🍩 Database of Original & NonTheoretical Uses of Topology
(found 3 matches in 0.001s)


Statistical Topological Data Analysis  A Kernel Perspective (2015)
Roland Kwitt, Stefan Huber, Marc Niethammer, Weili Lin, Ulrich BauerAbstract
We consider the problem of statistical computations with persistence diagrams, a summary representation of topological features in data. These diagrams encode persistent homology, a widely used invariant in topological data analysis. While several avenues towards a statistical treatment of the diagrams have been explored recently, we follow an alternative route that is motivated by the success of methods based on the embedding of probability measures into reproducing kernel Hilbert spaces. In fact, a positive definite kernel on persistence diagrams has recently been proposed, connecting persistent homology to popular kernelbased learning techniques such as support vector machines. However, important properties of that kernel enabling a principled use in the context of probability measure embeddings remain to be explored. Our contribution is to close this gap by proving universality of a variant of the original kernel, and to demonstrate its effective use in twosample hypothesis testing on synthetic as well as realworld data. 
Statistical Inference for Persistent Homology Applied to Simulated fMRI Time Series Data (2023)
Hassan Abdallah, Adam Regalski, Mohammad Behzad Kang, Maria Berishaj, Nkechi Nnadi, Asadur Chowdury, Vaibhav A. Diwadkar, Andrew SalchAbstract
Timeseries data are amongst the most widelyused in biomedical sciences, including domains such as functional Magnetic Resonance Imaging (fMRI). Structure within time series data can be captured by the tools of topological data analysis (TDA). Persistent homology is the mostly commonly used dataanalytic tool in TDA, and can effectively summarize complex highdimensional data into an interpretable 2dimensional representation called a persistence diagram. Existing methods for statistical inference for persistent homology of data depend on an independence assumption being satisfied. While persistent homology can be computed for each time index in a timeseries, timeseries data often fail to satisfy the independence assumption. This paper develops a statistical test that obviates the independence assumption by implementing a multilevel block sampled Monte Carlo test with sets of persistence diagrams. Its efficacy for detecting taskdependent topological organization is then demonstrated on simulated fMRI data. This new statistical test is therefore suitable for analyzing persistent homology of fMRI data, and of nonindependent data in general.