Bayesian Method Of The Low Rank Learning For Sparse-spike Covariance Matrix

For a long time,the estimation of high-dimensional multivariate normal distri-bution covariance matrix is a basic problem in statistics.In practical problems such as abnormal ECG analysis,only a few components of the high-dimensional data we get are pulse,and all of them are sparse.We call the covariance matrix in this case is Sparse-Spike covariance matrix.In high-dimensional problems,the number of samples we get is often much smaller than the matrix dimension.We call the estimation and feature extraction of covariance matrix in the case of small samples as its low rank learning.The partial diagonal elements of the sparse spike covariance matrix studied in this paper are also sparse,which is different from many predecessors.For its estimation,most predecessors directly use the sample covariance matrix or its sparse improvement.Considering and using Bayesian method to estimate the sparse spike covariance matrix in the case of small samples is a blank in the research of sparse spike covariance matrix.The main work of this paper is to apply the modified reference prior and uniform prior to the estimation of sparse spike covariance matrix.Firstly,this paper introduces several non Bayesian methods for covariance matrix estimation,which can deal with the estimation problem in the case of small samples.Then this paper introduces a kind of prior of covariance matrix,which can compare with the traditional prior.At the same time,some priors need only a few samples to make the posterior exist.At the same time,this paper also introduces a new sampling method of covariance matrix.This paper then applies this kind of a prior and sampling methods to the sparse spike model.We prove that for the modified reference prior,we can get the estimation regardless of the dimension,even if there are only two samples.At the same time,it also shows that our Bayesian method will be faster in the sparse spike model.Moreover,our method does not need to add sparsity assumptions and selection parameters,which many methods do not have.Finally,we compare the Bayesian method and non Bayesian method of covariance matrix estimation under the sparse spike model.We find that the modified reference prior and the uniform prior have faster estimation efficiency for the high-dimensional sparse spike covariance matrix.At the same time,when the sample size is only 5,the modified reference a prior is better than other methods in any dimension.Therefore,for the estimation of sparse spike covariance matrix,we recommend Bayesian method.