Parameter Estimation of Locally Compensated Ridge-Geographically Weighted Regression Model

Abstract

Geographically weighted regression (GWR) is a spatial data analysis method where spatially varying relationships are explored between explanatory variables and a response variable. One unresolved problem with spatially varying coefficient regression models is local collinearity in weighted explanatory variables. The consequence of local collinearity is: estimation of GWR coefficients is possible but their standard errors tend to be large. As a result, the population values of the coefficients cannot be estimated with great precision or accuracy. In this paper, we propose a recently developed method to remediate the collinearity effects in GWR models using the Locally Compensated Ridge Geographically Weighted Regression (LCR-GWR). Our focus in this study was on reviewing the estimation parameters of LCR-GWR model. And also discussed an appropriate statistic for testing significance of parameters in the model. The result showed that Parameter estimation of LCR-GWR model using weighted least square method is \(\hat{\beta }({u}_{i},{v}_{i},{\lambda }_{i})={[{X}^{\ast T}W* ({u}_{i},{v}_{i}){X}^{\ast }+\lambda I({u}_{i},{v}_{i})]}^{-1}{X}^{\ast T}W* ({u}_{i},{v}_{i}){y}^{\ast }\), where the ridge parameter, λ, varies across space. The LCR-GWR is not necessarily calibrates the ridge regressions everywhere; only at locations where collinearity is likely to be an issue. And the parameter significance test using t-test, t = \(t=\frac{{\hat{\beta }}_{k}({u}_{i},{v}_{i},{\lambda }_{i})}{\hat{\sigma }\sqrt{{v}_{kk}}}\).