Keywords: TDA, Persistence, Wavelets, Change-Point Detection
Webpages:
https://github.com/speegled/cpbaywave Topological data analysis (TDA) offers a multi-scale method to represent, visualize and interpret complex data by extracting topological features using persistent homology. We will focus on persistence diagrams, which are a way of representing the persistent homology of a point cloud. At their most basic level, persistence diagrams can give something similar to clustering information, but they also can give information about loops or other topological structures within a data set.
Wavelets are another multi-scale tool used to represent, visualize and interpret complex data. Wavelets offer a way of examining the local changes of a data set while also estimating the global trends.
We will present two algorithms that combine wavelets and persistence. First, we use a wavelet based density estimator to bootstrap confidence intervals in persistence diagrams. Wavelets seem well-suited for this, since if the underlying data lies on a manifold, then the density should have discontinuities that will need to be detected. Additionally, the wavelet based algorithm is fast enough to allow some cross-validation of the tuning parameters. Second, we present an algorithm for detecting the most likely change point of the persistent homology of a time series.
The majority of this talk will consist of presenting examples which will illustrate persistence diagrams, the change point detection algorith, and the types of changes in geometric and/or topological structure in data that can be detected via this algorithm.
