Study On Anisotropic Geographically Weighted Regression Model And Its Algorithms

Regression analysis of geographical relations is a hot research topic in GIScience.Developing new spatial regression methods and enhancing the ability to analyze and mine geographical relations is of great theoretical and practical significance for a deeper understanding of natural phenomena and social processes.Spatial non-stationarity is an inherent characteristic of geospatial relations,with both distance and direction playing important roles in affecting its modeling accuracy.However,geographic relationship model represented by Geographically Weighted Regression(GWR)assumes that the variable relationship is spatially isotropic,while ignoring the significance of direction in spatial relationship modeling.Spatial anisotropy is later introduced instead of spatial isotropy for modeling spatial relationships by constructing new distance metrics,which improved the GWR model and its accuracy to some extent.Nevertheless,issues have remained with those new distance metrics as follows:(ⅰ)Insufficient expression of anisotropic relationships.The fact that numerical characteristics of different spatial positions that usually have different anisotropy features can be distorted due to the same distance parameters are used to characterize the anisotropy of the whole study area;(ⅱ)The disadvantage of the use of bandwidth to depict the spatial non-stationarity of geographical relationships.As current bandwidth parameters(whether single or multi-scale)only depict the "average" nonstationarity relationship across the entire study area,the local spatial variation characteristics of geographical relationships cannot be finely described.To address these issues,this paper attempts to conduct GWR modeling investigations on anisotropic relationships from the perspective of the spatial relationship gradient field.First,the spatial relationship gradient field is introduced to realize the anisotropic expression of the spatial relationship between geographical variables.Then,the expression of the anisotropic relationship of the spatial relationship gradient field is integrated into the Geographically Weighted Regression,and the anisotropic geographically weighted regression model is therefore constructed.Finally,algorithms for model parameter estimation and statistical inference are developed.Details are listed as follows.(1)Constructing Anisotropic Geographical Weighted Regression model based on spatial relationship gradient field.Considering the characteristics of the difference in the speed of geographical relations in terms of positions and directions,an anisotropic relationship expression framework is constructed based on the spatial relationship gradient field.To be specific,the spatial relationship gradient field is first used to depict the main direction and strength of anisotropic relationship of sampling points.Then,directional derivative is introduced to depict the difference in speed of geographical relationship in different directions.Next,Gradients and directional derivatives of different orders are used to express anisotropic relationships that adapt to different types of spatial relationships.Among them,the gradient is all 0 to express global smoothness of geographical space relationship,and the 1st,2nd order gradient expresses linear and nonlinear anisotropy of geographical space relationship respectively.Subsequently,isotropic hypothesis-based Geographically Weighted Regression model is extended in the developed anisotropic relationship expression framework,and a conceptual model of Anisotropic Geographically Weighted Regression model is constructed.Moreover,two types of spatial relationship analysis models,the anisotropic geographically weighted regression(GGWR)and the anisotropic multi-scale geographically weighted regression(GMGWR),are constructed in light of the scale characteristics of geographical relationships.(2)Algorithms of the GGWR/GMGWR model for parameter estimation and statistical inference are developed.The key to restricting the parameter estimation of GGWR and GMGWR is the adaptive determination of anisotropic relationship types.we first transform it into a problem of determining the order of spatial relationship gradient,which can later be transformed into a "feature" selection problem,and then the idea of sparse optimization is proposed to achieve the recovery of anisotropic relationship types.Therefore,the primary task is to develop a model with high accuracy in feature selection.(ⅰ)The l0 sparse representation is introduced and l0-GWR model is proposed to realize the feature selection.Additionally,we propose a heuristic method(Splicing algorithm)to implement the parameter estimation of l0-GWR,and the MBIC(modified Bayesian information criteria)is introduced to determine the feature quantity adaptively.Finally,the coefficient sign consistency index(ICS)is proposed to quantitatively evaluate the interpretability of the model parameter results.(ⅱ)The parameter estimation method of l0-GWR is fused with weighted least squares and back-fitting methods,respectively,to achieve parameter estimation and statistical inference of GGWR and GMGWR.At the same time,for the case of global stationary relationships exist in GGWR or GMGWR,a two-stage estimation method is proposed to achieve separate solutions for global stationary and local non-stationary relationships.(3)Model integration and validation.Anisotropic Geographically Weighted Regression models have been integrated with classic models(i.e.,GWR,sGWR,and MGWR)for the development of FGWR software.Further,the developed FGWR software has been used to evaluate the effectiveness of the proposed models as follows.(ⅰ)Validation of the l0-GWR model has been conducted through both simulation and real experiments(Boston housing price modeling).Results show that the l0-GWR outperforms other benchmark models in terms of multiple evaluation metrics,including the accuracy of variable selection,coefficient estimation accuracy,ICS,and fitting accuracy.In particular,the ICS of the l0-GWR is significantly higher than that of GWL(geographically weighted LASSO regression),GWRR(geographically weighted ridge regression),and GWR.It can be concluded that the regression coefficients of the l0-GWR are a more friendly interpretation in line with the intuition in daily practice.(ⅱ)Validation of the GGWR and GMGWR models has also been conducted through both simulation and real experiments(the number of bachelor’s degrees in Georgia and housing prices in Shenzhen).Results demonstrate that GGWR and GMGWR outperform GWR,MGWR,LGWR(local linear geographically weighted regression),DGWR(directional geographically weighted regression),l0-GWR,and minKowGWR(geographically weighted regression based on Minkowski distance)in terms of fitting accuracy,coefficient estimation accuracy,and ICS under different types of anisotropic relationships.Additionally,the spatial relationship gradient field obtained by GGWR and GMGWR can identify global stationary relationships and finely characterize the local variation characteristics of geographical regression relationships.Moreover,a practical application-based case study,i.e.,land surface temperature in the Hengyang region,has been introduced to further validate the effectiveness of the GGWR and GMGWR models.This paper conducts innovative research on the theoretical and methodological aspects of the anisotropic representation of geographic relationships from the perspective of spatial relationship gradient field.The research findings show the strong potential for intuitive and in-depth modeling of geographic relationships in complex scenarios,consequently improve people’s cognition and prediction ability of geographical problems.