Thesis of Fei Zheng


Subject:
Learning and Smoothing in Switching Markov Models with Copulas

Summary:

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

Jury:
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)