  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.
