Functional Linear Models With Spatial Dependency And Missing Response Variables

With the gradual improvement of modern measurement technology and storage equipment,the collection of functional data with spatial dependent has become more and more convenient,but due to improper operation,machine failure and other reasons,there are missing data,and the existing functional linear model(FLM)does not consider spatial dependence,and the form of independent variable in ordinary spatial econometric model is ordinary vector data.Therefore,there is no corresponding model to deal with spatially dependent functional independent variables and random missing response variables.Thus,this article constructed two functional linear models with spatial dependence and missing response variables,namely,the functional linear spatial autoregressive model(FLSARM)and the functional linear spatial error model(FLSEM).When estimating the model,first use FPCA to process the functional independent variables,and select the number of eigenfunctions when BIC reach the minimum,so as to obtain the truncated FLSARM and FLSEM,and finally use the EM algorithm to achieve simultaneously imputation of missing data and estimation of model parameters.In the simulation analysis of FLSARM and FLSEM,whether from the data analysis of 500 simulation results or the curve comparison analysis of the slope function,it showed that FLSARM and FLSEM based on EM algorithm are better than those based on CC method.Moreover,no matter what distribution the residual term follows,the unknown parameters of the model can be well estimated.At the same time,the simulation results showed that the estimation effect of FLSARM and FLSEM is much better than that of FLM when dealing with functional data with spatial dependent.In order to verify the practical application significance of FLSARM and FLSEM,this article used FLSARM,FLSEM and FLM based on the EM algorithm and the CC method to analyze the annual average data of the temperature and the number of sunny days obtained by 73 meteorological stations in Spain.The results showed that FLSARM and FLSEM based on the EM algorithm are better than FLM when dealing with spatially dependent functional data.The real data analysis verified the rationality of FLSARM and FLSEM,and also reflected the application value of these two models.