Online Expectation Maximization Algorithm And Its Application To Parametric Estimation Of A Class Of Hidden Markov Models

Hidden Markov models(HMMs)are widely used in many fields such as finance,biology,statistics and computer science,and thus the study of them is of great significance in both theory and application.As one of the most popular applications among them,applying the HMM to investigate the financial market,such as stock prices,and predict the trend,has attracted great attention of many scholars.Therefore,parametric estimation of the HMM based on observational data is of significant importance to guide the actual industrial production,control and decision-making.This paper mainly studies the parametric estimation problem of the HMM,that is,to es-timate the parameters in the model based on observation data.Since the hidden variables are not observable,a classic algorithm for solving this problem is the Expected Maximization(EM)algorithm.However,the traditional offline EM algorithms require to use all samples at each iteration,which leads to the problems that the computation is inefficient,memory intensive,and cannot be updated online.Therefore,many big data scenarios present serious challenges to of-fline EM algorithms.In order to solve the problems of offline EM algorithm in such scenarios,we propose a block online EM algorithm for the HMM,which updates the estimates of parameters in real time according to batch observation data,so it is more suitable for big data applications such as streaming data and high frequency data.The main contributions of the paper are summarized as follows:On the one hand,a block online EM algorithm for general HMMs is proposed and applied to parametric estimation of finite Gaussian mixture models and exponential family distribution models.In view of the HMM with different characteristics,recursive algorithms for computation of the Q functions of the models are presented separately,and specific algorithm flow of the block online EM algorithm is provided for parametric estimation of the two models.On the other hand,numerical experiments and empirical analysis of the online EM algorithm are given to demonstrate its feasibility and effectiveness.Through the numerical experiments of the above two HMMs,namely,the double Gaussian mixture model and the finite state Markov chain model observed with Gaussian noise,the feasibility and accuracy of the online algorithm are explained.Meanwhile,through the comparison of the experimental results of the online EM algorithm and the offline EM algorithm on the same model,the advantages of the proposed algorithm are demonstrated.Finally,application of the algorithm to actual scenarios of stock prices is explained through an empirical analysis.