Scalar Variance and Scalar Correlation for Functional Data

Cristhian Leonardo Urbano-Leon (), Manuel Escabias, Diana Paola Ovalle-Muñoz and Javier Olaya-Ochoa
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Cristhian Leonardo Urbano-Leon: Department of Statistics and Operations Research, University of Granada, 18071 Granada, Spain
Manuel Escabias: Department of Statistics and Operations Research, University of Granada, 18071 Granada, Spain
Diana Paola Ovalle-Muñoz: Department of Statistics and Operations Research, University of Granada, 18071 Granada, Spain
Javier Olaya-Ochoa: School of Statistics, University of Valle, Cali 760042, Colombia

Mathematics, 2023, vol. 11, issue 6, 1-20

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.

Keywords: correlation for functional data; covariance for functional data; FDA; summary statistics in functional data; variance for functional data (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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