Distance Measures for Probabilistic Patterns

Abstract

Numerical measures of pattern dissimilarity are at the heart of pattern recognition and classification. Applications of pattern recognition grow more sophisticated every year, and consequently we require distance measures for patterns not easily expressible as feature vectors. Examples include strings, parse trees, time series, random spatial fields, and random graphs [79] [117].

Distance measures are not arbitrary. They can only be effective when they incorporate information about the problem domain; this is a direct consequence of the Ugly Duckling theorem [37].

This thesis poses the question: how can the principles of information theory and statistics guide us in constructing distance measures? In this thesis, I examine distance functions for patterns that are maximum-likelihood model estimates for systems that have random inputs, but are observed noiselessly. In particular, I look at distance measures for histograms, stationary ARMA time series, and discrete hidden Markov models.

I show that for maximum likelihood model estimates, the L2 distance involving the information matrix at the most likely model estimate minimizes the type II classification error, for a fixed type I error. I also derive explicit L2 distance measures for ARMA(p, q) time series and discrete hidden Markov models, based on their respective information matrices.