Persistence Norms and the Datasaurus

TL;DR

This paper demonstrates that persistence norms from Topological Data Analysis can distinguish datasets with identical summary statistics, highlighting their usefulness as additional data shape measures.

Contribution

It shows that persistence norms can differentiate datasets with similar traditional statistics, emphasizing their value in data shape analysis.

Findings

Persistence norms vary across datasets with identical summary statistics.

Multivariate distributions with same covariance can have different persistence norms.

Persistence norms are sensitive to the distribution of point clouds.

Abstract

Topological Data Analysis (TDA) provides a toolkit for the study of the shape of high dimensional and complex data. While operating on a space of persistence diagrams is cumbersome, persistence norms provide a simple real value measure of multivariate data which is seeing greater adoption within finance. A growing literature seeks links between persistence norms and the summary statistics of the data being analysed. This short note targets the demonstration of differences in the persistence norms of the Datasaurus datasets of Matejka and Fitzmaurice. We show that persistence norms can be used as additional measures that often discriminate datasets with the same collection of summary statistics. Treating each of the data sets as a point cloud we construct the $L_1$ and $L_2$ persistence norms in dimensions 0 and 1. We show multivariate distributions with identical covariance and correlation matrices can have considerably different persistence norms. Through the example, we remind users of persistence norms of the importance of checking the distribution of the point clouds from which the norms are constructed.