Maximum-likelihood estimation for multivariate ….pdf

Maximum-likelihood estimation for multivariatespatial linear coregionalization models Hao Zhang * , y Department of Statistics, Washington State University, Pullman, WA 99164-3144, USA SUMMARY A multivariate spatial linear coregionalization model is considered that incorporates the Mate ′rn class of covariograms. An EM algorithm is developed for maximum-likelihood estimation that has a few desirable properties and is capable of handling high-dimensional data. Most estimates in the EM algorithm are updated through closed form expressions and these estimates automatically satisfy necessary constraints. The model and algorithm are illustrated through a real example. Copyright # 2006 John Wiley & Sons, Ltd. key words : cokriging; EM algorithm; linear coregionalization model; multivariate covariogram; spatial correlation 1. INTRODUCTION Multiple spatial variables are often observed in many studies in environmental, agricultural, and ecological sciences. The observed values of these spatial variables are referred to as multivariate spatial data, which often possess two kinds of spatial correlation: spatial autocorrelation that exists between observations of an individual variable at different locations, and spatial cross-correlation that describes the correlation between two different variables measures at either the same or different locations. It is an important problem to model both kinds of spatial correlation. By appropriately accounting for and modeling the spatial correlation, ef?cient estimation, and better prediction can be achieved. For example, cokriging is a technique for linear prediction of one variable by making use of observed values of other variables, and can result in more precise prediction than the kriging methods that utilize only the spatial auto-correlation of this particular variable being predicted. Cross correlation has been modeled through a multivariate covariogram and a cross-variogram (Wackernagel, 1998; Chile ′s and Del?ner, 1999), and a