Spatial regression estimation for functional data with spatial dependency

Ternynck, Camille

Journal de la société française de statistique, Tome 155 (2014) no. 2, pp. 138-160.

Nous proposons un estimateur non paramétrique de la fonction de régression d’une variable spatiale, $Y_{\mathbf{i}}$, scalaire conditionnellement à une variable, $X_{\mathbf{i}}$, fonctionnelle. La spécificité de l’estimateur proposé est de dépendre de deux noyaux permettant de contrôler à la fois la distance entre les observations et les sites. La convergence en moyenne quadratique de cet estimateur est obtenue quand l’échantillon considéré est une séquence $\alpha$-mélangeante. Pour terminer, des résultats numériques illustrent le comportement de notre estimateur.

We propose a nonparametric estimator of the regression function of a scalar spatial variable $Y_{\mathbf{i}}$ given a functional variable $X_{\mathbf{i}}$. The specificity of the proposed estimator is to depend on two kernels in order to control both the distance between observations and spatial locations. Mean square consistency of this estimator is obtained when the sample considered is an $\alpha$-mixing sequence. Lastly, numerical results are provided to illustrate the behavior of our estimator.

[1] Anselin, Luc; Florax, Raymond J G M New Directions in Spatial Econometrics, Advances in Spatial Science, Springer, 1995 | Zbl

[2] Attouch, Mohammed K; Gheriballah, Abdelkader; Laksaci, Ali Robust Nonparametric Estimation for Functional Spatial Regression, Recent Advances in Functional Data Analysis and Related Topics (Ferraty, Frédéric, ed.) (Contributions to Statistics), Physica-Verlag HD, 2011, pp. 27-31

[3] Biau, Gérard; Cadre, Benoît Nonparametric spatial prediction, Statistical Inference for Stochastic Processes, Volume 7 (2004) no. 3, pp. 327-349 | Zbl

[4] Bosq, Denis Linear Processes in Function Spaces: Theory and Applications, Lecture Notes in Statistics, 149, Springer-Verlag New York Inc, 2000 | Zbl

[5] Chilès, Jean-Paul; Delfiner, Pierre Geostatistics: Modeling Spatial Uncertainty, Wiley Series in Applied Probability and Statistics, John Wiley & Sons, Inc, 1999 | Zbl

[6] Carbon, Michel; Francq, Christian; Tran, Lanh Tat Kernel regression estimation for random fields, Journal of Statistical Planning and Inference, Volume 137 (2007) no. 3, pp. 778-798 | Zbl

[7] Chamizo, Lorente Fernando; Iwaniec, Henryk On the sphere problem, Revista matemática iberoamericana, Volume 11 (1995) no. 2, pp. 417-430 | Zbl

[8] Cressie, Noel A C Statistics for Spatial Data, Wiley Series in Probability and Statistics, 110, Wiley-Interscience, 1993 | Zbl

[9] Carbon, Michel; Tran, Lanh Tat; Wu, Berlin Kernel density estimation for random fields (density estimation for random fields), Statistics & Probability Letters, Volume 36 (1997) no. 2, pp. 115-125 | Zbl

[10] Delsol, Laurent Régression sur variable fonctionnelle: Estimation, tests de structure et Applications., Université Paul Sabatier-Toulouse III (2008) (Ph. D. Thesis)

[11] Delicado, Pedro; Giraldo, Ramon; Comas, Carlos; Mateu, Jorge Statistics for spatial functional data: some recent contributions, Environmetrics, Volume 21 (2010) no. 3-4, pp. 224-239

[12] Dabo-Niang, Sophie; Hamdad, Leila; Ternynck, Camille; Yao, Anne-Françoise A kernel spatial density estimation allowing for the analysis of spatial clustering: application to Monsoon Asia Drought Atlas data, In revision (2013)

[13] Dabo-Niang, Sophie; Kaid, Zoulikha; Laksaci, Ali On spatial conditional mode estimation for a functional regressor, Statistics & Probability Letters, Volume 82 (2012) no. 7, pp. 1413-1421 | Zbl

[14] Dabo-Niang, Sophie; Rachdi, Mustapha; Yao, Anne-Françoise Kernel regression estimation for spatial functional random variables, Far East Journal of Theoretical Statistics, Volume 37 (2011) no. 2, pp. 77-113 | Zbl

[15] Dabo-Niang, Sophie; Ternynck, Camille; Yao, Anne-Françoise Spatial regression for multivariate data taking spatial characteristics into consideration and applications, Preprint (2014)

[16] Dabo-Niang, Sophie; Yao, Anne-Françoise Kernel regression estimation for continuous spatial processes, Mathematical Methods of Statistics, Volume 16 (2007) no. 4, pp. 298-317 | Zbl

[17] Dabo-Niang, Sophie; Yao, Anne-Françoise Kernel spatial density estimation in infinite dimension space, Metrika, Volume 76 (2013) no. 1, pp. 19-52 | Zbl

[18] Dabo-Niang, Sophie; Yao, Anne-Françoise; Pischedda, Laura; Cuny, Philippe; Gilbert, Franck Spatial mode estimation for functional random fields with application to bioturbation problem, Stochastic Environmental Research and Risk Assessment, Volume 24 (2010) no. 4, pp. 487-497 | Zbl

[19] El Machkouri, Mohamed Asymptotic normality of the Parzen–Rosenblatt density estimator for strongly mixing random fields, Statistical Inference for Stochastic Processes, Volume 14 (2011) no. 1, pp. 73-84 | Zbl

[20] Ferraty, Frédéric; Vieu, Philippe Nonparametric Functional Data Analysis: Theory and Practice, Springer Series in Statistics, Springer, 2006 | Zbl

[21] García-Soidán, Pilar; Menezes, Raquel Estimation of the spatial distribution through the kernel indicator variogram, Environmetrics, Volume 23 (2012) no. 6, pp. 535-548

[22] Guyon, Xavier Random Fields on a Network: Modeling, Statistics, and Applications, Probability and its Applications, Springer-Verlag, 1995 | Zbl

[23] Hallin, Marc; Lu, Zudi; Tran, Lanh T Local linear spatial regression, The Annals of Statistics, Volume 32 (2004) no. 6, pp. 2469-2500 | Zbl

[24] Hallin, Marc; Lu, Zudi; Yu, Keming Local linear spatial quantile regression, Bernoulli, Volume 15 (2009) no. 3, pp. 659-686 | Zbl

[25] Journel, Andre G Nonparametric estimation of spatial distributions, Journal of the International Association for Mathematical Geology, Volume 15 (1983) no. 3, pp. 445-468

[26] Kelejian, Harry H; Prucha, Ingmar R HAC estimation in a spatial framework, Journal of Econometrics, Volume 140 (2007) no. 1, pp. 131-154 | Zbl

[27] Laksaci, Ali; Maref, Fouzia Estimation non paramétrique de quantiles conditionnels pour des variables fonctionnelles spatialement dépendantes, Comptes Rendus Mathematique, Volume 347 (2009) no. 17-18, pp. 1075-1080 | Zbl

[28] Laksaci, Ali; Mechab, Boubakeur Estimation non paramétrique de la fonction de hasard avec variable explicative fonctionnelle: cas des données spatiales, Revue Roumaine de Mathématiques Pures et Appliquées, Volume 55 (2010) no. 1, pp. 35-51 | Zbl

[29] Meyer, Annika On the number of lattice points in a small sphere, WCC 2011 - Workshop on coding and cryptography (2011), pp. 463-472

[30] Menezes, Raquel; García-Soidán, Pilar; Ferreira, Célia Nonparametric spatial prediction under stochastic sampling design, Journal of Nonparametric Statistics, Volume 22 (2010) no. 3, pp. 363-377 | Zbl

[31] Mitchell, WC The Number of Lattice Points in a $k$-Dimensional Hypersphere, Mathematics of Computation, Volume 20 (1966) no. 94, pp. 300-310 | Zbl

[32] Nerini, David; Monestiez, Pascal; Manté, Claude Cokriging for spatial functional data, Journal of Multivariate Analysis, Volume 101 (2010) no. 2, pp. 409-418 | Zbl

[33] Ripley, Brian D Spatial Statistics, Wiley Series in Probability and Statistics, John Wiley & Sons, Inc, 1981 | Zbl

[34] Rosenblatt, Murray Remarks on some nonparametric estimates of a density function, The Annals of Mathematical Statistics, Volume 27 (1956) no. 3, pp. 832-837 | Zbl

[35] Ramsay, Jim O; Silverman, Bernard W Functional Data Analysis, Springer Series in Statistics, Springer, 2005

[36] Tran, Lanh Tat Kernel density estimation on random fields, Journal of Multivariate Analysis, Volume 34 (1990) no. 1, pp. 37-53 | Zbl

[37] Tsang, Kai-Man Counting lattice points in the sphere, Bulletin of the London Mathematical Society, Volume 32 (2000) no. 6, pp. 679-688 | Zbl

[38] Wang, Hongxia; Wang, Jinde Estimation of the trend function for spatio-temporal models, Journal of Nonparametric Statistics, Volume 21 (2009) no. 5, pp. 567-588 | Zbl

[39] Wang, Hongxia; Wang, Jinde; Huang, Bo Prediction for spatio-temporal models with autoregression in errors, Journal of Nonparametric Statistics, Volume 24 (2012) no. 1, pp. 217-244 | Zbl