False-alarm probability in relation to over-sampled power spectra, with application to Super-Kamiokande solar neutrino data

TL;DR

This paper revises false-alarm probability calculations for over-sampled power spectra, providing more accurate estimates and applying them to analyze Super-Kamiokande solar neutrino data.

Contribution

It introduces a more conservative false-alarm probability formula for over-sampled spectra and integrates it with Bayesian methods for hypothesis testing.

Findings

New formula for false-alarm probability in over-sampled spectra

Application to Super-Kamiokande neutrino data demonstrates method effectiveness

Improved false-alarm estimates reduce false detections in spectral analysis

Abstract

The term "false-alarm probability" denotes the probability that at least one out of M independent power values in a prescribed search band of a power spectrum computed from a white-noise time series is expected to be as large as or larger than a given value. The usual formula is based on the assumption that powers are distributed exponentially, as one expects for power measurements of normally distributed random noise. However, in practice one typically examines peaks in an over-sampled power spectrum. It is therefore more appropriate to compare the strength of a particular peak with the distribution of peaks in over-sampled power spectra derived from normally distributed random noise. We show that this leads to a formula for the false-alarm probability that is more conservative than the familiar formula. We also show how to combine these results with a Bayesian method for estimating the probability of the null hypothesis (that there is no oscillation in the time series), and we discuss as an example the application of these procedures to Super-Kamiokande solar neutrino data.