Hierarchical clustering of spatially correlated functional data

Abstract

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.

Spatially Weighted Functional Clustering of River Network Data

“…This idea has been extended to the functional clustering case by Giraldo et al . (2010) who proposed incorporating spatial covariance in hierarchical functional clustering by weighting the functional distance matrix, defined in equation (1), using a functional covariance matrix that has been estimated by using an appropriate variogram.…”

“…In addition, Giraldo et al . (2010) extended existing ideas used for clustering to include spatial correlation between curves. Hierarchical clustering methods are adapted via weighting the dissimilarity matrix by a measure of spatial functional covariance.…”

Clustering Spatially Correlated Functional Data

“…In this framework a central role is played by the measure of similarity/dissimilarity among the object. The hierarchical method we refer (Giraldo et al, 2009) for spatial functional data is a natural extension to the functional framework of the approaches proposed for geostatistical data, where the the L 2 norm between curves χ s i , χ s j is replaced by a weighted norm among the georeferenced functions. Especially two alternatives are proposed, respectively for univariate and multivariate context.…”

“…The purpose of clustering methods in the spatial functional framework is to find subgroups of spatial homogeneous curves. To the best of our knowledge, very few clustering methods incorporating spatial dependence information between curves exist (see Giraldo et al, 2009, Romano et al, 2010a, 2010b, Jiang and Serban, 2010.…”

“…The first (Giraldo et al, 2009) is a hierarchical approach based on a spatial weighted dissimilarity measure between curves. The weight is the trace-variogram function or the multivariable variogram calculated on the coefficients of the basis function.…”