Separating Persistent Homology of Noise from Time Series Data Using Topological Signal Processing.
Audun Myers, Firas A. Khasawneh, Brittany Terese Fasy · arXiv (Cornell University) · 2020
We introduce a novel method for separating significant features in the sublevel set persistence diagram based on a statistics analysis of the sublevel set persistence of a noise distribution. Specifically, the statistical analysis of the sublevel set persistence of additive noise distributions are leveraged to provide a noise cutoff or confidence interval in the sublevel set persistence diagram. This analysis is done for several common noise models including Gaussian, uniform, exponential and Rayleigh distributions. We then develop a framework implementing this statistical analysis of sublevel set persistence for signals contaminated by an additive noise distribution to separate the sublevel sets associated to noise and signal. This method is computationally efficient, does not require any signal pre-filtering, is widely applicable, and has open-source software available. We demonstrate the functionality of the method with both numerically simulated examples and an experimental data set. Additionally, we provide an efficient $O(nlog(n))$ algorithm for calculating the zero-dimensional sublevel set persistence homology.