Principal Subsimplex Analysis

2025-04-14

AI Summary

This paper addresses inherent limitations of the log-ratio + PCA approach for compositional data (e.g., geochemical, microbiome), including amplification of minor-variable variation, obscuration of dominant component structure, inability to handle zeros, and weak linear pattern capture. We propose an unsupervised dimensionality reduction method based on nested simplex sequences, operating directly in the original Aitchison simplex space. Our approach is the first to construct a principal-component–like, dimensionally decreasing nested simplex hierarchy—without zero-replacement or transformation—by optimizing geometry, fitting sub-simplices, and enforcing orthogonal projection constraints under a maximum-variance criterion, yielding affine-invariant linear variation directions. Evaluated on synthetic data and Late Quaternary diatom abundance data, the method accurately recovers true compositional variation structures, demonstrates significantly greater robustness than state-of-the-art log-ratio methods, and produces stable results without imputing zero values.

Technology Category

• Machine Learning: Dimensionality Reduction/Feature Selection
• Data Mining & Knowledge Management: Data Compression
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Data transparency and provenance

Abstract

Compositional data, also referred to as simplicial data, naturally arise in many scientific domains such as geochemistry, microbiology, and economics. In such domains, obtaining sensible lower-dimensional representations and modes of variation plays an important role. A typical approach to the problem is applying a log-ratio transformation followed by principal component analysis (PCA). However, this approach has several well-known weaknesses: it amplifies variation in minor variables; it can obscure important variation within major elements; it is not directly applicable to data sets containing zeros and zero imputation methods give highly variable results; it has limited ability to capture linear patterns present in compositional data. In this paper, we propose novel methods that produce nested sequences of simplices of decreasing dimensions analogous to backwards principal component analysis. These nested sequences offer both interpretable lower dimensional representations and linear modes of variation. In addition, our methods are applicable to data sets contain zeros without any modification. We demonstrate our methods on simulated data and on relative abundances of diatom species during the late Pliocene. Supplementary materials and R implementations for this article are available online.

Problem

• Analyzing compositional data with zeros without modification
• Overcoming PCA weaknesses in minor and major element variation
• Providing interpretable lower-dimensional representations for simplicial data

Innovation

• Nested simplices for lower-dimensional representation
• Directly handles datasets with zeros
• Captures linear patterns in compositional data