The shifted wavelet $ (q, q') $-entropy and the classification of stationary fractal signals

Julio César Ramírez Pacheco; Joel Antonio Trejo-Sánchez; Luis Rizo-Domínguez; Julio César Ramírez Pacheco; Joel Antonio Trejo-Sánchez; Luis Rizo-Domínguez

doi:10.3934/nhm.2025006

Networks and Heterogeneous Media

2025, Volume 20, Issue 1: 89-103. doi: 10.3934/nhm.2025006

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The shifted wavelet $ (q, q') $-entropy and the classification of stationary fractal signals

1.
Cancun Multidisciplinary Sciences Division, Autonomous University of Quintana Roo, UQRoo, 77519, Cancun, Quintana Roo, Mexico
2.
SECIHTI - Center for Research in Mathematics
3.
Department of Electronics, Systems and Informatics, ITESO, Western Institute of Technology and Higher Education, Guadalajara, Mexico

Received: 08 October 2024 Revised: 23 December 2024 Accepted: 25 December 2024 Published: 22 January 2025

In this article, a wavelet entropy, which behaves as a shifted version of the standard wavelet $ (q, q') $-entropy of fractal signals, is presented. The shifted wavelet $ (q, q') $-entropy is obtained by computing the standard $ (q, q') $-entropy functional on a weighted relative-wavelet-energy (RWE) representation of fractal signals; it is shown that the weight within the RWE plays the role of a shifting factor in the characteristics of the standard wavelet $ (q, q') $-entropy. Therefore the shifted wavelet $ (q, q') $-entropy relocates the wavelet entropy values to any point of the fractality index range, which allows us to analyze a wide variety of fractal signal families thus improving on previously proposed entropies in the literature. Information planes for these entropies are obtained using different shifts and values of parameters $ q $ and $ q' $, which allow us to highlight the potential applications for a fractal signal analysis. Moreover, an experimental study using synthesized exact fractal signals shows that the shifted wavelet entropy can classify stationary long-memory signals from short-memory ones and can also be used to differentiate other fractal signal families.
- fractals,
- fractal analysis,
- entropy,
- wavelet entropy,
- fractal signal classification
Citation: Julio César Ramírez Pacheco, Joel Antonio Trejo-Sánchez, Luis Rizo-Domínguez. The shifted wavelet $ (q, q') $-entropy and the classification of stationary fractal signals[J]. Networks and Heterogeneous Media, 2025, 20(1): 89-103. doi: 10.3934/nhm.2025006

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Abstract

In this article, a wavelet entropy, which behaves as a shifted version of the standard wavelet $ (q, q') $-entropy of fractal signals, is presented. The shifted wavelet $ (q, q') $-entropy is obtained by computing the standard $ (q, q') $-entropy functional on a weighted relative-wavelet-energy (RWE) representation of fractal signals; it is shown that the weight within the RWE plays the role of a shifting factor in the characteristics of the standard wavelet $ (q, q') $-entropy. Therefore the shifted wavelet $ (q, q') $-entropy relocates the wavelet entropy values to any point of the fractality index range, which allows us to analyze a wide variety of fractal signal families thus improving on previously proposed entropies in the literature. Information planes for these entropies are obtained using different shifts and values of parameters $ q $ and $ q' $, which allow us to highlight the potential applications for a fractal signal analysis. Moreover, an experimental study using synthesized exact fractal signals shows that the shifted wavelet entropy can classify stationary long-memory signals from short-memory ones and can also be used to differentiate other fractal signal families.

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