Spatially relaxed inference on high-dimensional linear models

TL;DR

This paper introduces a spatially relaxed inference framework for high-dimensional linear models with spatially correlated covariates, enabling more accurate detection within a specified spatial uncertainty.

Contribution

It proposes ensembled clustered inference algorithms that control the $oldsymbol{ ext{δ-FWER}}$ in high-dimensional spatial models, accounting for spatial correlation and uncertainty.

Findings

Algorithms control δ-FWER under standard assumptions.

Empirical results show accurate δ-FWER control.

Decent power achieved in spatial inference tasks.

Abstract

We consider the inference problem for high-dimensional linear models, when covariates have an underlying spatial organization reflected in their correlation. A typical example of such a setting is high-resolution imaging, in which neighboring pixels are usually very similar. Accurate point and confidence intervals estimation is not possible in this context with many more covariates than samples, furthermore with high correlation between covariates. This calls for a reformulation of the statistical inference problem, that takes into account the underlying spatial structure: if covariates are locally correlated, it is acceptable to detect them up to a given spatial uncertainty. We thus propose to rely on the $\delta$-FWER, that is the probability of making a false discovery at a distance greater than $\delta$ from any true positive. With this target measure in mind, we study the properties of ensembled clustered inference algorithms which combine three techniques: spatially constrained clustering, statistical inference, and ensembling to aggregate several clustered inference solutions. We show that ensembled clustered inference algorithms control the $\delta$-FWER under standard assumptions for $\delta$ equal to the largest cluster diameter. We complement the theoretical analysis with empirical results, demonstrating accurate $\delta$-FWER control and decent power achieved by such inference algorithms.