Statistical Inference Of Longitudinal Functional Varying Coefficient Mixed Effect Model

Since the outbreak of the third industrial revolution in the last century,human race has gradually entered an informational era.With the advancement of the Internet,researchers found a new data type to process the massive amounts of data,which is in a more structured and quantifiable form than ever before.However,how to dig out valuable information from massive amounts of data is still a challenge task.As a result,the functional data type which proposed at the end of the last century gradually entered the vision of researchers.Its innate ability to process high-dimensional and complex data has been recognized,and is applying in medical treatment,meteorological,economic and many other fields.The essence of functional data analysis(FDA)is to treat dense observation data as elements in an infinite-dimensional function space,and to approximate discrete points as continuous curves.The general functional linear regression model(FLRM)is to establish a connection between responses and functional covariates.And then analyzing the estimated the coefficients based on this to dig out the information behind the data.As an extension of the classic FLRM,the longitudinal functional linear mixed-effects model(LFLMEM)has the ability to capture and distinguish the commonalities and characteristics between samples.However,on the one hand,due to the addition of the random effect slope function of functional covariates(RE-SFFC)to LFLMEM,the dimensionality of the model is further increased,which greatly increases the risk of "dimensionality disaster".On the other hand,the limitation of application is always an important problem.The existing model does not conduct in-depth discussion on the relationship between covariates with nonlinear relationships.Therefore,this article attempts to promote and research the existing general LFLMEM in both theoretical and empirical aspects,and further expand the application field of the model.The main work of this paper can be organized as the following points:First,this article improves longitudinal functional linear mixed-effects model(LFLMEM)and proposes a longitudinal functional varying coefficient mixed-effects model(LFVCMEM).That is,a varying coefficient part is added to the model Fixed Effect Slope Function of Functional Covariates(FE-SFFC).Perform linear expansion of the functional data,reduce the dimensionality of the model and obtain a low-order linear model,to establish a generalized penalty likelihood function,and minimize it to obtain estimates of fixed effects and random effects at the same time,and prove the estimated Asymptotic nature.Perform statistical simulations on the model,and the simulation results show the validity of the estimates.Second,the LFVCMEM studied in this article will estimate the random effect functions of continuous covariates and functional covariates at the same time.The idea of basis expansion is mainly used to reduce the dimensionality of RE-SFFC,which effectively avoids the risk of encountering the "curse of dimensionality".Third,based on the data from the NMMAPS,the impact of urban temperature on urban ozone concentration and the combined effect of the two on urban non-accidental mortality were discussed.Studies have found that high temperature has a promoting effect on the level of urban ozone concentration.Higher urban ozone concentration will increase the non-accidental death rate of local residents,which is not conducive to the health of residents.In addition,comparing the model proposed in this paper with the model of Liu et al.(2017),it is found that the estimation's MSE of the model proposed in this paper is significantly reduced,which enhances the model's ability to interpret instance data.Therefore,the model proposed in this paper is effective.