Modeling and replicating statistical topology and evidence for CMB nonhomogeneity.

Under the banner of "big data," the detection and classification of structure in extremely large, high-dimensional, data sets are two of the central statistical challenges of our times. Among the most intriguing new approaches to this challenge is "TDA," or "topological data analysis," one of the primary aims of which is providing nonmetric, but topologically informative, preanalyses of data which make later, more quantitative, analyses feasible. While TDA rests on strong mathematical foundations from topology, in applications, it has faced challenges due to difficulties in handling issues of statistical reliability and robustness, often leading to an inability to make scientific claims with verifiable levels of statistical confidence.

MULTIPLE TESTING OF LOCAL MAXIMA FOR DETECTION OF PEAKS IN 1D.

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.