Model reduction for Hidden Markov models

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 2006.Includes bibliographical references (leaves 57-60).The contribution of this thesis is the development of tractable computational methods for reducing the complexity of two classes of dynamical systems, finite alphabet Hidden Markov Models and Jump Linear Systems with finite parameter space. The reduction algorithms employ convex optimization and numerical linear algebra tools and do not pose any structural requirements on the systems at hand. In the Jump Linear Systems case, a distance metric based on randomization of the parametric input is introduced. The main point of the reduction algorithm lies in the formulation of two dissipation inequalities, which in conjunction with a suitably defined storage function enable the derivation of low complexity models, whose fidelity is controlled by a guaranteed upper bound on the stochastic L2 gain of the approximation error. The developed reduction procedure can be interpreted as an extension of the balanced truncation method to the broader class of Jump Linear Systems. In the Hidden Markov Model case, Hidden Markov Models are identified with appropriate Jump Linear Systems that satisfy certain constraints on the coefficients of the linear transformation. This correspondence enables the development of a two step reduction procedure.(cont.) In the first step, the image of the high dimensional Hidden Markov Model in the space of Jump Linear Systems is simplified by means of the aforementioned balanced truncation method. Subsequently, in the second step, the constraints that reflect the Hidden Markov Model structure are imposed by solving a low dimensional non convex optimization problem. Numerical simulation results provide evidence that the proposed algorithm computes accurate reduced order Hidden Markov Models, while achieving a compression of the state space by orders of magnitude.by Georgios Kotsalis.Ph.D

Balanced truncation for linear switched systems

In this paper, we present a theoretical analysis of the model reduction algorithm for linear switched systems. This algorithm is a reminiscence of the balanced truncation method for linear parameter varying systems. Specifically in this paper, we provide a bound on the approximation error in L2 norm for continuous-time and l2 norm for discrete-time linear switched systems. We provide a system theoretic interpretation of grammians and their singular values. Furthermore, we show that the performance of bal- anced truncation depends only on the input-output map and not on the choice of the state-space representation. For a class of stable discrete-time linear switched systems (so called strongly stable systems), we define nice controllability and nice observability grammians, which are genuinely related to reachability and controllability of switched systems. In addition, we show that quadratic stability and LMI estimates of the L2 and l2 gains depend only on the input-output map.Comment: We have corrected a number of typos and inconsistencies. In addition, we added new results in Theorem

Model Reduction of Linear Switched Systems by Restricting Discrete Dynamics

We present a procedure for reducing the number of continuous states of discrete-time linear switched systems, such that the reduced system has the same behavior as the original system for a subset of switching sequences. The proposed method is expected to be useful for abstraction based control synthesis methods for hybrid systems

Subspace estimation and prediction methods for hidden Markov models

Hidden Markov models (HMMs) are probabilistic functions of finite Markov chains, or, put in other words, state space models with finite state space. In this paper, we examine subspace estimation methods for HMMs whose output lies a finite set as well. In particular, we study the geometric structure arising from the nonminimality of the linear state space representation of HMMs, and consistency of a subspace algorithm arising from a certain factorization of the singular value decomposition of the estimated linear prediction matrix. For this algorithm, we show that the estimates of the transition and emission probability matrices are consistent up to a similarity transformation, and that the $m$-step linear predictor computed from the estimated system matrices is consistent, i.e., converges to the true optimal linear $m$-step predictor.Comment: Published in at http://dx.doi.org/10.1214/09-AOS711 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Bayesian Nonparametric Inference of Switching Linear Dynamical Systems

Many complex dynamical phenomena can be effectively modeled by a system that switches among a set of conditionally linear dynamical modes. We consider two such models: the switching linear dynamical system (SLDS) and the switching vector autoregressive (VAR) process. Our Bayesian nonparametric approach utilizes a hierarchical Dirichlet process prior to learn an unknown number of persistent, smooth dynamical modes. We additionally employ automatic relevance determination to infer a sparse set of dynamic dependencies allowing us to learn SLDS with varying state dimension or switching VAR processes with varying autoregressive order. We develop a sampling algorithm that combines a truncated approximation to the Dirichlet process with efficient joint sampling of the mode and state sequences. The utility and flexibility of our model are demonstrated on synthetic data, sequences of dancing honey bees, the IBOVESPA stock index, and a maneuvering target tracking application.Comment: 50 pages, 7 figure

Model Reduction by Moment Matching for Linear Switched Systems

Two moment-matching methods for model reduction of linear switched systems (LSSs) are presented. The methods are similar to the Krylov subspace methods used for moment matching for linear systems. The more general one of the two methods, is based on the so called "nice selection" of some vectors in the reachability or observability space of the LSS. The underlying theory is closely related to the (partial) realization theory of LSSs. In this paper, the connection of the methods to the realization theory of LSSs is provided, and algorithms are developed for the purpose of model reduction. Conditions for applicability of the methods for model reduction are stated and finally the results are illustrated on numerical examples.Comment: Sent for publication in IEEE TAC, on October 201

Bayesian Recurrent Neural Network Models for Forecasting and Quantifying Uncertainty in Spatial-Temporal Data

Recurrent neural networks (RNNs) are nonlinear dynamical models commonly used in the machine learning and dynamical systems literature to represent complex dynamical or sequential relationships between variables. More recently, as deep learning models have become more common, RNNs have been used to forecast increasingly complicated systems. Dynamical spatio-temporal processes represent a class of complex systems that can potentially benefit from these types of models. Although the RNN literature is expansive and highly developed, uncertainty quantification is often ignored. Even when considered, the uncertainty is generally quantified without the use of a rigorous framework, such as a fully Bayesian setting. Here we attempt to quantify uncertainty in a more formal framework while maintaining the forecast accuracy that makes these models appealing, by presenting a Bayesian RNN model for nonlinear spatio-temporal forecasting. Additionally, we make simple modifications to the basic RNN to help accommodate the unique nature of nonlinear spatio-temporal data. The proposed model is applied to a Lorenz simulation and two real-world nonlinear spatio-temporal forecasting applications