Asymptotic analysis of synchrosqueezing transform -- toward statistical inference with nonlinear-type time-frequency analysis

TL;DR

This paper offers a statistical framework for the synchrosqueezing transform (SST), enabling rigorous inference in nonlinear time-frequency analysis by deriving distributional properties and a bootstrap testing method.

Contribution

It provides the first comprehensive statistical analysis of SST, including distributional results and a bootstrap scheme for testing oscillatory components in time series.

Findings

Derived the quotient distribution of complex Gaussian variables in SST.

Established a central limit theorem for SST.

Proposed a bootstrap test for oscillatory components.

Abstract

We provide a statistical analysis of a tool in nonlinear-type time-frequency analysis, the synchrosqueezing transform (SST), for both the null and non-null cases. The intricate nonlinear interaction of different quantities in SST is quantified by carefully analyzing relevant multivariate complex Gaussian random variables. Specifically, we provide the quotient distribution of dependent and improper complex Gaussian random variables. Then, a central limit theorem result for SST is established. {As an example}, we provide a block bootstrap scheme based on the established SST theory to test if a given time series contains oscillatory components.