Hidden Markov Models: Theory and Implementation Using MATLAB®

Description

This book presents, in an integrated form, both the analysis and synthesis of three different types of hidden Markov models. Unlike other books on the subject, it is generic and does not focus on a specific theme, e.g. speech processing. Moreover, it presents the translation of hidden Markov models' concepts from the domain of formal mathematics into computer codes using MATLAB(R). The unique feature of this book is that the theoretical concepts are first presented using an intuition-based approach followed by the description of the fundamental algorithms behind hidden Markov models using MATLAB(R). This approach, by means of analysis followed by synthesis, is suitable for those who want to study the subject using a more empirical approach. Key Selling Points: Presents a broad range of concepts related to Hidden Markov Models (HMM), from simple problems to advanced theory Covers the analysis of both continuous and discrete Markov chains Discusses the translation of HMM concepts from the realm of formal mathematics into computer code Offers many examples to supplement mathematical notation when explaining new concepts Cover Title Page Copyright Page Preface Table of Contents Glossary 1: Introduction 1.1 System Models 1.2 Markov Chains 1.3 Book Outline 2: Probability Theory and Stochastic Processes 2.1 Introduction 2.2 Introduction to Probability Theory 2.2.1 Events and Random Variables 2.2.1.1 Types of variables 2.2.2 Probability Definition 2.2.3 Axioms and Properties 2.3 Probability Density Function 2.4 Statistical Moments 2.5 Summary 3: Discrete Hidden Markov Models 3.1 Introduction 3.2 Hidden Markov Model Dynamics 3.2.1 The Forward Algorithm 3.2.2 The Backward Algorithm 3.2.3 The Viterbi Algorithm 3.3 Probability Transitions Estimation 3.3.1 Maximum Likelihood Definition 3.3.2 The Baum-Welch Training Algorithm 3.3.2.1 Operation conditions for the Baum-Welch algorithm 3.3.2.2 Parameter estimation using multiple trials 3.3.2.3 Baum-Welch algorithm numerical stability 3.4 Viterbi Training Algorithm 3.5 Gradient-based Algorithms 3.5.1 Partial Derivative of Lk 3.5.1.1 Partial derivative of Lk in order to aij 3.5.1.2 Partial derivative of Lk in order to bij 3.5.2 Partial Derivative of LK in order to c 3.5.3 Performance Analysis of the Re-estimation Formulas 3.5.4 Parameters Coercion by Re-parameterization 3.5.5 Rosen’s Algorithm 3.5.5.1 Linear equality constraints 3.5.5.2 Lagrange multipliers and Karush-Kuhn-Tucker conditions 3.5.5.3 Linear inequality constraints 3.5.5.4 Putting it all together 3.5.5.5 Rosen’s method applied to hidden Markov Models 3.6 Architectures for Markov Models 3.7 Summary 4: Continuous Hidden Markov Models 4.1 Introduction 4.2 Probability Density Functions and Gaussian Mixtures 4.2.1 Gaussian Functions in System Modeling 4.2.2 Gaussian Function and Gaussian Mixture 4.3 Continuous Hidden Markov Model Dynamics 4.3.1 Forward, Backward and Viterbi Algorithms Revisited 4.4 Continuous Observations Baum-Welch Training Algorithm 4.5 Summary 5: Autoregressive Markov Models 5.1 Introduction 5.2 ARMM Structure 5.3 Likelihood and Probability Density for AR Models 5.3.1 AR Model Probability Density Function 5.3.2 Autoregressive Model Likelihood 5.4 Likelihood of an ARMM 5.5 ARMM Parameters Estimations 5.5.1 Parameters Estimation 5.6 Time Series Prediction with ARMM 5.6.1 One Step Ahead Time Series Prediction 5.6.2 Multiple Steps Ahead Time Series Prediction 5.7 Summary 6: Selected Applications 6.1 Cardiotocography Classification 6.2 Solar Radiation Prediction 6.3 Summary References Index Color Figures Section