Published
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.v45n2.98957Keywords:
Functional dependent data, fixed design, non-parametric prediction, Supervised classification (en)Clasificación supervisada, Datos funcionales dependientes, Diseño fijo, Predicción no paramétrica (es)
Downloads
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.
Biau, G. & Devroye, L. (2015), Lectures on the nearest neighbor method, Springer.
Bosq, D. (1998), Nonparametric Statistics for Stochastic Processes: Estimation and prediction, Vol. 110 of Lecture Notes in Statist., 2nd edn, Springer-Verlag, New York.
Carbon, M., Tran, L. T. & Wu, B. (1997), ‘Kernel density estimation for random fields’, Statistics & Probability Letters 36(2), 115–125.
Cressie, N. A. C. (1993), Statistics for Spatial Data, Vol. 110 of Wiley Series in Probability and Statistics, revised edn, Wiley-Interscience.
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.
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.
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.
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 .
Dabo-Niang, S. & Yao, A.-F. (2007), ‘Kernel regression estimation for continuous spatial processes’, Mathematical Methods of Statistics 16(4), 298–317.
Dabo-Niang, S. & Yao, A.-F. (2013), ‘Kernel spatial density estimation in infinite dimension space’, Metrika 76(1), 19–52.
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.
Devroye, L., Gyorfi, L., Krzyzak, A. & Lugosi, G. (1994), ‘On the strong universal consistency of nearest neighbor regression function estimates’, The Annals of Statistics .
Devroye, L. & Wagner, T. J. (1982), ‘8 nearest neighbor methods in discrimination’, Handbook of Statistics .
El Machkouri, M. (2007), ‘Nonparametric regression estimation for random fields in a fixed-design’, Stat. Inference Stoch. Process. 10(1), 29–47.
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.
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.
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.
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
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.
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.
Giraldo, R., Delicado, P. & Mateu, J. (2011), ‘Ordinary kriging for function-valued spatial data’, Environmental and Ecological Statistics 18(3), 411–426.
Hallin, M., Lu, Z. & Tran, L. T. (2004), ‘Local linear spatial regression’, The Annals of Statistics 32(6), 2469–2500.
Hastie, T. & Tibshirani, R. (1996), ‘Discriminant adaptive nearest neighbor classification and regression’, Advances in Neural Information Processing Systems .
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.
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
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.
Li, X., Ghosal, S. et al. (2018), ‘Bayesian classification of multiclass functional data’, Electronic Journal of
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.
Masry, E. (2005), ‘Nonparametric regression estimation for dependent functional data: asymptotic normality’, Stochastic Process. Appl. 115(1), 155–177.
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.
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).
Menezes, R., García-Soidán, P. & Ferreira, C. (2010), ‘Nonparametric spatial prediction under stochastic sampling design’, Journal of Nonparametric Statistics 22(3), 363–377.
Neaderhouser, C. C. (1980), ‘Convergence of block spins defined by a random field’, J. Statist. Phys. 22(6), 673–684.
Paredes, R. & Vidal, E. (2006), ‘Learning weighted metrics to minimize nearest-neighbor classification error’, IEEE Transactions on Pattern Analysis and Machine Intelligence .
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.
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.
Ripley, B. (1987), Spatial point pattern analysis in ecology, in ‘Develoments in Numerical Ecology’, Springer, pp. 407–429.
Rivoirard, J., Simmonds, J., Foote, K., Fernandes, P. & Bez, N. (2000), Geostatistics for estimating fish abundance, Wiley Online Library.
Rosenblatt, M. (1985), Stationary sequences and random fields, Birkhauser, Boston.
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 .
Ruiz-Medina, M. D. (2011), ‘Spatial autoregressive and moving average hilbertian processes’, Journal of Multivariate Analysis 102(2), 292–305.
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 .
Sørensen, H., Goldsmith, J. & Sangalli, L. M. (2013), ‘An introduction with medical applications to functional data analysis’, Statistics in medicine 32(30), 5222–5240.
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.
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.
Tran, L. T. (1990), ‘Kernel density estimation on random fields’, Journal of Multivariate Analysis 34(1), 37–53.
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.
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 .
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.
Younso, A. (2017), ‘On the consistency of a new kernel rule for spatially dependent data’, Statistics & Probability Letters .
Zhang, H. (2019), Topics in functional data analysis and machine learning predictive inference, PhD thesis, Iowa State University.
Dimensions
PlumX
Article abstract page views
Downloads
How to Cite
License
This work is licensed under a Creative Commons Attribution 4.0 International License.
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
