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Fractal Fract
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Recent Advances in Fractal Interpolation Functions and Their Applications in...
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Recent Advances in Fractal Interpolation Functions and Their Applications in AI

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

Deadline for manuscript submissions: closed (15 January 2024) | Viewed by 4507

Special Issue Editors

Prof. Dr. Da-Chin Lour

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Guest Editor
Department of Data Science and Analytics, School of Intelligent Science and Technology, I-Shou University, Dashu District, Kaohsiung City 84001, Taiwan
Interests: fractal interpolation functions; approximation theory; harmonic analysis
Dr. Liang-Yu Hsieh

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Guest Editor
Department of Data Science and Analytics, I-Shou University, Kaohsiung, Taiwan
Interests: matrix theory; linear algebra
Prof. Dr. María A. Navascués

E-Mail Website
Guest Editor
Departamento de Matemática Aplicada, Universidad de Zaragoza, 50018 Zaragoza, Spain
Interests: fractal functions; fixed point theory; interpolation; approximation; computational mathematics
Special Issues, Collections and Topics in MDPI journals
Dr. Vasileios Drakopoulos

E-Mail Website
Guest Editor
Department of Computer Science and Biomedical Informatics, University of Thessaly, 35131 Lamia, Greece
Interests: chaos theory; computer graphics; fractal and computational geometry; mathematical modelling; computational complex analysis; nonlinear dynamics
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

To date, Artificial Intelligence (AI) has been used for classification, prediction, and optimization. However, there is still room for improving the application of AI in many fields. In situations where the problem to be modelled is too intricate, the approach of incorporating AI can lead to promising results. There is a consistent body of literature demonstrating that AI-based methods provide greater reliability, accuracy, and predictability, while helping to detect hidden nonlinear chaotic patterns in big data applications. To this end, AI methods are often used to complement traditional classical methods in many fields, especially in complicated hybrid chaotic systems.

Although many studies in recent years have used AI in applied mathematics, the application of AI methods to fractal theory is less common. Fractal, fractional calculus, and wavelets have been the most widely-used methods in approximation theory and signal processing. Combining AI methods and fractal-based analysis is a new direction of research area.

Fractal-AI allows for the derivation of new mathematical tools that are designed to provide efficient solutions to problems in which many entities are interconnected. Fractal-AI is also used to extract hidden information from the dynamics of complicated systems and speed up experiments by performing massive data analysis.

In this special issue, the focus will be on recent developments of fractal functions and their applications in AI and data science, including theoretical and numerical aspects. The purpose is to develop the research area of differential equations, integral equations, approximation theory, wavelets, curve fitting, and reproducing kernel Hilbert spaces. We also aim to extend applications of fractal functions in machine learning and data modelling. Applications of AI in fractal functions are also welcome to this special issue.

Prof. Dr. Da-Chin Lour
Dr. Liang-Yu Hsieh 
Prof. Dr. María Antonia Navascués
Dr. Vasileios Drakopoulos
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

  • fractal interpolation functions
  • fractional calculus
  • fractal interpolation functions and wavelets
  • fractal interpolation functions and approximation
  • fractal interpolation functions and reproducing kernel spaces
  • fractal interpolation and signal processing
  • fractal interpolation functions and curve fitting problems
  • fractal modeling of data ai applications in fractal interpolation functions
  • applications of fractal interpolation functions in AI
  • applications of fractal interpolation functions in statistics and data science

Published Papers (3 papers)

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Research

11 pages, 724 KiB  
Article
Reproducing Kernel Hilbert Spaces of Smooth Fractal Interpolation Functions
by Dah-Chin Luor and Liang-Yu Hsieh
Fractal Fract. 2023, 7(5), 357; https://doi.org/10.3390/fractalfract7050357 - 27 Apr 2023
Viewed by 749
Abstract
The theory of reproducing kernel Hilbert spaces (RKHSs) has been developed into a powerful tool in mathematics and has lots of applications in many fields, especially in kernel machine learning. Fractal theory provides new technologies for making complicated curves and fitting experimental data. [...] Read more.
The theory of reproducing kernel Hilbert spaces (RKHSs) has been developed into a powerful tool in mathematics and has lots of applications in many fields, especially in kernel machine learning. Fractal theory provides new technologies for making complicated curves and fitting experimental data. Recently, combinations of fractal interpolation functions (FIFs) and methods of curve estimations have attracted the attention of researchers. We are interested in the study of connections between FIFs and RKHSs. The aim is to develop the concept of smooth fractal-type reproducing kernels and RKHSs of smooth FIFs. In this paper, a linear space of smooth FIFs is considered. A condition for a given finite set of smooth FIFs to be linearly independent is established. For such a given set, we build a fractal-type positive semi-definite kernel and show that the span of these linearly independent smooth FIFs is the corresponding RKHS. The nth derivatives of these FIFs are investigated, and properties of related positive semi-definite kernels and the corresponding RKHS are studied. We also introduce subspaces of these RKHS which are important in curve-fitting applications. Full article
(This article belongs to the Special Issue Recent Advances in Fractal Interpolation Functions and Their Applications in AI)
10 pages, 744 KiB  
Article
Fractal Perturbation of the Nadaraya–Watson Estimator
by Dah-Chin Luor and Chiao-Wen Liu
Fractal Fract. 2022, 6(11), 680; https://doi.org/10.3390/fractalfract6110680 - 17 Nov 2022
Cited by 6 | Viewed by 1268
Abstract
One of the main tasks in the problems of machine learning and curve fitting is to develop suitable models for given data sets. It requires to generate a function to approximate the data arising from some unknown function. The class of kernel regression [...] Read more.
One of the main tasks in the problems of machine learning and curve fitting is to develop suitable models for given data sets. It requires to generate a function to approximate the data arising from some unknown function. The class of kernel regression estimators is one of main types of nonparametric curve estimations. On the other hand, fractal theory provides new technologies for making complicated irregular curves in many practical problems. In this paper, we are going to investigate fractal curve-fitting problems with the help of kernel regression estimators. For a given data set that arises from an unknown function m, one of the well-known kernel regression estimators, the Nadaraya–Watson estimator m^, is applied. We consider the case that m is Hölder-continuous of exponent β with 0<β≤1, and the graph of m is irregular. An estimation for the expectation of |m^−m|2 is established. Then a fractal perturbation f[m^] corresponding to m^ is constructed to fit the given data. The expectations of |f[m^]−m^|2 and |f[m^]−m|2 are also estimated. Full article
(This article belongs to the Special Issue Recent Advances in Fractal Interpolation Functions and Their Applications in AI)
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15 pages, 350 KiB  
Article
Scale-Free Fractal Interpolation
by María A. Navascués, Cristina Pacurar and Vasileios Drakopoulos
Fractal Fract. 2022, 6(10), 602; https://doi.org/10.3390/fractalfract6100602 - 16 Oct 2022
Cited by 13 | Viewed by 1745
Abstract
An iterated function system that defines a fractal interpolation function, where ordinate scaling is replaced by a nonlinear contraction, is investigated here. In such a manner, fractal interpolation functions associated with Matkowski contractions for finite as well as infinite (countable) sets of data [...] Read more.
An iterated function system that defines a fractal interpolation function, where ordinate scaling is replaced by a nonlinear contraction, is investigated here. In such a manner, fractal interpolation functions associated with Matkowski contractions for finite as well as infinite (countable) sets of data are obtained. Furthermore, we construct an extension of the concept of α-fractal interpolation functions, herein called R-fractal interpolation functions, related to a finite as well as to a countable iterated function system and provide approximation properties of the R-fractal functions. Moreover, we obtain smooth R-fractal interpolation functions and provide results that ensure the existence of differentiable R-fractal interpolation functions both for the finite and the infinite (countable) cases. Full article
(This article belongs to the Special Issue Recent Advances in Fractal Interpolation Functions and Their Applications in AI)
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