Spatially relaxed inference on high-dimensional linear models
This addresses the challenge of statistical inference in high-dimensional, spatially structured data like imaging, where traditional methods fail due to high correlation and dimensionality, offering a novel approach for researchers in statistics and applied fields.
The paper tackles the inference problem in high-dimensional linear models with spatially correlated covariates, such as in imaging, by proposing ensembled clustered inference algorithms that control the δ-FWER, a measure allowing false discoveries beyond a spatial distance δ, and demonstrates accurate control and decent power empirically.
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 $δ$-FWER, that is the probability of making a false discovery at a distance greater than $δ$ 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 $δ$-FWER under standard assumptions for $δ$ equal to the largest cluster diameter. We complement the theoretical analysis with empirical results, demonstrating accurate $δ$-FWER control and decent power achieved by such inference algorithms.