Parameter Estimation Of High-Dimensional State Spapce Model With Applications

The state space model(SSM)is used to infer the relevant information of unobservable variables from the observed information.The problem of state estimation and parameter estimation is a hot topic of research.In biological and economic data,there are many time series data with high dimension observation variables.In linear SSM,maximum likelihood method and Expectation Maximization(EM)algorithm are used to estimate parameters.Because Kalman filtering involves the inverse operation of high-dimensional matrix,the computational burden of the model is heavy and the numerical value is unstable.In addition,the estimation results of state matrix and observation matrix are often non-sparse.It is difficult to extract the important relationship between the observed variables and the state variables,as well as the transition of the state variables.Therefore,it is necessary to study the sparse estimation methods of state matrix and observation matrix in high-dimensional case.The contents of this paper are as follows:(1)In the high-dimensional state space model,this paper combines EM algorithm with coefficient shrinkage method to give sparse estimation of the matrix,and builds network model and dynamic factor model based on sparse matrix.Firstly,based on the general form of linear Gauss SSM,we deduce the specific steps of Expectation Regularization Maximization(ERM)algorithm.For the inverse operation of high-dimensional matrices in Kalman filtering,the recursive method is used to solve it.For the problem that there are many parameters to be estimated in the matrix,in the “pseudo-regression”,row by row estimation and matrix estimation are used respectively according to the dimension of the matrix to be estimated.Secondly,two recognizable forms are simulated to verify the effectiveness of the ERM method.Among all the penalty functions,the Adaptive Elastic Net penalty is optimal.Finally,we use stock market data to do empirical analysis.Based on the sparse estimation of the first recognizable form state matrix,we establish a stock network.The yield network depicts the return relationship of the stock market,and the volatility network depicts the risk relationship.In Shanghai,Shenzhen,Hong Kong and the United States,we construct networks respectively,and find that the yield network is more clustered and the volatility network is more divergent.Based on the second recognizable form,the dynamic factor model is established,and the sparse component matrix is obtained.Combining with the policy and market factors,this paper gives a possible explanation for the change of the return of some stocks in Shanghai stock market.(2)For multi-individual and multi-variable time series data with exogenous variables,this paper modifies the recognizable form of SSM.The sparse estimation method of state matrix is deduced by adding exogenous variables varying with time into the model.In view of the characteristics of vaginal microbial abundance data,we first cluster the samples and then cluster the females.For each group of women,the exogenous variable of menstruation is added to the model,and the sparse state matrix is used to construct the microbial interaction network.Taking LC and LI women as examples,this paper discusses the relationship between microbials,microbials and bacterial vaginitis.