高维成分数据的精度矩阵估计

Large Precision Matrix Estimation for Compositional Data

摘要: 高维成分数据在许多应用中均有出现，由于定和约束，统计方法通常不能产生合理的结果。高维协方差矩阵和精度(逆协方差)矩阵的估计是现代多元分析的基本问题，本文考虑高维成分数据的精度矩阵估计问题。已知样本协方差矩阵求逆对于估计精度矩阵是不稳定的，由于数据的样本量小于变量个数，高维数据矩阵的逆很难估计。本文利用中心对数比变换方法，处理高维成分数据，然后解决协方差矩阵奇异性问题，得到高维成分数据的精度矩阵估计。模拟实验和实际数据可以验证提出方法的合理性。

Abstract: High-dimensional compositional data arise in many applications, and statistical methods often fail to produce sensible results due to the unit-sum constraints. The estimation of high dimensional covariance matrix or precision (inverse covariance) matrix is the basic problem of modern multivariate analysis. In this paper, the precision matrix estimation problem for high-dimensional compositional data is considered. 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, we use the centered log-ratio transformation method to process high-dimensional compositional data, and then solve the singularity problem of covariance matrix, and obtain the precision matrix estimation of high-dimensional compositional data. Simulation experiments and actual data can verify the rationality of the proposed method.