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Fractal Fract
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Advanced Modeling and Methods of Statistical Processing of Stochastic Signals...
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Advanced Modeling and Methods of Statistical Processing of Stochastic Signals in Fractional Dynamic Systems

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

Deadline for manuscript submissions: 13 June 2025 | Viewed by 6459

Special Issue Editors

Prof. Dr. Serhii Lupenko

E-Mail Website
Guest Editor
1. Faculty of Electrical Engineering, Automatic Control and Informatics, Opole University of Technology, 45-758 Opole, Poland
2. Institute of Telecommunications and Global Information Space, National Academy of Sciences of Ukraine, 02000 Kyiv, Ukraine
Interests: random processes; signal processing; data science; artificial intelligence; fractional calculus; medical signal processing; fractional stochastic processes; fractional-order machine learning
Prof. Dr. Jacek Leśkow

E-Mail Website
Guest Editor
1. College of Computer Sciences, Cracow University of Technology, PL-31155 Craków, Poland
2. School of Digital Technologies, American University Kyiv, 03056 Kyiv, Ukraine
Interests: nonstationary signal statistical processing; cybersecurity; artificial intelligence

Special Issue Information

Dear Colleagues,

The aim of this Special Issue is to publish original research articles and critical reviews covering advances in the theory, applications, modeling and statistical processing of non-stationary stochastic signals in fractional, non-fractional and hybrid dynamic systems, which take place in different scientific domains, including telecommunications, cybersecurity, energy, economics, biology and medicine. Scientific works related, but not limited, to fractional modeling and the processing of non-stationary signals, including both a model-based approach and a data-based (data-driven, data-oriented, data-centric) approach, as well as a combination of these approaches, are invited.

Potential topics include, but are not limited to, the following:

General theory of random processes;

Fractional stochastic processes;

Cyclostationary random processes;

Cyclic random processes;

Fractional-order signal processing;

Transformation of stochastic signals in fractional and hybrid dynamic systems;

Model-based signal statistical processing techniques;

Fractional-order machine learning and deep learning techniques in signal processing;

Signal computer simulation techniques.

Prof. Dr. Serhii Lupenko
Prof. Dr. Jacek Leśkow
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Fractal and Fractional is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2700 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • fractional stochastic processes
  • cyclostationary random processes
  • fractional-order signal processing
  • fractional and hybrid dynamic systems
  • signal statistical processing techniques
  • fractional-order machine learning techniques in signal processing
  • signal computer simulation techniques

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Published Papers (4 papers)

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Research

15 pages, 1806 KiB  
Article
FLRNN-FGA: Fractional-Order Lipschitz Recurrent Neural Network with Frequency-Domain Gated Attention Mechanism for Time Series Forecasting
by Chunna Zhao, Junjie Ye, Zelong Zhu and Yaqun Huang
Fractal Fract. 2024, 8(7), 433; https://doi.org/10.3390/fractalfract8070433 - 22 Jul 2024
Cited by 1 | Viewed by 1000
Abstract
Time series forecasting has played an important role in different industries, including economics, energy, weather, and healthcare. RNN-based methods have shown promising potential due to their strong ability to model the interaction of time and variables. However, they are prone to gradient issues [...] Read more.
Time series forecasting has played an important role in different industries, including economics, energy, weather, and healthcare. RNN-based methods have shown promising potential due to their strong ability to model the interaction of time and variables. However, they are prone to gradient issues like gradient explosion and vanishing gradients. And the prediction accuracy is not high. To address the above issues, this paper proposes a Fractional-order Lipschitz Recurrent Neural Network with a Frequency-domain Gated Attention mechanism (FLRNN-FGA). There are three major components: the Fractional-order Lipschitz Recurrent Neural Network (FLRNN), frequency module, and gated attention mechanism. In the FLRNN, fractional-order integration is employed to describe the dynamic systems accurately. It can capture long-term dependencies and improve prediction accuracy. Lipschitz weight matrices are applied to alleviate the gradient issues. In the frequency module, temporal data are transformed into the frequency domain by Fourier transform. Frequency domain processing can reduce the computational complexity of the model. In the gated attention mechanism, the gated structure can regulate attention information transmission to reduce the number of model parameters. Extensive experimental results on five real-world benchmark datasets demonstrate the effectiveness of FLRNN-FGA compared with the state-of-the-art methods. Full article
(This article belongs to the Special Issue Advanced Modeling and Methods of Statistical Processing of Stochastic Signals in Fractional Dynamic Systems)
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36 pages, 12501 KiB  
Article
Isomorphic Multidimensional Structures of the Cyclic Random Process in Problems of Modeling Cyclic Signals with Regular and Irregular Rhythms
by Serhii Lupenko and Roman Butsiy
Fractal Fract. 2024, 8(4), 203; https://doi.org/10.3390/fractalfract8040203 - 30 Mar 2024
Cited by 1 | Viewed by 1034
Abstract
This paper is devoted to the research of the isomorphic multidimensional cyclic structure and multidimensional phase structure of the cyclic random process (CRP) and to its formation method, which enables a rigorous formalization of intuitive ideas concerning cyclic stochastic motion. The fundamental properties [...] Read more.
This paper is devoted to the research of the isomorphic multidimensional cyclic structure and multidimensional phase structure of the cyclic random process (CRP) and to its formation method, which enables a rigorous formalization of intuitive ideas concerning cyclic stochastic motion. The fundamental properties of the cyclic random process and analytical dependencies between the multidimensional cyclic structure, multidimensional phase structure and rhythm structure of the CRP have been established. This work shows that the CRP is able to take into account the cyclicity of multidimensional distribution functions of cyclic signals as well as the variability in the rhythm of the investigated signals. A subclass of the CRP is the periodic random process, which allows for the use of classical processing methods of cyclic signals with a regular rhythm. Based on a series of experiments, significant advantages of the CRP as a mathematical model of electrocardiographic signals (ECG) compared to the periodic random process are shown. Full article
(This article belongs to the Special Issue Advanced Modeling and Methods of Statistical Processing of Stochastic Signals in Fractional Dynamic Systems)
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14 pages, 6550 KiB  
Article
Research on Active Repetitive Control for Tracking Lissajous Scan Trajectories with Voice Coil Motors Actuated Fast Steering Mirror
by Lin Wang, Shijiao Liu, Shuning Liang, Xuelian Liu and Chunyang Wang
Fractal Fract. 2024, 8(3), 128; https://doi.org/10.3390/fractalfract8030128 - 22 Feb 2024
Cited by 1 | Viewed by 1449
Abstract
The performance of laser beams in tracking Lissajous scan trajectories is severely limited by beam jitter. To enhance the performance of fast steering mirror (FSM) control in tracking Lissajous scan trajectories, this paper proposed a fractional order active disturbance rejection controller (FOADRC) and [...] Read more.
The performance of laser beams in tracking Lissajous scan trajectories is severely limited by beam jitter. To enhance the performance of fast steering mirror (FSM) control in tracking Lissajous scan trajectories, this paper proposed a fractional order active disturbance rejection controller (FOADRC) and verified its effectiveness in improving system scanning tracking accuracy. A dynamic mathematical model of a fast steering mirror was studied, and the design of parameters for the control mode of the closed-loop system was determined. A reduced-order linear active disturbance rejection controller suitable for FSM systems was designed, and the corresponding fractional-order proportional differentiation (FOPD) controller was determined according to the mathematical model. The use of the designed controller enabled high-performance tracking of high-frequency Lissajous scanning curves (X-axis 500 Hz, Y-axis 350 Hz) and met the need for high-frequency repetitive scanning. The controller has the characteristics of simple implementation and low computational complexity and is suitable for closed-loop control applications in engineering. Full article
(This article belongs to the Special Issue Advanced Modeling and Methods of Statistical Processing of Stochastic Signals in Fractional Dynamic Systems)
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14 pages, 4522 KiB  
Article
Multivariate Multiscale Higuchi Fractal Dimension and Its Application to Mechanical Signals
by Yuxing Li, Shuai Zhang, Lili Liang and Qiyu Ding
Fractal Fract. 2024, 8(1), 56; https://doi.org/10.3390/fractalfract8010056 - 15 Jan 2024
Cited by 8 | Viewed by 1834
Abstract
Fractal dimension, as a common nonlinear dynamics metric, is extensively applied in biomedicine, fault diagnosis, underwater acoustics, etc. However, traditional fractal dimension can only analyze the complexity of the time series given a single channel at a particular scale. To characterize the complexity [...] Read more.
Fractal dimension, as a common nonlinear dynamics metric, is extensively applied in biomedicine, fault diagnosis, underwater acoustics, etc. However, traditional fractal dimension can only analyze the complexity of the time series given a single channel at a particular scale. To characterize the complexity of multichannel time series, multichannel information processing was introduced, and multivariate Higuchi fractal dimension (MvHFD) was proposed. To further analyze the complexity at multiple scales, multivariate multiscale Higuchi fractal dimension (MvmHFD) was proposed by introducing multiscale processing algorithms as a technology that not only improved the use of fractal dimension in the analysis of multichannel information, but also characterized the complexity of the time series at multiple scales in the studied time series data. The effectiveness and feasibility of MvHFD and MvmHFD were verified by simulated signal experiments and real signal experiments, in which the simulation experiments tested the stability, computational efficiency, and signal separation performance of MvHFD and MvmHFD, and the real signal experiments tested the effect of MvmHFD on the recognition of multi-channel mechanical signals. The experimental results show that compared to other indicators, A achieves a recognition rate of 100% for signals in three features, which is at least 17.2% higher than for other metrics. Full article
(This article belongs to the Special Issue Advanced Modeling and Methods of Statistical Processing of Stochastic Signals in Fractional Dynamic Systems)
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