Special Issue "Fractals, Fractional Calculus and Applied Statistics"

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Mathematical Physics".

Deadline for manuscript submissions: 30 November 2021.

Special Issue Editor

Prof. Dr. Raoul R. Nigmatullin
E-Mail Website
Guest Editor
Kazan National Research Technical University, Kazan 420111, Russia
Interests: mathematical statistics; electrochemistry; signal processing; dielectric spectroscopy

Special Issue Information

Dear Colleagues,

This Special Issue provides a recent collection of the papers that are related to the following topics:

  1. Fractals and especially of these applications in applied mathematics and statistics
  2. Fractional calculus and its application in the fractal dynamics.
  3. Applied statistics as a “specific” lens to concentrate and direct corresponding mathematical tools in the solution of some applied problems.

These topics are becoming “hotspots” in the further development of fractional calculus, which covers different regions of mathematics. Many conferences have been trying to solve the following problem: is it true that there is a relationship between fractional integral and some types of fractals that can be observed in nature? This important question has not been solved and there are a variety of fractional integrals that fill in gaps between normal integral and derivative.

Prof. Dr. Raoul R. Nigmatullin
Guest Editor

Manuscript Submission Information

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Keywords

  • Regular fractals
  • Random fractals
  • Fractal physics
  • Fractional integrals/derivatives
  • Fractional statistics
  • Fractional distributions
  • Hurst exponent

Published Papers (3 papers)

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Research

Article
ML-LME: A Plant Growth Situation Analysis Model Using the Hierarchical Effect of Fractal Dimension
Mathematics 2021, 9(12), 1322; https://doi.org/10.3390/math9121322 - 08 Jun 2021
Viewed by 317
Abstract
Rice plays an essential role in agricultural production as the most significant food crop. Automated supervision in the process of crop growth is the future development direction of agriculture, and it is also a problem that needs to be solved urgently. Productive cultivation, [...] Read more.
Rice plays an essential role in agricultural production as the most significant food crop. Automated supervision in the process of crop growth is the future development direction of agriculture, and it is also a problem that needs to be solved urgently. Productive cultivation, production and research of crops are attributed to increased automation of supervision in the growth. In this article, for the first time, we propose the concept of rice fractal dimension heterogeneity and define it as rice varieties with different fractal dimension values having various correlations between their traits. To make a comprehensive prediction of the rice growth, Machine Learning and Linear Mixed Effect (ML-LME) model is proposed to model and analyze this heterogeneity, which is based on the existing automatic measurement system RAP and introduces statistical characteristics of fractal dimensions as novel features. Machine learning algorithms are applied to distinguish the rice growth stages with a high degree of accuracy and to excavate the heterogeneity of rice fractal dimensions with statistical meaning. According to the information of growth stage and fractal dimension heterogeneity, a precise prediction of key rice phenotype traits can be received by ML-LME using a Linear Mixed Effect model. In this process, the value of the fractal dimension is divided into groups and then rices of different levels are respectively fitted to improve the accuracy of the subsequent prediction, that is, the heterogeneity of the fractal dimension. Afterwards, we apply the model to analyze the rice pot image. The research results show that the ML-LME model, which possesses the hierarchical effect of fractal dimension, performs more excellently in predicting the growth situation of plants than the traditional regression model does. Further comparison confirmed that the model we proposed is the first to consider the hierarchy structure of plant fractal dimension, and that consideration obviously strengthens the model on the ability of variation interpretation and prediction precision. Full article
(This article belongs to the Special Issue Fractals, Fractional Calculus and Applied Statistics)
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Article
Multivariate Multifractal Detrending Moving Average Analysis of Air Pollutants
Mathematics 2021, 9(7), 711; https://doi.org/10.3390/math9070711 - 25 Mar 2021
Viewed by 412
Abstract
One of the most challenging endeavors of contemporary research is to describe and analyze the dynamic behavior of time series arising from real-world systems. To address the need for analyzing long-range correlations and multifractal properties of multivariate time series, we generalize the multifractal [...] Read more.
One of the most challenging endeavors of contemporary research is to describe and analyze the dynamic behavior of time series arising from real-world systems. To address the need for analyzing long-range correlations and multifractal properties of multivariate time series, we generalize the multifractal detrended moving average algorithm (MFDMA) to the multivariate case and propose a multivariate MFDMA algorithm (MV-MFDMA). The validity and performance of the proposed algorithm are tested by conducting numerical simulations on synthetic multivariate monofractal and multifractal time series. The MV-MFDMA algorithm is then utilized to analyze raw, seasonally adjusted, and remainder components of five air pollutant time series. Results from all three cases reveal multifractal properties with persistent long-range correlations. Full article
(This article belongs to the Special Issue Fractals, Fractional Calculus and Applied Statistics)
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Article
Generalized Hurst Hypothesis: Description of Time-Series in Communication Systems
Mathematics 2021, 9(4), 381; https://doi.org/10.3390/math9040381 - 14 Feb 2021
Viewed by 664
Abstract
In this paper, we focus on the generalization of the Hurst empirical law and suggest a set of reduced parameters for quantitative description of long-time series. These series are usually considered as a specific response of a complex system (economic, geophysical, electromagnetic and [...] Read more.
In this paper, we focus on the generalization of the Hurst empirical law and suggest a set of reduced parameters for quantitative description of long-time series. These series are usually considered as a specific response of a complex system (economic, geophysical, electromagnetic and other systems), where successive fixations of external factors become impossible. We consider applying generalized Hurst laws to obtain a new set of reduced parameters in data associated with communication systems. We analyze three hypotheses. The first one contains one power-law exponent. The second one incorporates two power-law exponents, which are in many cases complex-conjugated. The third hypothesis has three power-law exponents, two of which are complex-conjugated as well. These hypotheses describe with acceptable accuracy (relative error does not exceed 2%) a wide set of trendless sequences (TLS) associated with radiometric measurements. Generalized Hurst laws operate with R/S curves not only in the asymptotic region, but in the entire domain. The fitting parameters can be used as the reduced parameters for the description of the given data. The paper demonstrates that this general approach can also be applied to other TLS. Full article
(This article belongs to the Special Issue Fractals, Fractional Calculus and Applied Statistics)
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