Representational Rényi heterogeneity
This enables heterogeneity measurement in fields like biodiversity and economics where data do not fit existing indices, though it is an incremental improvement over existing methods.
The authors tackled the problem of measuring heterogeneity in non-categorical data by introducing representational Rényi heterogeneity (RRH), which transforms data into a latent space to compute heterogeneity without requiring prior binning or distance definitions, and demonstrated its effectiveness on beta-mixture distributions and natural images.
A discrete system's heterogeneity is measured by the Rényi heterogeneity family of indices (also known as Hill numbers or Hannah--Kay indices), whose units are {the numbers equivalent}. Unfortunately, numbers equivalent heterogeneity measures for non-categorical data require {a priori} (A) categorical partitioning and (B) pairwise distance measurement on the observable data space, thereby precluding application to problems with ill-defined categories or where semantically relevant features must be learned as abstractions from some data. We thus introduce representational Rényi heterogeneity (RRH), which transforms an observable domain onto a latent space upon which the Rényi heterogeneity is both tractable and semantically relevant. This method requires neither {a priori} binning nor definition of a distance function on the observable space. We show that RRH can generalize existing biodiversity and economic equality indices. Compared with existing indices on a beta-mixture distribution, we show that RRH responds more appropriately to changes in mixture component separation and weighting. Finally, we demonstrate the measurement of RRH in a set of natural images, with respect to abstract representations learned by a deep neural network. The RRH approach will further enable heterogeneity measurement in disciplines whose data do not easily conform to the assumptions of existing indices.