IDEAS home Printed from https://ideas.repec.org/a/eee/phsmap/v389y2010i5p909-920.html
   My bibliography  Save this article

Fractional calculus approach to the statistical characterization of random variables and vectors

Author

Listed:
  • Cottone, Giulio
  • Di Paola, Mario
  • Metzler, Ralf

Abstract

Fractional moments have been investigated by many authors to represent the density of univariate and bivariate random variables in different contexts. Fractional moments are indeed important when the density of the random variable has inverse power-law tails and, consequently, it lacks integer order moments. In this paper, starting from the Mellin transform of the characteristic function and by fractional calculus method we present a new perspective on the statistics of random variables. Introducing the class of complex moments, that include both integer and fractional moments, we show that every random variable can be represented within this approach, even if its integer moments diverge. Applications to the statistical characterization of raw data and in the representation of both random variables and vectors are provided, showing that the good numerical convergence makes the proposed approach a good and reliable tool also for practical data analysis.

Suggested Citation

  • Cottone, Giulio & Di Paola, Mario & Metzler, Ralf, 2010. "Fractional calculus approach to the statistical characterization of random variables and vectors," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(5), pages 909-920.
    • Handle: RePEc:eee:phsmap:v:389:y:2010:i:5:p:909-920
    DOI: 10.1016/j.physa.2009.11.018
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0378437109009285
    Download Restriction: Full text for ScienceDirect subscribers only. Journal offers the option of making the article available online on Science direct for a fee of $3,000
    File URL: https://libkey.io/10.1016/j.physa.2009.11.018?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Marta C. González & César A. Hidalgo & Albert-László Barabási, 2009. "Understanding individual human mobility patterns," Nature, Nature, vol. 458(7235), pages 238-238, March.
    2. D. Brockmann & L. Hufnagel & T. Geisel, 2006. "The scaling laws of human travel," Nature, Nature, vol. 439(7075), pages 462-465, January.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Alotta, G. & Di Paola, M., 2015. "Probabilistic characterization of nonlinear systems under α-stable white noise via complex fractional moments," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 420(C), pages 265-276.
    2. Xu, Jun & Song, Jinheng & Yu, Quanfu & Kong, Fan, 2023. "Generalized distribution reconstruction based on the inversion of characteristic function curve for structural reliability analysis," Reliability Engineering and System Safety, Elsevier, vol. 229(C).

    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. Chaogui Kang & Yu Liu & Diansheng Guo & Kun Qin, 2015. "A Generalized Radiation Model for Human Mobility: Spatial Scale, Searching Direction and Trip Constraint," PLOS ONE, Public Library of Science, vol. 10(11), pages 1-11, November.
    2. Li, Ze-Tao & Nie, Wei-Peng & Cai, Shi-Min & Zhao, Zhi-Dan & Zhou, Tao, 2023. "Exploring the topological characteristics of urban trip networks based on taxi trajectory data," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 609(C).
    3. Toru Nakamura & Toru Takumi & Atsuko Takano & Fumiyuki Hatanaka & Yoshiharu Yamamoto, 2013. "Characterization and Modeling of Intermittent Locomotor Dynamics in Clock Gene-Deficient Mice," PLOS ONE, Public Library of Science, vol. 8(3), pages 1-8, March.
    4. Barmak, D.H. & Dorso, C.O. & Otero, M., 2016. "Modelling dengue epidemic spreading with human mobility," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 447(C), pages 129-140.
    5. Christensen, Claire & Albert, István & Grenfell, Bryan & Albert, Réka, 2010. "Disease dynamics in a dynamic social network," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(13), pages 2663-2674.
    6. Li, Jun & Fu, Siyao & He, Haibo & Jia, Hongfei & Li, Yanzhong & Guo, Yi, 2015. "Simulating large-scale pedestrian movement using CA and event driven model: Methodology and case study," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 437(C), pages 304-321.
    7. Qianqian Liu & Qun Wang, 2017. "A comparative study on uncooperative search models in survivor search and rescue," Natural Hazards: Journal of the International Society for the Prevention and Mitigation of Natural Hazards, Springer;International Society for the Prevention and Mitigation of Natural Hazards, vol. 89(2), pages 843-857, November.
    8. Li, Yan & Ye, Hang & Zhang, Hong, 2016. "Evolution of cooperation driven by social-welfare-based migration," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 445(C), pages 48-56.
    9. Saberi, Meead & Ghamami, Mehrnaz & Gu, Yi & Shojaei, Mohammad Hossein (Sam) & Fishman, Elliot, 2018. "Understanding the impacts of a public transit disruption on bicycle sharing mobility patterns: A case of Tube strike in London," Journal of Transport Geography, Elsevier, vol. 66(C), pages 154-166.
    10. Paul Peeters & Martin Landré, 2011. "The Emerging Global Tourism Geography—An Environmental Sustainability Perspective," Sustainability, MDPI, vol. 4(1), pages 1-30, December.
    11. Tong Zhou & Xintao Liu & Zhen Qian & Haoxuan Chen & Fei Tao, 2019. "Dynamic Update and Monitoring of AOI Entrance via Spatiotemporal Clustering of Drop-Off Points," Sustainability, MDPI, vol. 11(23), pages 1-20, December.
    12. Daniel Austin & Robin M Cross & Tamara Hayes & Jeffrey Kaye, 2014. "Regularity and Predictability of Human Mobility in Personal Space," PLOS ONE, Public Library of Science, vol. 9(2), pages 1-8, February.
    13. Fernando Santa & Roberto Henriques & Joaquín Torres-Sospedra & Edzer Pebesma, 2019. "A Statistical Approach for Studying the Spatio-Temporal Distribution of Geolocated Tweets in Urban Environments," Sustainability, MDPI, vol. 11(3), pages 1-29, January.
    14. Rezapour, Shabnam & Baghaian, Atefe & Naderi, Nazanin & Sarmiento, Juan P., 2023. "Infection transmission and prevention in metropolises with heterogeneous and dynamic populations," European Journal of Operational Research, Elsevier, vol. 304(1), pages 113-138.
    15. Pengjun Zhao & Hao Wang & Qiyang Liu & Xiao-Yong Yan & Jingzhong Li, 2024. "Unravelling the spatial directionality of urban mobility," Nature Communications, Nature, vol. 15(1), pages 1-11, December.
    16. Shilin Xiao & Liming Zhang & Haihong Li & Qionglin Dai & Junzhong Yang, 2022. "Environment-driven migration enhances cooperation in evolutionary public goods games," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 95(4), pages 1-9, April.
    17. Yanyan Chen & Zheng Zhang & Tianwen Liang, 2019. "Assessing Urban Travel Patterns: An Analysis of Traffic Analysis Zone-Based Mobility Patterns," Sustainability, MDPI, vol. 11(19), pages 1-15, October.
    18. Fan Yang & Zhenxing Yao & Fan Ding & Huachun Tan & Bin Ran, 2019. "Understanding Urban Mobility Pattern with Cellular Phone Data: A Case Study of Residents and Travelers in Nanjing," Sustainability, MDPI, vol. 11(19), pages 1-17, October.
    19. Huang, Zhiren & Wang, Pu & Zhang, Fan & Gao, Jianxi & Schich, Maximilian, 2018. "A mobility network approach to identify and anticipate large crowd gatherings," Transportation Research Part B: Methodological, Elsevier, vol. 114(C), pages 147-170.
    20. Sun, Li & Zhao, Juanjuan & Zhang, Jun & Zhang, Fan & Ye, Kejiang & Xu, Chengzhong, 2024. "Activity-based individual travel regularity exploring with entropy-space K-means clustering using smart card data," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 636(C).

    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:eee:phsmap:v:389:y:2010:i:5:p:909-920. 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: Catherine Liu (email available below). General contact details of provider: http://www.journals.elsevier.com/physica-a-statistical-mechpplications/ .

    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.