Filtering the Wright-Fisher diffusion

Mireille Chaleyat-Maurel; Valentine Genon-Catalot

ESAIM: Probability and Statistics (2009)

  • Volume: 13, page 197-217
  • ISSN: 1292-8100

Abstract

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We consider a Wright-Fisher diffusion (x(t)) whose current state cannot be observed directly. Instead, at times t1 < t2 < ..., the observations y(ti) are such that, given the process (x(t)), the random variables (y(ti)) are independent and the conditional distribution of y(ti) only depends on x(ti). When this conditional distribution has a specific form, we prove that the model ((x(ti),y(ti)), i≥1) is a computable filter in the sense that all distributions involved in filtering, prediction and smoothing are exactly computable. These distributions are expressed as finite mixtures of parametric distributions. Thus, the number of statistics to compute at each iteration is finite, but this number may vary along iterations.

How to cite

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Chaleyat-Maurel, Mireille, and Genon-Catalot, Valentine. "Filtering the Wright-Fisher diffusion." ESAIM: Probability and Statistics 13 (2009): 197-217. <http://eudml.org/doc/250672>.

@article{Chaleyat2009,
abstract = { We consider a Wright-Fisher diffusion (x(t)) whose current state cannot be observed directly. Instead, at times t1 < t2 < ..., the observations y(ti) are such that, given the process (x(t)), the random variables (y(ti)) are independent and the conditional distribution of y(ti) only depends on x(ti). When this conditional distribution has a specific form, we prove that the model ((x(ti),y(ti)), i≥1) is a computable filter in the sense that all distributions involved in filtering, prediction and smoothing are exactly computable. These distributions are expressed as finite mixtures of parametric distributions. Thus, the number of statistics to compute at each iteration is finite, but this number may vary along iterations. },
author = {Chaleyat-Maurel, Mireille, Genon-Catalot, Valentine},
journal = {ESAIM: Probability and Statistics},
keywords = {Stochastic filtering; partial observations; diffusion processes; discrete time observations; hidden Markov models; prior and posterior distributions; stochastic filtering},
language = {eng},
month = {6},
pages = {197-217},
publisher = {EDP Sciences},
title = {Filtering the Wright-Fisher diffusion},
url = {http://eudml.org/doc/250672},
volume = {13},
year = {2009},
}

TY - JOUR
AU - Chaleyat-Maurel, Mireille
AU - Genon-Catalot, Valentine
TI - Filtering the Wright-Fisher diffusion
JO - ESAIM: Probability and Statistics
DA - 2009/6//
PB - EDP Sciences
VL - 13
SP - 197
EP - 217
AB - We consider a Wright-Fisher diffusion (x(t)) whose current state cannot be observed directly. Instead, at times t1 < t2 < ..., the observations y(ti) are such that, given the process (x(t)), the random variables (y(ti)) are independent and the conditional distribution of y(ti) only depends on x(ti). When this conditional distribution has a specific form, we prove that the model ((x(ti),y(ti)), i≥1) is a computable filter in the sense that all distributions involved in filtering, prediction and smoothing are exactly computable. These distributions are expressed as finite mixtures of parametric distributions. Thus, the number of statistics to compute at each iteration is finite, but this number may vary along iterations.
LA - eng
KW - Stochastic filtering; partial observations; diffusion processes; discrete time observations; hidden Markov models; prior and posterior distributions; stochastic filtering
UR - http://eudml.org/doc/250672
ER -

References

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  1. O. Cappé, E. Moulines and T. Rydèn, Inference in hidden Markov models. Springer (2005).  Zbl1080.62065
  2. M. Chaleyat-Maurel and V. Genon-Catalot, Computable infinite-dimensional filters with applications to discretized diffusion processes. Stoch. Process. Appl.116 (2006) 1447–1467.  Zbl1122.93079
  3. F. Comte, V. Genon-Catalot and M. Kessler, Multiplicative Kalman filtering, Pré-publication 2007-16, MAP5, Laboratoire de Mathématiques Appliquées de Paris Descartes, submitted (2007).  Zbl1274.62549
  4. V. Genon-Catalot, A non-linear explicit filter. Statist. Probab. Lett.61 (2003) 145–154.  Zbl1041.62079
  5. V. Genon-Catalot and M. Kessler, Random scale perturbation of an AR(1) process and its properties as a nonlinear explicit filter. Bernoulli(10) (2004) 701–720.  Zbl1055.62102
  6. S. Karlin and H.M. Taylor, A Second Course in Stochastic Processes. Academic Press (1981).  Zbl0469.60001
  7. N.N. Lebedev, Special functions and their applications. Dover publications, Inc. (1972).  Zbl0271.33001
  8. A. Nikiforov and V. Ouvarov, Fonctions spéciales de la physique mathématique. Editions Mir, Moscou (1983).  
  9. W. Runggaldier and F. Spizzichino, Sufficient conditions for finite dimensionality of filters in discrete time: A Laplace transform-based approach. Bernoulli7 (2001) 211–221.  Zbl0981.62077
  10. G. Sawitzki, Finite dimensional filter systems in discrete time. Stochastics5 (1981) 107–114.  Zbl0471.60052
  11. T. Wai-Yuan, Stochastic models with applications to genetics, cancers, AIDS and other biomedical systems. Series on Concrete and Applicable Mathematics, Vol. 4. World Scientific (2002).  Zbl0991.92001
  12. M. West and J. Harrison, Bayesian forecasting and dynamic models. Springer Series in Statistics, second edition. Springer (1997).  Zbl0871.62026

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