Fast smoothing in switching approximations of non-linear and non-Gaussian models

International audienceStatistical smoothing in general non-linear non-Gaussian systems is a challenging problem. A new smoothing method based on approximating the original system by a recent switching model has been introduced. Such switching model allows fast and optimal smoothing. The new algorithm is validated through an application on stochastic volatility and dynamic beta models. Simulation experiments indicate its remarkable performances and low processing cost. In practice, the proposed approach can overcome the limitations of particle smoothing methods and may apply where their usage is discarded

Estimation bayésienne dans les modèles de Markov partiellement observés

This thesis addresses the Bayesian estimation of hybrid-valued state variables in time series. The probability density function of a hybrid-valued random variable has a finite-discrete component and a continuous component. Diverse general algorithms for state estimation in partially observable Markov processesare introduced. These algorithms are compared with the sequential Monte-Carlo methods from a theoretical and a practical viewpoint. The main result is that the proposed methods require less processing time compared to the classic Monte-Carlo methodsCette thèse porte sur l'estimation bayésienne d'état dans les séries temporelles modélisées à l'aide des variables latentes hybrides, c'est-à-dire dont la densité admet une composante discrète-finie et une composante continue. Des algorithmes généraux d'estimation des variables d'états dans les modèles de Markov partiellement observés à états hybrides sont proposés et comparés avec les méthodes de Monte-Carlo séquentielles sur un plan théorique et appliqué. Le résultat principal est que ces algorithmes permettent de réduire significativement le coût de calcul par rapport aux méthodes de Monte-Carlo séquentielles classique

Unsupervised learning of Markov-switching stochastic volatility with an application to market data

International audienceWe introduce a new method for estimating the regime-switching stochastic volatility models from the historical prices. Our methodology is based on a novel version of the assumed density filter (ADF). We estimate the switching model by maximizing the quasi-likelihood function of our ADF. The simulation experiments show the efficiency of our method. Then we analyze different market price histories for consistency with a regime-shifting mode

EXACT FAST SMOOTHING IN SWITCHING MODELS WITH APPLICATION TO STOCHASTIC VOLATILITY

International audienceWe consider the problem of statistical smoothing in nonlin-ear non-Gaussian systems. Our novel method relies on a Markov-switching model to operate recursively on series of noisy input data to produce an estimate of the underlying system state. We show through a set of experiments that our technique is efficient within the framework of the stochastic volatility model

Triplet Markov models and fast smoothing in switching systems

The aim of the paper is twofold. The first aim is to present a mini tutorial on « pairwise Markov models » (PMMs) and " triplet Markov models " (TMMs) which extend the popular " hidden Markov models " (HMMs). The originality of these extensions is due to the fact that the hidden data does not need to be Markov. More precisely, for X hidden data and Y observed ones, the originality of PMMs is that X does not need to be Markov, and the originality of TMMs is that even (X, Y) does not need to be Markov. In spite of these lacks of Markovianity fast processing methods, similar to those applied in HMMs or their other extensions, remain workable. The second goal is to present an original switching model approximation allowing fast smoothing. The method we propose, called " double filtering based smoothing " (DFBS), uses a particular TMM in which the pair (X, R), where R models switches, is not Markov. It is based on two filters, and uses a class of models, known as conditionally Gaussian observed Markov switching models (CGOMSMs), where exact fast filtering is feasible. The original model is approximated by two CGOMSMs in order to process the past data and the future data in direct and reverse order, respectively. Then state estimates produced by these two models are fused to provide a smoothing estimate. The DFBS is insensitive to the dimensions of the hidden and observation space and appears as an alternative to the classic particle smoothing in the situations where the latter cannot be applied due to its high processing cost

Lissage rapide dans des modèles non linéaires et non gaussiens

International audienceWe consider here the problem of statistical ltering and smoothing in nonlinear non-Gaussian systems. The noveltyconsists in approximating the nonlinear system by a recent switching system, in which exact fast optimal ltering and smoothingare workable. Our methods are applied to an asymetric stochastic volatility model and some experiments show their eciency.Nous nous intéressons a la question du filtrage et du lissage statistique dans les syst emes non linéaires et non gaussiens. La nouveauté réside dans l'approximation du syst eme non-linéaire par un mod elè a sauts dans lequel un calcul rapide et optimal du filtrage et du lissage est possible. Ces méthodes montrent leur intérêt , en particulier dans les récents mod eles de volatilité stochastique asymétrique. Abstract – We consider here the problem of statistical filtering and smoothing in nonlinear non-Gaussian systems. The novelty consists in approximating the nonlinear system by a recent switching system, in which exact fast optimal filtering and smoothing are workable. Our methods are applied to an asymetric stochastic volatility model and some experiments show their efficiency

Fast Filtering in Switching Approximations of Non-linear Markov Systems with Applications to Stochastic Volatility

International audienceWe consider the problem of optimal statistical filtering in general non-linear non-Gaussian Markov dynamic systems. The novelty of the proposed approach consists in approximating the non-linear system by a recent Markov switching process, in which one can perform exact and optimal filtering with a linear time complexity. All we need to assume is that the system is stationary (or asymptotically stationary), and that one can sample its realizations. We evaluate our method using two stochastic volatility models and results show its efficiency

Fast smoothing in switching approximations of non-linear and non-Gaussian models

International audienceStatistical smoothing in general non-linear non-Gaussian systems is a challenging problem. A new smoothing method based on approximating the original system by a recent switching model has been introduced. Such switching model allows fast and optimal smoothing. The new algorithm is validated through an application on stochastic volatility and dynamic beta models. Simulation experiments indicate its remarkable performances and low processing cost. In practice, the proposed approach can overcome the limitations of particle smoothing methods and may apply where their usage is discarded

Optimal filtering in hidden and pairwise Gaussian Markov systems

International audienceIn a hidden Markov model (HMM), the system goes through a hidden Markovian sequence of states (X) and produces a sequence of emissions (Y). We define the hidden Gaussian Markov model (HGMM) as the HMM where the hidden process is Gaussian and is affected by a normal white noise. The Kalman filter (KF) is a fast optimal statistical estimation method for the HGMMs and is very popular among the practitioners. However, the classic HGMM formulation is too restrictive. It extends to recent pairwise Gaussian Markov model (PGMM) where we assume that the pair (X, Y) is Gaussian Markovian. Moreover, there exists a KF version for the PGMM. The PGMM is more general than HGMM and in particular, the PGMM hidden process is not necessarily Markovian. The authors share their findings on about enhancing the KF when improving HGMM to PGMM. We discover singular cases where HGMM is at least ten times less accurate than the PGMM. On average, PGMM outperforms HGMM by twenty percen