Advanced sequential Monte Carlo methods and their applications to sparse sensor network for detection and estimation

The general state space models present a flexible framework for modeling dynamic systems and therefore have vast applications in many disciplines such as engineering, economics, biology, etc. However, optimal estimation problems of non-linear non-Gaussian state space models are analytically intractable in general. Sequential Monte Carlo (SMC) methods become a very popular class of simulation-based methods for the solution of optimal estimation problems. The advantages of SMC methods in comparison with classical filtering methods such as Kalman Filter and Extended Kalman Filter are that they are able to handle non-linear non-Gaussian scenarios without relying on any local linearization techniques. In this thesis, we present an advanced SMC method and the study of its asymptotic behavior. We apply the proposed SMC method in a target tracking problem using different observation models. Specifically, a distributed SMC algorithm is developed for a wireless sensor network (WSN) that incorporates with an informative-sensor detection technique. The novel SMC algorithm is designed to surmount the degeneracy problem by employing a multilevel Markov chain Monte Carlo (MCMC) procedure constructed by engaging drift homotopy and likelihood bridging techniques. The observations are gathered only from the informative sensors, which are sensing useful observations of the nearby moving targets. The detection of those informative sensors, which are typically a small portion of the WSN, is taking place by using a sparsity-aware matrix decomposition technique. Simulation results showcase that our algorithm outperforms current popular tracking algorithms such as bootstrap filter and auxiliary particle filter in many scenarios

Probabilistic sequential matrix factorization

We introduce the probabilistic sequential matrix factorization (PSMF) method for factorizing time-varying and non-stationary datasets consisting of high-dimensional time-series. In particular, we consider nonlinear Gaussian state-space models where sequential approximate inference results in the factorization of a data matrix into a dictionary and time-varying coefficients with potentially nonlinear Markovian dependencies. The assumed Markovian structure on the coefficients enables us to encode temporal dependencies into a low-dimensional feature space. The proposed inference method is solely based on an approximate extended Kalman filtering scheme, which makes the resulting method particularly efficient. PSMF can account for temporal nonlinearities and, more importantly, can be used to calibrate and estimate generic differentiable nonlinear subspace models. We also introduce a robust version of PSMF, called rPSMF, which uses Student-t filters to handle model misspecification. We show that PSMF can be used in multiple contexts: modeling time series with a periodic subspace, robustifying changepoint detection methods, and imputing missing data in several high-dimensional time-series, such as measurements of pollutants across London.Comment: Accepted for publication at AISTATS 202

Langevin and Hamiltonian based Sequential MCMC for Efficient Bayesian Filtering in High-dimensional Spaces

Nonlinear non-Gaussian state-space models arise in numerous applications in statistics and signal processing. In this context, one of the most successful and popular approximation techniques is the Sequential Monte Carlo (SMC) algorithm, also known as particle filtering. Nevertheless, this method tends to be inefficient when applied to high dimensional problems. In this paper, we focus on another class of sequential inference methods, namely the Sequential Markov Chain Monte Carlo (SMCMC) techniques, which represent a promising alternative to SMC methods. After providing a unifying framework for the class of SMCMC approaches, we propose novel efficient strategies based on the principle of Langevin diffusion and Hamiltonian dynamics in order to cope with the increasing number of high-dimensional applications. Simulation results show that the proposed algorithms achieve significantly better performance compared to existing algorithms

IDENTIFICATION AND ESTIMATION OF MULTI-MODAL COMPLEX DYNAMIC SYSTEM

In this dissertation we study identification of complex dynamic systems as well as hybrid system estimation. For the identification part, we propose a scheme to identify an autonomous complex stochastic dynamic system based on a black-box model, that is, the system is modeled based on output data only. The system under study is a system whose underlying space is the union of strong attraction domains. The system exhibits a behavior such that it spends a long time in one strong attraction domain before transitioning to another one. Systems showing this behavior can be found in many applications ranging from biology to power systems to chemical processes. Considering the nature of this type of a system, we model it as a hybrid system. In particular, it is a strong attraction domain featured hybrid system (SAFHS). Two principal features of this type of a hybrid system are that the boundaries between the modes (strong attraction domains) are nonlinear and the dynamic behavior within each mode can be highly nonlinear, e.g. limit cycle. Identification algorithms for this kind of hybrid system are not well developed. In this dissertation we propose our first result for identification of this type of system. The resulting model is hybrid in nature. We detect the multi-modal dynamics as well as local dynamics within each mode, thus providing a complete unified approach of identification of the system dynamics. The approach developed in this dissertation is based on finite dimensional approximations of compact operators, spectral theory for non-reversible Markov chains, identification techniques for hidden Markov models (HMM), and identification techniques for linear and non-linear dynamics. Examples are carried out to verify our analysis and to illustrate the effectiveness of the proposed algorithms. In the estimation part, we present a high accuracy, low computational load method for a nonlinear/non-Gaussian hybrid system, motivated by the need to get a better trade-off between efficiency and accuracy which is a crucial issue in real time estimation problems. The efficiency and accuracy of the proposed algorithm are illustrated by examples. Moreover, its good performance makes it practical and robust for tracking a target in a complex situation, as we demonstrate by a simulated maneuvering target tracking example

A Spectral Algorithm for Learning Hidden Markov Models

Hidden Markov Models (HMMs) are one of the most fundamental and widely used statistical tools for modeling discrete time series. In general, learning HMMs from data is computationally hard (under cryptographic assumptions), and practitioners typically resort to search heuristics which suffer from the usual local optima issues. We prove that under a natural separation condition (bounds on the smallest singular value of the HMM parameters), there is an efficient and provably correct algorithm for learning HMMs. The sample complexity of the algorithm does not explicitly depend on the number of distinct (discrete) observations---it implicitly depends on this quantity through spectral properties of the underlying HMM. This makes the algorithm particularly applicable to settings with a large number of observations, such as those in natural language processing where the space of observation is sometimes the words in a language. The algorithm is also simple, employing only a singular value decomposition and matrix multiplications.Comment: Published in JCSS Special Issue "Learning Theory 2009