Study On Spatially Varying Coefficient Modeling Via Least Squares Support Vector Machine Method

Spatially varying coefficient models(SVC)and geographically weighted regression(GWR),as powerful tools in exploring spatial nonstationarity,have been applied extensively to the fields of ecology,environment,meteorology,economics,epidemiology etc.These models could reflect the spatial varying relationship through assuming that regression coefficients as functions of spatial location.On the other hand,one of the methods in Machine Learning,called support vector machine(SVM)has the properties such as flexible and highly accurate.And the improved form of the model called least squares support vector machines(LSSVM)could transfer the quadratic programming in SVM to a set of linear equations in order to enhance the computability.Considering the advantages of classic SVC models and GWR,and combines the basic idea with the operability and practicality of LS-SVM on the frameworks of both regression and classification problems,a novel way of spatially varying coefficient modeling based on LS-SVM were constructed.The basic idea is to apply the optimization process and the least squares framework of LS-SVM to the coefficient estimation of SVC models.The proposed methods improves the computability and the flexibility of the models,and at the same time,remains the interpretability as in classic statistical methods.The whole procedures of the coefficient estimation had been given for both continuous and binomial respond variables under the same framework.And parameter selection,simulation as well as case studies were also presented.As it shown in the results of the simulations and case studies,the proposed regression methods has shown high precision as well as high stability.the MMSE of the estimated coefficients with relatively larger sample sizes(n=625)under 2 cases were both below 0.01,and the standard deviations were less than 0.003;and the estimations of the variance of the error terms were also high comparing with the real value 0.25.The estimations of the variances were 0.250 and 0.241 respectively;the estimated coefficient surfaces were also highly consistent;in the case study,the MSE of the estimated precipitation of spring and summer of Xinjiang area in 2016 were 0.986 and 0.734 respectively,and the R2 were 0.997 and 0.999.In simulation of the proposed classification method,the classification accuracy were also fairly high in both cases,the overall accuracy were 0.893 and 0.935 with relatively larger sample sizes(n=625)under 2 cases;And in the case studies,the overall accuracy of relative precipitation of the two seasons were 0.939 and 0.955.