Data-driven Local Spatial Correlation Index

Measuring the spatial association is always an important topic in spatial statistics and spatial econometrics.There has been a good number of well-behaved indices proposed by the previous research.The two significant features of spatial or spatio-temporal data,spatial association and heterogeneity,are widely existed simultaneously.Besides spatial heterogeneity would always lead to the nonstationarity in process.Previous research majorly use local indicator of spatial association to reflect the existence of spatial heterogeneity.Foundation of this kind of indicators is “local stationary” and this assumption of narrow the range of usage of indicators.Because this assumption requires that the quantity of observed individuals in a local is sufficiently large to remain the asymptotic properties,which is sometimes unpractical.On the other hand,these indicators generally rely on spatial weight matrix.But,as we know,problem of the subjectivity of weight matrix designing has not been well solved yet.Additionally,almost all of the previous research does not take a widely-seen phenomenon into consideration,asymmetry in association.Hence,in this paper,based on spatio-temporal data which is timely stationary and spatially nonstationary,we propose a more general definition of the word “local”.Based on that,a new entropy-based,totally data-driven(no spatial weight matrix),local indicator of spatial association is built in this paper.This indicator includes almost all aspects of association,symmetry or asymmetry,linear or nonlinear.By making well use of the stationarity of time series to tackle the nonstationarity of spatial series,a relatively robust nonparametric estimator of indicator is also proposed.Meanwhile,we obtain the consistency and convergence rate of this estimator,as well as the approximate normality.Then we offer the framework of a set of feasible hypothesis tests of local spatial association.An interesting connection between linear correlation and asymmetrical dependence is discovered and theoretically discussed.In simulation work,this connection is also strongly proved due to the evidence of simulation.Another key point is also discovered by simulation,which is,when sample sizeis larger than 500,convergence of our estimator can be generally guaranteed.Finally,according to the empirical research based on the data of PM2.5 from 30 cities around the country,the analytic result of our methods match up with the information from descriptive statistics and previous knowledge,which means our method is effective.