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

Delay Parameter Selection in Permutation Entropy Using Topological Data Analysis

Permutation Entropy (PE) is a powerful tool for quantifying the predictability of a sequence which includes measuring the regularity of a time series. Despite its successful application in a variety of scientific domains, PE requires a judicious choice of the delay parameter $\tau$. While another parameter of interest in PE is the motif dimension $n$, Typically $n$ is selected between $4$ and $8$ with $5$ or $6$ giving optimal results for the majority of systems. Therefore, in this work we focus solely on choosing the delay parameter. Selecting $\tau$ is often accomplished using trial and error guided by the expertise of domain scientists. However, in this paper, we show that persistent homology, the flag ship tool from Topological Data Analysis (TDA) toolset, provides an approach for the automatic selection of $\tau$. We evaluate the successful identification of a suitable $\tau$ from our TDA-based approach by comparing our results to a variety of examples in published literature

ABJM on ellipsoid and topological strings

It is known that the large N expansion of the partition function in ABJM theory on a three-sphere is completely determined by the topological string on local Hirzebruch surface F_0. In this note, we investigate the ABJM partition function on an ellipsoid, which has a conventional deformation parameter b. Using 3d mirror symmetry, we find a remarkable relation between the ellipsoid partition function for b^2=3 (or b^2=1/3) in ABJM theory at k=1 and a matrix model for the topological string on another Calabi-Yau threefold, known as local P^2. As in the case of b=1, we can compute the full large N expansion of the partition function in this case. This is the first example of the complete large N solution in ABJM theory on the squashed sphere. Using the obtained results, we also analyze the supersymmetric Renyi entropy.Comment: 29 page

Towards Emotion Recognition: A Persistent Entropy Application

Emotion recognition and classification is a very active area of research. In this paper, we present a first approach to emotion classification using persistent entropy and support vector machines. A topology-based model is applied to obtain a single real number from each raw signal. These data are used as input of a support vector machine to classify signals into 8 different emotions (calm, happy, sad, angry, fearful, disgust and surprised)

Characterising epithelial tissues using persistent entropy

In this paper, we apply persistent entropy, a novel topological statis- tic, for characterization of images of epithelial tissues. We have found out that persistent entropy is able to summarize topological and geomet- ric information encoded by -complexes and persistent homology. After using some statistical tests, we can guarantee the existence of signi cant di erences in the studied tissues.Ministerio de Economía y Competitividad MTM2015-67072-