A review and application of hidden Markov models and double chain Markov models

A Dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in ful lment of the requirements for the degree of Master of Science. Johannesburg, 2016.Hidden Markov models (HMMs) and double chain Markov models (DCMMs) are classical Markov model extensions used in a range of applications in the literature. This dissertation provides a comprehensive review of these models with focus on i) providing detailed mathematical derivations of key results - some of which, at the time of writing, were not found elsewhere in the literature, ii) discussing estimation techniques for unknown model parameters and the hidden state sequence, and iii) discussing considerations which practitioners of these models would typically take into account. Simulation studies are performed to measure statistical properties of estimated model parameters and the estimated hidden state path - derived using the Baum-Welch algorithm (BWA) and the Viterbi Algorithm (VA) respectively. The effectiveness of the BWA and the VA is also compared between the HMM and DCMM. Selected HMM and DCMM applications are reviewed and assessed in light of the conclusions drawn from the simulation study. Attention is given to application in the field of Credit Risk.LG201

Factorial Hidden Markov Models

We present a framework for learning in hidden Markov models with distributed state representations. Within this framework, we derive a learning algorithm based on the Expectation--Maximization (EM) procedure for maximum likelihood estimation. Analogous to the standard Baum-Welch update rules, the M-step of our algorithm is exact and can be solved analytically. However, due to the combinatorial nature of the hidden state representation, the exact E-step is intractable. A simple and tractable mean field approximation is derived. Empirical results on a set of problems suggest that both the mean field approximation and Gibbs sampling are viable alternatives to the computationally expensive exact algorithm

Model-based clustering with Hidden Markov Models and its application to financial times-series data

We have developed a method to partition a set of data into clusters by use of Hidden Markov Models. Given a number of clusters, each of which is represented by one Hidden Markov Model, an iterative procedure finds the combination of cluster models and an assignment of data points to cluster models which maximizes the joint likelihood of the clustering. To reflect the non-Markovian nature of some aspects of the data we also extend classical Hidden Markov Models to employ a non-homogeneous Markov chain, where the non-homogeneity is dependent not on the time of the observation but rather on a quantity derived from previous observations. We present the method, a proof of convergence for the training procedure and an evaluation of the method on simulated time-series data as well as on large data sets of financial time-series from the Public Saving and Loan Banks in Germany

Model-based clustering with Hidden Markov Models and its application to financial times series data

We have developed a method to partition a set of data into clusters by use of Hidden Markov Models. Given a number of clusters, each of which is represented by one Hidden Markov Model, an iterative procedure finds the combination of cluster models and an assignment of data points to cluster models which maximizes the joint likelihood of the clustering. To reflect the non-Markovian nature of some aspects of the data we also extend classical Hidden Markov Models to employ a non-homogeneous Markov chain, where the non-homogeneity is dependent not on the time of the observation but rather on a quantity derived from previous observations. We present the method, a proof of convergence for the training procedure and an evaluation of the method on simulated time-series data as well as on large data sets of financial time-series from the Public Saving and Loan Banks in Germany

Probabilistic Constraint Logic Programming

This paper addresses two central problems for probabilistic processing models: parameter estimation from incomplete data and efficient retrieval of most probable analyses. These questions have been answered satisfactorily only for probabilistic regular and context-free models. We address these problems for a more expressive probabilistic constraint logic programming model. We present a log-linear probability model for probabilistic constraint logic programming. On top of this model we define an algorithm to estimate the parameters and to select the properties of log-linear models from incomplete data. This algorithm is an extension of the improved iterative scaling algorithm of Della-Pietra, Della-Pietra, and Lafferty (1995). Our algorithm applies to log-linear models in general and is accompanied with suitable approximation methods when applied to large data spaces. Furthermore, we present an approach for searching for most probable analyses of the probabilistic constraint logic programming model. This method can be applied to the ambiguity resolution problem in natural language processing applications.Comment: 35 pages, uses sfbart.cl