Entropy and Extropy of  the Wavelet Energy Distribution in Reciprocal Fractal Systems

Pakgohar, Alireza; Shokri, Soheil

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Entropy and Extropy of the Wavelet Energy Distribution in Reciprocal Fractal Systems

Alireza Pakgohar ^*

, Soheil Shokri

Abstract: (19 Views)

This study investigates the wavelet energy distribution in high-frequency fractal systems and analyzes its characteristics using information-theoretic measures. The main innovation of this paper lies in modeling the wavelet energy distribution ($p_j$) using a truncated geometric distribution and incorporating the concept of extropy to quantify system complexity. It is demonstrated that this distribution is strongly influenced by the fractal parameter $alpha$ and the number of decomposition levels $M$. By computing wavelet entropy and extropy as measures of disorder and information, respectively—the study provides a quantitative analysis of the complexity of these systems. The paper further examines key properties of this distribution, including its convergence to geometric, uniform, and degenerate distributions under limiting conditions (e.g., $M to infty$ or $alpha to 0$). Results indicate that entropy and extropy serve as complementary tools for a comprehensive description of system behavior: while entropy measures disorder, extropy reflects the degree of information and certainty. This approach establishes a novel framework for analyzing real-world signals with varying parameters and holds potential applications in the analysis of fractal signals and modeling of complex systems in fields such as finance and biology.

To validate the theoretical findings, synthetic fractal signals (fractional Brownian motion) with varying fractal parameters ($alpha$) and decomposition levels ($M$) were simulated. Numerical results show that wavelet entropy increases significantly with the number of decomposition levels ($M$), whereas extropy exhibits slower growth and saturates at higher decomposition levels. These findings underscore the importance of selecting an appropriate decomposition level. The proposed combined framework offers a powerful tool for analyzing and modeling complex, non-stationary systems in domains such as finance and biology.

Keywords: Wavelet Energy, Wavelet Entropy, Wavelet Extropy, Truncated Geometric Distribution, Reciprocal Fractal Systems.

Full-Text [PDF 246 kb] (17 Downloads)

Type of Study: Research | Subject: Reliability
Received: 2025/01/19 | Accepted: 2026/09/1

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