Assessing time series predictability by extending wavelet energy-based entropy and divergence measures using the Sharma–Mittal framework

In this article, we introduce and analyze an extension of existing wavelet energy-based measures for assessing the predictability of a time series: the Sharma–Mittal wavelet energy entropy measure (WSEEM) and the Sharma–Mittal wavelet energy divergence measure (WSEDM). Unlike classical entropy-based measures, such as Shannon’s, which do not have adjustable parameters, and extensions such as those of Rényi and Tsallis, which incorporate a single parameter to modulate the emphasis on high- or low-frequency components, the Sharma–Mittal measure stands out for its greater flexibility. It incorporates two parameters: the first, similar to those of Rényi and Tsallis entropies, controls the focus on different frequency ranges, while the second governs the degree of additivity of entropy for independent systems, i.e., it determines how energy contributions are aggregated. In this work, we study and prove the mathematical properties of the proposed WSEEM and WSEDM, and evaluate their effectiveness on real-world data. The results show that the dual-parameter formulation provides a richer and more adaptable description of energy distribution compared to traditional approaches. • Wavelet analysis enables decomposition of time series in the time–frequency domain. • Wavelet energy-based measures can assess the predictability of time series. • Dual-parameter Sharma–Mittal framework enhances flexibility in entropy-based analysis. • Adjustable sensitivity to frequency components and entropy additivity.