Thesis of Fei Zheng

Learning and Smoothing in Switching Markov Models with Copulas


Switching Markov Models, also called Jump Markov Systems (JMS), are widely used in many fields such as target tracking, seismic signal processing and finance, since they can approach non-Gaussian non-linear systems. A considerable amount of related work studies linear JMS in which data restoration is achieved by Markov Chain Monte-Carlo (MCMC) methods. In this dissertation, we try to find alter- native restoration solution for JMS to MCMC methods. The main contribution of our work includes two parts. Firstly, an algorithm of unsupervised restoration for a recent linear JMS known as Conditionally Gaussian Pairwise Markov Switching Model (CGPMSM) is proposed. This algorithm combines a parameter estima- tion method named Double EM, which is based on the Expectation-Maximization (EM) principle applied twice sequentially, and an efficient approach for smoothing with estimated parameters. Secondly, we extend a specific sub-model of CGPMSM known as Conditionally Gaussian Observed Markov Switching Model (CGOMSM) to a more general one, named Generalized Conditionally Observed Markov Switch- ing Model (GCOMSM) by introducing Copulas. Comparing to CGOMSM, the pro- posed GCOMSM adopts inherently more flexible distributions and non-linear struc- tures, while optimal restoration is feasible. In addition, an identification method called GICE-LS based on the Generalized Iterative Conditional Estimation (GICE) and the Least-Square (LS) principles is proposed for GCOMSM to approximate any non-Gaussian non-linear systems from their sample data set. All proposed methods are tested by simulation. Moreover, the performance of GCOMSM is discussed by application on other generable non-Gaussian non-linear Markov models, for exam- ple, on stochastic volatility models which are of great importance in finance.

Advisor: Stéphane Derrode

Defense date: monday, december 18, 2017

Jean-Yves TourneretProfesseur(e)ENSEEIHT ToulousePrésident(e)
Séverine DubuissonMaître de conférenceUPMC Sorbonnes UniversitésRapporteur(e)
François RoueffProfesseur(e)Telecom ParisTechRapporteur(e)
Stéphane DerrodeProfesseur(e)É́cole Centrale de LyonDirecteur(trice) de thèse
Wojciech PieczynskiProfesseur(e)Telecom SudParisCo-directeur (trice)