Abstract
This paper combines optimal spatial sampling designs with geostatistical analysis of functional data. We propose a methodology and design criteria to find the set of spatial locations that minimizes the variance of the spatial functional prediction at unsampled sites for three functional predictors: ordinary kriging, simple kriging and simple cokriging. The last one is a modification of an existing predictor that uses ordinary cokriging based on the basis coefficients. Instead, we propose to use a simple cokriging predictor with the scores resulting from a representation of the functional data with the empirical functional principal components, allowing to remove restrictions and complexity of the covariance models and constraints on the estimation procedure. The methodology is applied to a network of air quality in Bogotá city, Colombia.
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References
Angulo J, Bueso M, Alonso F (2000) A study on sampling design for optimal prediction of space–time stochastic processes. Stoch Environ Res Risk Assess 14(6):412–427
Bohorquez M, Mateu J, Diaz L (2014) A note on smoothness measures for space–time surfaces. Stoch Environ Res Risk Assess 28(4):1011–1022
Bosq D (2000) Linear processes in function spaces: theory and applications, vol 149. Springer, Berlin
Brooks S, Morgan B (1995) Optimization using simulated annealing. Statistician 44(2):241–257
Caselton W, Zidek J (1984) Optimal monitoring network designs. Stat Probab Lett 2(4):223–227
Ferraty F, Vieu P (2006) Nonparametric functional data analysis: theory and practice. Springer, Berlin
Giraldo R (2014) Cokriging based on curves, prediction and estimation of the prediction variance. InterStat 2:1–30
Giraldo R, Delicado P, Mateu J (2010) Continuous time-varying kriging for spatial prediction of functional data: an environmental application. J Agric Biol Environ Stat 15(1):66–82
Giraldo R, Delicado P, Mateu J (2011) Ordinary kriging for function-valued spatial data. Environ Ecol Stat 18(3):411–426
Giraldo R, Mateu J (2013) Kriging for functional data. Wiley, Hoboken
Goulard M, Voltz M (1993) Geostatistical interpolation of curves: a case study in soil science. In: Soares A (ed) Geostatistics Tróia’92, vol 2. Springer, Berlin, pp 805–816
Harville DA, Jeske DR (1992) Mean squared error of estimation or prediction under a general linear model. J Am Stat Assoc 87(419):724–731
Horvath L, Kokoszka P (2012) Inference for functional data with applications. Springer, Berlin
Hubert M, Rousseeuw P, Segaert P (2015) Multivariate functional outlier detection. Stat Methods Appl 24(2):177–202
Ignaccolo R, Ghigo S, Bande S (2013) Functional zoning for air quality. Environ Ecol Stat 20(1):109–127
Ignaccolo R, Mateu J, Giraldo R (2014) Kriging with external drift for functional data for air quality monitoring. Stoch Environ Res Risk Assess 28(5):1171–1186
Le N, Zidek J (2006) Statistical analysis of environmental space–time processes. Springer, Berlin
Müller W (2007) Collecting spatial data: optimum design of experiments for random fields. Springer, Berlin
Myers D (1982) Matrix formulation of co-kriging. Math Geol 14(3):249–257
Nerini D, Monestiez P, Manté C (2010) Cokriging for spatial functional data. J Multivar Anal 101(2):409–418
Ramsay J, Silverman B (2005) Functional data analysis. Springer, New York
Schabenberger O, Gotway C (2004) Statistical methods for spatial data analysis. CRC Press, Boca Raton
Secchi P, Vantini S, Vitelli V (2015) Analysis of spatio-temporal mobile phone data: a case study in the metropolitan area of milan. Stat Methods Appl 24(2):279–300
Secretaria Distrital de Ambiente de Bogotá (SDA) (2015) Informe anual de calidad del aire de bogotá 2014. Alcaldía Mayor de Bogotá. http://201.245.192.252:81/
Wackernagel H (1998) Multivariate geostatistics: an introduction with applications. Springer, Berlin
Zhu Z, Stein M (2006) Spatial sampling design for prediction with estimated parameters. J Agric Biol Environ Stat 11(1):24–44
Zimmerman D (2006) Optimal network design for spatial prediction, covariance parameter estimation, and empirical prediction. Environmetrics 17(6):635–652
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Bohorquez, M., Giraldo, R. & Mateu, J. Optimal sampling for spatial prediction of functional data. Stat Methods Appl 25, 39–54 (2016). https://doi.org/10.1007/s10260-015-0340-9
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DOI: https://doi.org/10.1007/s10260-015-0340-9