Topological Hidden Markov Models

Adam B Kashlak, Prachi Loliencar, Giseon Heo

Abstract:

The Hidden Markov Model (HMM) is a classic modelling tool with a wide swath of ap
plications. Its inception considered observations restricted to a nite alphabet, but it was
quickly extended to multivariate continuous distributions. In this article, we further extend
the HMM from mixtures of normal distributions in d-dimensional Euclidean space to gen
eral Gaussian measure mixtures in locally convex topological spaces, and hence, we christen
this method the Topological Hidden Markov Model (THMM). The main innovation is the
use of the Onsager-Machlup functional as a proxy for the probability density function in
innite dimensional spaces. This allows for choice of a Cameron-Martin space suitable for
a given application. We demonstrate the versatility of this methodology by applying it to
simulated di usion processes such as Brownian and fractional Brownian sample paths as
well as the Ornstein-Uhlenbeck process. Our methodology is applied to the identi cation
of sleep states from overnight polysomnography time series data with the aim of diagnos
ing Obstructive Sleep Apnea in pediatric patients. It is also applied to a series of annual
cumulative snowfall curves from 1940 to 1990 in the city of Edmonton, Alberta.