Spatial $k$NN-Local linear estimation for semi-functional partial linear regression

Year 2025, Volume: 54 Issue: 3, 1164 - 1186, 24.06.2025

Mohamed El Ouard Baouche , Benchikh Tawfik , Omar Fetitah , Toufik Guendouzi

https://doi.org/10.15672/hujms.1640600

Abstract

The objective of this paper is to investigate a semi-functional partial linear regression model for spatial data. The estimators are constructed using a $k$-nearest neighbors local linear method.Then, under suitable regularity conditions, we establish the asymptotic distribution of the parametric component and derive the uniform almost sure convergence rate for the nonparametric component. To assess the performance of the proposed estimators, we performed both simulation studies and real-data analyses. The results are compared with existing methods for semi-functional partial linear regression models using cross-validation. Specifically, we evaluate the predictive performance in terms of mean squared error and compare it against several benchmark estimators, including the kernel estimator, the local linear estimator and the $k$NN estimator. This practical study clearly demonstrates the feasibility and superiority of the local linear method estimator $k$-nearest neighbors over competing methods. This is evidenced by the lower mean squared error achieved by this estimator in both the simulation study and the real data application. These results indicate that this hybrid approach effectively addresses the common issue of bandwidth selection and yields estimators with reduced bias.

Keywords

Asymptotic normality , functional data analysis , {\it k}NN estimation , local linear estimation , partial linear regression

Ethical Statement

The authors contributed approximately equally to this work. All authors have read and agreed to the final version of the manuscript. Formal analysis, M.O. Baouche; Validation, O. Fetitah; Writing – review \& editing, T. Benchikh and T. Guendouzi.

Supporting Institution

This research was funded by Thematic Research Agency in Science and Technology (ATRST) for funding this work through research groups program under the project number PRFU, C00L03UN220120220002.

Thanks

The authors are indebted to the Editor-in-Chief and the referees for their very valuable comments and suggestions which led to a considerable improvement of the manuscript.

References

• [1] M. Abeidallah, B. Mechab and T. Merouan, Local linear estimate of the point at high risk: spatial functional data case, Commun. Stat. Theory Methods 49, 25612584, 2020.
• [2] M.S. Ahmed, M. N’diaye, M. Kadi Attouch and S. Dabo-Niange, k-nearest neighbors prediction and classification for spatial data, J. Spat. Econ. 4 (12), 2023.
• [3] I. M. Almanjahie, K. A. Assiri, A. Laksaci and Z. Chikr Elmezouar, The k nearest neighbors smoothing of the relative-error regression with functional regressor, Commun. Stat. Theory Methods 51, 41964209, 2022.
• [4] I.M Almanjahie, W.M. Alahmari and A. Laksaci, The k nearest neighbors local linear estimator of functional conditional density when there are missing data. Hacet. J. Math. Stat. 51 (3), 914-931, 2022.
• [5] I.M. Almanjahie, Z. Chikr-Elmezouar, A. Laksaci and M. Rachdi, kNN local linear estimation of the conditional cumulative distribution function: Dependent functional data case. C. R. Acad. Sci. Paris, Ser. I 356, 10361039, 2018.
• [6] I. M. Almanjahie, W. Mesfer, A. Laksaci and M. Rachdi Computational aspects of the k NN local linear smoothing for some conditional models in high dimensional statistics, Commun. Stat. Simul. Comput. 52 (7), 2985-3005, 2023.
• [7] F. Alshahrani, I.M. Almanjahi, T. Benchikh, O. Fetitah and M.K. Attouch, Asymptotic normality of nonparametric kernel regression estimation for missing at random functional spatial data, Journal of Mathematics, 2023, https://doi.org/10.1155/2023/8874880.
• [8] F. Alshahrani, W. Bouabsa, I.M. Almanjahie and M.K. Attouch kNN local linear estimation of the conditional density and mode for functional spatial high dimensional data. AIMS Math. 8 (7), 1584415875, 2023.
• [9] G. Aneiros-Pérez, I. Horov´a, M. Hu˜skov´a and P. Vieu, Editorial for the Special Issue on Functional Data Analysis and Related Fields, J. Multivariate Anal. 189, 2022. M
• [10] G. Aneiros-Pérez, R. Cao and P. Vieu, Editorial on the special issue on functional data analysis and related topics. Computational Statistics, 34, 447-450, 2019.
• [11] G. Aneiros-Pérez G. and P. Vieu, Nonparametric time series prediction. A semifunctional partial linear modeling. J. Multivariate Anal. 99, 834-857, 2008.
• [12] G. Aneiros-Pérez G. and P. Vieu, Semi-functional partial linear regression, Stat. Probab. Lett. 76 (11), 1102-1110, 2006.
• [13] M.K Attouch M and T. Benchikh, Asymptotic distribution of robust k-nearest neighbour estimator for functional nonparametric models, Mat. Vesnik 644, 275-285, 2012.
• [14] M.K. Attouch, B. Chouaf and A. Laksaci, Nonparametric M-estimation for functional spatial data, Commun. Stat. Appl. Methods 19, 193-211, 2012.
• [15] M.K. Attouch, A. Gheriballah and A. Laksaci, Robust nonparametric estimation for functional spatial regression. In F. Ferraty editor, Recent Advances in Functional Data Analysis and Related Topics, Contrib. Stat. 27-31. Physica-Verlag HD, 2011.
• [16] M.K. Attouch, A. Laksaci and F. Rafaa, Estimation locale linéaire de la régression non paramétrique fonctionnelle par la méthode des k plus proches voisins. Comptes Rendus. Mathématique 355 (7), 824-829, 2017.
• [17] A. Baíllo and A. Grané, Local linear regression for functional predictor and scalar response, J. Multivariate Anal. 100 (1), 102-111, 2009.
• [18] J. Barrientos-Marin J, F. Ferraty and P. Vieu, Locally modelled regression and functional data. J Nonparametr Stat. 22 (5-6), 617-632, 2009.
• [19] F. Belarbi, S. Chemikh and A. Laksaci, Local linear estimate of the nonparametric robust regression in functional data, Stat. Probabil. Lett. 134, 128133, 2018.
• [20] M. Benallou, M.K. Attouch, T. Benchikh and O. Fetitah, Asymptotic results of semifunctional partial linear regression estimate under functional spatial dependency. Commun. Stat. - Theory Methods 51 (20), 7172-7192, 2021.
• [21] T. Benchikh, I.M. Almanjahie, O. Fetitah and M.K. Attouch, Estimation for spatial semi-functional partial linear regression model with missing response at random, Demonstr. Math. 58, 2025, doi:10.1515/dema-2025-0108.
• [22] G. Biau, F. Cérou, and A. Guyader. Rates of convergence of the functional k-Nearest Neighbor estimate. IEEE Trans. Inf. Theory 56 (4), 2034-2040, 2010.
• [23] G. Boente and A. Vahnovan, Robust estimators in semi-functional partial linear regression models. J. Multivariate Anal. 154 (C), 59-84, 2017.
• [24] F. Burba, F. Ferraty and P. Vieu, k-Nearest Neighbour method in functional nonparametric regression. J. Nonparametr. Stat. 21 (4), 453-469, 2009.
• [25] M. Cameletti, R. Ignaccolo and S. Bande, Comparing spatio-temporal models for particulate matter in Piemonte. Environmetrise 22 (8), 985-996, 2011.
• [26] M. Carbon, L.T. Tran and B. Wu, Kernel density estimation for random fields (density estimation for random fields). Stat Probab Lett. 36 (2), 115-125, 1997.
• [27] A. Chahad, L. Ait-Hennani, A. Laksaci, Functional local linear estimate for functional relative-error regression, J. Stat. Theory Pract. 11, 771789, 2017.
• [28] Z. Chikr-Elmezouar, I.M. Almanjahie, A. Laksaci and M. Rachdi M. FDA: strong consistency of the k nn local linear estimation of the functional conditional density and mode. J. Nonparametric Stat. 31, 175195, 2019.
• [29] A. Chouaf and A. Laksaci, On the functional local linear estimate for spatial regression, Stat. Risk Model 29, 189-214, 2013.
• [30] S. Dabo-Niang, M. Rachdi M. and A.F. Yao, Kernel regression estimation for spatial functional random variables, Far East J. Theor. Stat. 37 (2), 77-113, 2011.
• [31] J. Demongeot, A. Naceri, A. Laksaci and M. Rachdi, Local linear regression modelization when all variables are curves. Statist. Probab. Lett. 121, 37-44, 2017.
• [32] H. Ding, Z. Lu, J. Zhang and R.n Zhang, Semi-functional partial linear quantile regression, Stat Probab Lett. 142, 92-101, 2018.
• [33] P. Doukhan, Mixing Properties and Examples. In: Lecture Notes in Statistics, 85, Springer-Verlag, New York, 1994.
• [34] J. Fan and I. Gijbels. Local polynomial modelling and its applications, London: Chapman and Hall, 1996.
• [35] S. Feng and L. Xue, Partially functional linear varying coefficient model. Statistics 50 (4), 717-732, 2016.
• [36] Ferraty F, Laksaci A, Tadj A, Vieu P. Rates of uniform consistency for nonparametric estimates with functional variables. J Stat Plan Infer 140 (2), 335352, 2010.
• [37] F. Ferraty and P. Vieu, Nonparametric functional data analysis. Theory and Practice. Springer Series in Statistics. New York, 2006.
• [38] J. T. Gao and, Z. Lu and D. Tj'stheim D., Estimation in semiparametric spatial regression. Ann. Stat. 34 (3), 1395-1435, 2066.
• [39] S. Greven and F. Scheipl, A general framework for functional regression modelling. Stat. Model. 17 (1-2), 1-35, 2017.
• [40] X. Guyon, Random Fields on a Network-Modeling, Statistics and Applications, Springer, New-York, 1995.
• [41] M. Hallin, Z. Lu and K. Yu, Local Linear Spatial Quantile Regression, Bernoulli 15 (3), 659-686, 2009.
• [42] T. Hsing and R.L. Eubank, Theoretical Foundations of Functional Data Analysis, with an Introduction to Linear Operators, John Wiley and Sons, 2015.
• [43] L. Kara-Zaitri, A. Laksaci, M. Rachdi and P. Vieu, Uniform in bandwidth consistency for various kernel estimators involving functional data, J. Nonparametr. Stat. 29 (1), 85-107, 2017.
• [44] N. Kudraszow and P. Vieu P, Uniform consistency of kNN regressors for functional variables, Statist. Probab. Lett. 83 (8), 1863-1870, 2013.
• [45] N.H. Kedir, T. Benchikh, A. Naceri and O. Fetitah, Local linear-kNN smoothing for semi-functional partial linear regression. Hacet. J. Math. Stat. 53 (2):537-555, 2024.
• [46] A. Laksaci, M. Rachdi, S. Rahmani, Spatial modelization: local linear estimation of the conditional distribution for functional data, Spatial Stat. 6, 123, 2013.
• [47] T. Laloë, A k-nearest neighbor approach for functional regression. Stat. Probab. Lett. 78, 11891193, 2008.
• [48] N. Ling N, G. Aneiros-Pérez and P. Vieu, knn estimation in functional partial linear modeling. Statist. Papers 61 (1), 423-444, 2020.
• [49] N. Ling, R. Kan, P. Vieu and S. Meng, Semi-functional partially linear regression model with responses missing at random. Metrika 82 (1), 39-70, 2019.
• [50] N. Ling, S. Meng and P. Vieu, Uniform consistency rate of kNN regression estimation for functional time series data, J. Nonparametr. Stat. 31 (2),451-468, 2019.
• [51] N. Ling and P. Vieu, Nonparametric modelling for functional data: selected survey and tracks for future. Statistics 52 (4), 934-949, 2018.
• [52] N. Ling and P. Vieu, On semiparametric regression in functional data analysis. WIRES Comput. Stat. 12 (6), 20-30, 2020.
• [53] J. Mateu and E. Romano, Advances in spatial functional statistics. Stoch. Environ. Res. Risk. Assess. 31, 1-6, 2017.
• [54] J. Mateu, R. Giraldo, Geostatistical Functional Data Analysis, Wiley Series in Probability and Statistics, 1st Edition, 2021.
• [55] A. Naceri, A. Laksaci and M. Rachdi, Exact quadratic error of the local linear regression operator estimator for functional covariates. In Functional statistics and applications, 79-90, Springer Cham Heidelberg, New York, 2019.
• [56] M. Ndiaye, S. Dabo-Niang, P. Ngom, Nonparametric Prediction for Spatial Dependent Functional Data Under Fixed Sampling Design, Rev. Colomb. Estad. 45 (2), 391-428, 2022.
• [57] S. Novo, G. Aneiros, and P. Vieu. A kNN procedure in semiparametric functional data analysis, Statist. Probab. Lett. 171, 2021.
• [58] M. Rachdi M. Functional Data Analysis: Theory and Applications to Different Scenarios. Mathematics, an Open Access Journal by MDPI 2023. https://www.mdpi.com/journal/mathematics/special issues/45POZ9BG9S.
• [59] M. Rachdi, A. Laksaci, K. Kaid, A. Benchiha A and F. Al-Awadh, k-Nearest neighbors local linear regression for functional and missing data at random. Stat. Neerl. 75 (1), 42-65, 2021.
• [60] M. Rachdi, A. Laksaci and N. M. Al-Kandari, Expectile regression for spatial functional data analysis (sFDA), Metrika 85, 627655, 2022.
• [61] J. Ramsay and B. Silverman, Functional Data Analysis, Second Edition, Spinger- Verlag, New York, 2005.
• [62] A. Saadaoui, F Benaissa and A. Chouaf, On the local linear estimation of a generalized regression function with spatial functional data, Commun. Stat. - Theory Methods 52 (21), 2023.
• [63] J. L. Wang, J. M. Chiou, and H. G. Müller, Functional data analysis. Annu. Rev. Stat. Appl. 3 (1), 257-295, 2016.
• [64] Y. Li and C. Ying, Semi-functional partial linear spatial autoregressive model, Commun. Stat. - Theory Methods 50 (24), 2021.
• [65] Y. Hu, S. Wu and S. Feng, Estimation in functional partially linear spatial autoregressive model, Hacet. J. Math. Stat. 53 (4), 1196 1217, 2024.