Precision Matrix Estimation And Discriminant Analysis Based On High-dimensional Compositional Data

In many fields,such as geology,economy,microorganism,etc.,the compositional data are widely used.For example,the chemical composition of rocks,the species composition of biological communities,and household expenditure patterns.Now the componsitional data are mainly used to study the proportion relationship between the parts that make up a whole.High-dimensional compositional data appears in many applications,and statistical methods usually do not produce reasonable results due to the constraints.The estimation of high dimensional covariance matrix or precision(inverse covariance)matrix is the basic problem of modern multivariate analysis.The covariance matrix reveals marginal correlations between variables,while the precision matrix encodes conditional correlations between pairs of variables given the remaining variables.Covariance and precision matrices are used in principal components analysis,linear/quadratic discriminant analysis,and graphical models.This paper considers the precision matrix estimation problem for high-dimensional compositional data.It is known that the inverse of the sample covariance matrix is unstable for the estimate precision matrix.Since the sample size of the data is smaller than the number of variables,the inverse of the high-dimensional data matrix is difficult to estimate.In this paper,the central logarithmic ratio transformation method is used to process high-dimensional compositional data,and the precision matrix estimation of high-dimensional compositional data is obtained.Simulation experiments and actual data can verify the rationality of the proposed method.