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Scalar Variance and Scalar Correlation for Functional Data

Author

Listed:
  • Cristhian Leonardo Urbano-Leon

    (Department of Statistics and Operations Research, University of Granada, 18071 Granada, Spain
    These authors contributed equally to this work.)

  • Manuel Escabias

    (Department of Statistics and Operations Research, University of Granada, 18071 Granada, Spain
    These authors contributed equally to this work.)

  • Diana Paola Ovalle-Muñoz

    (Department of Statistics and Operations Research, University of Granada, 18071 Granada, Spain
    These authors contributed equally to this work.)

  • Javier Olaya-Ochoa

    (School of Statistics, University of Valle, Cali 760042, Colombia
    These authors contributed equally to this work.)

Abstract

In Functional Data Analysis (FDA), the existing summary statistics so far are elements in the Hilbert space L 2 of square-integrable functions. These elements do not constitute an ordered set; therefore, they are not sufficient to solve problems related to comparability such as obtaining a correlation measurement or comparing the variability between two sets of curves, determining the efficiency and consistency of a functional estimator, among other things. Consequently, we present an approach of coherent redefinition of some common summary statistics such as sample variance, sample covariance and correlation in Functional Data Analysis (FDA). Regarding variance, covariance and correlation between functional data, our summary statistics lead to numbers instead of functions which is helpful for solving the aforementioned problems. Furthermore, we briefly discuss the functional forms coherence of some statistics already present in the FDA. We formally enumerate and demonstrate some properties of our functional summary statistics. Then, a simulation study is presented briefly, with evidence of the consistency of the proposed variance. Finally, we present the implementation of our statistics through two application examples.

Suggested Citation

  • Cristhian Leonardo Urbano-Leon & Manuel Escabias & Diana Paola Ovalle-Muñoz & Javier Olaya-Ochoa, 2023. "Scalar Variance and Scalar Correlation for Functional Data," Mathematics, MDPI, vol. 11(6), pages 1-20, March.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:6:p:1317-:d:1091925
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    References listed on IDEAS

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    1. Aguilera, Ana M. & Acal, Christian & Aguilera-Morillo, M. Carmen & Jiménez-Molinos, Francisco & Roldán, Juan B., 2021. "Homogeneity problem for basis expansion of functional data with applications to resistive memories," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 186(C), pages 41-51.
    2. Th. Gasser & P. Hall & B. Presnell, 1998. "Nonparametric estimation of the mode of a distribution of random curves," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 60(4), pages 681-691.
    3. J. Ramsay, 1982. "When the data are functions," Psychometrika, Springer;The Psychometric Society, vol. 47(4), pages 379-396, December.
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