The Basic Theoretics And Application Research On Geographically Weighted Regression

Geographically weighted regression (GWR) is a new presented analysis methodto explore spatially varying relationships recently, which expands ordinarily linearityregression by embeding spatial data structure into the regression model. This methodis studied and applied more and more because it is not only easy to construct a modeland to calculate estimates with explicit analytic expressions, but also able to choose aset of statistical inferential approaches to do significance test and give confidenceintervals. In this paper the basic theoretics and application on GWR is discussed,especial statistical inference on GWR, mixed GWR and spatial scale effect for GWRanalysis are laid a strong emphasis on the study, and a case of the average prices ofsaled houses in Shanghai is adopted to prove the validity of GWR to explore thespatial relationship non-stationarity.Firstly, the basic principle of GWR and the locally weighted least squareapproach are illustrated, then the spatial weighting functions, including Gaussfunction and bi-square function, and their bandwidth optimization are discussed indetial. Considering samples are usually sparse and distribute uneven in fact,self-adaptive weighting functions with varying bandwidth by meeting certianconstraints are introduced to improve estimate accuracy of GWR parameters.The relative statistics of regression model and model parameters are modeled inthis paper to test the significance of the spatial nonstationarity based on thedistribution of quadratic forms of normal variables and under some conditions. Anexact method and two simple approximate approaches, i.e. the three-momentx~2 approximation and the F distribution, for computing the p-value of the teststatistics are derived. The results of simulation experiments validate the validity ofproposed test statistics and calculating algorithm. Furthermore, the estimate of modelparameters and the prediction values of dependent variable are presented based onabove hypothesis and approaches. Beside classic statistical inference techniques, theAkaike Information Criterion (AIC) measurement is also used to test the significance of the spatial nonstationarity of regression model, and the experiments indicate aconsistency between these two methods.There is normally a mixture of constant and varying parameters in theregression models in practical application, so the model and estimate of mixed GWRare addressed deeply in this paper. Based on estimate methods of the linearsemi-parameter regression model and the additive model, a two-step approach and aback-fitting approach for constant parameters estimate are deducted. The simulationexperiments prove that both methods are capable to estimate constant parameters, andmore the accuracy and robustness of the two-step approach are a little better thanback-fitting approach. About how to select the fit bandwidth of weighting function soas to decide which patameters are constant parameters correctly, we suggest based ona lot of experiments that a larger bandwidth, not the optimized bandwidth selected bygeneralized cross-validation criterion (GCV), is more appropriate to obtain right testresults.Scale effect exists popular in spatial analysis. In this paper the concepts of scale,scale effects and scaling methods for the social economic data are detailed. Then theeffect to GWR analysis by spatial extent change and spatial grain change isinvestigated. The simulation experiments prove that the GWR analysis is sensitive tospatial extent change (the bandwidth of weighting fuction), but this impact can bedecreased greatly by optimizing the bandwidth of weighting functions. On thecontrary, the GWR analysis is relative robust to spatial grain change, and this meansthat the GWR is favor of solving modifiable areal unit problems in some degree.At the last of this paper, the average price of saled houses in Shanghai are usedto validate the theory proposed above, and the case demonstrates that (mixed) GWRcan be used to explore the spatial raring relationship among the spatial data well, andthe analysis results are well fitted with the actual fact.