Debiasing Welch's Method for Spectral Density Estimation

Abstract

Welch's method provides an estimator of the power spectral density that is statistically consistent. This is achieved by averaging over periodograms calculated from overlapping segments of a time series. For a finite length time series, while the variance of the estimator decreases as the number of segments increase, the magnitude of the estimator's bias increases: a bias-variance trade-off ensues when setting the segment number. We address this issue by providing a a novel method for debiasing Welch's method which maintains the computational complexity and asymptotic consistency, and leads to improved finite-sample performance. Theoretical results are given for fourth-order stationary processes with finite fourth-order moments and absolutely continuous fourth-order cumulant spectrum. The significant bias reduction is demonstrated with numerical simulation and an application to real-world data, where several empirical metrics indicate our debiased estimator compares favourably to Welch's. Our estimator also permits irregular spacing over frequency and we demonstrate how this may be employed for signal compression and further variance reduction. Code accompanying this work is available in the R and python languages.