论文标题
拓扑隐藏的马尔可夫模型
Topological Hidden Markov Models
论文作者
论文摘要
隐藏的马尔可夫模型（HMM）是一种经典的建模工具，具有广泛的应用程序。它的成立考虑的观察限于有限字母，但很快将其扩展到多元连续分布。在本文中，我们将HMM从$ d $二维欧几里得空间中的正常分布的混合物中延伸到本地凸形拓扑空间中的一般高斯度量混合物。主要创新是使用Onsager-Machlup用作无限尺寸空间概率密度函数的代理。这允许选择适合给定应用的Cameron-Martin空间。我们通过将其应用于模拟扩散过程（例如布朗和布朗样本路径）以及Ornstein-Uhlenbeck过程来证明该方法的多功能性。我们的方法应用于从隔夜多摄影时间序列数据中识别睡眠状态，目的是诊断小儿患者的阻塞性睡眠呼吸暂停。它也适用于1940年至1990年在艾伯塔省埃德蒙顿市的一系列年度累积降雪曲线。
The hidden Markov model (HMM) is a classic modeling 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 HMM from mixtures of normal distributions in $d$-dimensional Euclidean space to general Gaussian measure mixtures in locally convex topological spaces. 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.
