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A topological multiple testing scheme for one-dimensional domains is proposedwhere, rather than testing every spatial or temporal location for the presenceof a signal, tests are performed only at the local maxima of the smoothedobserved sequence. Assuming unimodal true peaks with finite support andGaussian stationary ergodic noise, it is shown that the algorithm withBonferroni or Benjamini--Hochberg correction provides asymptotic strong controlof the family wise error rate and false discovery rate, and is powerconsistent, as the search space and the signal strength get large, where thesearch 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 andbandwidth to the signal peaks, akin to the matched filter theorem in signalprocessing. The methods are illustrated in an analysis of electrical recordingsof neuronal cell activity.

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