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Fractal Fract
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New Trends in Fractional Stochastic Processes
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New Trends in Fractional Stochastic Processes

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

Deadline for manuscript submissions: closed (20 November 2022) | Viewed by 13905

Special Issue Editors

Dr. Wanqing Song

E-Mail Website
Guest Editor
1. School of Electric and Electronic Engineering, Shanghai University of Engineering Science, Shanghai 201620, China
2. School of Electronic and Electrical Engineering, Minnan University of Science and Technology, Quanzhou 362700, China
Interests: remaining useful life prediction; feature extrection of stochastic series; reliability analysis; nonlinear dynamic; prediction of stochastic series; long-range dependence; fractional modelling of stochastic series; stochastic signal process
Special Issues, Collections and Topics in MDPI journals
Dr. Francesco Villecco

E-Mail Website
Guest Editor
Department of Industrial Engineering, University of Salerno, Via Giovanni Paolo II 132, 84084 Fisciano, Italy
Interests: industrial design; entropy; fuzzy logic; computer-aided design (CAD); axiomatic design; MaxInf principle
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The aim of this Special Issue is to present a multidisciplinary state-of-the-art collection on fractional stochastic processes, with reference to theoretical and real-world engineering applications. We invite the submission of high-quality research papers dealing with nonlinear time series, statistical methods, data analysis tools, mathematical and statistical approaches, data mining techniques in mechanics and long-range fractal processes. Particular attention is paid to fractal time series and fractal long-range processes in mechanics and engineering applications. Fractal time series substantially differ from conventional time series in terms of their statistical properties. For instance, they may have a heavy-tailed probability distribution function (PDF), a slowly decayed autocorrelation function (ACF), and a power spectrum function (PSD) of 1/f type. Fractal time series may have statistical dependence—either long-range dependence (LRD) or short-range dependence (SRD)—and global or local self-similarity. In engineering applications, such as mechanical or electronics engineering, engineers usually consider fractal time series as the output or response of a differential system or filter of integer order under the excitation of white noise. In this Special Issue, a fractal time series is taken as the solution to a differential equation of fractional order or a response of a fractional system or a fractional filter driven with a white noise in the domain of stochastic processes. This Special Issue encourages both original research articles and review articles, both theories and applications in advanced statistical and mathematical modeling and in-depth examinations of the physical and mechanical systems. The research papers may incorporate one or a combination of analytical, numerical, statistical and experimental methodologies. This Issue, “New Trends in Fractional Stochastic Processes”, focuses on a wide range of topics in statistical physics and mechanics, including but not limited to:

  • classical and quantum mechanics; equilibrium and non-equilibrium fluids;
  • granular and soft matter; fractional calculus in statistical mechanics;
  • fractional calculus in statistical physics; interdisciplinary statistical mechanics;
  • interdisciplinary statistical physics; advanced methods for mechanical system fault diagnosis and life prediction;
  • advanced methods for signal processing of mechanical systems; neuronal signal analysis (EEG, BCI);
  • mathematical modeling of diseases; fractal theories in cities development;
  • computer simulation in artificial intelligence;
  • mathematical modeling in economics, management and engineering.

Dr. Wanqing Song
Dr. Francesco Villecco
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.

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Published Papers (7 papers)

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Research

26 pages, 6479 KiB  
Article
Tool Degradation Prediction Based on Semimartingale Approximation of Linear Fractional Alpha-Stable Motion and Multi-Feature Fusion
by Yuchen Yuan, Jianxue Chen, Jin Rong, Piercarlo Cattani, Aleksey Kudreyko and Francesco Villecco
Fractal Fract. 2023, 7(4), 325; https://doi.org/10.3390/fractalfract7040325 - 12 Apr 2023
Viewed by 1383
Abstract
Tool wear will reduce workpieces’ quality and accuracy. In this paper, the vibration signals of the milling process were analyzed, and it was found that historical fluctuations still have an impact on the existing state. First of all, the linear fractional alpha-stable motion [...] Read more.
Tool wear will reduce workpieces’ quality and accuracy. In this paper, the vibration signals of the milling process were analyzed, and it was found that historical fluctuations still have an impact on the existing state. First of all, the linear fractional alpha-stable motion (LFSM) was investigated, along with a differential iterative model with it as the noise term is constructed according to the fractional-order Ito formula; the general solution of this model is derived by semimartingale approximation. After that, for the chaotic features of the vibration signal, the time-frequency domain characteristics were extracted using principal component analysis (PCA), and the relationship between the variation of the generalized Hurst exponent and tool wear was established using multifractal detrended fluctuation analysis (MDFA). Then, the maximum prediction length was obtained by the maximum Lyapunov exponent (MLE), which allows for analysis of the vibration signal. Finally, tool condition diagnosis was carried out by the evolving connectionist system (ECoS). The results show that the LFSM iterative model with semimartingale approximation combined with PCA and MDFA are effective for the prediction of vibration trends and tool condition. Further, the monitoring of tool condition using ECoS is also effective. Full article
(This article belongs to the Special Issue New Trends in Fractional Stochastic Processes)
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13 pages, 2188 KiB  
Article
Hybrid Approach of Fractional Generalized Pareto Motion and Cosine Similarity Hidden Markov Model for Solar Radiation Forecasting
by Wanqing Song, Wujin Deng, Dongdong Chen, Rong Jin and Aleksey Kudreyko
Fractal Fract. 2023, 7(1), 93; https://doi.org/10.3390/fractalfract7010093 - 13 Jan 2023
Cited by 5 | Viewed by 1826
Abstract
Power from solar energy is not reliable, due to weather-related factors, which diminishes the power system’s reliability. Therefore, this study suggests a way to predict the intensity of solar irradiance using various statistical algorithms and artificial intelligence. In particular, we suggest the use [...] Read more.
Power from solar energy is not reliable, due to weather-related factors, which diminishes the power system’s reliability. Therefore, this study suggests a way to predict the intensity of solar irradiance using various statistical algorithms and artificial intelligence. In particular, we suggest the use of a hybrid predictive model, combining statistical properties and historical data training. In order to evaluate the maximum prediction steps of solar irradiance, the maximum Lyapunov exponent was applied. Then, we used the cosine similarity algorithm in the hidden Markov model for the initial prediction. The combination of the Hurst exponent and tail parameter revealed the self-similarity and long-range dependence of the fractional generalized Pareto motion, which enabled us to consider the iterative predictive model. The initial prediction was substituted into a stochastic differential equation to achieve the final prediction, which prevents error propagation. The effectiveness of the hybrid model was demonstrated in the case study. Full article
(This article belongs to the Special Issue New Trends in Fractional Stochastic Processes)
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22 pages, 482 KiB  
Article
Estimation and Testing of Random Effects Semiparametric Regression Model with Separable Space-Time Filters
by Shuangshuang Li, Jianbao Chen and Bogui Li
Fractal Fract. 2022, 6(12), 735; https://doi.org/10.3390/fractalfract6120735 - 11 Dec 2022
Cited by 8 | Viewed by 1340
Abstract
This paper focuses on studying a random effects semiparametric regression model (RESPRM) with separable space-time filters. The model cannot only capture the linearity and nonlinearity existing in a space-time dataset, but also avoid the inefficient estimators caused by ignoring spatial correlation and serial [...] Read more.
This paper focuses on studying a random effects semiparametric regression model (RESPRM) with separable space-time filters. The model cannot only capture the linearity and nonlinearity existing in a space-time dataset, but also avoid the inefficient estimators caused by ignoring spatial correlation and serial correlation in the error term of a space-time data regression model. Its profile quasi-maximum likelihood estimators (PQMLE) for parameters and nonparametric functions, and a generalized F-test statistic for checking the existence of nonlinear relationships are constructed. The asymptotic properties of estimators and asymptotic distribution of test statistic are derived. Monte Carlo simulations imply that our estimators and test statistic have good finite sample performance. The Indonesian rice farming data are used to illustrate our methods. Full article
(This article belongs to the Special Issue New Trends in Fractional Stochastic Processes)
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17 pages, 6801 KiB  
Article
An Adaptive Generalized Cauchy Model for Remaining Useful Life Prediction of Wind Turbine Gearboxes with Long-Range Dependence
by Wanqing Song, Dongdong Chen, Enrico Zio, Wenduan Yan and Fan Cai
Fractal Fract. 2022, 6(10), 576; https://doi.org/10.3390/fractalfract6100576 - 10 Oct 2022
Cited by 1 | Viewed by 1663
Abstract
Remaining useful life (RUL) prediction is important for wind turbine operation and maintenance. The degradation process of gearboxes in wind turbines is a slowly and randomly changing process with long-range dependence (LRD). The degradation trend of the gearbox is constantly changing, and a [...] Read more.
Remaining useful life (RUL) prediction is important for wind turbine operation and maintenance. The degradation process of gearboxes in wind turbines is a slowly and randomly changing process with long-range dependence (LRD). The degradation trend of the gearbox is constantly changing, and a single drift coefficient is not accurate enough to describe the degradation trend. This paper proposes an original adaptive generalized Cauchy (GC) model with LRD and randomness to predict the RUL of wind turbine gearboxes. The LRD is explained jointly by the fractal dimension and the Hurst exponent, and the randomness is explained by the diffusion term driven by the GC difference time sequence. The estimated value of the unknown parameter of adaptive GC model is deduced, and the specific expression of the RUL estimation is deduced. The adaptability is manifested in the time-varying drift coefficient of the GC model: by continuously updating the drift coefficient to adapt to the change in the degradation trend, the adaptive GC model offers high accuracy in the prediction of the degradation trend. The performance of the proposed model is analyzed using real wind turbine gearbox data. Full article
(This article belongs to the Special Issue New Trends in Fractional Stochastic Processes)
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11 pages, 4295 KiB  
Article
Averaging Principle for a Class of Time-Fractal-Fractional Stochastic Differential Equations
by Xiaoyu Xia, Yinmeng Chen and Litan Yan
Fractal Fract. 2022, 6(10), 558; https://doi.org/10.3390/fractalfract6100558 - 30 Sep 2022
Cited by 4 | Viewed by 1454
Abstract
In this paper, we study a class of time-fractal-fractional stochastic differential equations with the fractal–fractional differential operator of Atangana under the meaning of Caputo and with a kernel of the power law type. We first establish the Hölder continuity of the solution of [...] Read more.
In this paper, we study a class of time-fractal-fractional stochastic differential equations with the fractal–fractional differential operator of Atangana under the meaning of Caputo and with a kernel of the power law type. We first establish the Hölder continuity of the solution of the equation. Then, under certain averaging conditions, we show that the solutions of original equations can be approximated by the solutions of the associated averaged equations in the sense of the mean square convergence. As an application, we provide an example with numerical simulations to explore the established averaging principle. Full article
(This article belongs to the Special Issue New Trends in Fractional Stochastic Processes)
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20 pages, 13011 KiB  
Article
Finite Iterative Forecasting Model Based on Fractional Generalized Pareto Motion
by Wanqing Song, Shouwu Duan, Dongdong Chen, Enrico Zio, Wenduan Yan and Fan Cai
Fractal Fract. 2022, 6(9), 471; https://doi.org/10.3390/fractalfract6090471 - 26 Aug 2022
Cited by 5 | Viewed by 1319
Abstract
In this paper, an efficient prediction model based on the fractional generalized Pareto motion (fGPm) with Long-Range Dependent (LRD) and infinite variance characteristics is proposed. Firstly, we discuss the meaning of each parameter of the generalized Pareto distribution (GPD), and the LRD characteristics [...] Read more.
In this paper, an efficient prediction model based on the fractional generalized Pareto motion (fGPm) with Long-Range Dependent (LRD) and infinite variance characteristics is proposed. Firstly, we discuss the meaning of each parameter of the generalized Pareto distribution (GPD), and the LRD characteristics of the generalized Pareto motion are analyzed by taking into account the heavy-tailed characteristics of its distribution. Then, the mathematical relationship H=1⁄α between the self-similar parameter H and the tail parameter α is obtained. Also, the generalized Pareto increment distribution is obtained using statistical methods, which offers the subsequent derivation of the iterative forecasting model based on the increment form. Secondly, the tail parameter α is introduced to generalize the integral expression of the fractional Brownian motion, and the integral expression of fGPm is obtained. Then, by discretizing the integral expression of fGPm, the statistical characteristics of infinite variance is shown. In addition, in order to study the LRD prediction characteristic of fGPm, LRD and self-similarity analysis are performed on fGPm, and the LRD prediction conditions H>1⁄α is obtained. Compared to the fractional Brownian motion describing LRD by a self-similar parameter H, fGPm introduces the tail parameter α, which increases the flexibility of the LRD description. However, the two parameters are not independent, because of the LRD condition H>1⁄α. An iterative prediction model is obtained from the Langevin-type stochastic differential equation driven by fGPm. The prediction model inherits the LRD condition H>1⁄α of fGPm and the time series, simulated by the Monte Carlo method, shows the superiority of the prediction model to predict data with high jumps. Finally, this paper uses power load data in two different situations (weekdays and weekends), used to verify the validity and general applicability of the forecasting model, which is compared with the fractional Brown prediction model, highlighting the “high jump data prediction advantage” of the fGPm prediction model. Full article
(This article belongs to the Special Issue New Trends in Fractional Stochastic Processes)
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12 pages, 491 KiB  
Article
Galerkin Approximation for Stochastic Volterra Integral Equations with Doubly Singular Kernels
by Yuyuan Li, Wanqing Song, Yanan Jiang and Aleksey Kudreyko
Fractal Fract. 2022, 6(6), 311; https://doi.org/10.3390/fractalfract6060311 - 1 Jun 2022
Cited by 1 | Viewed by 1656
Abstract
This paper is concerned with the more general nonlinear stochastic Volterra integral equations with doubly singular kernels, whose singular points include both s=t and s=0. We propose a Galerkin approximate scheme to solve the equation numerically, and we [...] Read more.
This paper is concerned with the more general nonlinear stochastic Volterra integral equations with doubly singular kernels, whose singular points include both s=t and s=0. We propose a Galerkin approximate scheme to solve the equation numerically, and we obtain the strong convergence rate for the Galerkin method in the mean square sense. The rate is min{2−2(α1+β1),1−2(α2+β2)} (where α1,α2,β1,β2 are positive numbers satisfying 0<α1+β1<1, 0<α2+β2<12), which improves the results of some numerical schemes for the stochastic Volterra integral equations with regular or weakly singular kernels. Moreover, numerical examples are given to support the theoretical result and explain the priority of the Galerkin method. Full article
(This article belongs to the Special Issue New Trends in Fractional Stochastic Processes)
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