Dirichlet Component Analysis: Feature Extraction for Compositional Data

Abstract

We consider feature extraction (dimensionality reduction) for compositional data, where the data vectors are constrained to be positive and constant-sum. In real-world problems, the data components (variables) usually have complicated "correlations" while their total number is huge. Such scenario demands feature extraction. That is, we shall de-correlate the components and reduce their dimensionality. Traditional techniques such as the Principle Component Analysis (PCA) are not suitable for these problems due to unique statistical properties and the need to satisfy the constraints in compositional data. This paper presents a novel approach to feature extraction for compositional data. Our method first identifies a family of dimensionality reduction projections that preserve all relevant constraints, and then finds the optimal projection that maximizes the estimated Dirichlet precision on projected data. It reduces the compositional data to a given lower dimensionality while the components in the lower-dimensional space are de-correlated as much as possible. We develop theoretical foundation of our approach, and validate its effectiveness on some synthetic and real-world datasets.