Analysis of High Dimensional Compositional Data Containing Structural Zeros with Applications to Microbiome Data

TL;DR

This paper develops a new statistical framework for analyzing high-dimensional microbiome data with structural zeros, enabling accurate estimation of covariance structures and providing theoretical error bounds supported by simulations and real data application.

Contribution

It introduces a novel model accommodating structural zeros in microbiome data and proposes methods with proven error bounds for estimating covariance and precision matrices.

Findings

Proposed estimators achieve favorable error bounds in spectral and Frobenius norms.

Simulation studies validate the theoretical error bounds.

Application to gut microbiome data demonstrates practical utility.

Abstract

This paper is motivated by the recent interest in the analysis of high dimen- sional microbiome data. A key feature of this data is the presence of `structural zeros' which are microbes missing from an observation vector due to an underlying biological process and not due to error in measurement. Typical notions of missingness are insufficient to model these structural zeros. We define a general framework which allows for structural zeros in the model and propose methods of estimating sparse high dimensional covariance and precision matrices under this setup. We establish error bounds in the spectral and frobenius norms for the proposed esti- mators and empirically support them with a simulation study. We also apply the proposed methodology to the global human gut microbiome data of Yatsunenko (2012).