Describing High-Order Statistical Dependence Using 'Concurrence Topology,' with Application to Functional MRI Brain Data

• Activity Number: 410
• Type: Topic Contributed
• Date/Time: Tuesday, August 5, 2014 : 2:00 PM to 3:50 PM
• Sponsor: IMS
• Author(s): Steven Ellis*+ and Arno Klein
• Companies: NYSPI at Columbia University and Sage Bionetworks
• Keywords: dichotomous data ; high-order dependence ; Fourier analysis of time series ; computational homology ; fMRI ; ADHD

Abstract

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

"Concurrence topology (CT)" is an apparently new method for describing high-order dependence among up to dozens of binary variables (e.g., seventh-order dependence in 32 variables). This method generally produces summaries of dependence of manageable size. For time series, CT 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 shape.

Holes in the shape indicate relatively weak or negative association among the variables. The pattern of the holes in the shape can be analyzed using computational topology.

As a demonstration, CT is used to investigate brain functional connectivity based on dichotomized functional MRI data. The data set includes subjects diagnosed with ADHD and healthy controls. An exploratory analysis finds a number of differences between ADHD subjects and controls in their data topology.