Bayesian Inference In Functional Mixed Effects Models Under Heavy-tailed Processes

Due to the complexity,quantification,multi-classification,and heterogeneity of data,traditional statistical analysis models and methods are difficult to cope with Therefore,statistical analysis of functional data in meteorology,medicine,biology,and many other fields has crucial theoretical and applied value.At the same time,the data does not always satisfy the normal hypothesis,that is,when the analysis data has heavy tail,skew or multi-peak distribution characteristics,the analysis result based on Gaussian process lacks robustness.In this paper,the mixed-effects model is used to process functional data,and the random error is independent with the regression function.This paper discusses the statistical inference of functional mixed effect model based on scale mixed normal distribution and gives the parameter estimation method under Bayesian framework.Through statistical simulation and traffic flow data,the efficiency of the estimation method and the effectiveness of the model are verifiedThe main work of this master's thesis is as followsChapter 1 introduces the research background and research status of functional mixed effects model,the basic concepts of functional data and normal mixed scale distribution,and the Bayesian methods and their basic principlesThe second chapter studies the parameter estimation problem of the functional mixed effect model under the heavy tail process.Firstly,the structure of the model is proposed.Secondly,we discuss the prior distribution of each parameter under the general structural variance and under the assumption of independent homoskedastic-ity,and derive the detailed MCMC parameter estimation algorithm.Again,through three Numerical simulations show that the model based on the heavy tail process is more robust than the Gaussian process when there are abnormal points or abnor-mal curves in the data.Finally,the example analysis of traffic flow data verifies the validity of the model and the effectiveness of the Bayesian inferenceChapter 3 uses the Kernel function to describe the variance of the random effect in order to avoid the strong assumption of the variance.It not only simplifies the structure of the variance,but also ensures the effective characterization of the curve shape,and reduces the dimension of the variance parameter.The parameter esti-mation problem of the functional mixed effect model based on the kernel function is discussed.First,the usage of the type of kernel function is introduced.Secondly,we set a priori for each parameter and derive the specific steps of the MCMC algorithm.Then,numerical simulations show that the inference based on the heavy-tail process is more robust than the Gaussian process in two cases.Finally,the case analysis of traffic flow data verifies the validity and the robustness of Bayesian inference based on the heavy-tailed process and kernel function.Chapter 4 summarizes a series of models and methods discussed in this paper,and points out the direction that can be further researched.