Wavelet Based Cross Correlations with Applications

TL;DR

This paper explores wavelet-based methods for analyzing how correlations between signals vary across different frequency and time scales, offering a more detailed understanding of their relationships.

Contribution

It extends the theory of wavelet-based correlations, introduces new correlation measures, and compares different wavelet transforms and bases through simulations and real-world applications.

Findings

Wavelet correlograms reveal scale-dependent correlation patterns.

Wavelet correlation measures are robust across different wavelet bases.

Applications demonstrate practical utility in real datasets.

Abstract

Wavelet Transforms are a widely used technique for decomposing a signal into coefficient vectors that correspond to distinct frequency/scale bands while retaining time localization. This property enables an adaptive analysis of signals at different scales, capturing both temporal and spectral patterns. By examining how correlations between two signals vary across these scales, we obtain a more nuanced understanding of their relationship than what is possible from a single global correlation measure. In this work, we expand on the theory of wavelet-based correlations already used in the literature and elaborate on wavelet correlograms, partial wavelet correlations, and additive wavelet correlations using the Pearson and Kendall definitions. We use both Orthogonal and Non-decimated discrete Wavelet Transforms, and assess the robustness of these correlations under different wavelet bases. Simulation studies are conducted to illustrate these methods, and we conclude with applications to real-world datasets.