Geographically local regression and linear mixed models for tree height-diameter relationship

Spatial heterogeneity and autocorrelation are two aspects of spatial effects existent in the geo-referenced data collected in forest and ecological studies. Commonly used spatial models such as linear mixed models and spatial autoregressive models are not locally adaptive in order to account for spatial heterogeneity. On the other hand, geographically weighted regression (GWR) is not designed to handle spatial autocorrelation, although it has been proved to be powerful in dealing with spatial heterogeneity. In this study, local spatial regression models, which include GWR-spatial error model (GWR-SEM) and GWR-spatial lag model (GWR-SLM), and geographically local linear mixed model (GLLMM) were developed to overcome the drawbacks of GWR. They were applied to the height-diameter relationship of individual trees in softwood stand located near Sault Ste. Marie, Ontario, Canada. Next, different parameter estimation methods for spatial regression models were compared using observed tree height-diameter data and simulation study, including maximum likelihood estimation, Bayesian methods, two-stage least squares (2SLS) (for SLM) and generalized method of moments (GMM) (for SEM)). Further, three types of spatial autoregressive models (spatial lag, spatial error, and spatial Durbin), with different specifications of spatial weight matrix, were applied to predict the tree heights using observed diameter at unsampled locations. The results showed that both local spatial regression models and GLLMM fitted the data better and produced less spatially autocorrelated model residuals than GWR. GWR tended to overstate the spatial heterogeneity by ignoring the local spatial autocorrelation. GLLMM enabled the estimation of local variograms which were used in local empirical best linear unbiased prediction (EBLUP). Variogram was superior to other specification methods of spatial weight matrix in this study. For spatial autoregressive models, GMM (2SLS) was robust to the nonnormality of data distributions as well as the influence of outliers, indicating that it was an effective model parameter estimation method. Spatial autoregressive models were useful in prediction and interpolation of the tree heights at unsampled locations.;Key Words. spatial heterogeneity, spatial autocorrelation, height-diameter equation, geographically weighted regression, linear mixed model, spatial autoregressive models, local variogram, spatial weight matrix.