Estimation of Error Variance-Covariance Parameters Using Multivariate Geographically Weighted Regression Model

Abstract

The Multivariate Geographically Weighted Regression (MGWR) model is a development of the Geographically Weighted Regression (GWR) model that takes into account spatial heterogeneity and autocorrelation error factors that are localized at each observation location. The MGWR model is assumed to be an error vector $\left(\mathbf{\epsilon }\right)$ that distributed as a multivariate normally with zero vector mean and variance-covariance matrix $\mathbf{\Sigma }$ at each location $\left(u_i,v_i\right)$, which $\mathbf{\Sigma }$ is sized $qxq$ for samples at the $i$-location. In this study, the estimated error variance-covariance parameters is obtained from the MGWR model using Maximum Likelihood Estimation (MLE) and Weighted Least Square (WLS) methods. The selection of the WLS method is based on the weighting function measured from the standard deviation of the distance vector between one observation location and another observation location. This test uses a statistical inference procedure by reducing the MGWR model equation so that the estimated error variance-covariance parameters meet the characteristics of unbiased. This study also provides researchers with an understanding of statistical inference procedures.