Published

2022-07-14

Nonparametric Prediction for Spatial Dependent Functional Data Under Fixed Sampling Design

Predicción no paramétrica para datos funcionales dependientes del espacio bajo un diseño de muestreo fijo

DOI:

https://doi.org/10.15446/rce.v45n2.98957

Keywords:

Functional dependent data, fixed design, non-parametric prediction, Supervised classification (en)
Clasificación supervisada, Datos funcionales dependientes, Diseño fijo, Predicción no paramétrica (es)

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Authors

  • Mamadou Ndiaye Higher Polytechnic School (ESP), Cheikh Anta Diop University (UCAD), Dakar, Senegal
  • Sophie Dabo-Niang Department of Mathematics, University of Lille, Villeneuve d'ascq, France
  • Papa Ngom Department of Mathematics and Computer Science, Faculty of Sciences and Technologies, Cheikh Anta Diop University, Dakar, Senegal

In this work, we consider a nonparametric prediction of a spatiofunctional process observed under a non-random sampling design. The proposed predictor is based on functional regression and depends on two kernels, one of which controls the spatial structure and the other measures the proximity between the functional observations. It can be considered, in particular, as a supervised classification method when the variable of interest belongs to a predefined discrete finite set. The mean square error and almost complete (or sure) convergence are obtained when the sample considered is a locally stationary α-mixture sequence. Numerical studies were performed to illustrate the behavior of the proposed predictor. The finite sample properties based on simulated data show that the proposed prediction method outperformsthe classical predictor which not taking into account the spatial structure.

En este trabajo consideramos una predicción no paramétrica de un proceso espacial y funcional observado bajo un diseño de muestreo no aleatorio. El predictor propuesto se basa en la regresión funcional y depende de dos núcleos, uno de los cuales controla la estructura espacial y el otro mide la proximidad entre las observaciones funcionales. Esta metodología puede considerarse, en particular, como una nueva herramienta de clasificación supervisada cuando la variable de interés pertenece a un conjunto finito discreto predefinido. El error cuadrático medio y la convergencia casi completa (o certera) se obtienen cuando la muestra considerada es una secuencia α-mixta localmente estacionaria. Además, en este estudio se han realizado estudios numéricos para ilustrar el comportamiento de nuestro predictor. Esta aplicación mediante simulación de un modelo numérico muestra que el método de predicción propuesto supera al predictor clásico que no tiene en cuenta la estructura espacial.

References

Ahmed, M. S., Ndiaye, M., Attouch, M. & Dabo-Niang, S. (2019), ‘k-nearest neighbors prediction and classification for spatial data’, preprinted.

Baouche, R. (2015), Prédiction des Paramètres Physiques des Couches Pétrolifères par Analyse des Réseaux de Neurones et Analyse Faciologique., PhD thesis, université M’hamed Bougara. Boumerdès.

Biau, G. & Cadre, B. (2004), ‘Nonparametric spatial prediction’, Statistical Inference for Stochastic Processes 7(3), 327–349. DOI: https://doi.org/10.1023/B:SISP.0000049116.23705.88

Biau, G. & Devroye, L. (2015), Lectures on the nearest neighbor method, Springer. DOI: https://doi.org/10.1007/978-3-319-25388-6

Bosq, D. (1998), Nonparametric Statistics for Stochastic Processes: Estimation and prediction, Vol. 110 of Lecture Notes in Statist., 2nd edn, Springer-Verlag, New York. DOI: https://doi.org/10.1007/978-1-4612-1718-3

Carbon, M., Tran, L. T. & Wu, B. (1997), ‘Kernel density estimation for random fields’, Statistics & Probability Letters 36(2), 115–125. DOI: https://doi.org/10.1016/S0167-7152(97)00054-0

Cressie, N. A. C. (1993), Statistics for Spatial Data, Vol. 110 of Wiley Series in Probability and Statistics, revised edn, Wiley-Interscience. DOI: https://doi.org/10.1002/9781119115151

Cuesta-Albertos, J. A., Febrero-Bande, M. & de la Fuente, M. O. (2017), ‘The ddG-classifier in the functional setting’, Test 26(1), 119–142. DOI: https://doi.org/10.1007/s11749-016-0502-6

Cuevas, A., Febrero, M. & Fraiman, R. (2007), ‘Robust estimation and classification for functional data via projection-based depth notions’, Computational Statistics 22(3), 481–496. DOI: https://doi.org/10.1007/s00180-007-0053-0

Dabo-Niang, S., Hamdad, L., Ternynck, C. & Yao, A.-F. c. c. (2014), ‘A kernel spatial density estimation allowing for the analysis of spatial clustering: application to Monsoon Asia Drought Atlas data’, Stoch. Environ. Res. Risk Assess 28(8), 2075–2099. DOI: https://doi.org/10.1007/s00477-014-0903-6

Dabo-Niang, S., Rachdi, M. & Yao, A.-F. (2011), ‘Kernel regression estimation for spatial functional random variables’, Far East Journal of Theoretical Statistics 37(2), 77–113.

Dabo-Niang, S., Ternynck, C. & Yao, A.-F. (2016), ‘Nonparametric prediction of spatial multivariate data’, Nonparametric Statistics., 2., 428-458 . DOI: https://doi.org/10.1080/10485252.2016.1164313

Dabo-Niang, S. & Yao, A.-F. (2007), ‘Kernel regression estimation for continuous spatial processes’, Mathematical Methods of Statistics 16(4), 298–317. DOI: https://doi.org/10.3103/S1066530707040023

Dabo-Niang, S. & Yao, A.-F. (2013), ‘Kernel spatial density estimation in infinite dimension space’, Metrika 76(1), 19–52. DOI: https://doi.org/10.1007/s00184-011-0374-4

Dabo-Niang, S., Yao, A.-F. c. c., Pischedda, L., Cuny, P. & Gilbert, F. (2010), ‘Spatial mode estimation for functional random fields with application to bioturbation problem’, Stochastic Environmental Research and Risk Assessment 24(4), 487–497. DOI: https://doi.org/10.1007/s00477-009-0339-6

Devroye, L., Gyorfi, L., Krzyzak, A. & Lugosi, G. (1994), ‘On the strong universal consistency of nearest neighbor regression function estimates’, The Annals of Statistics . DOI: https://doi.org/10.1214/aos/1176325633

Devroye, L. & Wagner, T. J. (1982), ‘8 nearest neighbor methods in discrimination’, Handbook of Statistics . DOI: https://doi.org/10.1016/S0169-7161(82)02011-2

El Machkouri, M. (2007), ‘Nonparametric regression estimation for random fields in a fixed-design’, Stat. Inference Stoch. Process. 10(1), 29–47. DOI: https://doi.org/10.1007/s11203-005-7332-6

El Machkouri, M. (2011), ‘Asymptotic normality of the Parzen–Rosenblatt density estimator for strongly mixing random fields’, Statistical Inference for Stochastic Processes 14(1), 73–84. DOI: https://doi.org/10.1007/s11203-011-9052-4

El Machkouri, M. & Stoica, R. (2010), ‘Asymptotic normality of kernel estimates in a regression model for random fields’, J. Nonparametr. Stat. 22(8), 955–971. DOI: https://doi.org/10.1080/10485250903505893

Escabias, M., Aguilera, A. & Valderrama, M. (2005), ‘Modeling environmental data by functional principal component logistic regression’, Environmetrics: The official journal of the International Environmetrics Society 16(1), 95–107. DOI: https://doi.org/10.1002/env.696

Ferraty, F. & Vieu, P. (2006), Nonparametric Functional Data Analysis: Theory and Practice, Springer Series in Statistics, Springer.

Francisco-Fernández, M. & Opsomer, J. D. (2005), ‘Smoothing parameter selection methods for nonparametric regression with spatially correlated errors’, Canad. J. Statist. 33(2), 279–295. http://dx.doi.org/10.1002/cjs.5550330208 DOI: https://doi.org/10.1002/cjs.5550330208

Francisco-Fernández, M., Quintela-del Río, A. & Fernández-Casal, R. (2012), ‘Nonparametric methods for spatial regression. an application to seismic events’, Environmetrics 23(1), 85–93. DOI: https://doi.org/10.1002/env.1146

Gardner, B., Sullivan, P. J., Morreale, S. J. & Epperly, S. P. (2008), ‘Spatial and temporal statistical analysis of bycatch data: patterns of sea turtle bycatch in the north atlantic’, Canadian Journal of Fisheries and Aquatic Sciences 65(11), 2461–2470. DOI: https://doi.org/10.1139/F08-152

Giraldo, R., Delicado, P. & Mateu, J. (2011), ‘Ordinary kriging for function-valued spatial data’, Environmental and Ecological Statistics 18(3), 411–426. DOI: https://doi.org/10.1007/s10651-010-0143-y

Hallin, M., Lu, Z. & Tran, L. T. (2004), ‘Local linear spatial regression’, The Annals of Statistics 32(6), 2469–2500. DOI: https://doi.org/10.1214/009053604000000850

Hastie, T. & Tibshirani, R. (1996), ‘Discriminant adaptive nearest neighbor classification and regression’, Advances in Neural Information Processing Systems . DOI: https://doi.org/10.1109/34.506411

Heppell, S. S., Crowder, L. B. & Menzel, T. R. (1999), Life table analysis of long-lived marine species with implications for conservation and management, in ‘American Fisheries Society Symposium’, Vol. 23, pp. 137–148.

Ignaccolo, R., Ghigo, S. & Bande, S. (2013), ‘Functional zoning for air quality’, Environmental and ecological statistics 20(1), 109–127. DOI: https://doi.org/10.1007/s10651-012-0210-7

Jorge, M. & Romano, E. (2016), ‘Advances in spatial functional statistics’, STOCHASTIC ENVIRONMENTAL RESEARCH AND RISK ASSESSMENT .

Klemelä, J. (2008), ‘Density estimation with locally identically distributed data and with locally stationary data’, J. Time Ser. Anal. 29(1), 125–141. *http://dx.doi.org/10.1111/j.1467-9892.2007.00547.x DOI: https://doi.org/10.1111/j.1467-9892.2007.00547.x

Lefort, R., Fablet, R., Berger, L. & Boucher, J.-M. (2011), ‘Spatial statistics of objects in 3-d sonar images: application to fisheries acoustics’, IEEE Geoscience and Remote Sensing Letters 9(1), 56–59. DOI: https://doi.org/10.1109/LGRS.2011.2160328

Li, X., Ghosal, S. et al. (2018), ‘Bayesian classification of multiclass functional data’, Electronic Journal of DOI: https://doi.org/10.1214/18-EJS1522

Statistics 12(2), 4669–4696.

Luan, J., Zhang, C., Xu, B., Xue, Y. & Ren, Y. (2018), ‘Modelling the spatial distribution of three portunidae crabs in haizhou bay, china’, PloS one 13(11), e0207457. DOI: https://doi.org/10.1371/journal.pone.0207457

Masry, E. (2005), ‘Nonparametric regression estimation for dependent functional data: asymptotic normality’, Stochastic Process. Appl. 115(1), 155–177. DOI: https://doi.org/10.1016/j.spa.2004.07.006

Menafoglio, A. (2021), Spatial statistics for distributional data in bayes spaces: From object-oriented kriging to the analysis of warping functions, in ‘Advances in Compositional Data Analysis’, Springer International Publishing, pp. 207–224. DOI: https://doi.org/10.1007/978-3-030-71175-7_11

Menafoglio, A., Davide, P., Secchi, P. et al. (2019), ‘Mathematical foundations of functional kriging in hilbert spaces and riemannian manifolds’.

Menafoglio, A., Secchi, P. & Rosa, M. D. (2013), ‘A universal kriging predictor for spatially dependent functional data of a hilbert space’, Electronic Journal of Statistics 7(none). DOI: https://doi.org/10.1214/13-EJS843

Menezes, R., García-Soidán, P. & Ferreira, C. (2010), ‘Nonparametric spatial prediction under stochastic sampling design’, Journal of Nonparametric Statistics 22(3), 363–377. DOI: https://doi.org/10.1080/10485250903094294

Neaderhouser, C. C. (1980), ‘Convergence of block spins defined by a random field’, J. Statist. Phys. 22(6), 673–684. DOI: https://doi.org/10.1007/BF01013936

Paredes, R. & Vidal, E. (2006), ‘Learning weighted metrics to minimize nearest-neighbor classification error’, IEEE Transactions on Pattern Analysis and Machine Intelligence . DOI: https://doi.org/10.1109/TPAMI.2006.145

Pollock, L. J., Tingley, R., Morris, W. K., Golding, N., O’Hara, R. B., Parris, K. M., Vesk, P. A. & McCarthy, M. A. (2014), ‘Understanding co-occurrence by modelling species simultaneously with a joint species distribution model (jsdm)’, Methods in Ecology and Evolution 5(5), 397–406. DOI: https://doi.org/10.1111/2041-210X.12180

Rachdi, M., Laksaci, A. & Al-Awadhi, F. A. (2021), ‘Parametric and nonparametric conditional quantile regression modeling for dependent spatial functional data’, Spatial Statistics 43, 100498. DOI: https://doi.org/10.1016/j.spasta.2021.100498

Ripley, B. (1987), Spatial point pattern analysis in ecology, in ‘Develoments in Numerical Ecology’, Springer, pp. 407–429. DOI: https://doi.org/10.1007/978-3-642-70880-0_11

Rivoirard, J., Simmonds, J., Foote, K., Fernandes, P. & Bez, N. (2000), Geostatistics for estimating fish abundance, Wiley Online Library. DOI: https://doi.org/10.1002/9780470757123

Rosenblatt, M. (1985), Stationary sequences and random fields, Birkhauser, Boston. DOI: https://doi.org/10.1007/978-1-4612-5156-9

Ruiz-Medina M, Anh V, E. R. A. J. F. M. (2015), ‘Least squares estimation of multifractional random fields in a hilbertvalued context’, J Optim Theory Appl 167(3):888–911 . DOI: https://doi.org/10.1007/s10957-013-0423-4

Ruiz-Medina, M. D. (2011), ‘Spatial autoregressive and moving average hilbertian processes’, Journal of Multivariate Analysis 102(2), 292–305. DOI: https://doi.org/10.1016/j.jmva.2010.09.005

Ruiz-Medina M, E. R. (2012), ‘Spatial autoregressive functional plug-in prediction of ocean surface temperature’, Stoch Environ Res Risk Assess 26(3):335–344 . DOI: https://doi.org/10.1007/s00477-012-0559-z

Sørensen, H., Goldsmith, J. & Sangalli, L. M. (2013), ‘An introduction with medical applications to functional data analysis’, Statistics in medicine 32(30), 5222–5240. DOI: https://doi.org/10.1002/sim.5989

Takahata, H. (1983), ‘On the rates in the central limit theorem for weakly dependent random fields’, Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 64(4), 445–456. DOI: https://doi.org/10.1007/BF00534950

Ternynck, C. (2014), ‘Spatial regression estimation for functional data with spatial dependency’, SFDS,155, 2 .

Torres, J. M., Nieto, P. G., Alejano, L. & Reyes, A. (2011), ‘Detection of outliers in gas emissions from urban areas using functional data analysis’, Journal of hazardous materials 186(1), 144–149. DOI: https://doi.org/10.1016/j.jhazmat.2010.10.091

Tran, L. T. (1990), ‘Kernel density estimation on random fields’, Journal of Multivariate Analysis 34(1), 37–53. DOI: https://doi.org/10.1016/0047-259X(90)90059-Q

Xiaoying, W., Qian, S. & Jialiang, G. (2021), Research on nonparametric classification method of functional data, in ‘2021 2nd International Conference on Education, Knowledge and Information Management (ICEKIM)’, IEEE. DOI: https://doi.org/10.1109/ICEKIM52309.2021.00129

Yen JDL, Thomson JR, P. D. K. J. M. N. R. (2014), ‘Function regression in ecology and evolution’, free. Methods Ecol Evol 6, 17–26 . DOI: https://doi.org/10.1111/2041-210X.12290

Young, M. & Carr, M. H. (2015), ‘Application of species distribution models to explain and predict the distribution, abundance and assemblage structure of nearshore temperate reef fishes’, Diversity and Distributions 21(12), 1428–1440. DOI: https://doi.org/10.1111/ddi.12378

Younso, A. (2017), ‘On the consistency of a new kernel rule for spatially dependent data’, Statistics & Probability Letters . DOI: https://doi.org/10.1016/j.spl.2017.08.008

Zhang, H. (2019), Topics in functional data analysis and machine learning predictive inference, PhD thesis, Iowa State University.

How to Cite

APA

Ndiaye, M., Dabo-Niang, S. and Ngom, P. . (2022). Nonparametric Prediction for Spatial Dependent Functional Data Under Fixed Sampling Design. Revista Colombiana de Estadística, 45(2), 391–428. https://doi.org/10.15446/rce.v45n2.98957

ACM

[1]
Ndiaye, M., Dabo-Niang, S. and Ngom, P. 2022. Nonparametric Prediction for Spatial Dependent Functional Data Under Fixed Sampling Design. Revista Colombiana de Estadística. 45, 2 (Jul. 2022), 391–428. DOI:https://doi.org/10.15446/rce.v45n2.98957.

ACS

(1)
Ndiaye, M.; Dabo-Niang, S.; Ngom, P. . Nonparametric Prediction for Spatial Dependent Functional Data Under Fixed Sampling Design. Rev. colomb. estad. 2022, 45, 391-428.

ABNT

NDIAYE, M.; DABO-NIANG, S.; NGOM, P. . Nonparametric Prediction for Spatial Dependent Functional Data Under Fixed Sampling Design. Revista Colombiana de Estadística, [S. l.], v. 45, n. 2, p. 391–428, 2022. DOI: 10.15446/rce.v45n2.98957. Disponível em: https://revistas.unal.edu.co/index.php/estad/article/view/98957. Acesso em: 13 jan. 2025.

Chicago

Ndiaye, Mamadou, Sophie Dabo-Niang, and Papa Ngom. 2022. “Nonparametric Prediction for Spatial Dependent Functional Data Under Fixed Sampling Design”. Revista Colombiana De Estadística 45 (2):391-428. https://doi.org/10.15446/rce.v45n2.98957.

Harvard

Ndiaye, M., Dabo-Niang, S. and Ngom, P. . (2022) “Nonparametric Prediction for Spatial Dependent Functional Data Under Fixed Sampling Design”, Revista Colombiana de Estadística, 45(2), pp. 391–428. doi: 10.15446/rce.v45n2.98957.

IEEE

[1]
M. Ndiaye, S. Dabo-Niang, and P. . Ngom, “Nonparametric Prediction for Spatial Dependent Functional Data Under Fixed Sampling Design”, Rev. colomb. estad., vol. 45, no. 2, pp. 391–428, Jul. 2022.

MLA

Ndiaye, M., S. Dabo-Niang, and P. . Ngom. “Nonparametric Prediction for Spatial Dependent Functional Data Under Fixed Sampling Design”. Revista Colombiana de Estadística, vol. 45, no. 2, July 2022, pp. 391-28, doi:10.15446/rce.v45n2.98957.

Turabian

Ndiaye, Mamadou, Sophie Dabo-Niang, and Papa Ngom. “Nonparametric Prediction for Spatial Dependent Functional Data Under Fixed Sampling Design”. Revista Colombiana de Estadística 45, no. 2 (July 14, 2022): 391–428. Accessed January 13, 2025. https://revistas.unal.edu.co/index.php/estad/article/view/98957.

Vancouver

1.
Ndiaye M, Dabo-Niang S, Ngom P. Nonparametric Prediction for Spatial Dependent Functional Data Under Fixed Sampling Design. Rev. colomb. estad. [Internet]. 2022 Jul. 14 [cited 2025 Jan. 13];45(2):391-428. Available from: https://revistas.unal.edu.co/index.php/estad/article/view/98957

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CrossRef citations4

1. Salim Bouzebda, Inass Soukarieh. (2022). Non-Parametric Conditional U-Processes for Locally Stationary Functional Random Fields under Stochastic Sampling Design. Mathematics, 11(1), p.16. https://doi.org/10.3390/math11010016.

2. Yoba Kande, Ndague Diogoul, Patrice Brehmer, Sophie Dabo-Niang, Papa Ngom, Yannick Perrot. (2024). Demonstrating the relevance of spatial-functional statistical analysis in marine ecological studies: The case of environmental variations in micronektonic layers. Ecological Informatics, 81, p.102547. https://doi.org/10.1016/j.ecoinf.2024.102547.

3. Mamadou Ndiaye, Sophie Dabo-Niang, Papa Ngom, Ndiaga Thiam, Patrice Brehmer, Yeslem El Vally. (2024). Nonlinear Analysis, Geometry and Applications. Trends in Mathematics. , p.69. https://doi.org/10.1007/978-3-031-52681-7_3.

4. Salim Bouzebda. (2024). Limit Theorems in the Nonparametric Conditional Single-Index U-Processes for Locally Stationary Functional Random Fields under Stochastic Sampling Design. Mathematics, 12(13), p.1996. https://doi.org/10.3390/math12131996.

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