Power Transformations of Relative Count Data as a Shrinkage Problem

TL;DR

This paper introduces an information-geometric framework for power transformations of compositional count data, framing them as a shrinkage problem on the simplex, and provides an analytic solution for optimal exponential shrinkage parameters.

Contribution

It presents a novel geometric interpretation of power transformations as exponential shrinkage on the simplex, with an analytic method for optimal parameter estimation.

Findings

Power transformations can be viewed as exponential shrinkage on the tangent space.

An analytic solution for optimal exponential shrinkage parameter is derived.

Exponential shrinkage offers an alternative to zero imputation in compositional data analysis.

Abstract

Here we show an application of our recently proposed information-geometric approach to compositional data analysis (CoDA). This application regards relative count data, which are, e.g., obtained from sequencing experiments. First we review in some detail a variety of necessary concepts ranging from basic count distributions and their information-geometric description over the link between Bayesian statistics and shrinkage to the use of power transformations in CoDA. We then show that powering, i.e., the equivalent to scalar multiplication on the simplex, can be understood as a shrinkage problem on the tangent space of the simplex. In information-geometric terms, traditional shrinkage corresponds to an optimization along a mixture (or m-) geodesic, while powering (or, as we call it, exponential shrinkage) can be optimized along an exponential (or e-) geodesic. While the m-geodesic corresponds to the posterior mean of the multinomial counts using a conjugate prior, the e-geodesic corresponds to an alternative parametrization of the posterior where prior and data contributions are weighted by geometric rather than arithmetic means. To optimize the exponential shrinkage parameter, we use mean-squared error as a cost function on the tangent space. This is just the expected squared Aitchison distance from the true parameter. We derive an analytic solution for its minimum based on the delta method and test it via simulations. We also discuss exponential shrinkage as an alternative to zero imputation for dimension reduction and data normalization.