IDEAS home Printed from https://ideas.repec.org/p/hal/journl/hal-04506343.html
   My bibliography  Save this paper

An abelian way approach to study random extended intervals and their ARMA processes

Author

Listed:
  • Babel Raïssa Guemdjo Kamdem

    (Université de Douala)

  • Jules Sadefo Kamdem

    (MRE - Montpellier Recherche en Economie - UM - Université de Montpellier)

  • Carlos Ogouyandjou

    (IMSP - Institut de Mathématiques et de Sciences Physiques - UAC - Université d’Abomey-Calavi = University of Abomey Calavi)

Abstract

An extended interval is a range A = [A, A] where A may be bigger than A. This is not really natural, but is what has been used as the definition of an extended interval so far. In the present work we introduce a new, natural, and very intuitive way to see an extended interval. From now on, an extended interval is a subset of the Cartesian product R × Z2, where Z2 = {0, 1} is the set of directions; the direction 0 is for increasing intervals, and the direction 1 for decreasing ones. For instance, [3, 6] × {1} is the decreasing version of [6, 3]. Thereafter, we introduce on the set of extended intervals a family of metrics dγ, depending on a function γ(t), and show that there exists a unique metric dγ for which γ(t)dt is what we have called an "adapted measure". This unique metric has very good properties, is simple to compute, and has been implemented in the software R. Furthermore, we use this metric to define variability for random extended intervals. We further study extended interval-valued ARMA time series and prove the Wold decomposition theorem for stationary extended interval-valued times series.

Suggested Citation

  • Babel Raïssa Guemdjo Kamdem & Jules Sadefo Kamdem & Carlos Ogouyandjou, 2024. "An abelian way approach to study random extended intervals and their ARMA processes," Post-Print hal-04506343, HAL.
  • Handle: RePEc:hal:journl:hal-04506343
    DOI: 10.3934/dsfe.2024005
    Note: View the original document on HAL open archive server: https://hal.science/hal-04506343v1
    as

    Download full text from publisher

    File URL: https://hal.science/hal-04506343v1/document
    Download Restriction: no

    File URL: https://libkey.io/10.3934/dsfe.2024005?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
    ---><---

    References listed on IDEAS

    as
    1. Billard L. & Diday E., 2003. "From the Statistics of Data to the Statistics of Knowledge: Symbolic Data Analysis," Journal of the American Statistical Association, American Statistical Association, vol. 98, pages 470-487, January.
    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. Miguel de Carvalho & Gabriel Martos, 2022. "Modeling interval trendlines: Symbolic singular spectrum analysis for interval time series," Journal of Forecasting, John Wiley & Sons, Ltd., vol. 41(1), pages 167-180, January.
    2. Gil, Maria Angeles & Gonzalez-Rodriguez, Gil & Colubi, Ana & Montenegro, Manuel, 2007. "Testing linear independence in linear models with interval-valued data," Computational Statistics & Data Analysis, Elsevier, vol. 51(6), pages 3002-3015, March.
    3. Drago, Carlo, 2015. "Exploring the Community Structure of Complex Networks," MPRA Paper 81024, University Library of Munich, Germany.
    4. Philip Hans Franses & Max Welz, 2022. "Evaluating heterogeneous forecasts for vintages of macroeconomic variables," Journal of Forecasting, John Wiley & Sons, Ltd., vol. 41(4), pages 829-839, July.
    5. Dias, Sónia & Brito, Paula & Amaral, Paula, 2021. "Discriminant analysis of distributional data via fractional programming," European Journal of Operational Research, Elsevier, vol. 294(1), pages 206-218.
    6. A. Pedro Duarte Silva & Peter Filzmoser & Paula Brito, 2018. "Outlier detection in interval data," 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. 12(3), pages 785-822, September.
    7. J. Le-Rademacher & L. Billard, 2013. "Principal component histograms from interval-valued observations," Computational Statistics, Springer, vol. 28(5), pages 2117-2138, October.
    8. Sun, Yuying & Han, Ai & Hong, Yongmiao & Wang, Shouyang, 2018. "Threshold autoregressive models for interval-valued time series data," Journal of Econometrics, Elsevier, vol. 206(2), pages 414-446.
    9. Lima Neto, Eufrásio de A. & de Carvalho, Francisco de A.T., 2010. "Constrained linear regression models for symbolic interval-valued variables," Computational Statistics & Data Analysis, Elsevier, vol. 54(2), pages 333-347, February.
    10. Antonio Balzanella & Antonio Irpino, 2020. "Spatial prediction and spatial dependence monitoring on georeferenced data streams," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 29(1), pages 101-128, March.
    11. Paolo Giordani, 2015. "Lasso-constrained regression analysis for interval-valued data," 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. 9(1), pages 5-19, March.
    12. Hao, Peng & Guo, Junpeng, 2017. "Constrained center and range joint model for interval-valued symbolic data regression," Computational Statistics & Data Analysis, Elsevier, vol. 116(C), pages 106-138.
    13. Fei Liu & L. Billard, 2022. "Partition of Interval-Valued Observations Using Regression," Journal of Classification, Springer;The Classification Society, vol. 39(1), pages 55-77, March.
    14. Christiane Guinot & Denis Malvy & Jean-François Schémann & Filipe Afonso & Raja Haddad & Edwin Diday, 2015. "Strategies evaluation in environmental conditions by symbolic data analysis: application in medicine and epidemiology to trachoma," 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. 9(1), pages 107-119, March.
    15. Maia, André Luis Santiago & de Carvalho, Francisco de A.T., 2011. "Holt's exponential smoothing and neural network models for forecasting interval-valued time series," International Journal of Forecasting, Elsevier, vol. 27(3), pages 740-759, July.
    16. Yan Sun & Guanghua Lian & Zudi Lu & Jennifer Loveland & Isaac Blackhurst, 2020. "Modeling the Variance of Return Intervals Toward Volatility Prediction," Journal of Time Series Analysis, Wiley Blackwell, vol. 41(4), pages 492-519, July.
    17. Sun, Yuying & Zhang, Xinyu & Wan, Alan T.K. & Wang, Shouyang, 2022. "Model averaging for interval-valued data," European Journal of Operational Research, Elsevier, vol. 301(2), pages 772-784.
    18. Carlo Drago, 2021. "The Analysis and the Measurement of Poverty: An Interval-Based Composite Indicator Approach," Economies, MDPI, vol. 9(4), pages 1-17, October.
    19. Massimo Aria & Antonio D’Ambrosio & Carmela Iorio & Roberta Siciliano & Valentina Cozza, 2020. "Dynamic recursive tree-based partitioning for malignant melanoma identification in skin lesion dermoscopic images," Statistical Papers, Springer, vol. 61(4), pages 1645-1661, August.
    20. Nataša Kejžar & Simona Korenjak-Černe & Vladimir Batagelj, 2021. "Clustering of modal-valued symbolic data," 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. 15(2), pages 513-541, June.

    More about this item

    Keywords

    Random set; Random extended interval; Distance; Measure; Time series;
    All these keywords.

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    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:hal:journl:hal-04506343. 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: CCSD (email available below). General contact details of provider: https://hal.archives-ouvertes.fr/ .

    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.