lead to the Markov property for the solution of these equations. The boundary problems for the linear equations of the type:

Lx ( t )= f ( t ), t E s1

(1)

where L is a linear operator , lead to search the distribution x (t), t E T , in T 2 E d , containing the domain T C E d , such as 2

( t )= Y ( t ), t

E T\S2

(2)

where T\S is the additional of S in T. So that the probleme admits a (only) solution, the distributions f ( t )and v(t) as the solution x(t) have to belong to a certain class which we shall describe more low. The solution x(t) is bound to the operator L verifiying the equation:

L[ ( t )= [* ( t ), t E T3 229

(3)

230


where ( ( t )and (* ( t )are distributions with values in Hilbert's space H . By distribution with values in H , we understands a linear continuous application: € : cp E c r (TI ((3: €1 E H4 (4)

-

In first, we are interested in the properties of the solution { ( t )of the boundary problem (3) that we prolong in W' = C r (T)space of distributions u = (cp, u),cp E C r (T) . The distribution u E C r (T) is,

we define for a some set S ( S C T) the space

The solution E ( t )of the equation (3) for a local operator L possesses the following markov property: for everything S C T with border I' = dS, the . projection of the space H+ (T\S) in H+ ( S ) coincides with H+ (r) One notices that an equation of the type (3) with L = l*l and E* ( t ) = 1'17 ( t )occupies an important place in the theory of the stochastic differential equations of the type L ( ( t ) = t E T,where 1 is a differential linear operator, 71 (t)a distribution in H (Hilbert's space) called "white noise" in t E T. For every u E C r (T) , the distribution

L u = ( v , L u ) , cpECoOO(T) is a positive linear application u --+ (cp, L

Lu : 07

(7)

with 11 cp 11 = (cp, L u ) ; and one considers the espace de Hilbert's W,completed of C r (T) by the scalar product

(u, v) = ( u l L v ) ,u, E C r ( T ) 8

(8)

By using (8) ,one corresponds to every v E W the distribution Lv :

Lv

=

(cp,Lv) = (cp,v), cp E c r (T)9

(9)

and one introduces W* (set of distributions Lv prolonged by continuance on W ) : Lv = (ul Lv)= (u, v) 10 (10) It is evident that W' is the dual of W and exactely W * is the set of functional linear on W deescribed by (9), with

Markov Property of Solution of Generalized Stochastic Equations

11 Lv 11

231

= ( L v ,Lvu)i = 11 cp 11 = l S U p p (cp, v)

The equation (1)est undderstood inthe sense

f

(cp, Lx) = (9, ), cp E

c,- (S)11

(11)

aand the boundary conditions (2) as

( u , x )=

(U,V),

u E W * ,Suppu C T\S12

(12)

For the search for the solution of the problem (1) - (2), one considers the Hilbert's space H defining the isometric application [* : u E W

-+ (u,

t*)E H13

(13)

defined by ( 3 ) . Thc application (4)maybc idcritificd with thc isomctric application

E

:u E

W'

-+

(u, [) =

(L-lu, [*) E H14

(14)

We define H (S)and H* (S)respectively by H (S)= ((cp, E ) , cp E C r ( S ) ) and H* (S) = ( ( p , t*),(cp E C r (S))).It is evident that H (T) = H* (2"). Alterward, one deEne H = H ( T ). That is to say q } the scalar product of [ and 7 E H . One has so

{c,

The condition ( T L , t )I(cp,

H * (I') = H* (T\S)l = H+ (S) and H+ (r)@ H* (T\S) = H * (S)l = H+ (T\S) . By indicating by LI+ (S) and D+ (T\q the orthogonal projections on H+ ( S ) and H+ (T\S) respectively, one has IT+ ( S )H+ (T\S) = H+ (T\S) lf+ ( S ) = 11, (I'). What means that the distribution ( ( t ) possesses, in the espace of Hilbert, the markov

.

H

Markou Property of Solution of Generalized Stochastic Equations 233 References [l]G. KALLIANPUR and U. MANDREKAR. The markov property for gcncrdizcd gniissian random ficlds. Ann. Inst. Fourier, 1974, V.2, N2, p.143-167. [2]K.KHALD1. Boundary problems for local limited operators in Hilbert’s spaces. Stochastics processes , mathematical statistics and applications. University of Moscou. 1989, p.31-33. [3] K.KHALD1. Boundary problems for stochastics equations and stochastics markov ficlds. Proc. 3nd. International Conference on Applied Mathematics and Engineering Sciences. CIMASI 2000. Casablanca. 2000. [4] P. LEVY. A special problem of gaussian random functions. Proc. 3nd Berkeley Symp. Math. Stat. Prob. 1956, V.2, p.133-175. 131 Y . ROZANOV. Sorric bouiidary problciiis for pour diffcrciitial gciicralized equations. Math. Zametki, 1987, T.l, N.l, p.llO-118. 161 Y . ROZANOV. Boudaiy problcrris for stochastic partial diffacritial equations. Universitat Bielfeld. Forschungszentrum Bielfeld Bochum - Stochastic. 1987, N.23. [7] Y. ROZANOV. Stochastics markov fields. Naouka. Moscow. 1981.

Partially Markov models and unsupervised segmentation of semi-Markov chains hidden with long dependence noise JerGme Lapuyade-Lahorgueand Wojciech Pieczynski GET/INT, CITI Department, CNRS UMR 5 157 9, rue Charles Fourier, 91000 Evry, France

Abstract. The hidden Markov chain (HMC) model is a couple of random sequences ( X , Y ) , in which X is an unobservable Markov chain, and Y is its observable noisy version. Classically, the distribution p(ylx) is simple enough to ensure the Markovianity of p ( x l y ) , that enables one to use different Bayesian restoration techniques. HMC model has recently been extended to “triplet Markov chain” (TMC) model, which is obtained by adding a third chain U and considering the Markovianity of the triplet T = ( X , U , Y ) . When U is not too complex, X can still be recovered from Y . In particular, a semi-Markov hidden chain is a particular TMC. Otherwise, the recent triplet partially Markov chain (TPMC) is a triplet T = ( X , U , Y ) such that p ( x , u ( y )is a Markov distribution, which still allows one to recover X from Y . The aim of this paper is to introduce, using a particular TPMC, semi-Markov chains hidden with long dependence noise. The general iterative conditional estimation (ICE) method is then used to estimate the model parameters, and the interest of the new model in unsupervised data segmentation is validated through experiments. Keywords: hidden Markov chains, triplet Markov chains, unsupervised segmentation, image segmentation, iterative conditional estimation.

1. Introduction Let X = (Xn),sn6N and Y = ( < ) , s n s N two stochastic processes, where X is hidden and Y is observable. In the whole paper, each X n takes its values in a finite set of classes i2 = {u,,. ..,w,} and each takes its values in R . The problem of estimating X from Y , which occurs in numerous applications, can be solved with Bayesian methods once one has chosen some accurate distribution p(x,y ) for 2 = (X, Y ) . The hidden Markov chain (HMC) model is the simplest and most well known model. Its applications cover numerous fields, and can be seen in recent books or general papers [Koski, 20011, [CappC et al., 20051. [Ephraim and Merhav, 20021. However, it is insufficient in some situations and thus it has been extended to “hidden semi-Markov chains” models [Faisan et al., 20051, [Guedon, 20031, [Moore and Savic, 234

Partially Markov Models and Unsupervised Segmentation

235

20041, [Yu and Kobayashi, 20031. Otherwise, a long dependence noise does exist in some situations [Doukhan et al., 20031, but can not be taken into account in the classical HMC. The aim of this paper is to propose a new model in which the hidden chain is a Markov one, and in which the noise is a long dependence one. On the one hand, we exploit the fact, already mentioned in [Pieczynski and Desbouvries, 20051 that an HSMC is a particular “triplet Markov chain” (TMC, [Pieczynski et al., 20021, [Ait-elFquih and Desbouvries, 20061). On the other hand, we exploit the ideas proposed in [Pieczynski, 20041. We also propose a parameter estimation method of “iterative conditional estimation” (ICE) kind [Fjortoft et al., 20031, and show that the new model can be of non negligible interest in unsupervised data segmentation.

2. Triplet Markov chains and hidden semi-Markov chains

(x)lsncN

Let X = (X,1)ILn6N and Y = be two stochastic processes mentioned above. The problem of estimating X = x from Y = y can be solved once the marginal posterior distributions p ( x , Iy) are calculable. Let us consider an taking its values in a finite state space A = {l,...,L } and auxiliary process such that ( X , U ) is a Markov chain whose distribution, given by

The chain X

is then called “semi-Markov chain”. If we consider

N

p(ylx) = n p ( y n l x n ) the , triplet ( X , U , Y ) is the classical “hidden semi“=l

Markov chain” (HSMC). In this paper, we consider a more sophisticated noise distribution p ( y l x ) , which is a “long dependence” one. To introduce it, let us consider stationary Gaussian process Y = (q,...,Y N ). It will be called “long-dependence’’ if its covariance function y ( k ) = E( 1) and u,:+, = u,:

(if

+ 1. Otherwise, as the noise is a long dependence one,

the distribution p(ylx) is not a Markov one. In the model we propose, the distribution of y,,, conditional on X,,, = x,,+,, U:+, = k , and Y,= yi , ..., = y,, , depends on X,,+,= xn+,and Yn-k = yn-k, ..., q, = y , . Therefore, for

0.

Assumption 6 has t o be connected with the integrability condition in [Stute, 19951 and [Stute, 19961, which permits to obtain a central limit theorem for Kaplan-Meier integrals. In our case, we were forced to introduce an extra E > 0, which makes our Assumption slightly more restrictive, but permits to extend the i.i.d. representations of [Stute, 19951 t o kernel smoothing. 4.2

Behavior of the tests under the null hypothesis

The following theorem gives an asymptotic representation of the test statistics under Ha stated in (3).

T : ”

Nonparametric Regression Checks with Censored Data

265

Theorem 1. Under Assumptions 1 to 6, under Ho,

an probability, where 2 2 [ V 3 6 o ) ] = n ( n- 1 ) h P

c[u,””(eo,]z[u;”(Bo,]2

K2 (Xi

t#j

Moreover, under H o ,

Corollary 1. Under Assumptions 1 to ti, the test defined in the p,revious section has asymptotic level a . 4.3

Behavior of the tests under the alternatives

Consider a sequence of measurable functions X,(z), of alternatives

n 2 1, and the sequence

We study consistency against three types of alternatives: fixed alteriiatives, Pitman local alternatives, and smooth alternatives. The first case is obtained when the function A, is the same function for each n . Pitman alternatives consist in considering X,(z) = r,X(z), where X is a fixed function, aid r, a deterministic sequence tending to zero. The last case we consider is the consistency uniformly over alternatives that belong to a Holder smoothness class and vanish as n + co.The regularity s of the Holder class is supposed known. It can be shown that a representation similar to that of Theorem 1 holds under the alternatives. Hence our test will have the same property as the test in the uncensored case. In particular, it will be consistent against any fixed alterndive and against Pitman alternatives with rn decreasing to zero slower than n-1/2h-P/4. For the smooth alternatives, the fastest rate at which A, can approach to zero while permitting consistent testing uniformly over the Holder class of functions considered is arbitrarily close to the optimal rate of testing n - 2 s / ( 4 s + P ) , provided that s > 5p/4.

References [Delecroix et al., 2006lM. Delecroix, -, and et al. Nonlinear censored regression using synthetic data. Working Paper No. 2006-1 0, CREST-INSEE, 2006.

266


[Dette, 19991H. Dette. A consistent test for the functional form of a regression based on a difference of variance estimators. Ann. Statist. 27:1012-1040, 1999. [Fan and Huang, 2001]J. Fan and L.S. Huang. Goodness-of-fit tests for parametric regression models. J. Amer. Statist. Assoc., 96:640-652, 2001. [Guerre and Lavergne, 2005lE. Guerre and P. Lavergne. Data-driven rate-optimal specification testing in regression models. Ann. Statist., 33:840-870, 2005. [HBdle and Mammen, 1993lW. Hardle and E. Mammen. Comparing nonparametric versus parametric regression fits. Ann. Statist., 21:1296-1947, 1993. [Hjort, 199OlN.L. Hjort. Goodness of fit test in models for life history data based on cumulative hazard rates. Ann. Statist. 18:1221-1258, 1990. [Horowitz and Spokoiny, 2001lJ.L. Horowitz and V.G. Spokoiny. An adaptive, rateoptimal test of a parametric mean-regression model against a nonparametric alternative. Econometrica, 69:599-631, 2001. [Koul et al., 1981lH. Koul, -, and et al. Regression analysis with randomly right censored data. Ann. Statist., 9:1276-1288, 1981. [Leurgans, 19871s. Leurgans. Linear models, random censoring and synthetic data. Biometrika, 74:301-309, 1987. [SBncliez-Sellero et al., 20051C. Srincliez-Sellero, -, and et al. Uniform representation of product-limit integrals with applications. Scand. J . Statist., 32:563-581, 2005. [Stute, 19931W. Stute. Consistent estimation under random censorship when CDvariables are present. J . Multivariate Anal., 45:89-103, 1993. [Stute, 1995lW. Stute. The central limit theorem under random censorship. Ann. Statist., 23:422-439, 1995. [Stute, 1996lW. Stute. Distributional convergence under random censorship when covariables are present. Scand. J. Statist., 23:461-471, 1996. [Stute, 1997lW. Stute. Nonparametric models checks for regression. Ann. Statist., 25:6 13-641, 1997. [Stute et al., 2OOO]W. Stute, -, and et al. Nonparametric model checks in censored regression. Comm. Statist. Theory Methods, 29:1611-1629, 2000. [Zheng, 198712. Zheng. A class of estimators of the parameters in linear regression with censored data. Acta Mathematicae Applicatae Sinica, 3:231-241, 1987. [Zheng, 19961J.X. Zheng. A consistent test of functional form via nonparametric estimation techniques. J . Econometrics, 75:263-289, 1996.

CHAPTER 7

Dynamical Systems / Forecasting

Application of the single index model for forecasting of the inland conveyances Eugene Kopytov’, Diana Santalova’ 1

Transport and Telecommunication Institute 1 Lomonosov Str., LV-1019, Riga, Latvia (e-mail: kopitovOtsi.1~) * Faculty of Transport and Mechanical Engineering Riga Technical University 1 Kalku Str., LV-1658, Riga, Latvia (e-mail: [email protected]) Abstract. In this paper the regression models used for description and forecasting of the inland rail passenger conveyances of the regions of Latvia were considered. Two estimation approaches were compared: the classical linear regression model and the single index model. Various tests for hypothesis of explanatory variables insignificance and model correctness have been lead, and the cross-validation approach has been carried out as well. The analysis has shown obvious preference of the single index model. Keywords: Passenger conveyances, Forecasting, Regression model, Single index model

1

Introduction

Comprehensive planning of transport company activity demands presence of set of the models that adequately describe functioning of the railway and application of various mathematical methods for forecasting of rail passenger conveyances. The models considered in the given paper allow analyzing passenger flows and forecasting the demand for passenger transportation along some particular sectors of the railway on the whole territory of Latvia [Kopytov and Demidovs, 20061. In the course of the suggested research linear regression model and the single index model (SIM) have been received. The object of consideration is inland rail passenger conveyances in the regions of Latvia In the present research different statistical information about passenger transportation and social-economic development of Latvian regions in the period of 2000 - 2003 is used [Latvijas dzelzcejS 2003, Statistical Yearbook of Latvia 20031. The calculations have been made in the packages STATISTICA 6.0 and MathCad 12. Principal aim of the research is to construct the linear model and the single index model, and then to choose from them one, which gives the better forecasts ~

268

Application of Single Index Model for Forecasting of Inland Conveyances

269

of conveyances. For that the described below cross-validation approach is used. We have used the residual sum of squares RSS for comparing the elaborating models [Srivastava 20021. Especially for the single index model the series of experiments is carried out with the aim to determine the optimal value of bandwidth h [Hardle et al. 2004, Santalova 20061. The paper is organized as follow. First of all the used regression models are considered from theoretical point of view, then the used experimental data are described. After that we consider the suggested group models for conveyances forecasting and results of carried out estimation and comparing analysis of these models. 2

Description of the Models

In present research all investigated models are group models [Andronov et al., 19831. In other words we are able to forecast inland rail passenger conveyances for all considered regions using the same models. With respect to used mathematical model we consider linear regression models and semiparametric regression models. The linear regression model can be represented as follows:

where

xi

= (1

PT =(Po xi,, ...

...

Pd)

is a vector of unknown coefficients,

is a vector of values of independent variables in i-th

observation. One of semiparametric models is used as well model [Hardle et al. 20041:

-

the single index regression

where g(.) is an unknown link function of one dimensional variable and

zi = P T x i is called an index. Here we assume only that unknown function m(x) is a smooth function. As g ( . ) function the kerneZ function usually is considered [Hardle et al. 20041. Therefore we need to estimate the unknown coefficients vector p and the

270


link function g ( . ) . For the latter, the Nadaraya-Watson kernel estimator can be applied:

i=l

where zi = (x -

p

is the value of index for the i -th observation, Y, is the

value of dependent variable for the i-th observation and Kh(-)is so called kernel function. As K , (.) we use the Gaussian function

where h is a bandwidth. The unknown parameter vector criterion:

p

is estimated by use of the least squares

For that we use the gradient method. The corresponding gradient is the following:

Application of Single Index Model f o r Forecasting of Inland Conveyances

271

and

is the first order derivative of the Gaussian kernel. We are able to compare single index models by the residual sum of squares RSS only. We calculate the residual sum of squares as follow:

where n is the number of observations, d is the number of estimated coefficients, is observed value and g ( x i ) is estimated value.

3

Creation the Models for the Inland Rail Passenger Conveyances Forecasting

As an example that shows an opportunity of the suggested models, research of influence of social and economic parameters of Latvian regions development on demand of passenger transport service is carried out. All the needed data have been obtained from the reports [Latvijas dzelzcelS 2003, Statistical Yearbook of Latvia 20031. For the experiments statistical information about 28 Latvian regions (districts and cities) in the period of 2000 - 2003 have been chosen. First of all, the forecasted variable is the inland rail passenger conveyances, expressed in hundreds of passengers. Let us denote it by Y. The considered explanatory factors, or predictors, are [Kopytov and Demidovs, 20061: xl - the population density; x 2 - the number of enterprises per a unit of territory; x3-the number of enterprises per 1000 residents; x4-the density of the unemployed population; x5 - the number of schools per a unit of territory; x6 - the number of buses per a unit of territory; x7- the number of buses per 1000 residents; x8 -the number of railway stations.

272


Now we describe the investigated regression models. The first model is the simple linear regression model (2) and the second model is the single index model (3). The dependent variables

Y(') in linear

model and Y ( 2 )in SIM are inland rail passenger conveyances. Explanatory variables are all eight mentioned above. So, we have two regression models and our task is to estimate the unknown coefficients j? for the models, to compare the suggested models and to prove the preference of semiparametric model. Taking into account the fact that absolute preference of SIM in case of data smoothing has been shown in [Andronov et al. 2006, Santalova 20061, let us demonstrate only the obtained results for smoothing. We obtain the best variant of the linear model by using Backward Stepwise mode of Statistica 6.0 package. In the process of carrying out calculations the number of prediction variables was decreased from 8 to 5. At last, only five most significant predictors are included in this model, such as x2, x3, x4, x5 and xg. RSS for this model is 42 430, coefficient R2 is equal to 0.48 and the calculated value of Fisher criterion is 16.58. The theoretical value of Fisher criterion for 5 and 89 degrees of freedom and level of significance ct = 5% is 2.32. So, this model is adequate. Equation for the estimated linear model is such:

h(Y(')(x))=-255-82x2 +16x, -34x4 +13033x, +18xg.

(10)

Now we discuss results of estimation of investigated single index model. Estimation of coefficients j? has carried out with different bandwidths. Note that in the single index model only the most significant predictors are included, i.e. x2,x3, x4, x5 and x8 . This restriction can be explained also by calculation possibility of MathCadl2 package. The best chosen single index model with h = 0.5 and RSS = 8 877 can be written as:

i=l

Application of Single Index Model for Forecasting

of

Inland Conveyances

273

p'

where vector of estimated coefficients = (12 265 78 0.3 0.1). So, we can conclude that the chosen single index model with h = 0.5 in the case of smoothing gives more precise results than the linear model in sense of RSS.

4

Cross-Validation Analysis

Now we consider the suggested models from the other point of view. We have used the cross-validation approach. That means we estimate the unknown coefficients p for the models on the basis of statistical data from period 2000 2002. Then using the obtained estimates of p we forecast the conveyances for the 2003 and compare these forecasted conveyances with real ones, i.e. we calculate RSS for both models. The optimum value of bandwidth h is found for the single index model as well. The Table 1 contains the estimates of p f o r considered linear regression model. Coefficient R2 is equal to 0.42 and the Fisher criterion is 10.29, the residual sum of squares is 79 029. So, the investigated linear model is adequate in the case of cross-validation as well. The observed conveyances and the corresponding forecasts for the tested period are displayed at Figure 1.

Variable Intercept x2 x3 x4

~5 X8

h.

Std.Err. t(70) 121,94 -1,83 -223,30 -74,35 29,50 -2,52 14,24 9,43 1,51 4,55 -6,13 -27,89 11258,lO 2229,49 5,05 1537 9,47 1,64

p-level 0,071314 0,014008 0,135366 0,000000 0,000003 0,104623

Fig. 1. Forecasting by the linear model Now let us analyze the single index model in details. We begin with a choice of the bandwidth size. Our task is to find the optimal value of bandwidth ho that gives a minimal value of RSS (see [Hardle et al. 20041). The series of experiments was performed and the different estimates of p and values of RSS depending of various h were obtained as well. The corresponding results for the analysed single index model are shown in Table 2. We can see all the estimates of p differs from each other depending on h in spite of they were obtained for the same begin value of p . The values of RSS corresponding to various h are presented in Table 3. Thus, the best result for RSS is achieved for ho = 2. As it was supposed the residual sum of squares increases if h is bigger and smaller than optimal value of h. The forecasted conveyances by the single index model with ho = 2 and observed conveyances as well are shown on the Figure 2. Coefficients

A x2 x3 x4

X5 x8

Bandwidth h

10 6253 -15700 62 860 619 25750

5 2462 7730 35 180 3 13380

3

2

121 3 177 148 0.7 3 110

-38 2555 180 0.05 1561

Table 2. The estimates of p for the SIM

I -6 330 12 0.2 0.1

Application of Single Index Model for Forecasting of Inland Conveyances r

10 79453

I

5

46 194

Bandwidth h 3 I 50454

I

2 40992

1

275

1 166 415

Table 3. The values of RSS for the SIM

Obviously, the forecasted values are very close to the observed values almost in all the observations. Moreover, the SIM shows more precise forecasts for regions with large conveyances that the linear one.

Figure 2. Forecasting by the single index model The values of RSS for both investigated models in cases of smoothing and cross-validation analysis are collected in Table 4. As we can see, the values of RSS are less in all the considered cases.

Linear Model SIM

Smoothing 42 430 8 877

Cross-Validation 79 029 40 992

Table 4. The values of RSS

Conclusions In the course of the suggested research several regression models, which allow evaluating the influence of the main social-economic factors on the volumes of passenger transportation by the railway transport in the regions of Latvia have

276


been received. Two group models were compared: the classical linear regression model and the single index model. Various tests for hypothesis of explanatory variables insignificance and model correctness have been lead, and the crossvalidation approach has been carried out as well. The results of analysis show the obvious preference of the single index model, in other words the semiparametric approach has given better results than classical parametric approach in cases of smoothing and forecasting.

References [Andronov et al., 19831 A. Andronov et al. Forecasting ofpassenger conveyances on the air transport. Transport, Moscow, 1983 (in Russian). [Andronov et al., 20061 A. Andronov, C. Zhukovskaya and D. Santalova. On Mathematical Models for Analysis and Forecasting of the Europe Union Countries Conveyances. In RTU zindtniskie raksti, Informdcijas tehnologijas un vadtbas zindtne, 2006 (in printing). [Hardle et al., 20041 W. Htirdle et al. Nonparametric and Semiparametric Models. Springer-Verlag, Berlin, 2004. [Kopytov and Demidovs, 20061 E. Kopytov and V. Demidovs. Virtual Anticipatory Models in Decision Support System of Railway Transportation. In Daniel M. Dubois, editor, International Journal of Computing Anticipatoy Systems, Published by CHAOS, Institute of Mathematics, University of Liege, 2006. [Latvijas dzelzcej9, 20031 Annual Report of State Joint-stock Company “Latvijas dzelzcel?’ for 2003. Published by SJSC “Latvijas dzelzcelS”, Riga, 2004, (in Latvian). [Santalova 20061 D. Santalova. Forecasting of Railway Freight Conveyances in EU Countries on the Base of the Single Index Model, In Proceedings of RelStat ’06, Riga, 2006 (in printing). [Srivastava 20021 M.S. Srivastava. Methods of Multivariate Statistics. John Wiley & Sons Inc, New York, 2002. [Statistical Yearbook of Latvia, 20031 Statistical Yearbook of Latvia 2003. Published by Central Statistical Bureau of Latvia, Riga, 2003.

Development and Application of Mathematical Models for Internet Access Technology Substitution Apostolos N. Giovanis’ and Christos H. Skiadas’ IUniversity of Teesside, Dept. of Business Studies 8, Pellinis St. & 107, Patission Ave., 1 12 5 1, Athens, Greece

*Technical University of Crete, Dept. of Production Engineering & Management 73 100, Chania, Greece

Abstract Technological change is a dynamic and multidimensional phenomenon involving substitution and diffusion of technologies. In this process new technologies are entered into the market while the old ones are displaced. Various models have been constructed to explore the principles of technology substitutions. In general there are two approaches in tackling the problem. The first approach is using the replacing factor as the measure of technology substitution assuming that it is a function of market share captured by the technology or/and the time elapsed since technology introduction into the economic system. The second approach models the competitive technologies penetration separately where their intrinsic and competitive effects are explicitly represented into the relevant models. The purpose of this paper is to review the existent modeling approaches of technology substitution and analyze the factors affecting the interrelationship between the competing technologies. Finally, the models under review will be used to describe the substitution of the dial-up by the broadband technology for internet connectivity purposes and the models’ fitting and forecasting performance is illustrated. Keywords: Substitution Technologies.

Models,

Technology

Marketing,

Internet

Access

Introduction Rai (1999) defined technology substitution as the process by which an innovation is replaced partially or completely by another in terms of its market share over a period of time. The replacement of the old technology may be instantaneous o r it m a y take a lot of time. The new technology m a y seem to be evolutionary or revolutionary depending upon the take-over time period and each successive generation m a y have a new niche market b y creating new customers. Both the new and the old technology in the market influence the penetration of each other. The final result of this battle depends on the attacker’s and the defender’s strengths and weaknesses. The study of new technologies’ diffusion is the object o f interest for researches of market development and planners of technology development at the firm, regional and national levels. Considering the problems of 277

278


reaching relevant data on technological development the majority of the modeling approaches are focused on the time pattern of the technology development considering the market shares of competing technologies. The aim is the development of easy to handle models of technology substitution able to support the activities of explaining and forecasting the competing products penetration levels. Ryan and Gross (1943) note two important features of this process: the very wide range of rates at which different innovations spread over the time and the general S-shaped diffusion of most innovations. Broadly speaking the diffusion process consists of an introductory phase when the diffusion rate is not very high, followed by the growth phase of relatively quick penetration. The third phase is called the maturity phase in which the market share of the technology reaches its maximum value. In the fourth phase, the market share of that technology declines, caused by emergence of new, better technology. The history showed that the new technology emerges in the second phase of the old technology (Marchetti, Nackisenovic, 1979). The current paper will refer to models with two competing technologies. In the first part of the paper, the most popular technology substitution models are presented and commented. The models under investigation are separated in two categories, those which are using the technology replacement factor as dependent variable and those in which both the intrinsic and competitive effects are present in the relevant models. In the second section, the models are used to represent the substitution of dial-up by broadband technology using monthly data from UK. The models fitting and forecasting performance is illustrated and several conclusions are provided.

1 Review of Technology Substitution Models Most approaches aiming to describe technology substitution are based on the analogy to dissemination of information or to epidemic processes (diffusion of disease through contagion andor external influence). This paper deals with two types of technology substitution models. The first approach is using the replacing factor as the measure of technology substitution assuming that it is a function of market share captured by the technology or/and the time elapsed since technology introduction into the economic system (Fisher-Pry Type Models). The second approach models the competitive technologies penetration separately where their intrinsic and competitive effects are explicitly represented into the relevant models using the appropriate parameters into the respective mathematical forms (Lotka-Volterra Type Models). 1.1

Fisher-Pry Type of Models

The first group includes the models which assume that as the total population remains constant, the evolution of competing technologies is renormalized in

Models for Internet Access Technology Substitution

279

considering only the relevant market share in the total population. To analyze such cases technology substitution models have been proposed by many researches including Fisher-Pry, Blackman and Rai. They studied substitution on the basis of measuring the relative market share of the old versus the new technology competing in the market. The model proposed by Fisher-Pry (1971), which is the most widely used model to describe technology substitution processes, is based on the following assumptions: the substitution process is competitive once substitution has progressed as far as few percent, it will proceed until a complete takeover takes place and the rate of substitution is proportional to the remaining market potential It may be described by the following mathematical formula d f / d t = b .f .( I - f ) , which on integration yields ln(f/(I - f))= a + b .t . Here f denotes the market share of the new technology, (I - f ) denotes the market share of the old technology, b is the intrinsic diffusion rate of the new technology and a represents the constant of integration which value is given by the initial fraction. Recently a class of new substitution models has been proposed by Rai (1999) to describe the substitution process. These models relax the third hypotheses of the Fisher-Pry model assuming that the replacement factor is nonlinear in nature and are based on the arguments that influence is a function of both market share and time. The mathematical formulas describing these models along with the Fisher-Pry model are given below:

..

.

Fisher-Pry Parabolic

df = b .f .(1 - f ) dt

df = (a + b st). f .(I - f) dt

Power

( i,)

In - - a + b .t('+')

(3

Exponential 1.2

Lotka-Volterra Type of Models

The base of the second group of technology substitution models is the traditional diffusion laws such as the logistic and gompertzian ones. This group of models relaxes all the hypotheses posed earlier for the Fisher-Pry

280


type of models since several forms of technologies coexistence in the economic environment can be represented by them. The first model in this group is the well-known Lotka-Volterra model (LVM) which is based on logistic growth. Logistic growth has been popularly used as modeling intraspecific (within one species) competition. The LVM models interspecific (between species) competition since it includes the effects of intraspecific competition, but adds the competitive effect of another species. The interaction between two species (or technologies) can be expressed in general terms via LVM. When two species, X and Y, are competing together in the same environment, the LVM equations become:

This system of equations contains all fundamental parameters that impact the rate of growth of both species. Namely, a j ,i = X,Y is the logistic parameter for the species i when it is living alone and bi,i = X,Y is the limitation parameter of the niche capacity related to the niche size for the species i. The values of cH and cyx are related to the overlap. In other words, they represent the sales that one technology may lose due to fact that the other technology won one. The system of equation (2), formulates a measurement for one’s ability to attack, counterattack or retreat, as the case may be. The discrete forms of equation (2) have been worked out by Lesli (1957) as follows:

where

a, =ear

The discrete form of LVM has been used by Modis (1997) and Hwang et al. (2005) in order to examine competition phenomena in technology marketing and financial markets respectively. Assuming that X is the incumbent and Y the attacker, A represents the attacker’s advantage and D the defender’s

Models for Internet Access Technology Substitution 281

counterattack (Farrell, 1993). Smitalova and Sujan (1991) distinguished and labeled six types of competition according to the sign of the former two coupling parameters involved in LVM which are given in Table 1.

Table 1: Competition Types based on LVM

I

cXy

I

cYx

I

TypeofInteraction

I

Pure competition

+ +

+

Mutualisdsymbiosis

+

0

CommensalisdParasitism

0

Amenalism

0

Neutralism

0

Predator - Prey

The second model in this group is called GCM and is based on the Gompertz Growth Model. The Gompertz equation can be written as:

where F is the saturation level of the diffusion process and c is the diffusion rate. A discrete form of for the Gompertz equation can be written as (Dennis and Taper, 1994): ~ (+ I t) = ~ ( t exp(a ) . - c .ln(X(t)))

(9)

In equation (9) the carrying capacity corresponding to the process saturation level is given by F = exp(a/c) . The value of constant a is determined by the size of population. Assuming that two species X and Y, are competing together in the same environment, the equations for the Gompertz Competition Model (GCM) is a direct extension of equation (9) and are given by the following formulas:

where c, denotes the diffusion rate if technology i assuming no competition, a, denotes the limitation parameter of the niche capacity related to the niche size of technology i and n, m denote the interaction effect with other technologies. Environmental stochasticity can be introduced as follows:

282

Recent Advances i n Stochastic Modeling and Data Analysis

~ (+ 1)t = ~ ( t > exp(al . - c1+1n(X) + n .ln(Y(t))+ wl.~ ( t ) ) (1 1) ~ (+ t1) = ~ ( t )exp(a2 . - c2 .ln(Y(t)) + m . ln(X(t>)+ w2 .~ ( t ) ) where w land w 2 are positive constants and Z(t) is normally distributed with mean 0 and variance 1. The two forms of equation (1 1) can be transformed to a logarithmic scale as follows: ln(x(t + 1))= ln(X(t>)+ a, - c, .ln(x(t)) + n .ln(Y(t)) + w, . ~ ( t ) (12) ln(Y(t + I ) ) = ln(Y(t>)+ a2 - c, .ln(Y(t>)+ m .ln(X(t))+ w, .~ ( t ) Baretto (2003) distinguished and labeled four types of competition according to the value of model’s parameters involved in GCM which are given in Table 2.

Condition

Type of Interaction

1/m < al/a2 > n Technology X wins; technology Y is wiped out l / m > al/a2 < n Technology Y wins; technology X is wiped out I / m > al /a2 > n Both technologies coexist in stable equilibrium l / m < al /a2 < n Both technologies coexist in unstable equilibrium

2 Empirical Analysis The six models under review will be used to explain the substitution mechanism of dial-up by the broadband technology for internet connectivity purpose. Dial-up is the first internet access service that uses a standard modem over the regular copper wiring, with the usual modem speed of 56K, though actual speeds vary greatly. Broadband is a higher-speed transmission of data over a connection that is “always-on (upload speeds: 384 Kbps download speeds: up to 24 mbps). This means a broadband connection is online and ready to go 24 hours a day. Direct Subscriber Line (DSL) and cable modems are two examples of always-on technologies. The available data are concern the monthly market shares of the two technologies in UK from Jan 2001 until Dec 2005. The data for the two technologies’ market shares retrieved form http://www.statistics.gov.uk. The market shares of the two technologies and the relevant replacement factors are given in Figure 1. It can be seen that the market share of broadband overcome those of dial-up during the second quarter of 2005 and that the replacement factor is not a linear function of time. The first 48 months will be used for models’ calibration while the last 12 months will be used to compare their forecasting performance. The forecasting time-horizon is enough considering the dynamism of the broadband technology.

Models f o r Internet Access Technology Substitution

283

For the Fisher-Pry type of models the dependent variable is the replacement factor and the models’ calibration outcomes are given in Table 3 . All models’ parameters are statistically significant.

Figure 1: Internet Access Technologies Market Shares in UK

ZOO,

-a-Dial-up

MS

+Broadband

MS

+Replcement

Factor

From the results it is obvious that the Exponential model presents the best fitting performance followed by Power and Parabolic models. The original Fisher-Pry model failed to explain effectively the substitution process since the technology replacement factor is not a linear function of time. Table 3: Fitting Performance of the Fisher-Pry Type of Models

284


The calibration statistics for the LVM andGCM are given in Table 4. The IV/2SLS method (Pindyck and Rubinfield, 1981) was used for the models’ parameter estimation. The method showed quick convergence for both substitution models. Both models’ parameters are statistically significant and supports the Predator-Prey type of competition for the specific technology substitution process since for the LVM cH and cyx have different signs ( c H >O and c y x < 0) while for the GCM the ratio al/a2 is greater than llm and less than n.

Table 4: Fitting Performance of the LVM and GCM LOTKA-VOLTERRA MODEL Standard Error

0,002 0,021 0,395

0,006

1 1 11: 0,026

CXY

I

GOMPERTZ COMPETITIONMODEL

I

Coefficients

0,064

0,024

-0,014

0,007

0,085

0,042

BB: Broadband - DU: Dial-up It can be seen that the LVM produce better results than the GCM since the respective model’s Sum of Squared Errors (SSE) is lower for both technologies. The resulted values of cH and cKy shows that every time Broadband gains a sale, Dial-up would lose 0,026, showing the raise revolutionary technology. The 12-month ahead forecasting performance for all six technology substitution models is given in Table 5.

Table 5: Forecasting Performance of the Six Reviewed Models Forecast

SSE Dial-u

Models for Internet Access Technology Substitution

285

From Table 5 it can been seen that overall the LVM produce better results for both technologies while the Power extension of the Fisher-Pry models predicts better the Broadband penetration. Both the calibration and forecasting performance of six reviewed models are illustrated in Figure 2.

Figure 2: Fitting & Forecasting Performance of Reviewed Models

Conclusions In this paper six models were presented expressing the early and the latest approaches in technology substitution process representation. The early modeling approaches are using the technology replacement function as substitution measure which depends on the market share of the technology and/or the time elapsed since the new technology launching. The latest modeling approaches are measuring the market share of competing technologies with respect to its intrinsic diffusion rate as well as to competition effects. The models were applied to the dial-up technology

286


substitution by the broadband technology in UK. The latest modeling approaches are more useful on understanding the how and when of the substitution mechanism, while the early ones are simpler and easier to be applied in real conditions. The outcomes of the empirical analysis yields that the Lotka-Volterra Model shows overall greater exploratory and forecasting capabilities in representing substitution processes since it is able to represent several types of competition types

References Barreto, L. S. A Gompertzian Discrete Model for Tree Competition, Silva Lusitana, 11(1):77-89, 2003. Dennis, B. and Taper, M. L. Density Dependence in Time Series Observations of Natural Populations: Estimation and Testing, Ecological Monographs, 64:205-224. 1994. Farrell, C. Survival of the Fittest Technologies, New Scientist, 137: 35-39, 1993. Fisher, J. S. and Pry, H. A Simple Substitution Model for Technology Change, Technological Forecasting and Social Change, 2:75-88, 1971. Hwang, S., Lee, S. and Oh, H. A Dynamic Competition Analysis of Stock Markets, Journal of Emerging Market Finance, 4: 1-25,2005. Leslie, P. H. A Stochastic Model for Studying the Properties of Certain Biological Systems by Numerical Methods, Biometrica, 45: 16-31, 1957. Marchetti, C. and Nakicenivic, N. The Dynamics of Energy Systems and Logistic Substitution Model, RR-79, IIASA, Laxenburg, Austria. Modis, T. Genetic Reengineering of Corporations, Technological Forecasting and Social Change, 56: 107-118, 1997. Pindyck and Rubinfield. Econometric Models and Economic Forecasts, McGraw-Hill, 1981. Rai, L. P. Appropriate Models for Technology Substitution, Journal of Scientific and Industrial Research, 58: 14-18, 1999. Ryan, B. and Gross, N. The Diffusion of Hybrid Seed Corn in two Iowa Communities, Rural Sociology, 7:15-24, 1943. Smitalova, K. and Sujan, S. A Mathematical Treatment of Dynamical Models in Biological Science, Ellis Honvood, West Sussex, England, 1991.

Exploring and Simulating Chaotic Advection: A Difference Equations Approach C. H. Skiadas Technical University of Crete, Chania, Crete, Greece Abstract: This paper explores the chaotic properties of an advection system expressed in difference equations form. In the beginning the Arefs blinking vortex system is examined. Then several new lines are explored related to the sink problem (one central sink, two symmetric sinks, eccentric sink and others). Chaotic forms with or without space contraction are presented, analyzed and simulated. Several chaotic objects are formulated especially when special rotation angles or a complex sinus rotation angle are introduced in the rotation-translation difference equations. Very interesting chaotic forms arise when elliptic rotation-translation equations are applied. The simulated chaotic images and attractors express several vortex-like forms resulting in various situations and especially in fluid dynamics. Keywords: Chaotic advection, The sink problem, Aref system, Rotationtranslation equations, Rotation angle, Vortex, Vortex flow, Chaotic simulation.

1. Introduction Questions addressed when dealing with chaotic advection turn back to nineteenth century and the development of Hydrodynamics and especially the introduction of the Navier-Stokes equations (Claude Navier, 1821 and George Stokes, 1845). The vortex flow case and the related forms including vortex-lines and filaments, vortex rings, vortex pair and vortex systems can be found in the classical book by Horace Lamb first edited in 1879[6]. However, the formulation of a theory that partially explains the vortex problem and gives results that coincide with the real life situations is only a matter of recent years, along with the use of computer experiments. The introduction of terms like chaotic advection and the blinking vortex system came only last decades in order to define and analyze specific vortex flow cases. In most cases the problem setting and solution followed the differential equations approach which mostly was directed to solve a boundary value problem of a Navier-Stokes equation formulation. Few interesting cases are based on a difference equations analogue in the direction to simply explain in more details the vortex flow problem. However, the formulation and analysis of vortex flow problems by using the difference equations analogue can be very useful for several cases if a systematic study is applied. In this paper we follow the difference equations methodology by introducing rotationtranslation difference equations and a non-linear rotating angle along with a space contraction parameter in order to study chaotic advection problems. The interconnections between the difference and the differential equations case is also studied in specific cases. 287

288


2. The Sink Problem 2.1. Central sink Consider a circular bath with a sink in the center at ( x , y ) = (0,O). The water inside the bath is rotating counterclockwise. A colored fluid is injected in the periphery of the bath. Find the shape of the fluid filaments if the sink is open. Geometrically the problem is that of rotation with contraction following a parameter b < 1. The rotation-translation model is applied with the translation parameter u = 0 . The equations of flow are:

x,+, = b(x, cos 4t - Yt sin 4t 1 Yt+l

= b(xt sin 4t + Yt cos 4t )

The contraction to the radial ( r =

d m )

direction is found

from the last relation and the equations of flow

yt+l = bJ(x2

+ y?) = br,

The rotation angle is assumed to follow a function of the form

4t = c + d l r ,2

The space contraction is given by estimating the Jacobian of the flow

J = b2 . When b C 1 a particle is moving from the periphery of the bath to the sink in the center of coordinates following spirals as is illustrated in Figurel. The parameters selected are b = 0 . 8 5 , ~ = 0,d = 0.4 and the initial point is at ( x , y ) = (1,O) . When the same case is simulated for particles entering from the periphery of the rotating system at time t = 0,1,2,..., the following Figure 2 results. The spiral forms start from the periphery and are directed toward the central sink. It is also interesting that while the spiraling flow continues, colored co centric circles appear. These circles have smaller diameter or disappear, as the rotation parameter d is smaller. The parameter b also influences the spiral. Next Figure 3 illustrates an advection case for parameters b = 0 . 9 5 , = ~ 0 and d = 0.01.

Figure 1. Spiral particle paths

Figure 2. Spiral forms directed to the sink

Figure 3. Spiral formation toward a central sink

Exploring and Simulating Chaotic Advection 289

2.2. The contraction process From the above rotation-contraction equations and the very simple relation yt+l

follows that the radial contraction is Ayt = yt+1 - y t

=br,

= byt

= -(l-b)yt

-yt

The differential equation for the contraction process is found by observing that dr Ar - ~ t + ~ - y t

_ --dt

At

-

(t+l)-t

= -(l-b)r

The resulting differential equation expressing the radial speed

t = -(l-b)r is solved to give

r = roe-( 1-b)t

ro is the initial radius. As the equation for the rotation angle is given earlier, the movement is totally explained. The paths are spiraling toward the center. When the movement covers a full circle the new radius will be where

r = roe-( l-b)2d@)

3. Eccentric Sink In the following the case of a circular bath with an eccentric sink is analyzed. The sink is located at ( x ,y ) = (a,()).The equations of flow are: Xt+l

= b((xt - a ) cos $4 - Y t sin 4t 1

Yt+l

= W x t - a )sin 4)t + Y t cos 4 t )

The rotation angle is assumed to follow an equation of the form = c dlr,2

qt

where yt = J ( x t -

+

+ yt2

The limit argument ( 4 + 1 , Y t + l ) = (xt 7 Yt ) = (x,Y

1

will give the relation

x2 + y 2 = b2((x-a)2+ y 2 ) or after transformation

290


[

X

2

+

+Y2 =

This is the equation of a circle with radius

(5) ab R = -centered at 1-b2

The flow is not symmetric. The colored fluid starting from the outer periphery of the bath approaches the sink in few time periods as is illustrated in Figure 4. The parameters selected are u = 0.15,b = 0 . 8 5 , = ~ 0 and

d = 0.1. To simplify the process it is assumed that the colored fluid is introduced simultaneously in the periphery of the bath. Then gradually the circular form of the original colored line is transformed to a chaotic attractor located at the sink's center (x,y ) = (a,()).The attractor is quite stable in form and location. Changes are possible by changing the parameter values. The attractor also appears even if the colored particles are introduced into a small region of the bath as is presented in the next Figure 5. The colored particles are introduced in a square region (0.1 0.1) at the right

*

(x,y ) = (1,o). The parameters selected are a = 0.15,b = 0 . 8 5 , = ~ 0 and d = 0.8. As the vortex parameter d is

end of the bath at

higher than the previous case the chaotic attractor appears at the 6th time step of the process. The attractor is also larger than the previous case.

Figure 4. Chaotic attractor in eccentric sink

Figure 5. Chaotic attractor in eccentric sink

4. Two Symmetric Sinks 4.1. Aref's blinking vortex system Chaotic mixing in open flows is usually modeled by using the 'blinking

Exploring and Simulating Chaotic Advection

291

vortex-sink system' invented by Aref,1983[1], 1984[2] and Aref and Balachandar,l986[3]. Arefs system models the out-flow from a large bath tub with two sinks that are opened in an alternating manner in order to take place a chaotic mixing in the course of the process. To model the velocity field due to a sink we assume the superposition of the potential flows of a point sink and a point vortex. If z = x iy is the complex coordinate in the plane of flow the complex potential for a sinking vortex point is

+

w(z) = -(Q+iK)lnlz-z,l

= (?a,O) and 2@ is the sink strength and 2nK the vortex strength. The imaginary part of w ( z ) is the stream function Y =-Klnr-Q@ where, Z,

And the streamlines are logarithmic spirals defined by the function

@ = -(K/Q) In r

+ const.

The differential equations of motion in polar coordinates are

i. = -Q/r

i-4

=

K/r

And their solutions are

r=dr:-2Qt and

4 = 41 -(K/Q)ln(?/ro) The flow of the system is fully characterized by the non-dimensional sink strength

r7 = QT/a2

and the ratio of vortex to sink strength

5 = K/Q .

T is the flow period and a is the distance of each sink from the center of coordinates. As it is indicated in the literature (KBrolyi and TC1,1997[5], KBrolyi et a1,2002[4]) chaotic flow appears for parameter values = 0.5 or

5

larger and = 10. More precisely when particles are injected into the flow in few time periods are attracted in a specific region (the attractor) of the flow system. Several studies appear last years investigating the phenomenon theoretically and experimentally. The theoretical studies include also simulations by using large grids (1000x1000) and arithmetic solution of the general equations of flow. These studies suggest that the attractors are time periodic according to the time periodicity of the flow. However, if only one sink is used a stable attractor could be present at least theoretically and following simulation experiments as is presented above. This is modeled by investigating the geometry of the flow. First of all Arefs blinking vortex system is applied in a rotating fluid. We select a counter-clockwise rotation. The symmetric sinks are located at (x,y ) = (-a,()) and (x,y ) = (a$) and the time period is

T = 1. According to this system the flow is not

292


stationary and there are jumps in the velocity field at each half period

In other words a particle located at

(-a,())

appears at

Tl2.

(a,()) the next time

period as is illustrated in the next graph of Figure 6.

Figure 6. The two symmetric sinks model The modeling we propose is to analyze a discrete time system, as is Arefs system by using the theory of difference equations and discrete systems. It looks more convenient and highly simpler considering the geometry of this system. The model we search must be a rotation-translation one with a parameter b < 1 expressing the gradual shortening of the radius r , which leads the particles to follow logarithmic spiral trajectories around the sinking vortices. Following to above theory a rotationtranslation model of this type is expressed by the difference equation: i @t

zt+l = a + b ( z ,-zZ,)e

The above complex equation can be written as

x,,, + i ~ , +=~a+b[(x,+ a ) + i y , ] ( c ~ s $+~i . ~ i n $ ~ ) The system of iterative difference relations for X and y is obtained by equating from both sides the real and the imaginary parts of the last complex formula

x,+, = a+b[(x,+a)cos$f - y t sin$t] Yt+l - b[(x,+a)sin$f + yt C O S @ ~ ] If a particle is located at position (X,y ) = (-a,()) the next point after time t = 1 will be located at (XI,y1) = (a,()).The next problem we have is to define the form of the function of the angle flow the differential equation for

$

4 . From the original differential equations of

is

$ = QK/r2 where

r=

d

As the value of

$ yields: And thus

m At

is equal to the periodic time

T = 1 the last equation for

Exploring and Simulating Chaotic Advection

@=&+,

293

175 r

Now it is very easy to find that for Arefs blinking vortex flow the value of = 7T in order to have a half-cycle rotation from one sink to the other. The last equation for @ is a ~ s owritten as

@ =&+ =,

d r

C+,

d

r

d = 175 is the vortex strength. For the experiments presented in the literature 17 = 0.5,5 = 10 and thus d = 5 . However, the chaotic region is Where

more wide as is illustrated in the next figures. The graph of Figure 7 of a chaotic attractor illustrates the two-sink case for parameter values a = 1,b = 0.8, c = 7T and d = 3 . There are two main vortex forms counter-balancing each other. The first form is located at the right hand side sink at (x,y) = (a,()). The second vortex form is centered at

(x,y) = ( a + 2ab cos(@),2absin(@)), where @ = d/(4a2).

The two main vortex forms can be separated when the parameter d expressing the vortex strength is relatively small. Such a case is presented in the next Figure 8. The parameter d = 1 while the other parameters remain the same with the previous example. The attractor is now completely separated into two chaotic vortex forms (attractors).

Figure 7. Chaotic attractor in the two-sink problem

Figure 8. Two distinct vortex forms ( d = 1)

Another idea is to give high values to the parameter d expressing the vortex strength. The selection of a value d = 2n for the vortex strength parameter leads to a more complicated vortex form as is presented in the next Figure 9. There are three equilibrium points for time t = 1,2,3 . The first of these points is the center of the right hand side sink.

294


Figure 9. The chaotic attractor with strong vortex strength parameter

d = 2n 5. Conclusions In this paper we examined the chaotic properties of chaotic advection systems starting from the classical Arefs blinking vortex system. The study followed a difference equation methodology which is, in several cases, more simple and more instructive from the differential equations analogue. W e analyzed and applied a rotation-translation set of difference equations with a dynamical non-linear rotation angle. The resulting chaotic images and attractors express several vortex-like forms resulting in various situations especially in fluid dynamics.

References [ 11 H. Aref. Integrable, chaotic, and turbulent vortex motion in two-dimensional

flows. Ann. Rev. Fluid Mech., 15:345, 1983. [2] H. Aref. Stirring by chaotic advection. J. Fluid mech., 143:1, 1984. [3] H. Aref and S. Balachandar. Chaotic advection in a Stokes flow. Phys. Fluids, 29~3515-3521,1986. [4] G. Khrolyi, I. Scheuring, and T. C z i r h . Metabolic network dynamics in open chaotic flow. Chaos, 12(2):460-469,2002, [ 5 ] G. Kkolyi and T. Ttl. Chaotic tracer scattering and fractal basin boundaries in a blinking vortex-sink system. Physics Reports, 290:125-147, 1997. [6] H. Lamb. Hydrodynamics. Cambridge University Press, Cambridge, 1879.

CHAPTER 8

Modeling and Stochastic Modeling

Likelihood Ratio Tests and Applications in 2D Lognormal Diffusions * Ram6n Gutikrrez' , Concepci6n RoldBn2, Ram6n Gutikrrez-SBnchez' , and Jos6 M. Angulo' Department of Statistics and Operations Research University of Granada. Campus de Fuentenueva, s/n, El8071 Granada, Spain (e-mail: rgjaimezmugr .e s , ramongsaugr .e s , jmanguloaugr .es) Department of Statistics and Operations Research University of J&n. Las Lagunillas, s/n, E-23071 J d n , Spain (e-mail: iroldanaujaen. e s )

Abstract. In a previous work by the authors, maximum likelihood estimators (MLEs) were obtained for the drift and diffusion coefficients characterizing 2D lognormal diffusion models involving exogenous factors affecting the drift term. Such models are well-known t o describe properly the behaviour of real phenomena of interest, for instance, in geophysical and environmental studies. The present paper provides the distribution of these MLEs, the Fisher information matrix, and the solution t o some likelihood ratio tests of interest for hypotheses on the parameters weighting the relative effect of the exogenous factors. Keywords: Diffusion Random Field, Likelihood Ratio Test, Lognormal Diffusion Process, Exogenous Factor.

1

Introduction

The usefulness of diffusion random fields in describing, for example, economic or environmental phenomena, has led to significant developments, particularly regarding inferential aspects. In that respect, from the contribution to theoretical foundations for 2D diffusions given in [Nualart, 19831, we considered 2D lognormal diffusions involving exogenous factors affecting the drift term. We obtained maximum likelihood estimators (MLEs) for the drift and diffusion coefficients, which characterize these diffusions under certain conditions (see [GutiQrez et al., 20051). Using these MLEs, we developed techniques for estimation, prediction and conditional simulation of 2D lognormal 20071). diffusions in [GutiCrrez et d., In this paper, the results obtained previously are completed with the obtaining of the distribution of the MLEs and the Fisher information matrix. This work has been partially supported by projects MTM2005-09209 and MTM2005-08597 of the DGI, Ministerio de Educacio'n y Ciencia, and projects FQM-2271 and FQM-990 of the Andalusian CICYE, Spain. 296

Inference in a 2 0 Lognormal Diffusion 297

However, the main interest of this work is to solve some likelihood ratio tests for the parameters weighting the effect of the exogenous factors, which are relevant for possible application. The contents are organized as follows. First, the 2D lognormal random field model is introduced. Second, estimation of the drift and diffusion coefficients based on a discrete finite set of data is given. Next, the distribution of the MLEs and the Fisher information matrix are calculated. Finally, some likelihood ratio tests for the parameters involved in the formulation of the drift term are solved.

2

Lognormal Diffusion Random Fields

[Nualart, 19831 considered a class of two-parameter random fields which are diffusions on each coordinate and satisfy a particular Markov property related : . Using this theory, we introduced a 2D lognormal to partial ordering in R diffusion random field as follows (see [GutiCrrez and RoldLn, 20071). Let { X (z) : z = (s, t ) E I = [0,S] x [0,T ]c R$} be a positive-valued Markov random field, defined on a probability space ( 0 ,A, P ) , where X (0,O) is assumed to be constant or a lognormal random variable with E [In X (0,O)] = q50 and V a r (In X (0,O)) = The distribution of the random field is determined by the following transition probabilities:

0-0".

+ +

p ( B ,(s h, t + k ) I (z1,z, zz) ,z) = P [ X (s h, t k ) E B I x (s, t k ) = 2 1 , x (z)

+

+

= z,

x (s + h, k ) = 221 ,

where z = ( s , t ) E I , h, k > 0, (z1,z,z2) E R t , and B is a Bore1 subset. We suppose that the transition densities exist and are given by

S(Y,(S+h,t+k)

Y for y E

{ -; JG

(In ( + h ; m z i h 3 k ) 2 }

1

-

I (z1,2,52),z)

R+,with

mz;h,k

=

J,'

s+h

.I

t+k 2

?L (0, 7)dad71

az;h,k =

LS"I"*

B (a,7)dad-r,

-

and 5 , B being continuous functions on I . Under these conditions we can assert that { X (z) : z E I } is a lognormal diffusion random field. The oneparameter drift and diffusion coefficients associated are given by a1

(z) z := 61 (z) (

+ -B1 (z) 2l -

)

(z) z2 := B 1 (z) 2 2 ,

z,

B1

2,

Bz (z) x2 := Bz (z) z2,

298


for all z = ( s , t ) E I , z E R+. The random field {Y (z) : z E I } defined as Y (z) = 1nX (z) is then a Gaussian diffusion random field, with SL and B being, respectively, the drift and diffusion coefficients, and 61,62, B 1 and B 2 being the corresponding oneparameter drift and diffusion coefficients. Furthermore, if z, z’ € I , z = (s, t ) , z’= (s’,t’) , then m y (2) := E

+

[Y (z)] = 40

1’I’ + 1’1

ii (a,T ) d a d T , t

a$ cy

(2) := Var (Y (z))=

a:

(z,z’) := COV (Y (2),Y (z’))= 0:

(Z

B (a,I-) d a d r , A 2’),

where we write z A z’ for (s A s’,t A t’),with ‘A’ denoting the minimum. Henceforth we will assume that the conditions usually considered for estimation of the drift and diffusion coefficients in the one-pacameter case hold; that is, P [lnX (0,O) = 401= 1 (i.e. a$ = 0) and o$ (z) = Bst, z = (s, t ) E I .

3

Inference in the 2D Lognormal Diffusion Model

Let {X (z) : z E I } be a lognormal diffusion random field. Data X =(X(zl), ...,X (z,))~ are assumed to be observed at known spatial locations z1 = (sl,tl), 22 = (s2,tz),..., z, = (sn,tn) E I. Let x = (21,22 ,...,z,)~ be a sample. Let us consider the log-transformed n-dimensional random vec...,Y (z,))~ = (InX (z1), I ~ (z2), X ...,l n x (z,))~ = tor, Y = (Y (z1),Y (221, l n X , and the log-transformed sample, y = (yl,y2, ...,Y,)~ = lnx. We denote

3.1

M L E s for the Drift and Diffusion Coefficients Using Exogenous Factors

Suppose that the drift coefficient 6 of Y is a linear combination of several known functions, set {hl (z) , ...,h, (z): z E I } , with real coefficients 41,...,4, : P

Z E I a=l

Inference in a 2 0 Lognormal Diffusion 299 Defining, for z = ( s , t ) E I ,

the mean of Y is given by

Thus, denoting F = (fo, f1, ...,fp), with fa = ( f a (z1) , f a t for (Y = 0,1, ...,p, and = (40,411..., 4p) , we have

+

my = (4ofo

,

( 2 2 ) ....,fa(z,))~,

+ 4lf1 + ... + 4pfp)= F+.

Let us write ( ~ A1 ~ 2 (tl ) A t2)

(si A

Ey = BM :=B

~ 2 (ti ) A t2)

S2t2

( ~ A1 sn) (tl A tn) ( ~ A2 s n ) ( t 2 A

. . (s1 A sn) (tl A tn) . ' . ( ~ A2 sn) ( t 2 A t n ) '

tn) . ' .

Sntn

With this notation, the MLEs for the drift and diffusion coefficients are, respectively,

+* and

=

(4;,4;, ...,4;lt -

=

( F ~ M - ~ FF ) -~~M - I1nx

(1)

1

B* = - (lnx - m;)t M-' (lnx - m;) , n

(2)

where m; = F+* (see [GutiCrrez et al., 20051).

Remark. I n m a n y practical applications, a polynomial trend provides a suitable representation f o r the drift surface,

m(z) =

c

fpklSktZ, z

=(Sit),

OgC+Z-F@

Using the previous notation,

Finally, the likelihood ratio statistic is written as

Inference in a 2 0 Lognormal Diflusion -

where

-

3,)t

(&a - &)

is

W1

(

->

pl, B

303

and distributes indepen-

dently of g; (see [Anderson, 20031). This means that the distribution of A f is the same as the distribution of

U U+V’ where U and V are independent random variables with distributions given n - p - 1,B and Wl

->

4.1

Some interesting contrasts

If we consider the null hypothesis HIJ :

= 0,

Pl

with Ot =

(0, that we have

1- Af n - p - 1

A?

5

+

Fp1,n-p-1.

Pl

Conclusions

Considering a 2D lognormal diffusion model, in this paper we have calculated the distribution of the MLEs of the drift and diffusion coefficients, the Fisher information matrix, and solved some likelihood ratio tests for hypotheses on the parameters weighting the relative effect of exogenous factors affecting the drift. The results obtained are important for real applications; in particular, for prediction and conditional simulation following the techniques described in [GutiQrez et al., 20071).

References [Anderson, 2003lT.W Anderson. A n Introduction to Multivariate Statistical Analysis. Wiley & Sons, New Jersey, 3rd edition, 2003. [GutiCrrez and Roldan, 2007lR. Gutikrrez and C. Roldan. Some analytical and statistical aspects related to 2D lognormal diffusion random fields. Scientiae Mathematicae Japonicae Journal Online, e-2007:341-360, 2007. [GutiCrrez et al., 2005lR. Gutikrrez, C . Roldin, R. Gutikrrez-Sanchez, and J.M. Angulo. Estimation and prediction of a 2D lognormal diffusion random field. Stochastic Environmental Research and Risk Assessment, 19:258-265, 2005. [Gutikrrez et al., 2007lR. GutiCrrez, C . Roldan, R. Gutikrrez-Sinchez, and J.M. Angulo. Prediction and conditional simulation of a 2D lognormal diffusion random field. Methodology and Computing in Applied Probability, DO1 10.1007/~11009007-9029-3, 2007. [Nualart, 1983lD. Nualart. Two-parameter diffusion processes and martingales. Stochastic Processes and their Applications, 15:31-57, 1983.

Cartographical Modeling as a statistical method for monitoring of a spatial behaviour of population Irina Pribytkova’ 1 Institute of Sociology of National Academy of Sciences of Ukraine Shovkovichnaja str., 12 Kiev, Ukraine (e-mail: Abstract. The Cartographical Modeling belongs to the system of common scientific methods we use in search of new knowledge and its proving. The study of spatial relations is based on a map providing the most complete description and comprehension of any territorial problems. A map gives a new information of more high order on mapping phenomena which is hidden in an initial figures. This new information one have got due to generalization of statistics is of particular value to scientific research and practical needs. The process of generalization results in discovery of the cartographical structures forming a certain system. Analysis of these structures enables the revelation of spatial regularities in disposition, proportion, combination and dynamics of sociodemographic and socioeconomical processes and phenomena. Besides, the cartographical modeling provides the transition from discrete to continuous knowledge. This is the only method to obtain the continuous picture of spatially unbroken phenomena on the basis of discrete factual information (Aslanicashvili A,, 1974). The importance of uninterrupted knowledge contained in the cartographical model is conditioned not only by its possibility to reveal the changes of investigated process or phenomena “from place to place” but also by its potentialities to bring to light a significant spatial relations between them and other social and natural processes and phenomena represented in the given model (map). The new knowledge obtained in the course of modeling serves as a basis for working out of the management decisions. The comparison of identical models for a few years in succession gives us the notion about the nature and rate of changes and development of spatial structures. The cartographical modeling may be regarded as one of the modification of latent structure analysis which pursues an object to reveal and distinguish the latent groups of population with peculiar social organization, material and cultural consumption, goals, preferences and behaviour. The permanent observation of current statistical information during a long time creates the necessary grounds for organization of data base. The collection of statistical data, their standardization and compiling of series of relevant maps are integral parts of monitoring as a system of supervision and control after the processes of spatial behaviour of population. 304

Cartographical Modeling as a Statistical Method

305

The scientific programme of monitoring includes also the working out of prognoses concerning eventual changes in the course of spatial self-organization of people, providing it with necessary information about possible unfavourable consequences, appraisals of regulation decisions and their efficiency. Present paper contains the analysis of a spatial behaviour of rural population in Ukraine since the seventies, carried out by means of cartographical modeling of statistical data in the monitoring regime. Keywords: Cartographical Modeling, spatial relations, map, factual information, continuous knowledge, latent structure analysis, statistical information, permanent observation, supervision, control.

1

The common conception of a method

The cartographical modeling belongs to the system of common scientific methods used in search of new knowledge and its proving. The study of spatial relations is based on a map providing the most complete description and comprehension of any territorial problems. Many researchers perceive the maps only as illustration for text or figures having no independent scientific value. Meanwhile the map is none other than information system, channel for transmitting of spatial information. Cartographical language can be regarded as the peculiar sign system: the cartographical images are the primary means to transfer the information. Giving rise to visual notions, the cartographical images enable to obtain the integral perceiving of spatial structures. When analyzing the figures in the tables the thought and attention of researcher are distracted from the general to the particular. Cartographical images, as a means of transmitting of spatial relations, proves to be immeasurably stronger than algebraic symbols because the object mode of thinking is more effective than Eormal. The cartographical language has many positive qualities. It is universal and gets over all speech barriers without difficulties. It has a laconic and capacious character and enables to express the judgments in the lapidary form. And at last it has the two-dimensional nature. All these properties of a map language enlarge to a considerable extent the information capacities of a cartographical model. Just the two-dimensional character of cartographical linguistic system affords the possibilities for the investigation of spatial relations. A map gives a new information of more high order on mapping phenomena which is hidden in an initial figures. This new information one have got due to generalization of statistics is of particular value to scientific research and practical needs. The process of generalization results in discovery of the cartographical structures forming a certain system. Analysis of these structures enables the revelation of spatial regularities in disposition, proportion, combination and dynamics of sociodemographic and socioeconomical processes and phenomena.

306


2 The cognition properties of cartographical model The cartographical model represents the scientific abstraction received in consequence of generalization of a concrete properties of studied objects. The abstract character of generalized cartographical model A. Berliant defines as one of a main positive qualities of a map: applying to its contents the researcher can observe not only separate or systematized factual statistics but also an integral spatial image. He meets with “system of notions”, appearing in the course of map composition and fixed in the cartographical model and its legend (Berliant, c. 19). Besides, the cartographical modeling provides the transition from discrete to continuous knowledge. This is the only method to obtain the continuous picture of spatially unbroken phenomena on the basis of discrete factual information (Aslanicashvili A., 1974). The importance of uninterrupted knowledge contained in the cartographical model is conditioned not only by its possibility to reveal the changes of investigated process or phenomena “from place to place” but also by its potentialities to bring to light a significant spatial relations between them and other social and natural processes and phenomena represented in the given model (map). A new knowledge obtained in the course of modeling serves as a basis for working out of the management decisions.

3 The methodological premises It is expendient from the methodological point of view to research the spatial structures and regularities of their development on the basis of cartographical analysis of fractional administrative units - rural administrative regions and within the bounds of the last - on a scale of village councils. The regions and provinces are too large-scale territorial units for such investigation. Cartographical modeling on a scale of provinces enables to carry out the macrodivision of territory into districts by examined sign and to reveal only the levels of its intensity in the spatial aspect. The preparation of initial data is a previous stage of cartographical modeling. Relevant methods are defined by goals and tasks of modeling in every concrete case . The method of data groups, for example, is used as a basic principle when composing the statistical maps known as cartograms that are instrumental for revealing of regularities in spatial distribution of mapping signs. The grouping of statistical data is simultaneously carried out by geographical sign and by size of indices. The correct choice of value intervals of statistical indices used for procedure of grouping is especially important stage when working out the cartogram.


307

Precision of spatial model depends on size of these intervals. It is expedient to select them by method of consecutive approximation. On the one hand, such an order helps to avoid the excessive, unnecessary detailing in the picture of spatial structure of modeling process or phenomena and, on the other hand, to ensure an adequate representation of its essential features. There is no need to use any standartizated scale of gradation or purely mechanical statistical methods/modes. In every concrete case when defining the limits of intervals, one must take into consideration not only the range of values distribution proper but also the necessity to show the existing differences in the spatial structure of investigated phenomena (processes, structures).

4 Cartographical modeling of latent structures The cartographical modeling may be regarded as one of the modification of latent structure analysis which pursues an object to reveal and distinguish the latent groups of population with peculiar social organization, material and cultural consumption, goals, preferences and behaviour. The analysis of latent structure is to be started with appraisal of empirical data and working out of hypothesis on the presence of a few definite groups of population forming the latent structure. And after that the proposed hypothesis has to be verified in the statistical respect on the basis of factual data. The model of latent structure tests the fact of presence of postulated groups, but more deep penetration into essence of problem calls for additional information. The revelation of latent structures as the instrument of analysis may be of use also when studying the population attitudes towards different problems, for statistical interpretation of regional distinctions in the structure of people’s consumption, for explaining of the intensity of population moving within the urban and rural areas, for estimation of life conditions inside of cities and their suburbs etcetera. The data for study of population behaviour or measuring of social structure parameters may be received in the course of sociological surveys of public opinion or be got from current statistical returns. The last source of information is more preferable. The cartographical model, worked out on the basis of a current statistics on size and structure of rural population, enables to reveal the system of regions with specific socio-group organization of their inhabitants. In other words, this model confirms the existence of latent groups of population in the countryside and indicates their localization in the space.

308


5 Monitoring of a spatial behaviour of population The permanent observation of current statistical information during a long time creates the necessary prerequisites for organization of data base. The collection of statistical data, their standardization and compiling of series of relevant maps are integral parts of monitoring as a system of supervision and control after the processes of spatial behaviour of population. The appropriate statistical data for analysis should to meet the demands of a highest possible spatial detailing, uniformity, simultaneity, authenticity, continuity and comparability. The infringement of least one of these demands results in lowering of cognitive value of initial information and obtaining rather probable than true knowledge of the research subject. The scientific programme of monitoring includes also the working out of prognoses concerning eventual changes in the course of spatial self-organization of people, providing it with necessary information on possible unfavorable consequences, appraisals of regulation decisions and their efficiency. The revealing of regularities in behaviour of rural population in Ukraine in process of spatial selforganization is based on the analysis of statistics for the last forty years. So far as territorial movement of population is closely bound up with alteration of place and character of labour (in other words it is the movement of mainly able-bodied contingents), only the rural population in working age was chosen as an object of modeling. The cartographical models of spatial disposition of rural population in ablebodied age in Ukraine give a clear view of the geographical location of rural inhabitants and their concentration or dispersion in definite regions of countryside. The comparison of identical models for a few years in succession gives us the notion about the nature and rate of changes and development of spatial structures and discovers the important spatial relations between movements of rural inhabitants and urbanization in Ukraine. We come to the conclusion that concentration of urban and concentration of rural population within the bounds of the Ukraine are two sides of one and the same process of urbanization. Towns and cities perform in this process the duties of peculiar nucleifor crystallizationfor a new sociospatial structures of rural resettling. The large cities with strong economical potential and diverse functions have the greatest influence on the level of concentration of rural able-bodied inhabitants and area of their location in suburbs. At the same time, one can observe the rise of separate hotbeds and then a whole zones where the dispersion of rural population and destructive demographic processes adliance with a high speed. The deep changes in the structure of spatial self-organization of rural population take place over a long period of time. The main point of this process lies in the permanent deepening of its territorial polarization.


309

The demographical consequences of spatial self-organization of countrypeople are highly various, closely tied and are evinced first of all in different types of dynamics and age structure of rural population in the regions of concentration and dispersion. Rise in the number of rural inhabitants at the ablebodied age and increase in amount of large villages in the suburbs are accompanied by improving of demographical structure of rural population in these areas. A social and territorial mobility of village inhabitants is gaining in strength and scope, the structure of employment changes for the better. At the same time destructive demographical processes are observed in the rural areas of dispersion. Reproductive activity is reduced, mortality rises rapidly, the life expectancy at birth is growing shorter, the age structure of rural population is getting worse and worse.

*** In conclusion we would like to remark that the spatial selforganization of rural inhabitants within the Ukraine is on-going process. It should be expected for subsequent deepening of age disproportion in age structure of rural population in the areas of dispersion, lessening of their number and decrease of labour potential in countryside.

References [Aslanikashvili, 1974lA. F. Aslanikashvili. Metacartography. Basic Problems Tbilisi, Metsniereba, 1974. (Russian). [A. M. Berliant, 1978lA. M. Berliant. Cartographical method of research. Moscow, Moscow State University. Publishing House, 1978. [Garner, 1967lGamer B. J. Models of Urban Geography and Dislocation of Populated Area. Edited by Richard J. Charley and Peter Hagget, London, 1967. [Et al., 1973]Mathematical Methods in Social Sciences. Moscow, Progress, 1973 (Russian). [Osipov and Andreyev, 1977lG. V. Osipov and E. P. Andreyev. Methods of Measurement in Sociology Moscow, Nauka, 1977. (Russian). [Stouffer, 19401s. A. Stouffer. Intervening Opportunities: A Theory Relation Mobility and Distance. The American Sociological Review, 1940, v. 5, No. 6, p. 845-867. [Ullman, 1956lEdward L. Ullman. The Role of Transportation and the Bases for Interaction in Man’s Role in Changing the Face of the Earth. University of Chicago Press, Chicago, 1956. [Zablotsky, 1975lG. A . Zablotsky. Estimation of Social Conditions of Population Settlement and Mathematical Simulation of Urban Development. In: Social Base for Urban Development. Moscow, 1975 (Russian).

310

Recent Advances in Stochastic Mo~elingand Data Analysis

6

Cartographical ~ o d e l i ~asg a Statistical Method 311

Learning and Inference in Switching Conditionally Heteroscedast ic Factor Models Using Variational Methods Mohamed Saidane and Christian Lavergne Department of Mathematics University of Montpellier I1 34095 Montpellier, France (e-mail: {saidane, Christian.Lavergne}Qmath.univ-montp2. fr) Abstract. A data-driven approach for modeling volatility dynamics and comovements in financial markets is introduced. Special emphasis is given to multivariate conditionally heteroscedastic factor models in which the volatilities of the latent factors depend on their past values, and t h e parameters are driven by regime switching in a latent state variable. We propose an innovative indirect estimation method based on the generalized EM algorithm principle combined with a structured variational approach, that can handle models with large cross-sectional dimensions. Extensive Monte Carlo simulations and preliminary experiments with financial data show promising results. Keywords: Factor Models, HMM, Conditional Heteroscedasticity, EM Algorithm, Variational Approximation.

1

Introduction

In the financial econometric literature, factor models have been developed and used widely in the area of asset pricing as an alternative t o the Capital Asset Pricing Model (CAPM) since the early 1960s. In this context, factor models have been used as a parsimonious means of describing the covariance matrix of returns since the single-index model of [Sharpe, 19631. Traditionally, these issues were considered in a static framework, but recently, the emphasis has shifted toward inter-temporal asset pricing models in which agents decisions are based on the distribution of returns conditional on the available information, which is obviously changing. It is now generally accepted that asset returns are heteroscedastic. A large body of empirical research (e.g., [Schwert, 19891) reports convincing evidence that the volatility of asset returns is time-varying. Yet, the literature that combines multi-factor models of asset returns with a time-varying covariance matrix is relatively small. This is probably due t o the econometric challenges associated with estimating and testing such models. Papers that examine a linear factor model of asset returns in which the covariance matrix of returns is heteroscedastic include [Jones, 20011 and [Demos and Sentana, 19881 and their references.

312

Variational Methods for Factor Models 313

This paper extends the different models proposed in the above literature to a multi-state model by allowing for model transitions that are governed by a Markov chain on a set of possible models describing the different states of volatility. The switching conditionally heteroscedastic latent factor model is presented in Section 2, followed by a description of a maximum likelihood estimation procedure based on the Expectation-Maximisation (EM) principle combined with a structured variational learning approach, the empirical results in Sections 4 and 5, and a summary in Section 6.

2

The Switching Factor Model

This specification supposes that excess returns depend both on unobservable factors that are common across the multivariate time series, and on unobservable different regimes that describe the different states of volatility. 2.1

Dynamic Factor Structure

Let y t denote the q-vector of excess asset returns and ft denote the k-vector of latent factor shocks in period t . In our switching conditionally heteroscedastic factor model, the realized excess return on an asset is the sum of its expected return, k systematic shocks and an idiosyncratic shock. In matrix notation, the switching factor model for the excess return vector is:

st

-

P(St = j / S t & l = i )

t = l , ..., n and i , j = l ,..., rn fs,

= Hii2f,*

Y t = &fs,

-

+ Est

f:

where with

Es,

- N(0,

Ik)

- N(&,

, *st)

where St P(St = j / S t - l = i) is a hidden Markov chain indicating the state or the regime at the date t . In an unspecified state St = j , 0, are the ( q x 1) mean vectors, fi the ( k x 1)vectors of unobserved common factors, &itthe ( q x 1) vectors of idiosyncratic noises, Xj the ( q x k ) factor loadings matrices, *j are ( q X q ) diagonal and definite positive matrices of idiosyncratic variances, and Hi the ( k x k ) diagonal and definite positive matrices whose elements are the variances of the common factors presumedly time varying and their parameters changes according to the regime. In particular, we suppose that the variances of the common factors follow univariate Generalized Quadratic Autoregressive Conditionally Heteroscedastic processes GQARCH(1,l ) , the I-th diagonal element of the matrix Hi under an unspecified regime St = j is given by hit = w i a 1jf ijt -2 6: hit-l for 1 = 1,..., k .

+$ f i - l +

+

314


To guarantee the positivity of the conditional common variances and the covariance stationarity, we impose the constraints w j , a;, 6; > 0 , yj2 5 4w$z~i and a:. 6; < 1, V j, I . For model identification we suppose that q 2 k and rank(Xj) = k , V j . We suppose also that the common and idiosyncratic factors are uncorrelated, and that ft and E t f are mutually independent for all

+

t , t’. 2.2

A State-Space Representation

The state-space representation of our model, with continuous state variable ft, is given by: [Measurement Equation] [Transition Equation]

y t = Os,

-

+ Xs,fs, + cst + fs,

fa, = O.fS,-,

-

where ESt/V1:t-1 N(0,Q8,) and fst/V1:t-l N(0,Hst). The information set available at time t is denoted by V1:t-l = { ~ l : t - l , F l : t - l , S ~ :where ~-l}, YlZT= {yl, ..., y T } , F1:T= {fl, ...,fT} and Sl:, = (5’1, ..., S T } . In a given state St = j , the prediction equations are given by:

E(ft+l/Vl:t) = f&

=0

v j = 1, ...,m

Var(fit+i/vi:t) = hit+,/, = a$+ $fZ,/,

+a$

[fi;,+ h:t/,] + 6$hit/t-l

where h;t/t is the 1-th diagonal element of Hilt. The updating equations are:

fi+l/t+l =

Hi+,/,

X’.xj - 1 3 t+l/t (Yt

- Qj)

x/,~ -1 i Hi+i/t+i = Hi+i/t - Hi+,/, 3 t+l/tX.iHi+l/t where xi+,/,= XjH:+,/,X; +\kj. Importantly, given the degenerate nature of the transition equation, smoothing is unnecessary in this case, so that fz/,, = f-” t / t and H:/n = Hi/,.

3

Learning

An efficient learning algorithm for the parameters of our switching factor model can be derived by generalizing the Expectation Maximization (EM) algorithm [Dempster et al., 19771. EM alternates between optimizing a distribution over the hidden states (the E-step) and optimizing the parameters given the distribution over hidden states (the M-step). Any distribution over the complete sequence of hidden states, Q ( S , F ) ,can be used to define a lower bound, B,on the log-probability of the observed data:

Variational Methods for Factor Models

315

where 0 denotes the parameters of the model and we have made use of Jensen’s inequality to establish (3). The E-step holds the parameters fixed and sets Q to be the posterior distribution over the hidden states given the parameters: Q ( S ,3)= P ( S ,F/y, 0).This maximizes B with respect to the distribution, turning the lower bound into an equality, which can be easily seen by substitution. The M-step holds the distribution fixed and computes the parameters that maximize B for that distribution. Given the change in the parameters produced by the M-step, the distribution produced by the previous E-step is typically no longer optimal, so the whole procedure must be iterated. Unfortunately, the exact E-step for our factor model is intractable, because the posterior probability of the real-valued states is a Gaussian mixture with mn terms. In order to derive an efficient learning algorithm for this system, we relax the EM algorithm by approximating the posterior probability of the hidden states. The basic idea is that, since expectations with respect to P are intractable, rather than setting Q ( S ,F)= P ( S ,F/y)in the E-step, a tractable distribution Q is used to approximate P. The difference between the bound B and the log likelihood is given by the Kullback-Liebler ( K L ) divergence between Q and P :

While there are many possible approximations to the posterior distribution of the hidden variables that one could use for learning and inference in switching factor models, we focus on the following:

where the P are unnormalized probabilities, which we will call potential functions and define soon, and ZQ is a normalization constant ensuring that Q integrates to one. The terms involving the switch variables St define a discrete

316


Markov chain and the terms involving the state vectors ft define m uncoupled factor models. Like in mean field approximations we have removed the stochastic coupling between the chains that results from the fact that the observation at time t depends on all the hidden variables at time t. However, we retain the coupling between the hidden variables at successive time steps since these couplings can be handled exactly using the forward-backward and Kalman smoothing recursions. The discrete switching process is defined by

where the q,(j) are variational parameters of the Q distribution. These parameters scale the probabilities of each of the states of the switch variable at each time step, so that q , ( j ) plays exactly the same role as the observation probability p(yt/St = j ) would play in a regular hidden Markov model. The uncoupled factor models in the approximation Q are also defined by potential functions which are related to probabilities in the original system. These potentials are the prior and transition probabilities for ft multiplied by a factor that changes these potentials to try to account for the data:

P(f,3)= [p(fl/Sl= j ) P ( Y l / f l , s 1 = j ) l C ? P(fl / ft -1) = [P (ftlft-1, st = j)P (Yt/ft,

st = j)l

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