TY  - JOUR
T1  - Spatial $k$NN-Local linear estimation for semi-functional partial linear regression
AU  - Tawfik, Benchikh
AU  - Baouche, Mohamed El Ouard
AU  - Fetitah, Omar
AU  - Guendouzi, Toufik
PY  - 2025
DA  - June
Y2  - 2025
DO  - 10.15672/hujms.1640600
JF  - Hacettepe Journal of Mathematics and Statistics
PB  - Hacettepe University
WT  - DergiPark
SN  - 2651-477X
SP  - 1164
EP  - 1186
VL  - 54
IS  - 3
LA  - en
AB  - 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.
KW  - Asymptotic normality
KW  - functional data analysis
KW  - {\it k}NN estimation
KW  - local linear estimation
KW  - partial linear regression
CR  - [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.
CR  - [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.
CR  - [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.
CR  - [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.
CR  - [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.
CR  - [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.
CR  - [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.
CR  - [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.
CR  - [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
CR  - [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.
CR  - [11] G. Aneiros-Pérez G. and P. Vieu, Nonparametric time series prediction. A semifunctional
partial linear modeling. J. Multivariate Anal. 99, 834-857, 2008.
CR  - [12] G. Aneiros-Pérez G. and P. Vieu, Semi-functional partial linear regression, Stat.
Probab. Lett. 76 (11), 1102-1110, 2006.
CR  - [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.
CR  - [14] M.K. Attouch, B. Chouaf and A. Laksaci, Nonparametric M-estimation for functional
spatial data, Commun. Stat. Appl. Methods 19, 193-211, 2012.
CR  - [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.
CR  - [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.
CR  - [17] A. Baíllo and A. Grané, Local linear regression for functional predictor and scalar
response, J. Multivariate Anal. 100 (1), 102-111, 2009.
CR  - [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.
CR  - [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.
CR  - [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.
CR  - [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.
CR  - [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.
CR  - [23] G. Boente and A. Vahnovan, Robust estimators in semi-functional partial linear
regression models. J. Multivariate Anal. 154 (C), 59-84, 2017.
CR  - [24] F. Burba, F. Ferraty and P. Vieu, k-Nearest Neighbour method in functional nonparametric
regression. J. Nonparametr. Stat. 21 (4), 453-469, 2009.
CR  - [25] M. Cameletti, R. Ignaccolo and S. Bande, Comparing spatio-temporal models for
particulate matter in Piemonte. Environmetrise 22 (8), 985-996, 2011.
CR  - [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.
CR  - [27] A. Chahad, L. Ait-Hennani, A. Laksaci, Functional local linear estimate for functional
relative-error regression, J. Stat. Theory Pract. 11, 771789, 2017.
CR  - [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.
CR  - [29] A. Chouaf and A. Laksaci, On the functional local linear estimate for spatial regression,
Stat. Risk Model 29, 189-214, 2013.
CR  - [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.
CR  - [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.
CR  - [32] H. Ding, Z. Lu, J. Zhang and R.n Zhang, Semi-functional partial linear quantile
regression, Stat Probab Lett. 142, 92-101, 2018.
CR  - [33] P. Doukhan, Mixing Properties and Examples. In: Lecture Notes in Statistics, 85,
Springer-Verlag, New York, 1994.
CR  - [34] J. Fan and I. Gijbels. Local polynomial modelling and its applications, London: Chapman
and Hall, 1996.
CR  - [35] S. Feng and L. Xue, Partially functional linear varying coefficient model. Statistics
50 (4), 717-732, 2016.
CR  - [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.
CR  - [37] F. Ferraty and P. Vieu, Nonparametric functional data analysis. Theory and Practice.
Springer Series in Statistics. New York, 2006.
CR  - [38] J. T. Gao and, Z. Lu and D. Tj'stheim D., Estimation in semiparametric spatial
regression. Ann. Stat. 34 (3), 1395-1435, 2066.
CR  - [39] S. Greven and F. Scheipl, A general framework for functional regression modelling.
Stat. Model. 17 (1-2), 1-35, 2017.
CR  - [40] X. Guyon, Random Fields on a Network-Modeling, Statistics and Applications,
Springer, New-York, 1995.
CR  - [41] M. Hallin, Z. Lu and K. Yu, Local Linear Spatial Quantile Regression, Bernoulli 15
(3), 659-686, 2009.
CR  - [42] T. Hsing and R.L. Eubank, Theoretical Foundations of Functional Data Analysis,
with an Introduction to Linear Operators, John Wiley and Sons, 2015.
CR  - [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.
CR  - [44] N. Kudraszow and P. Vieu P, Uniform consistency of kNN regressors for functional
variables, Statist. Probab. Lett. 83 (8), 1863-1870, 2013.
CR  - [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.
CR  - [46] A. Laksaci, M. Rachdi, S. Rahmani, Spatial modelization: local linear estimation of
the conditional distribution for functional data, Spatial Stat. 6, 123, 2013.
CR  - [47] T. Laloë, A k-nearest neighbor approach for functional regression. Stat. Probab. Lett.
78, 11891193, 2008.
CR  - [48] N. Ling N, G. Aneiros-Pérez and P. Vieu, knn estimation in functional partial linear
modeling. Statist. Papers 61 (1), 423-444, 2020.
CR  - [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.
CR  - [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.
CR  - [51] N. Ling and P. Vieu, Nonparametric modelling for functional data: selected survey
and tracks for future. Statistics 52 (4), 934-949, 2018.
CR  - [52] N. Ling and P. Vieu, On semiparametric regression in functional data analysis.
WIRES Comput. Stat. 12 (6), 20-30, 2020.
CR  - [53] J. Mateu and E. Romano, Advances in spatial functional statistics. Stoch. Environ.
Res. Risk. Assess. 31, 1-6, 2017.
CR  - [54] J. Mateu, R. Giraldo, Geostatistical Functional Data Analysis, Wiley Series in Probability
and Statistics, 1st Edition, 2021.
CR  - [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.
CR  - [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.
CR  - [57] S. Novo, G. Aneiros, and P. Vieu. A kNN procedure in semiparametric functional
data analysis, Statist. Probab. Lett. 171, 2021.
CR  - [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.
CR  - [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.
CR  - [60] M. Rachdi, A. Laksaci and N. M. Al-Kandari, Expectile regression for spatial functional
data analysis (sFDA), Metrika 85, 627655, 2022.
CR  - [61] J. Ramsay and B. Silverman, Functional Data Analysis, Second Edition, Spinger-
Verlag, New York, 2005.
CR  - [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.
CR  - [63] J. L. Wang, J. M. Chiou, and H. G. Müller, Functional data analysis. Annu. Rev.
Stat. Appl. 3 (1), 257-295, 2016.
CR  - [64] Y. Li and C. Ying, Semi-functional partial linear spatial autoregressive model, Commun.
Stat. - Theory Methods 50 (24), 2021.
CR  - [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.
UR  - https://doi.org/10.15672/hujms.1640600
L1  - https://dergipark.org.tr/en/download/article-file/4612960
ER  -