In various scientific fields properties are represented by functions varying over space. In this paper, we present a methodology to make spatial predictions at non-data locations when the data values are functions. In particular, we propose both\ud an estimator of the spatial correlation and a functional kriging predictor. We adapt an\ud optimization criterion used in multivariable spatial prediction in order to estimate the\ud kriging parameters. The curves are pre-processed by a non-parametric fitting, where\ud the smoothing parameters are chosen by cross-validation. The approach is illustrated\ud by analyzing real data based on soil penetration resistances.Postprint (published version

Principal curves have been defined as smooth curves passing through thè`m iddle'' of a multidimensional data set. They are nonlinear generalizations of the first principal component, a characterization of which is the basis of the definition of principal curves. We establish a new characterization of the first principal component and base our new definition of a principal curve on this property. We introduce the notion of principal oriented points and we prove the existence of principal curves passing through these points. We extend the definition of principal curves to multivariate data sets and propose an algorithm to find them. The new notions lead us to generalize the definition of total variance. Successive principal curves are recursively defined from this generalization. The new methods are illustrated on simulated and real data sets. Academic PressAMS 1991 subject classifications: 62H05, 62H25, 62G07.

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Classification problems of functional data arise naturally in many applications. Several approaches have been considered for solving the problem of finding groups based on functional data. In this paper we are interested in detecting groups when the functional data are spatially correlated. Our methodology allows to find spatially homogeneous groups of sites when the observations at each sampling location consist of samples of random functions. In univariable and multivariable geostatistics various methods of incorporating spatial information into the clustering analysis have been considered. Here we extend these methods to the functional context in order to fulfill the task of clustering spatially correlated curves. In our approach we initially use basis functions to smooth the observed data and then we weight the dissimilarity matrix among curves by either the trace-variogram or the multivariable variogram calculated with the coefficients of the basis functions. As an illustration the methodology is applied to a real data set corresponding to average daily temperatures measured at 35 Canadian weather stations.