<i>L^p</i>-Norm for Compositional Data: Exploring the CoDa <i>L</i>¹-Norm in Penalised Regression

Abstract

The Least Absolute Shrinkage and Selection Operator (LASSO) regression technique has proven to be a valuable tool for fitting and reducing linear models. The trend of applying LASSO to compositional data is growing, thereby expanding its applicability to diverse scientific domains. This paper aims to contribute to this evolving landscape by undertaking a comprehensive exploration of the L1-norm for the penalty term of a LASSO regression in a compositional context. This implies first introducing a rigorous definition of the compositional Lp-norm, as the particular geometric structure of the compositional sample space needs to be taken into account. The focus is subsequently extended to a meticulous data-driven analysis of the dimension reduction effects on linear models, providing valuable insights into the interplay between penalty term norms and model performance. An analysis of a microbial dataset illustrates the proposed approach.

Keywords

• Aitchison’s geometry
• compositional data
• <i>L^p</i>-norm
• balance selection