Inference for location and height of peaks of a standardized field after selection

TL;DR

This paper introduces a statistical method for detecting and accurately localizing peaks in noisy fields, providing valid confidence regions post-selection, with proven control over false discoveries.

Contribution

It develops a novel peak inference approach that combines significance testing with confidence region construction, supported by theoretical analysis using the Kac-Rice formula.

Findings

Asymptotic control of false discoveries and confidence region coverage.

A new approximation for the intensity of local maxima counts.

Method performs well under high-curvature asymptotic conditions.

Abstract

Peak inference concerns the use of local maxima ("peaks") of a noisy random field to detect and localize regions where underlying signal is present. We propose a peak inference method that first subjects observed peaks to a significance test of the null hypothesis that no signal is present, and then uses the peaks that are declared significant to construct post-selectively valid confidence regions for the location and height of nearby true peaks. We analyze the performance of this method in a smooth signal plus constant variance noise model under a high-curvature asymptotic assumption, and prove that it asymptotically controls both the number of false discoveries, and the number of confidence regions that do not contain a true peak, relative to the number of points at which inference is conducted. An important intermediate theoretical result uses the Kac-Rice formula to derive a novel approximation to the intensity function of a point process that counts local maxima, which is second-order accurate under the alternative, nearby high-curvature true peaks.