Principal components for multivariate spatiotemporal functional 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 of different pollutant levels recorded for a month/year at different sites. In this kind of dataset we can recognize time series along one dimension, spatial series along another and multivariate data along the third dimension. Statistical techniques aiming at handling huge amounts of information are very important in this context and classical dimension reduction techniques, such as Principal Components, are relevant, allowing to compress the information without much loss. Although time series, as well as spatial series, are recorded as discrete observations, to convert them into Functional Data presents the advantage of preserving their functional structure and reducing a great number of observations to a few coefficients. Consequently, PCA for Functional Data is here considered. In this paper we propose to take into account both the temporal and the spatial information inside the data. The main aim is to develop a spatial variant of the temporal Functional Principal Component Analysis (FPCA) approach treated in Ramsay and Silverman (2005). The possibility of extension of the temporal FPCA to spatial 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 Yao et al. (2003) and Yao et al. (2005). Nevertheless, the exact way the analysis is done is not carried out. Furthermore, up to our knowledge, software implementations are available only for the one-dimensional case. An approach to spatial FPCA is also proposed 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 considered and therefore functions of locations in d-dimensional Euclidean space Rd instead of functions of time measured in R. In particular, we deal with data measured on twodimensional domains D in R^d, d=2, considering both longitude and latitude.