Homology, Homotopy and Applications

Volume 16 (2014)

Number 1

Describing high-order statistical dependence using “concurrence topology,” with application to functional MRI brain data

Pages: 245 – 264

DOI: https://dx.doi.org/10.4310/HHA.2014.v16.n1.a14


Steven P. Ellis (New York State Psychiatric Institute, Columbia University, New York, N.Y., U.S.A.)

Arno Klein (Sage Bionetworks, Seattle, Washington, U.S.A.)


In multivariate data analysis dependence beyond pair-wise can be important. With many variables, however, the number of simple summaries of even third-order dependence can be unmanageably large.

“Concurrence topology” is an apparently new method for describing high-order dependence among up to dozens of dichotomous (i.e., binary) variables (e.g., seventh-order dependence in 32 variables). This method generally produces summaries of dependence of manageable size. (But computing time can be lengthy.) For time series, this method can be applied in both the time and Fourier domains.

Write each observation as a vector of 0’s and 1’s. A “concurrence” is a group of variables all labeled “1” in the same observation. The collection of concurrences can be represented as a filtered simplicial complex. Holes in the filtration indicate relatively weak or negative association among the variables. The pattern of the holes in the filtration can be analyzed using persistent homology.

We applied concurrence topology on binarized, resting-state, functional MRI data acquired from patients diagnosed with attention-deficit hyperactivity disorder and from healthy controls. An exploratory analysis finds a number of differences between patients and controls in the topologies of their filtrations, demonstrating that concurrence topology can find in data high-order structure of real-world relevance.


dichotomous data, high-order dependence, Fourier analysis of time series, computational homology, persistent homology, fMRI, ADHD

2010 Mathematics Subject Classification

62H17, 62M15, 92C55

Published 2 June 2014