The Hidden Markov Model is a classic modelling tool with a wide swath of applications. Its inception considered observations restricted to a finite alphabet, but it was quickly extended to multivariate continuous distributions. In this article, we further extend the Hidden Markov Model from mixtures of normal distributions in $d$-dimensional Euclidean space to general Gaussian measure mixtures in locally convex topological spaces, and hence, we christen this method the Topological Hidden Markov Model. The main innovation is the use of the Onsager-Machlup functional as a proxy for the probability density function in infinite 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 diffusion processes such as Brownian and fractional Brownian sample paths as well as the Ornstein-Uhlenbeck process. Our methodology is applied to the identification of sleep states from overnight polysomnography time series data with the aim of diagnosing 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.

Pediatric obstructive sleep apnea is a serious condition causing a litany of health problems, including developmental issues. The gold standard for diagnosis is an overnight sleep study, which is often expensive and inaccessible. Additionally, the standard metric for identifying sleep apnea, the apnea-hypopnea index, has been shown to be ineffective in characterizing severity or treatment outcomes in children. To address this, we take a first step in developing an alternative accessible diagnostic tool using 3D facial images. We consider persistent homology and geometric shape analysis towards identifying facial features associated with OSA. The methodology developed here will be used in the future for two different approaches towards diagnosis: supervised learning to classify severity of apnea, and unsupervised learning to explore clinical phenotypes.