Principal components for multivariate spatiotemporalfunctional data

Multivariate spatio-temporal data consist of a three way array with two dimensions’domains both structured, temporally and spatially; think for example to a set ofdifferent pollutant levels recorded for a month/year at different sites. In this kind ofdataset we can recognize time series along one dimension, spatial series along anotherand multivariate data along the third dimension.Statistical techniques aiming at handling huge amounts of information are veryimportant in this context and classical dimension reduction techniques, such asPrincipal Components, are relevant, allowing to compress the information withoutmuch loss. Although time series, as well as spatial series, are recorded as discreteobservations, to convert them into Functional Data presents the advantage ofpreserving their functional structure and reducing a great number of observations to afew coefficients. Consequently, PCA for Functional Data is here considered.In this paper we propose to take into account both the temporal and the spatialinformation inside the data. The main aim is to develop a spatial variant of thetemporal Functional Principal Component Analysis (FPCA) approach treated inRamsay and Silverman (2005). The possibility of extension of the temporal FPCA tospatial FPCA is mentioned by some authors, including Ramsay and Silverman (2005),and examined for regular grids, as well as for highly irregular and sparse data, by Yaoet al. (2003) and Yao et al. (2005). Nevertheless, the exact way the analysis is doneis not carried out. Furthermore, up to our knowledge, software implementations areavailable only for the one-dimensional case. An approach to spatial FPCA is alsoproposed by Winzenborg (2011), but ignoring the possible temporal aspect of data.In this paper the univariate spatial FPCA is generalized to multivariate case.According to this approach, spatial instead of temporal basis functions are consideredand therefore functions of locations in d-dimensional Euclidean space Rd instead offunctions of time measured in R. In particular, we deal with data measured on twodimensionaldomains D in R^d, d=2, considering both longitude and latitude.