A wavelet entropy-based concentration index for the analysis of multiscale interactions

This paper introduces a novel entropy-based measure for the analysis of multiscale interactions between time series in the wavelet domain. Starting from the cross-wavelet energy decomposition, we construct a normalized scale-dependent distribution that describes how the shared energy between two signals is allocated across scales over time. Based on this representation, we define the wavelet entropy-based concentration index, which quantifies the degree of localization of cross-wavelet energy and provides a global measure of multiscale structural organization. We investigate the main mathematical properties of the proposed index, including its bounds and interpretation in terms of concentration versus dispersion across scales. We further introduce a joint representation linking the scale-specific energy distribution with the entropy-based index, allowing for a unified characterization of multiscale interaction patterns. The methodology is validated using simulated signals with different dependence structures and applied to WTI and Brent crude oil prices. The empirical results reveal a structured broadband interaction with a dominant short-run component and time-varying organization. Overall, the proposed framework provides an effective tool for the analysis of complex multiscale dependence in time series.