Fitting the Linear Model of Coregionalization by Generalized Least Squares |
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| Authors: | Bernard Pelletier Pierre Dutilleul Guillaume Larocque and James W Fyles |
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| Institution: | (1) Department of Natural Resource Sciences, McGill University, Macdonald Campus, Ste-Anne-de-Bellevue, Québec, Canada, H9X 3V9;(2) Department of Plant Science, McGill University, Macdonald Campus, Ste-Anne-de-Bellevue, Québec, Canada, H9X 3V9;(3) Department of Plant Science, McGill University, Macdonald Campus, Ste-Anne-de-Bellevue, Québec, Canada, H9X 3V9 |
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| Abstract: | In geostatistical studies, the fitting of the linear model of coregionalization (LMC) to direct and cross experimental semivariograms is usually performed with a weighted least-squares (WLS) procedure based on the number of pairs of observations at each lag. So far, no study has investigated the efficiency of other least-squares procedures, such as ordinary least squares (OLS), generalized least squares (GLS), and WLS with other weighing functions, in the context of the LMC. In this article, we compare the statistical properties of the sill estimators obtained with eight least-squares procedures for fitting the LMC: OLS, four WLS, and three GLS. The WLS procedures are based on approximations of the variance of semivariogram estimates at each distance lag. The GLS procedures use a variance–covariance matrix of semivariogram estimates that is (i) estimated using the fourth-order moments with sill estimates (GLS1), (ii) calculated using the fourth-order moments with the theoretical sills (GLS2), and (iii) based on an approximation using the correlation between semivariogram estimates in the case of spatial independence of the observations (GLS3). The current algorithm for fitting the LMC by WLS while ensuring the positive semidefiniteness of sill matrix estimates is modified to include any least-squares procedure. A Monte Carlo study is performed for 16 scenarios corresponding to different combinations of the number of variables, number of spatial structures, values of ranges, and scale dependence of the correlations among variables. Simulation results show that the mean square error is accounted for mostly by the variance of the sill estimators instead of their squared bias. Overall, the estimated GLS1 and theoretical GLS2 are the most efficient, followed by the WLS procedure that is based on the number of pairs of observations and the average distance at each lag. On that basis, GLS1 can be recommended for future studies using the LMC. |
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| Keywords: | direct and cross semivariograms empirical variance and bias fourth-order moments multivariate nested semivariogram model positive semidefiniteness sill estimators |
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