Multiple testing of local maxima for detection of peaks in 1D

TL;DR

This paper introduces a topological multiple testing method for 1D data that tests only at local maxima, controlling error rates asymptotically and maximizing power with optimal smoothing, demonstrated on neuronal data.

Contribution

It proposes a novel local maxima testing approach with error control and power consistency, applicable to 1D signals with Gaussian noise, and illustrates its effectiveness on neuronal recordings.

Findings

Asymptotic control of error rates achieved

Power maximized with matched smoothing kernel

Method effective in neuronal data analysis

Abstract

A topological multiple testing scheme for one-dimensional domains is proposed where, rather than testing every spatial or temporal location for the presence of a signal, tests are performed only at the local maxima of the smoothed observed sequence. Assuming unimodal true peaks with finite support and Gaussian stationary ergodic noise, it is shown that the algorithm with Bonferroni or Benjamini--Hochberg correction provides asymptotic strong control of the family wise error rate and false discovery rate, and is power consistent, as the search space and the signal strength get large, where the search space may grow exponentially faster than the signal strength. Simulations show that error levels are maintained for nonasymptotic conditions, and that power is maximized when the smoothing kernel is close in shape and bandwidth to the signal peaks, akin to the matched filter theorem in signal processing. The methods are illustrated in an analysis of electrical recordings of neuronal cell activity.