Abstract
We address the problem of predicting spatially dependent functional data belonging to a Hilbert space, with a Functional Data Analysis approach. Having defined new global measures of spatial variability for functional random processes, we derive a Universal Kriging predictor for functional data. Consistently with the new established theoretical results, we develop a two-step procedure for predicting georeferenced functional data: first model selection and estimation of the spatial mean (drift), then Universal Kriging prediction on the basis of the identified model. The proposed methodology is applied to daily mean temperatures curves recorded in the Maritimes Provinces of Canada.
Highlights
Functional Data Analysis (FDA, Ramsay and Silverman (2005)) has recently received a great deal of attention in the literature because of the increasing need to analyze infinite-dimensional data, such as curves, surfaces and images
We address the problem of predicting spatially dependent functional data belonging to a Hilbert space, with a Functional Data Analysis approach
A relatively small body of literature has been produced on this topic: theoretical results in this direction have been recently derived (Giraldo et al (2008b, 2011); Delicado et al (2010); Giraldo et al (2010a); Monestiez and Nerini (2008); Nerini et al (2010)) moving from the pioneering work by Goulard and Voltz (1993), but this theory is still limited to stationary functional stochastic processes valued in L2
Summary
Functional Data Analysis (FDA, Ramsay and Silverman (2005)) has recently received a great deal of attention in the literature because of the increasing need to analyze infinite-dimensional data, such as curves, surfaces and images. In geophysical and environmental applications, natural phenomena are typically very complex and they rarely show a uniform behavior over the spatial domain: in these cases, non-stationary methods are needed To this end, two techniques for kriging non-stationary functional data belonging to L2 have been developed concurrently with the present work, proposing a Residual Kriging approach (Reyes et al, 2012) and a Universal Kriging approach (Caballero et al, 2013). The problem of spatial prediction of temperatures is of interest in microclimate prediction as well as in hydrological and forest ecosystem modeling It has been already faced in the literature about kriging for functional data by means of stationary techniques (e.g., (Giraldo et al, 2010a)); here a non-Euclidean distance is adopted for the spatial domain and a drift term is modeled. We will show that the introduction of a drift term has a strong influence on the analysis in terms of cross-validation performance and prediction accuracy, besides allowing a climatical interpretation of the results
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