A multifractal formalism for new general fractal measures

Achour, Rim; Li, Zhiming; Selmi, Bilel; Wang, Tingting

doi:10.1016/j.chaos.2024.114655

Cited by 15 publications

(2 citation statements)

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“…Fractal theory provides unique tools and methodologies that aid in understanding and analyzing the complex phenomena and structures within these fields. Subsequently, in order to obtain a more detailed description of a fractal, it is necessary to add parameters that can describe different fractal subsets, so multifractal theory should be introduced, which is a research hotspot [4][5][6][7].…”

Section: Introductionmentioning

confidence: 99%

Modified MF-DFA Model Based on LSSVM Fitting

Wang,

Zhong,

Yue

et al. 2024

Fractal Fract

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This paper proposes a multifractal least squares support vector machine detrended fluctuation analysis (MF-LSSVM-DFA) model. The system is an extension of the traditional MF-DFA model. To address potential overfitting or underfitting caused by the fixed-order polynomial fitting in MF-DFA, LSSVM is employed as a superior alternative for fitting. This approach enhances model accuracy and adaptability, ensuring more reliable analysis results. We utilize the p model to construct a multiplicative cascade time series to evaluate the performance of MF-LSSVM-DFA, MF-DFA, and two other models that improve upon MF-DFA from recent studies. The results demonstrate that our proposed modified model yields generalized Hurst exponents h(q) and scaling exponents τ(q) that align more closely with the analytical solutions, indicating superior correction effectiveness. In addition, we explore the sensitivity of MF-LSSVM-DFA to the overlapping window size s. We find that the sensitivity of our proposed model is less than that of MF-DFA. We find that when s exceeds the limited range of the traditional MF-DFA, h(q) and τ(q) are closer than those obtained in MF-DFA when s is in a limited range. Meanwhile, we analyze the performances of the fitting of the two models and the results imply that MF-LSSVM-DFA achieves a better outstanding performance. In addition, we put the proposed MF-LSSVM-DFA into practice for applications in the medical field, and we found that MF-LSSVM-DFA improves the accuracy of ECG signal classification and the stability and robustness of the algorithm compared with MF-DFA. Finally, numerous image segmentation experiments are adopted to verify the effectiveness and robustness of our proposed method.

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Section: Introductionmentioning

confidence: 99%

Modified MF-DFA Model Based on LSSVM Fitting

Wang,

Zhong,

Yue

et al. 2024

Fractal Fract

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“…This work popularized the term "multifractal analysis" among mathematicians and physicists. More specifically, from a mathematical standpoint, multifractal analysis refers to a technique used to precisely characterize the heterogeneity in the distribution of a set of measurements in a metric space at small scales [15][16][17][18]. This analysis aims to express the dimension of the set of measurements in terms of the Legendre transformation of a free energy function, drawing an analogy with thermodynamic theory.…”

Section: Introductionmentioning

confidence: 99%

Multifractal Properties of Human Chromosome Sequences

Correia,

Silva,

Anselmo

et al. 2024

Fractal Fract

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The intricacy and fractal properties of human DNA sequences are examined in this work. The core of this study is to discern whether complete DNA sequences present distinct complexity and fractal attributes compared with sequences containing exclusively exon regions. In this regard, the entire base pair sequences of DNA are extracted from the NCBI (National Center for Biotechnology Information) database. In order to create a time series representation for the base pair sequence {G,C,T,A}, we use the Chaos Game Representation (CGR) approach and a mapping rule f, which enables us to apply the metric known as the Complexity–Entropy Plane (CEP) and multifractal detrended fluctuation analysis (MF-DFA). To carry out our investigation, we divided human DNA into two groups: the first is composed of the 24 chromosomes, which comprises all the base pairs that form the DNA sequence, and another group that also includes the 24 chromosomes, but the DNA sequences rely only on the exons’ presence. The results show that both sets provide fractal patterns in their structure, as obtained by the CGR approach. Complete DNA sequences show a sharper visual fractal pattern than sequences composed only of exons. Moreover, the sequences occupy distinct areas of the complexity–entropy plane, and the complete DNA sequences lead to greater statistical complexity and lower entropy than the exon sequences. Also, we observed that different fractal parameters between chromosomes indicate diversity in genomic sequences. All these results occur in different scales for all chromosomes.

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