Statistical inference for persistent homology applied to simulated fMRI time series data

Abstract

<p style='text-indent:20px;'>Time-series data are amongst the most widely-used 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 data-analytic tool in TDA, and can effectively summarize complex high-dimensional data into an interpretable 2-dimensional representation called a <i>persistence diagram</i>. 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 time-series, time-series data often fail to satisfy the independence assumption. This paper develops a statistical test that obviates the independence assumption by implementing a multi-level block sampled Monte Carlo test with sets of persistence diagrams. Its efficacy for detecting task-dependent 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 non-independent data in general.</p>

Unified topological inference for brain networks in temporal lobe epilepsy using the Wasserstein distance

“…However, its application in statistical inference for brain network studies has been limited, largely due to computational constraints and scalability issues. Notable exceptions exist in the literature ( Abdallah et al, 2023 ; Kumar et al, 2023 ; Robinson and Turner, 2017 ; Salch et al, 2021 ; Songdechakraiwut et al, 2021 ). Instead, researchers have turned to the vectorization of persistence diagrams as a more practical and efficient alternative for statistical inference.…”