IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v9y2021i19p2429-d647090.html
   My bibliography  Save this article

Discretization of Fractional Operators: Analysis by Means of Advanced Computational Techniques

Author

Listed:
  • Jose Tenreiro Machado

    (Institute of Engineering, Polytechnic of Porto, Rua Dr. António Bernardino de Almeida, 431, 4249-015 Porto, Portugal
    These authors contributed equally to this work.)

  • Alexandra M. Galhano

    (Faculdade de Ciências Naturais, Engenharias e Tecnologias, Universidade Lusófona do Porto, Rua Augusto Rosa 24, 4000-098 Porto, Portugal
    These authors contributed equally to this work.)

  • Carla S. Cordeiro

    (Faculdade de Ciências Naturais, Engenharias e Tecnologias, Universidade Lusófona do Porto, Rua Augusto Rosa 24, 4000-098 Porto, Portugal)

Abstract

This paper studies the discretization of fractional operators by means of advanced clustering methods. The Grünwald–Letnikov fractional operator is approximated by series generated by the Euler, Tustin and generalized mean. The series for different fractional orders form the objects to be assessed. For this purpose, the several distances associated with the hierarchical clustering and multidimensional scaling computational techniques are tested. The Arc-cosine distance and the 3-dim multidimensional scaling produce good results. The visualization of the graphical representations allows a better understanding of the properties embedded in each type of approximation of the fractional operators.

Suggested Citation

  • Jose Tenreiro Machado & Alexandra M. Galhano & Carla S. Cordeiro, 2021. "Discretization of Fractional Operators: Analysis by Means of Advanced Computational Techniques," Mathematics, MDPI, vol. 9(19), pages 1-16, September.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:19:p:2429-:d:647090
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/9/19/2429/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/9/19/2429/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Machado, J. A. Tenreiro & Lopes, António M., 2016. "The N-link pendulum: Embedding nonlinear dynamics into the multidimensional scaling method," Chaos, Solitons & Fractals, Elsevier, vol. 89(C), pages 130-138.
    2. Roger Shepard, 1962. "The analysis of proximities: Multidimensional scaling with an unknown distance function. II," Psychometrika, Springer;The Psychometric Society, vol. 27(3), pages 219-246, September.
    3. J. Kruskal, 1964. "Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis," Psychometrika, Springer;The Psychometric Society, vol. 29(1), pages 1-27, March.
    4. Roger Shepard, 1962. "The analysis of proximities: Multidimensional scaling with an unknown distance function. I," Psychometrika, Springer;The Psychometric Society, vol. 27(2), pages 125-140, June.
    5. de Leeuw, Jan & Mair, Patrick, 2009. "Multidimensional Scaling Using Majorization: SMACOF in R," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 31(i03).
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. José A. Tenreiro Machado & Alexandra Galhano & Daniel Cao Labora, 2021. "A Clustering Perspective of the Collatz Conjecture," Mathematics, MDPI, vol. 9(4), pages 1-14, February.
    2. Groenen, P.J.F. & Borg, I., 2013. "The Past, Present, and Future of Multidimensional Scaling," Econometric Institute Research Papers EI 2013-07, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    3. Gruenhage, Gina & Opper, Manfred & Barthelme, Simon, 2016. "Visualizing the effects of a changing distance on data using continuous embeddings," Computational Statistics & Data Analysis, Elsevier, vol. 104(C), pages 51-65.
    4. Roger Shepard, 1974. "Representation of structure in similarity data: Problems and prospects," Psychometrika, Springer;The Psychometric Society, vol. 39(4), pages 373-421, December.
    5. Venera Tomaselli, 1996. "Multivariate statistical techniques and sociological research," Quality & Quantity: International Journal of Methodology, Springer, vol. 30(3), pages 253-276, August.
    6. Bijmolt, T.H.A. & Wedel, M., 1996. "A Monte Carlo Evaluation of Maximum Likelihood Multidimensional Scaling Methods," Other publications TiSEM f72cc9d8-f370-43aa-a224-4, Tilburg University, School of Economics and Management.
    7. Phipps Arabie & J. Carroll, 1980. "Mapclus: A mathematical programming approach to fitting the adclus model," Psychometrika, Springer;The Psychometric Society, vol. 45(2), pages 211-235, June.
    8. Dionisios Koutsantonis & Konstantinos Koutsantonis & Nikolaos P. Bakas & Vagelis Plevris & Andreas Langousis & Savvas A. Chatzichristofis, 2022. "Bibliometric Literature Review of Adaptive Learning Systems," Sustainability, MDPI, vol. 14(19), pages 1-18, October.
    9. Stephen Johnson, 1967. "Hierarchical clustering schemes," Psychometrika, Springer;The Psychometric Society, vol. 32(3), pages 241-254, September.
    10. Morales José F. & Song Tingting & Auerbach Arleen D. & Wittkowski Knut M., 2008. "Phenotyping Genetic Diseases Using an Extension of µ-Scores for Multivariate Data," Statistical Applications in Genetics and Molecular Biology, De Gruyter, vol. 7(1), pages 1-20, June.
    11. J. Kruskal, 1964. "Nonmetric multidimensional scaling: A numerical method," Psychometrika, Springer;The Psychometric Society, vol. 29(2), pages 115-129, June.
    12. Roger Girard & Norman Cliff, 1976. "A monte carlo evaluation of interactive multidimensional scaling," Psychometrika, Springer;The Psychometric Society, vol. 41(1), pages 43-64, March.
    13. J. Ramsay, 1969. "Some statistical considerations in multidimensional scaling," Psychometrika, Springer;The Psychometric Society, vol. 34(2), pages 167-182, June.
    14. Massimiliano Agovino & Maria Ferrara & Antonio Garofalo, 2017. "The driving factors of separate waste collection in Italy: a multidimensional analysis at provincial level," Environment, Development and Sustainability: A Multidisciplinary Approach to the Theory and Practice of Sustainable Development, Springer, vol. 19(6), pages 2297-2316, December.
    15. Jerzy Grobelny & Rafal Michalski & Gerhard-Wilhelm Weber, 2021. "Modeling human thinking about similarities by neuromatrices in the perspective of fuzzy logic," WORking papers in Management Science (WORMS) WORMS/21/09, Department of Operations Research and Business Intelligence, Wroclaw University of Science and Technology.
    16. Bijmolt, T.H.A. & Wedel, M., 1996. "A Monte Carlo Evaluation of Maximum Likelihood Multidimensional Scaling Methods," Research Memorandum 725, Tilburg University, School of Economics and Management.
    17. Giovanni De Luca & Paola Zuccolotto, 2011. "A tail dependence-based dissimilarity measure for financial time series clustering," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 5(4), pages 323-340, December.
    18. Raffaella Piccarreta, 2012. "Graphical and Smoothing Techniques for Sequence Analysis," Sociological Methods & Research, , vol. 41(2), pages 362-380, May.
    19. Xiaomeng Cao & Yuan Gao & Jingwei Cui & Shuangbiao Han & Lei Kang & Sha Song & Chengshan Wang, 2020. "Pore Characteristics of Lacustrine Shale Oil Reservoir in the Cretaceous Qingshankou Formation of the Songliao Basin, NE China," Energies, MDPI, vol. 13(8), pages 1-25, April.
    20. Pepermans, Roland & Verleye, Gino, 1998. "A unified Europe? How euro-attitudes relate to psychological differences between countries," Journal of Economic Psychology, Elsevier, vol. 19(6), pages 681-699, December.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:9:y:2021:i:19:p:2429-:d:647090. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.