Robust Estimation,Classification And Prediction Of Functional Data Under Bayesian Framework

For the analysis of one-dimensional functional data,this paper chooses to use the functional mixed effects model to model.However,in reality,data often produce outliers due to errors for some reasons,and some data do not conform to the hy-pothesis of normal distribution,but the traditional model is generally based on the assumption of Gaussian process,which is not robust to outliers and will have a wrong estimation.In order to improve the robustness of inference,we adopt the normal scale mixed distribution to construct the heavy-tailed distribution,and propose the assumption that the random-effects term and the random error term are independent,and used parameterized and nonparameterized methods to characterize the nonlinear correlation between repeated observations.For the complex hierarchical model struc-ture in this paper,the Bayesian method is used to estimate the parameters.Finally,the robustness and efficiency of the model are verified through detailed numerical simulation and the example analysis of the Britain traffic flow data.The main research of this paper is as follows;The first chapter introduces the research background,research status and main work of the functional mixed-effects model,and describes the concept of functional data,normal scale mixed distribution,MCMC and model selection method.In chapter 2,we first introduce the functional mixed-effects model GPFR based on Gaussian process,and then propose the HPFR model based on heavy-tailed pro-cess.The fixed-effects term is expanded by spline basis functions,and the nonlinear kernel functions are selected to describe the covariance of random-effects term.The conjugate prior distribution of each parameter is set,and the full conditional distri-bution of each parameter is deduced.For the estimation of the kernel function,the parameters in the kernel function are updated by Hybrid MCMC method.Then,this model is generalized to a classification model,and a mixture prediction method is proposed by combining classification probability and conditional distribution,and the information consistency of this model is proved.Two different types of outliers are designed in the numerical simulation to test the effect of the model.It is found that HPFR has more robust results when there were outliers or misspecification of distributions in the data.Finally,a case study of the British traffic flow data shows that the model is practical.The model in Chapter 3 is similar to that in Chapter 2,but the difference lies in the description of covariance in the random-effects term.In Chapter 2,the kernel function was used as covariance,but there are some problems,such as which kernel function to choose and whether the selected kernel function can effectively describe the nonlinear results of the data.To solve this problem,we use a data-driven method called FPCA to approximate the kernel function.Then Bayesian framework is still used for parameter estimation,and the selection of hyperparameters in Wishart dis-tribution was discussed in detail.In the numerical simulation in this chapter,we consider a variety of cases,including the types of outliers,the distribution of the generation of random-effects and random error terms,and design 14 Schemes,Firstly,we find that the selection of covariance has no obvious effects on the parameter esti-mation,and prove the robustness of HPFR when there are outliers in the data and misspecification of distributions.At the same time,compared with the traditional FPCA method,it is found that the performance of the methods presented in this chapter is better than that of FPCA method,and more importantly,the program running speed is greatly improved compared with that of the methods in the previous chapter.Finally,we still use the British traffic flow data for application analysis,through the final forecast results to prove our conclusion.In the fourth chapter,the methods and results of the model are summarized,and the future work is prospected.