Spatial Confidence Regions for Piecewise Continuous Processes

TL;DR

This paper develops a new method for constructing confidence regions for excursion sets of piecewise continuous processes, broadening applicability beyond differentiable processes in fields like neuroimaging and climatology.

Contribution

It introduces a novel convergence concept for piecewise continuous functions, enabling confidence region construction without requiring differentiability.

Findings

Allows confidence regions for non-differentiable processes

Generalizes continuous mapping theorem for piecewise functions

Enables analysis of symmetric differences of excursion sets

Abstract

In scientific disciplines such as neuroimaging, climatology, and cosmology it is useful to study the uncertainty of excursion sets of imaging data. While the case of imaging data obtained from a single study condition has already been intensively studied, confidence statements about the intersection, or union, of the excursion sets derived from different subject conditions have only been introduced recently. Such methods aim to model the images from different study conditions as asymptotically Gaussian random processes with differentiable sample paths. In this work, we remove the restricting condition of differentiability and only require continuity of the sample paths. This allows for a wider range of applications including many settings which cannot be treated with the existing theory. To achieve this, we introduce a novel notion of convergence on piecewise continuous functions over finite partitions. This notion is of interest in its own right, as it implies convergence results for maxima of sequences of piecewise continuous functions over sequences of sets. Generalizing well-known results such as the extended continuous mapping theorem, this novel convergence notion also allows us to construct for the first time confidence regions for mathematically challenging examples such as symmetric differences of excursion sets.