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Abstract: Two types of spatial regression models, a spatial lag model (SLM) and a spatial error model (SEM), were applied to fit the height-diameter relationship of trees. SEM had better model fitting and performance than both SLM and ordinary least squares. Moran's I coefficients showed that SEM effectively reduced the spatial autocorrelation in the model residuals. Both real data and Monte Carlo simulations were used to compare different parameter estimation methods for the two spatial regression models, including maximum likelihood estimation (MLE), Bayesian methods, two-stage least squares (for SLM) and generalized method of moments (GMM) (for SEM). Our results indicated that GMM was close to MLE in terms of model fitting, much easier in computation, and robust to non-normality and outliers. The Bayesian method with heteroscedasticity did not effectively estimate the spatial autoregressive parameters but produced very small biases for the regression coefficients of the model when few outliers existed. FOR. SCI. 56(5):505-514.
Keywords: tree height-diameter relationship, spatial autoregressive parameters, maximum likelihood estimation, Bayesian methods, two-stage least squares, generalized method of moments
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BECAUSE OF THE RAPID DEVELOPMENT of geographical information techniques, numerous data sets have been collected with references to locations measured as points in space in different study fields. Two problems arise for the geo-referenced sample data: spatial dependence existing among the observations and spatial heterogeneity occurring in the relationships among variables (Anselin and Griffith 1988, LeSage and Pace 2009). It is known that spatial dependence and heterogeneity cause the violations of traditional Gauss-Markov assumptions, which have drawn great attention from forest and ecological modelers (Zhang et al. 2005, 2009). In recent years, spatial regression models have been developed to take the spatial dependence into account and have been widely used in spatial econometrics (LeSage and Pace 2009), whereas local models such as geographically weighted regression and drifted analysis of regression parameters have been used to deal with the spatial heterogeneity (Fotheringham et al. 2002).
The spatial dependence or autocorrelation can be described in different ways in the context of regression modeling, including first-order autoregressive model, mixed regressive-spatial model, and regression model with spatial autocorrelation in the disturbances (LeSage 1998, Anselin 1999). The latter two are commonly called spatial lag model (SLM) and spatial error model (SEM), respectively....