Coregionalization analysis with a drift for multi-scale assessment of spatial relationships between ecological variables 1. Estimation of drift and random components

In this and a second article, we propose 'coregionalization analysis with a drift' (CRAD), as a method to assess the multi-scale variability of and relationships between ecological variables from a multivariate spatial data set. CRAD is carried out in two phases: (I) a deterministic component representing the large-scale pattern (called 'drift') and a random component modeled as a second-order stationary process are estimated for each variable separately; (II) a linear model of coregionalization is fitted to the direct and cross experimental variograms of residuals (i.e., after removing the estimated drifts) to assess relationships at smaller scales, while the estimated drifts are used to study relationships at large scale. In this article, we focus on phase I of CRAD, by addressing the questions of the choice of the drift estimation procedure, which is linked to the estimation of random components, and of the presence of a bias in the direct experimental variogram of residuals. In this phase, both the estimation of the drift and the fitting of a model to the direct experimental variogram of residuals are performed iteratively by estimated generalized least squares (EGLS). We use theoretical calculations and a Monte Carlo study to demonstrate that complex large-scale patterns, such as patchy drifts, are better captured with local drift estimation procedures using low-order polynomials within a moving window, than with global procedures. Furthermore, despite the bias in direct experimental variograms of residuals, good estimates of spatial autocovariance parameters are obtained with the double iterative EGLS procedure in the conditions of application of CRAD. An example with forest soil property and tree species diversity data is presented to discuss the choice of drift estimation procedure in practice.