Large Covariance Estimation for Compositional Data Via Composition-Adjusted Thresholding

ABSTRACT

High-dimensional compositional data arise naturally in many applications such as metagenomic data analysis. The observed data lie in a high-dimensional simplex, and conventional statistical methods often fail to produce sensible results due to the unit-sum constraint. In this article, we address the problem of covariance estimation for high-dimensional compositional data and introduce a composition-adjusted thresholding (COAT) method under the assumption that the basis covariance matrix is sparse. Our method is based on a decomposition relating the compositional covariance to the basis covariance, which is approximately identifiable as the dimensionality tends to infinity. The resulting procedure can be viewed as thresholding the sample centered log-ratio covariance matrix and hence is scalable for large covariance matrices. We rigorously characterize the identifiability of the covariance parameters, derive rates of convergence under the spectral norm, and provide theoretical guarantees on support recovery. Simulation studies demonstrate that the COAT estimator outperforms some existing optimization-based estimators. We apply the proposed method to the analysis of a microbiome dataset to understand the dependence structure among bacterial taxa in the human gut.

KEYWORDS:

• Adaptive thresholding
• Basis covariance
• Centered log-ratio covariance
• High dimensionality
• Microbiome
• Regularization

Supplementary Materials

The online supplementary materials contain the appendices for the article.

Acknowledgments

The authors thank Professor Hongmei Jiang and Dr. Huaying Fang for sharing R code and the Associate Editor and two reviewers for helpful comments.

Additional information

Funding

Cao and Li’s research was supported in part by NIH grants CA127334 and GM097505. Lin’s research was supported in part by NSFC grants 11671018 and 71532001.