Fractional Models and Statistical Applications

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

Deadline for manuscript submissions: 30 June 2024 | Viewed by 4238

Special Issue Editors


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Guest Editor
Department of Probability Theory, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, 01601 Kyiv, Ukraine
Interests: stochastic processes; stochastic analysis and stochastic differential equations; fractional processes; their financial and statistical applications
Special Issues, Collections and Topics in MDPI journals

E-Mail Website
Guest Editor
Department of Probability Theory, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, 01601 Kyiv, Ukraine
Interests: fractional and multifractional stochastic processes and fields; stochastic differential equations (ordinary and partial); statistical inference for stochastic processes

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Guest Editor
Department of Mathematics, University of Oslo, 0851 Oslo, Norway
Interests: stochastic processes; stochastic analysis and stochastic differential equations; mathematical finance

Special Issue Information

Dear Colleagues,

The aim of this issue is to present the modern theoretical and applied questions in fractional stochastic models. This includes a wide range of problems in the theory of random processes with long- and short-term dependence. Such processes arise in physics, computer science, economics, climatology, fluid mechanics and financial mathematics. Fractional stochastic models are widely used for modeling and forecasting exchange rates, stock prices, options and other securities, and can also be applied in image processing and as models for processes in computer, electrical and telecommunication networks. The focus of this Special Issue is on models with fractional Brownian motion, which are now becoming popular because (unlike classical models, driven by the Wiener process) they allow the description of processes with non-zero correlations of random noise, namely, processes with long-term dependence (correlations that decay slowly) or short-term dependence (correlations that decay rapidly). Special attention is paid to statistical inference for fractional and related models—a rapidly developing area of research that is highly important for practical applications. Topics that are invited for submission include (but are not limited to) the following:

  • Fractional and multifractional stochastic processes;
  • Fractional stochastic analysis;
  • Stochastic (partial) differential equations with fractional processes;
  • Applications of fractional models;
  • Statistical inference for fractional models.

Prof. Dr. Yuliya Mishura
Dr. Kostiantyn Ralchenko
Dr. Anton Yurchenko-Tytarenko
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.

Published Papers (4 papers)

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Research

10 pages, 1718 KiB  
Article
Critical Exponents and Universality for Fractal Time Processes above the Upper Critical Dimensionality
by Shaolong Zeng, Yangfan Hu, Shijing Tan and Biao Wang
Fractal Fract. 2024, 8(5), 294; https://doi.org/10.3390/fractalfract8050294 - 16 May 2024
Viewed by 576
Abstract
We study the critical behaviors of systems undergoing fractal time processes above the upper critical dimension. We derive a set of novel critical exponents, irrespective of the order of the fractional time derivative or the particular form of interaction in the Hamiltonian. For [...] Read more.
We study the critical behaviors of systems undergoing fractal time processes above the upper critical dimension. We derive a set of novel critical exponents, irrespective of the order of the fractional time derivative or the particular form of interaction in the Hamiltonian. For fractal time processes, we not only discover new universality classes with a dimensional constant but also decompose the dangerous irrelevant variables to obtain corrections for critical dynamic behavior and static critical properties. This contrasts with the traditional theory of critical phenomena, which posits that static critical exponents are unrelated to the dynamical processes. Simulations of the Landau–Ginzburg model for fractal time processes and the Ising model with temporal long-range interactions both show good agreement with our set of critical exponents, verifying its universality. The discovery of this new universality class provides a method for examining whether a system is undergoing a fractal time process near the critical point. Full article
(This article belongs to the Special Issue Fractional Models and Statistical Applications)
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17 pages, 686 KiB  
Article
Some Fractional Stochastic Models for Neuronal Activity with Different Time-Scales and Correlated Inputs
by Enrica Pirozzi
Fractal Fract. 2024, 8(1), 57; https://doi.org/10.3390/fractalfract8010057 - 15 Jan 2024
Viewed by 1074
Abstract
In order to describe neuronal dynamics on different time-scales, we propose a stochastic model based on two coupled fractional stochastic differential equations, with different fractional orders. For the specified choice of involved functions and parameters, we provide three specific models, with/without leakage, with [...] Read more.
In order to describe neuronal dynamics on different time-scales, we propose a stochastic model based on two coupled fractional stochastic differential equations, with different fractional orders. For the specified choice of involved functions and parameters, we provide three specific models, with/without leakage, with fractional/non-fractional correlated inputs. We give explicit expressions of the process representing the voltage variation in the neuronal membrane. Expectation values and covariances are given and compared. Numerical evaluations of the average behaviors of involved processes are presented and discussed. Full article
(This article belongs to the Special Issue Fractional Models and Statistical Applications)
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16 pages, 508 KiB  
Article
Advanced Mathematical Approaches in Psycholinguistic Data Analysis: A Methodological Insight
by Cecilia Castro, Víctor Leiva, Maria do Carmo Lourenço-Gomes and Ana Paula Amorim
Fractal Fract. 2023, 7(9), 670; https://doi.org/10.3390/fractalfract7090670 - 5 Sep 2023
Cited by 1 | Viewed by 1273
Abstract
In the evolving landscape of psycholinguistic research, this study addresses the inherent complexities of data through advanced analytical methodologies, including permutation tests, bootstrap confidence intervals, and fractile or quantile regression. The methodology and philosophy of our approach deeply resonate with fractal and fractional [...] Read more.
In the evolving landscape of psycholinguistic research, this study addresses the inherent complexities of data through advanced analytical methodologies, including permutation tests, bootstrap confidence intervals, and fractile or quantile regression. The methodology and philosophy of our approach deeply resonate with fractal and fractional concepts. Responding to the skewed distributions of data, which are observed in metrics such as reading times, time-to-response, and time-to-submit, our analysis highlights the nuanced interplay between time-to-response and variables like lists, conditions, and plausibility. A particular focus is placed on the implausible sentence response times, showcasing the precision of our chosen methods. The study underscores the profound influence of individual variability, advocating for meticulous analytical rigor in handling intricate and complex datasets. Drawing inspiration from fractal and fractional mathematics, our findings emphasize the broader potential of sophisticated mathematical tools in contemporary research, setting a benchmark for future investigations in psycholinguistics and related disciplines. Full article
(This article belongs to the Special Issue Fractional Models and Statistical Applications)
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16 pages, 663 KiB  
Article
Quasi-Cauchy Regression Modeling for Fractiles Based on Data Supported in the Unit Interval
by José Sérgio Casé de Oliveira, Raydonal Ospina, Víctor Leiva, Jorge Figueroa-Zúñiga and Cecilia Castro
Fractal Fract. 2023, 7(9), 667; https://doi.org/10.3390/fractalfract7090667 - 4 Sep 2023
Cited by 1 | Viewed by 940
Abstract
A fractile is a location on a probability density function with the associated surface being a proportion of such a density function. The present study introduces a novel methodological approach to modeling data within the continuous unit interval using fractile or quantile regression. [...] Read more.
A fractile is a location on a probability density function with the associated surface being a proportion of such a density function. The present study introduces a novel methodological approach to modeling data within the continuous unit interval using fractile or quantile regression. This approach has a unique advantage as it allows for a direct interpretation of the response variable in relation to the explanatory variables. The new approach provides robustness against outliers and permits heteroscedasticity to be modeled, making it a tool for analyzing datasets with diverse characteristics. Importantly, our approach does not require assumptions about the distribution of the response variable, offering increased flexibility and applicability across a variety of scenarios. Furthermore, the approach addresses and mitigates criticisms and limitations inherent to existing methodologies, thereby giving an improved framework for data modeling in the unit interval. We validate the effectiveness of the introduced approach with two empirical applications, which highlight its practical utility and superior performance in real-world data settings. Full article
(This article belongs to the Special Issue Fractional Models and Statistical Applications)
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