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

Hassan Abdallah; Adam Regalski; Mohammad Behzad Kang; Maria Berishaj; Nkechi Nnadi; Asadur Chowdury; Vaibhav A. Diwadkar; Andrew Salch

doi:10.3934/fods.2022014

Article Contents

2023, Volume 5, Issue 1: 1-25. Doi: 10.3934/fods.2022014

This issue Previous Article Next Article Persistent path Laplacian

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

1.
Department of Mathematics, Wayne State University, MI 48202, USA
2.
Dept. of Psychiatry & Behavioral Neuroscience, Wayne State University, MI 48202, USA
3.
Department of Mathematics, Northwestern University, IL 60208, USA

^* Corresponding author: Hassan Abdallah
^* Corresponding author: Hassan Abdallah

Received: January 2022

Revised: May 2022

Early access: July 2022

Published: March 2023

This work was supported by the National Institutes of Mental Health (MH111177), the Mark Cohen Neuroscience Endowment, the Jack Dorsey Endowment, and the Lycaki-Young Funds from the State of Michigan.

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Abstract

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 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 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.

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Mathematics Subject Classification: 62R40, 55N31.

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Figure 1. On the left is a plot of the point cloud with balls of radius $ \frac{1}{3} $ around each point. On the right is a visualization of the simplicial complex for filtration = $ \frac{1}{3} $

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Figure 2. The persistence diagram computed from the point cloud in Figure 1

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Figure 3. This shows the amplitude of a voxel outside of the embedded sphere that does not respond to the experimental task and has been simulated with physiological noise. Simulated with effect size = 5

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Figure 4. This shows the amplitude of a voxel within the embedded sphere that does respond to the periodic experimental task. Simulated with effect size = 5

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Figure 5. Simulation Volumes: Spheres of various sizes embedded in a mask of the right hippocampus (lateral view)

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Figure 6. Hippocampus mask overlaid onto a brain image

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Figure 7. On the left is a persistence diagram from a "rest" epoch of our simulation and on the right is a persistence diagram from an "activation" epoch of our simulation. This is for effect size = 5, sphere radius = 5, and SNR = 2

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Figure 8. Empirical power estimates by radius of embedded sphere

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Figure 9. Empirical power estimates by minimum persistence threshold of homological features

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Figure 10. Empirical power estimates by effect size for embedded sphere's response to task

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Figure 11. Empirical power estimates by SNR of simulated fMRI data

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