A Scalable Conditional Independence Test for Nonlinear, Non-Gaussian Data

TL;DR

This paper introduces a scalable conditional independence test for nonlinear, non-Gaussian data that is computationally efficient and accurate, enabling analysis of large high-dimensional datasets in scientific research.

Contribution

It presents a new class of O(N^2) tests based on conditional correlation independence, improving computational efficiency over existing kernel-based methods while maintaining accuracy.

Findings

The proposed tests are significantly faster, requiring seconds instead of hours.

They achieve similar accuracy to existing methods on complex nonlinear, non-Gaussian data.

The method is practical for large-scale high-dimensional data analysis.

Abstract

Many relations of scientific interest are nonlinear, and even in linear systems distributions are often non-Gaussian, for example in fMRI BOLD data. A class of search procedures for causal relations in high dimensional data relies on sample derived conditional independence decisions. The most common applications rely on Gaussian tests that can be systematically erroneous in nonlinear non-Gaussian cases. Recent work (Gretton et al. (2009), Tillman et al. (2009), Zhang et al. (2011)) has proposed conditional independence tests using Reproducing Kernel Hilbert Spaces (RKHS). Among these, perhaps the most efficient has been KCI (Kernel Conditional Independence, Zhang et al. (2011)), with computational requirements that grow effectively at least as O(N3), placing it out of range of large sample size analysis, and restricting its applicability to high dimensional data sets. We propose a class of O(N2) tests using conditional correlation independence (CCI) that require a few seconds on a standard workstation for tests that require tens of minutes to hours for the KCI method, depending on degree of parallelization, with similar accuracy. For accuracy on difficult nonlinear, non-Gaussian data sets, we also compare a recent test due to Harris & Drton (2012), applicable to nonlinear, non-Gaussian distributions in the Gaussian copula, as well as to partial correlation, a linear Gaussian test.