Discovering focal regions of slightly-aggregated sparse signals

Abstract

The characteristic aspects of dynamic distortions on a lengthy time series of i.i.d. pure noise when embedded with slightly-aggregating sparse signals are summarized into a significantly shorter recurrence time process of a chosen extreme event. We first employ the Kolmogorov---Smirnov statistic to compare the empirical recurrence time distribution with the null geometry distribution when no signal being present in the original time series. The power of such a hypothesis testing depends on varying degrees of aggregation of sparse signals: from a completely random distribution of singletons to batches of various sizes on the entire temporal span. We demonstrate the Kolmogorov---Smirnov statistic capturing the dynamic distortions due to slightly-aggregating sparse signals better than does Tukey's Higher Criticism statistic even when the batch size is as small as five. Secondly, after confirming the presence of signals in the pure noise time series, we apply the hierarchical factor segmentation (HFS) algorithm again based on the recurrence time process to compute focal segments that contain a significantly higher intensity of signals than do the rest of the temporal regions. In a computer experiment with a given fixed number of signals, the focal segments identified by the HFS algorithm afford many folds of signal intensity which also critically depend on the degree of aggregation of sparse signals. This ratio information can facilitate better sensitivity, equivalent to a smaller false discovery rate, if the signal-discovering protocol implemented within the computed focal regions is different from that used outside of the focal regions. We also numerically compute the specificity as the total number of signals contained in the computed collection of focal regions, which indicates the inherent difficulty in the task of sparse signal discovery.