Abstract
The models of crown area / length as a function of tree diameter at breast height (dbh) were fitted using geographically weighted regression (GWR) and Bayesian geographically weighted regression (BGWR), along with its robust version (robust BGWR), for two observed and eight synthetic data sets. The observed data included the measurements on 1666 trees from an uneven aged mature softwood stand and 1230 trees from a hardwood stand located in central Ontario. Similarly, the synthetic data included the measurements on trees ranging from 1474 to 715, including tree level (local) coefficients. The true local coefficients were simulated using the bivariate Gaussian copula. In addition, each synthetic data had few outliers and weak observations (i.e., any observation with small sum of its neighbor's geographic weights).
The crown area predictions made by the methods were similar for the observed data. The cumulative distributions of estimated coefficients by the methods were different from one another for both observed and synthetic data sets. Similarly, the estimated local coefficients by the robust BGWR had the larger local and regional spatial heterogeneity than those estimated by the BGWR and GWR methods. For the eight synthetic data sets, the 95% credible limits of each coefficient posteriors were compared with the corresponding true coefficient for all observations. This assessment indicated that the BGWR and robust BGWR methods estimated 71.32 % and 62.17% intercept; 55.74% and 52.10% slope coefficients, respectively, of crown length model were not significantly (α = 0.05) different from the respective true coefficients. For weak observations, the bias in estimated coefficients and crown length prediction errors was compatible for all methods. In summary, the accuracy of predicted crown area / length was similar across the methods. However, the BGWR and robust BGWR methods used a valid probability model which unified the parameter estimation across observations, provided accurate estimates of variance of estimated coefficients, and the estimated coefficients were robust to outlier(s). However, to achieve this efficiency using the Bayesian methods, a price (i.e., computation time) needs to be paid.