Kernel methods for compositional data and dimensionality reduction

Abstract

Compositional data analysis has been garnering more focus, particularly due to its significance in human microbiome studies. Traditional techniques often struggle with recent data sets as they are high-dimensional and constituted of a significant proportion of zeros. We approach this problem using kernel methods, which naturally handle zeros in data, and develop dimension reduction methods to alleviate the curse of dimensionality and enhance interpretability in subsequent analyses. In this thesis, we introduce three projects utilizing kernel methods for compositional data.

In Project 1, we demonstrate that the prevalent approach of log-ratio transformation, performed after zero-replacement, produces significant distortions in the marginal distribution of data. Instead, we suggest employing kernel methods based on geometric considerations, eliminating the need for zero replacements. In Project 2, we propose a kernel-based variable selection method of compositional data, arguing the use of amalgamation over subcomposition. In Project 3, we extend the methodology from the second project to develop a novel method for reducing the dimension of compositional data through a more relaxed version of amalgamation.