Persistent Homology as a Tool for Timeseries Analysis

Persistence is a fairly well established tool in topological data analysis used to infer geometric information form discrete data. The aim of our work is to modify the classical persistent workflow in order to study the behavior of time series. First we introduce the sliding window embedding technique in order to codify temporal regularity of the time series in geometric properties of the obtained point cloud, and we give an general overview on the historical approach to persistent homology. Secondly, we present the categorical approach to persistence in order to cover all the relevant theoretical background needed for our analysis. In order to compute the complete invariant (the barcode decomposition) of the persistent module, we used the algorithm introduced by de Silva, Morozov and Vejdemo-Johansson for persistent cohomology. Finally, we applied our methodology to time series of different nature; we studied orbits of both the logistic map and the Chirikov standard map on the torus. Finally, we test our method on a financial time series; precisely on the time series of the exchange rate BitCoin/USDollar