On the stability of persistent entropy and new summary functions for Topological Data Analysis

Persistent entropy of persistence barcodes, which is based on the Shannon entropy, has been recently defined and successfully applied to different scenarios: characterization of the idiotypic immune network, detection of the transition between the preictal and ictal states in EEG signals, or the classification problem of real long-length noisy signals of DC electrical motors, to name a few. In this paper, we study properties of persistent entropy and prove its stability under small perturbations in the given input data. From this concept, we define three summary functions and show how to use them to detect patterns and topological features