Abstract
Modeling of and inference on multivariate data that have been measured in space, such as temperature and pressure, are challenging tasks in environmental sciences, physics and materials science. We give an overview over and some background on modeling with crosscovariance models. The R package RandomFields supports the simulation, the parameter estimation and the prediction in particular for the linear model of coregionalization, the multivariate Matern models, the delay model, and a spectrum of physically motivated vector valued models. An example on weather data is considered, illustrating the use of RandomFields for parameter estimation and prediction.
Highlights
Spatial data very frequently have more than one component
Throughout the paper, a second order m-variate random field on T ⊂ Rd denotes a collection of real-valued random vectors Z(x) = (Z1(x), . . . , Zm(x)) indexed by x ∈ T with existing second moments
At least three different kinds of well known difficulties exist with the classical maximum likelihood (ML) approach: first, the number of data is rather limited as the multivariate Gaussian density function involves the inverse of the covariance matrix, so that the calculation of the likelihood needs of order N 3 operations
Summary
Spatial data very frequently have more than one component. In meteorology, for instance, temperature, rainfall and pressure are measured at the same locations at predefined instances in time. To name a further example, price developments of goods are spatio-temporal data consisting of various components at each instant of time and at each market. These kinds of data are ubiquitous, multivariate spatial models are largely under-. RandomFields: Multivariate Random Fields developed and multivariate space-time models are unknown, except for simple constructions. A multivariate random field is an Rm-valued random process on a subset of Rd with m ≥ 2 Such fields are called multivariable in the package gstat (Pebesma 2004).
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