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 ε that distributed as a multivariate normally with zero vector mean and variance-covariance matrix Σ at each location ui,vi, which Σ 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.

Spatially Varying Relation between Built Environment and Station-Level Subway Passenger-Distance

“…Ordinary least squares (OLS) regression models are nonspatial models and the most representative linear regression models were used to initially explain the complex relationship between the built environment and subway capacity [15,35,36], which assumes that the relationship between independent variables and the dependent variable is global and not spatially heterogeneous. In terms of spatial models, among the most common are spatial error models [37,38], spatial lag models [39], geographically weighted regression models [32,40,41], and derivative models related to geographically weighted regression models, such as geographic time-weighted regression (GTWR) models [42], geographically weighted negative binomial regression (GWNBR) models [43], geographically weighted Poisson regression (GWPR) models [44], and mixed geographically weighted regression (MGWR) models [13,18,45]. For example, Li et al [32] used a GWR model to refine the study of Guangzhou Subway.…”

Geographically Weighted Multivariate Logistic Regression Model and Its Application

“…Harini et al [1,2] introduced the multivariate GWR (MGWR) model and demonstrated the parameter estimation and hypothesis test procedures using the restricted maximum likelihood estimation (RMLE) and maximum likelihood ratio test (MLRT) methods, respectively. The form and properties of the estimated errors variance-covariance parameters of the MGWR model using the MLE and weighted least squares methods were investigated [3]. Triyanto et al [4,5] introduced the geographically weighted multivariate Poisson regression (GWMPR) model.…”