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Entropy
Volume 27
Issue 6
10.3390/e27060552
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This is an early access version, the complete PDF, HTML, and XML versions will be available soon.
Article

Scalar-on-Function Mode Estimation Using Entropy and Ergodic Properties of Functional Time Series Data

by
Mohammed B. Alamari
1,
Fatimah A. Almulhim
2,
Ibrahim M. Almanjahie
1,
Salim Bouzebda
3,* and
Ali Laksaci
1
1
Department of Mathematics, College of Science, King Khalid University, Abha 62223, Saudi Arabia
2
Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
3
Université de Technologie de Compiègne, LMAC (Laboratory of Applied Mathematics of Compiègne), 60203 Compiègne, France
*
Author to whom correspondence should be addressed.
Entropy 2025, 27(6), 552; https://doi.org/10.3390/e27060552
Submission received: 18 April 2025 / Revised: 14 May 2025 / Accepted: 22 May 2025 / Published: 24 May 2025
(This article belongs to the Section Information Theory, Probability and Statistics)
Versions Notes

Abstract

In this paper, we investigate the recursive L1 estimator of the conditional mode when the input variable takes values in a pseudo-metric space. The new proposed estimator is constructed under an ergodicity assumption, which provides a robust alternative to the standard mixing processes in various practical settings. The particular interest of this contribution arises from the difficulty in incorporating the mathematical properties of a functional mixing process. In contrast, ergodicity is characterized by the Kolmogorov–Sinai entropy, which measures the dynamics, the sparsity, and the microscopic fluctuations of the functional process. Using an observation sampled from ergodic functional time series (fts), we establish the asymptotic properties of this estimator. In particular, we derive its convergence rate and show Borel–Cantelli (BC) consistency. The general expression for the convergence rate is then specialized to several notable scenarios, including the independence case, the classical kernel method, and the vector-valued case. Finally, numerical experiments on both simulated and real-world datasets demonstrate the superiority of the L1-recursive estimator compared to existing competitors.
Keywords: L1-modal regression; functional data; ergodic data; recursive estimate; nonparametric prediction; complete consistency; conditional mode; quantile regression L1-modal regression; functional data; ergodic data; recursive estimate; nonparametric prediction; complete consistency; conditional mode; quantile regression

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MDPI and ACS Style

Alamari, M.B.; Almulhim, F.A.; Almanjahie, I.M.; Bouzebda, S.; Laksaci, A. Scalar-on-Function Mode Estimation Using Entropy and Ergodic Properties of Functional Time Series Data. Entropy 2025, 27, 552. https://doi.org/10.3390/e27060552

AMA Style

Alamari MB, Almulhim FA, Almanjahie IM, Bouzebda S, Laksaci A. Scalar-on-Function Mode Estimation Using Entropy and Ergodic Properties of Functional Time Series Data. Entropy. 2025; 27(6):552. https://doi.org/10.3390/e27060552

Chicago/Turabian Style

Alamari, Mohammed B., Fatimah A. Almulhim, Ibrahim M. Almanjahie, Salim Bouzebda, and Ali Laksaci. 2025. "Scalar-on-Function Mode Estimation Using Entropy and Ergodic Properties of Functional Time Series Data" Entropy 27, no. 6: 552. https://doi.org/10.3390/e27060552

APA Style

Alamari, M. B., Almulhim, F. A., Almanjahie, I. M., Bouzebda, S., & Laksaci, A. (2025). Scalar-on-Function Mode Estimation Using Entropy and Ergodic Properties of Functional Time Series Data. Entropy, 27(6), 552. https://doi.org/10.3390/e27060552

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