Modern Methods for Fractal and Multifractal Analysis of Time Series: Theoretical Frameworks and Practical Applications

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "Complexity".

Deadline for manuscript submissions: closed (30 April 2024) | Viewed by 2819

Special Issue Editor


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Guest Editor
Applied Mathematics Department, Kharkiv National University of Radio Electronics, 61166 Kharkiv, Ukraine
Interests: deterministic chaotic systems; stochastic self-similar and multifractal processes; time series modeling and forecasting; fractal and multifractal analysis

Special Issue Information

Dear Colleagues,

Fractal analysis of time series holds significant importance and finds wide-ranging applications across various domains today due to its ability to uncover hidden patterns and complex structures inherent in temporal data. It provides a powerful framework for capturing the long-term memory and scaling properties of time series, making it invaluable in understanding the underlying dynamics of diverse systems.

The Special Issue aims to explore the cutting-edge techniques employed and advancements made in analyzing time series data using self-similar and multifractal approaches. It delves into both the theoretical underpinnings and real-world applications of these methods, highlighting their relevance and potential impact in various scientific and practical domains.

This Special Issue serves as an essential resource for researchers, practitioners, and students interested in leveraging advanced techniques to analyze time series data. By combining theoretical foundations with diverse practical applications, this volume seeks to advance the understanding and utilization of fractal methods across various disciplines.

 The scope of this Special Issue includes, but is not limited to, the following topics:

  • Theoretical concepts of fractal analysis applied to time series.
  • Innovative methodologies and algorithms for evaluating self-similarity in time series.
  • Time series multifractal analysis techniques.
  • AI-enabled analysis of time series with fractal structure: forecasting, anomaly detection, clustering, classification, and others.
  • AI-driven fractal analysis: exploring the synergy of artificial intelligence and fractal investigations.
  • Applications: economics and finance; biomedical signal processing and enhanced medical diagnostics; study of anomalous diffusion; network traffic analysis; environmental and climate science; optimizing industrial processes; interdisciplinary research.

Prof. Dr. Lyudmyla Kirichenko
Guest Editor

Manuscript Submission Information

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Keywords

  • fractal/multifractal analysis
  • self-similarity
  • time series
  • long-term memory
  • scaling properties
  • artificial intelligence
  • applications

Published Papers (4 papers)

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Research

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22 pages, 8943 KiB  
Article
Modified MF-DFA Model Based on LSSVM Fitting
by Minzhen Wang, Caiming Zhong, Keyu Yue, Yu Zheng, Wenjing Jiang and Jian Wang
Fractal Fract. 2024, 8(6), 320; https://doi.org/10.3390/fractalfract8060320 - 28 May 2024
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Abstract
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 [...] Read more.
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. Full article
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26 pages, 395 KiB  
Article
Asymptotic Growth of Sample Paths of Tempered Fractional Brownian Motions, with Statistical Applications to Vasicek-Type Models
by Yuliya Mishura and Kostiantyn Ralchenko
Fractal Fract. 2024, 8(2), 79; https://doi.org/10.3390/fractalfract8020079 - 25 Jan 2024
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Abstract
Tempered fractional Brownian motion (TFBM) and tempered fractional Brownian motion of the second kind (TFBMII) modify the power-law kernel in the moving average representation of fractional Brownian motion by introducing exponential tempering. We construct least-square estimators for the unknown drift parameters within Vasicek [...] Read more.
Tempered fractional Brownian motion (TFBM) and tempered fractional Brownian motion of the second kind (TFBMII) modify the power-law kernel in the moving average representation of fractional Brownian motion by introducing exponential tempering. We construct least-square estimators for the unknown drift parameters within Vasicek models that are driven by these processes. To demonstrate their strong consistency, we establish asymptotic bounds with probability 1 for the rate of growth of trajectories of tempered fractional processes. Full article
22 pages, 1188 KiB  
Article
Multi-Signal Multifractal Detrended Fluctuation Analysis for Uncertain Systems —Application to the Energy Consumption of Software Programs in Microcontrollers
by Juan Carlos de la Torre, Pablo Pavón-Domínguez, Bernabé Dorronsoro, Pedro L. Galindo and Patricia Ruiz
Fractal Fract. 2023, 7(11), 794; https://doi.org/10.3390/fractalfract7110794 - 30 Oct 2023
Cited by 1 | Viewed by 1047
Abstract
Uncertain systems are those wherein some variability is observed, meaning that different observations of the system will produce different measurements. Studying such systems demands the use of statistical methods over multiple measurements, which allows overcoming the uncertainty, based on the premise that a [...] Read more.
Uncertain systems are those wherein some variability is observed, meaning that different observations of the system will produce different measurements. Studying such systems demands the use of statistical methods over multiple measurements, which allows overcoming the uncertainty, based on the premise that a single measurement is not representative of the system’s behavior. In such cases, the current multifractal detrended fluctuation analysis (MFDFA) method cannot offer confident conclusions. This work presents multi-signal MFDFA (MS-MFDFA), a novel methodology for accurately characterizing uncertain systems using the MFDFA algorithm, which enables overcoming the uncertainty of the system by simultaneously considering a large set of signals. As a case study, we consider the problem of characterizing software (Sw) consumption. The difficulty of the problem mainly comes from the complexity of the interactions between Sw and hardware (Hw), as well as from the high uncertainty level of the consumption measurements, which are affected by concurrent Sw services, the Hw, and external factors such as ambient temperature. We apply MS-MFDFA to generate a signature of the Sw consumption profile, regardless of the execution time, the consumption levels, and uncertainty. Multiple consumption signals (or time series) are built from different Sw runs, obtaining a high frequency sampling of the instant input current for each of them while running the Sw. A benchmark of eight Sw programs for analysis is also proposed. Moreover, a fully functional application to automatically perform MS-MFDFA analysis has been made freely available. The results showed that the proposed methodology is a suitable approximation for the multifractal analysis of a large number of time series obtained from uncertain systems. Moreover, analysis of the multifractal properties showed that this approach was able to differentiate between the eight Sw programs studied, showing differences in the temporal scaling range where multifractal behavior is found. Full article
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Review

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22 pages, 1823 KiB  
Review
Multifractal Properties of Human Chromosome Sequences
by J. P. Correia, R. Silva, D. H. A. L. Anselmo, M. S. Vasconcelos and L. R. da Silva
Fractal Fract. 2024, 8(6), 312; https://doi.org/10.3390/fractalfract8060312 - 24 May 2024
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Abstract
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 [...] Read more.
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. Full article
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