doi: 10.3934/fods.2022014
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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

Received  January 2022 Revised  May 2022 Early access July 2022

Fund Project: 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

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.

Citation: Hassan Abdallah, Adam Regalski, Mohammad Behzad Kang, Maria Berishaj, Nkechi Nnadi, Asadur Chowdury, Vaibhav A. Diwadkar, Andrew Salch. Statistical inference for persistent homology applied to simulated fMRI time series data. Foundations of Data Science, doi: 10.3934/fods.2022014
References:
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P. BubenikM. HullD. Patel and B. Whittle, Persistent homology detects curvature, Inverse Problems, 36 (2020), 025008.  doi: 10.1088/1361-6420/ab4ac0.

[4]

C. CericolaI. JohnsonJ. KiersM. KrockJ. Purdy and J. Torrence, Extending hypothesis testing with persistent homology to three or more groups, Involve, 11 (2018), 27-51.  doi: 10.2140/involve.2018.11.27.

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F. ChazalM. GlisseC. Labruère and B. Michel, Convergence rates for persistence diagram estimation in topological data analysis, Journal of Machine Learning Research, 16 (2015), 3603-3635. 

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F. Chazal and B. Michel, An introduction to topological data analysis: Fundamental and practial aspects for data scientists, Frontiers in Artificial Intelligence, 29 (2021). doi: 10.3389/frai.2021.667963.

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K. J. FristonK. H. PrellerC. MathysH. CagnanJ. HeinzleA. Razi and P. Zeidman, Dynamic causal modelling revisited, NeuroImage, 199 (2019), 730-744.  doi: 10.1016/j.neuroimage.2017.02.045.

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G. Lohmann, J. Stelzer, E. Lacosse, V. J. Kumar, K. Mueller, E. Kuehn, W. Grodd and K. Scheffler, LISA improves statistical analysis for fMRI, Nature Communications, 9 (2018). doi: 10.1038/s41467-018-06304-z.

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B. J. MacintoshR. MrazW. E. McIlroy and S. J. Graham, Brain activity during a motor learning task: An fMRI and skin conductance study, Human Brain Mapping, 28 (2007), 1359-1367.  doi: 10.1002/hbm.20351.

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C. OballeA. CherneD. BootheS. KerickP. J. Franaszczuk and V. Maroulas, Bayesian topological signal processing, Discrete Contin. Dyn. Syst. Ser. S, 15 (2022), 797-817.  doi: 10.3934/dcdss.2021084.

[19]

N. OtterM. A. PorterU. TillmannP. Grindrod and H. A. Harrington, A roadmap for the computation of persistent homology, EPJ Data Science, 6 (2017), 1-38.  doi: 10.1140/epjds/s13688-017-0109-5.

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J. A Perea, A brief history of persistence, Morfismos, 23 (2019), 1-16. 

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J. A. Perea, Topological times series analysis, Notices Amer. Math. Soc., 66 (2019), 686-694. 

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J. A. Perea and J. Harer, Sliding windows and persistence: An application of topological methods to signal analysis, Found. Comput. Math., 15 (2015), 799-838.  doi: 10.1007/s10208-014-9206-z.

[23]

N. Ravishanker and R. Chen, An introduction to persistent homology for time series, Wiley Interdiscip. Rev. Comput. Stat., 13 (2021), Paper No. e1548, 25 pp. doi: 10.1002/wics.1548.

[24]

A. Robinson and K. Turner, Hypothesis testing for topological data analysis, Journal of Applied and Computational Topology, 1 (2017), 241-261.  doi: 10.1007/s41468-017-0008-7.

[25]

A. Salch, A. Regalski, H. Abdallah, R. Suryadevara, M. J. Catanzaro and V. A. Diwadkar, From mathematics to medicine: A practical primer on topological data analysis (TDA) and the development of related analytic tools for the functional discovery of latent structure in fMRI data, PLoS ONE, 16 (2021), e0255859. doi: 10.1371/journal.pone.0255859.

[26]

L. M. Seversky, S. Davis and M. Berger, On time-series topological data analysis: New data and opportunities, In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition Workshops, (2016), 59–67. doi: 10.1109/CVPRW.2016.131.

[27]

K. E. Stephan and A. Roebroeck, A short history of causal modeling of fMRI data, NeuroImage, 62 (2012), 856-863.  doi: 10.1016/j.neuroimage.2012.01.034.

[28]

L. Wasserman, Topological data analysis, Annu. Rev. Stat. Appl., 5 (2018), 501-535.  doi: 10.1146/annurev-statistics-031017-100045.

[29]

M. WelvaertJ. DurnezB. MoerkerkeG. Verdoolaege and Y. Rosseel, neuRosim: An R package for generating fMRI data, Journal of Statistical Software, 44 (2011), 1-18.  doi: 10.18637/jss.v044.i10.

[30]

M. Welvaert and Y. Rosseel, On the definition of signal-to-noise ratio and contrast-to-noise ratio for fmri data, PLoS ONE, 8 (2013), e77089, 11. doi: 10.1371/journal.pone.0077089.

[31]

A. M. WinklerM. A. WebsterD. VidaurreT. E. Nichols and S. M. Smith, Multi-level block permutation, NeuroImage, 123 (2015), 253-268.  doi: 10.1016/j.neuroimage.2015.05.092.

show all references

References:
[1]

S. Benzekry, J. A. Tuszynski, E. A. Rietman and G. L. Klement, Design principles for cancer therapy guided by changes in complexity of protein-protein interaction networks, Biology Direct, 10 (2015), Article number: 32. doi: 10.1186/s13062-015-0058-5.

[2]

S. L. Bressler and A. K. Seth, Wiener-granger causality: A well established methodology, NeuroImage, 58 (2011), 323-329.  doi: 10.1016/j.neuroimage.2010.02.059.

[3]

P. BubenikM. HullD. Patel and B. Whittle, Persistent homology detects curvature, Inverse Problems, 36 (2020), 025008.  doi: 10.1088/1361-6420/ab4ac0.

[4]

C. CericolaI. JohnsonJ. KiersM. KrockJ. Purdy and J. Torrence, Extending hypothesis testing with persistent homology to three or more groups, Involve, 11 (2018), 27-51.  doi: 10.2140/involve.2018.11.27.

[5]

F. ChazalM. GlisseC. Labruère and B. Michel, Convergence rates for persistence diagram estimation in topological data analysis, Journal of Machine Learning Research, 16 (2015), 3603-3635. 

[6]

F. Chazal and B. Michel, An introduction to topological data analysis: Fundamental and practial aspects for data scientists, Frontiers in Artificial Intelligence, 29 (2021). doi: 10.3389/frai.2021.667963.

[7] T. K. Dey and Y. Wang, Computational Topology for Data Analysis, Cambridge University Press, Cambridge, 2022.  doi: 10.1017/9781009099950.
[8]

H. Edelsbrunner and J. L. Harer, Computational Topology: An Introduction, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/mbk/069.

[9]

A. Eklund, T. Nichols and H. Knutsson, Can parametric statistical methods be trusted for fMRI based group studies?, arXiv preprint, arXiv: 1511.01863, 11 (2015).

[10]

K. J. FristonK. H. PrellerC. MathysH. CagnanJ. HeinzleA. Razi and P. Zeidman, Dynamic causal modelling revisited, NeuroImage, 199 (2019), 730-744.  doi: 10.1016/j.neuroimage.2017.02.045.

[11] A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002. 
[12]

H. HonariA. S. ChoeJ. J. Pekar and M. A. Lindquist, Investigating the impact of autocorrelation on time-varying connectivity, NeuroImage, 197 (2019), 37-48.  doi: 10.1016/j.neuroimage.2019.04.042.

[13]

H. W. Kuhn, The Hungarian method for the assignment problem, Naval Res. Logist. Quart., 2 (1955), 83-97.  doi: 10.1002/nav.3800020109.

[14]

P.-J. LahayeJ.-B. PolineG. FlandinS. Dodel and L. Garnero, Functional connectivity: Studying nonlinear, delayed interactions between BOLD signals, NeuroImage, 20 (2003), 962-974.  doi: 10.1016/S1053-8119(03)00340-9.

[15]

G. Lohmann, J. Stelzer, E. Lacosse, V. J. Kumar, K. Mueller, E. Kuehn, W. Grodd and K. Scheffler, LISA improves statistical analysis for fMRI, Nature Communications, 9 (2018). doi: 10.1038/s41467-018-06304-z.

[16]

B. J. MacintoshR. MrazW. E. McIlroy and S. J. Graham, Brain activity during a motor learning task: An fMRI and skin conductance study, Human Brain Mapping, 28 (2007), 1359-1367.  doi: 10.1002/hbm.20351.

[17]

Y. NirI. DinsteinR. Malach and D. J. Heeger, Bold and spiking activity, Nature Neuroscience, 11 (2008), 523-524.  doi: 10.1038/nn0508-523.

[18]

C. OballeA. CherneD. BootheS. KerickP. J. Franaszczuk and V. Maroulas, Bayesian topological signal processing, Discrete Contin. Dyn. Syst. Ser. S, 15 (2022), 797-817.  doi: 10.3934/dcdss.2021084.

[19]

N. OtterM. A. PorterU. TillmannP. Grindrod and H. A. Harrington, A roadmap for the computation of persistent homology, EPJ Data Science, 6 (2017), 1-38.  doi: 10.1140/epjds/s13688-017-0109-5.

[20]

J. A Perea, A brief history of persistence, Morfismos, 23 (2019), 1-16. 

[21]

J. A. Perea, Topological times series analysis, Notices Amer. Math. Soc., 66 (2019), 686-694. 

[22]

J. A. Perea and J. Harer, Sliding windows and persistence: An application of topological methods to signal analysis, Found. Comput. Math., 15 (2015), 799-838.  doi: 10.1007/s10208-014-9206-z.

[23]

N. Ravishanker and R. Chen, An introduction to persistent homology for time series, Wiley Interdiscip. Rev. Comput. Stat., 13 (2021), Paper No. e1548, 25 pp. doi: 10.1002/wics.1548.

[24]

A. Robinson and K. Turner, Hypothesis testing for topological data analysis, Journal of Applied and Computational Topology, 1 (2017), 241-261.  doi: 10.1007/s41468-017-0008-7.

[25]

A. Salch, A. Regalski, H. Abdallah, R. Suryadevara, M. J. Catanzaro and V. A. Diwadkar, From mathematics to medicine: A practical primer on topological data analysis (TDA) and the development of related analytic tools for the functional discovery of latent structure in fMRI data, PLoS ONE, 16 (2021), e0255859. doi: 10.1371/journal.pone.0255859.

[26]

L. M. Seversky, S. Davis and M. Berger, On time-series topological data analysis: New data and opportunities, In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition Workshops, (2016), 59–67. doi: 10.1109/CVPRW.2016.131.

[27]

K. E. Stephan and A. Roebroeck, A short history of causal modeling of fMRI data, NeuroImage, 62 (2012), 856-863.  doi: 10.1016/j.neuroimage.2012.01.034.

[28]

L. Wasserman, Topological data analysis, Annu. Rev. Stat. Appl., 5 (2018), 501-535.  doi: 10.1146/annurev-statistics-031017-100045.

[29]

M. WelvaertJ. DurnezB. MoerkerkeG. Verdoolaege and Y. Rosseel, neuRosim: An R package for generating fMRI data, Journal of Statistical Software, 44 (2011), 1-18.  doi: 10.18637/jss.v044.i10.

[30]

M. Welvaert and Y. Rosseel, On the definition of signal-to-noise ratio and contrast-to-noise ratio for fmri data, PLoS ONE, 8 (2013), e77089, 11. doi: 10.1371/journal.pone.0077089.

[31]

A. M. WinklerM. A. WebsterD. VidaurreT. E. Nichols and S. M. Smith, Multi-level block permutation, NeuroImage, 123 (2015), 253-268.  doi: 10.1016/j.neuroimage.2015.05.092.

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} $
Figure 2.  The persistence diagram computed from the point cloud in Figure 1
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
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
Figure 5.  Simulation Volumes: Spheres of various sizes embedded in a mask of the right hippocampus (lateral view)
Figure 6.  Hippocampus mask overlaid onto a brain image
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
Figure 8.  Empirical power estimates by radius of embedded sphere
Figure 9.  Empirical power estimates by minimum persistence threshold of homological features
Figure 10.  Empirical power estimates by effect size for embedded sphere's response to task
Figure 11.  Empirical power estimates by SNR of simulated fMRI data
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