Clustering Analysis Of Longitudinal Data Based On Mixture Growth Curves Model

Longitudinal data is composed of multiple measurements of each individual in time or space.Longitudinal data has the characteristics of both time series data and cross-sectional data.Growth curve model(GCM)is an commonly important model for longitudinal data analysis.In the existing literatures,there are usually two important assumptions in the growth curve model :(1)the random error term followes the multivariate normal distribution;(2)the group matrix is known.However,the two assumptions may not be able to meet in practice.In this paper,we relax the above two assumptions,and assume that error term followes the multivariate power exponential distribution and group matrix is unknown.Based on these assumptions,we discuss parameter estimations of the growth curve model(we called the mixture growth curve model),and study the longitudinal data clustering based on the mixture growth curve model and random search method.General GCM has the above two strong assumptions,which are relaxed in this paper.Firstly,we extend the distribution of random error matrix from multivariate normal distribution to more general multivariate power exponential distribution,and research the parameter estimation under GCM.Secondly,we relax the assumption of the group matrix,which extend it from the known case to the unknown case.The model where the group matrix is unknown is called the mixure growth curve model.We deduce the parameter estimation under the mixure growth curve model when the error matrix follows multivariate power exponential distribution.This paper mainly promotes the work of Yating Pan et al.(2020)and Pan et al.(2021)about the mixture growth curve model.Based on the mixture growth curve model,they analyzed longitudinal data where the error matrix obeyed multivariate normal distribution and the covariance structure was a specific form,and derived the estimation formula of unknown parameters.In this paper,the distribution of error matrix is extended from multivariate normal distribution to multivariate power exponential distribution,and the parameter estimation of mixture growth curve model is discussed when the group matrix is unknown.Because the group matrix can realize the group control of each individual,the cluster analysis of longitudinal data can be carried out according to the estimated group matrix.According to this method,we can simplify the complex data clustering problem to the estimation of classification parameters,and provide a new perspective for the cluster analysis of longitudinal data.About longitudinal data clustering analysis,mixture growth curve model provides a new way of thinking and a new method.Namely,we can obtain longitudinal data grouping(clustering)according to the estimation of group matrix,but the parameter estimation of the mixture growth curve model is more complex.Aiming at this problem,this paper proposes a new method for longitudinal data clustering,which is clustering method based on random search.In this method,an information criterion of regression clustering is firstly proposed,then we construct a sampling probability mass function of the group matrix according to the observed data.The optimal estimation of the group matrix is the value that maximizes the function.When the number of individuals n is large,it is difficult or even impossible to search the optimal group matrix in the whole space.Thus we employ markov chain monte carlo methods to estimate it.Specifically,through iterative regression and clustering,we obtain a Gibbs sampler to generate markov samples of the group matrix from the sampling probability mass function,which is induced by the information criterion,and estimate the group matrix based on the samples.Here we use empirical BIC to determine the optimal number of clusters.Simulation analysis and real data analysis show that the parameter estimation and clustering method are effective,which are based on mixture growth curve model and random search.In a word,this paper studies longitudinal data clustering based on mixture growth curve model and random search method,and the main research contents are as follows:(1)Parameter estimation of mixture growth curve with multivariate power exponential distribution;(2)Longitudinal data clustering analysis based on mixture growth curve model;(3)Longitudinal data clustering analysis based on mixture growth curve model and random search method.The research contributions of this paper are as follows :(1)We extend the general growth curve model,where the distribution assumption of the error term is extended from multivariate normal distribution to multivariate power exponential distribution.Thus the growth curve model is expanded from the perspective of distribution,which makes the growth curve model more general and more applicable.(2)Based on the mixture growth curve model,the individuals are classified according to the estimation of the group matrix,which provides a new idea for clustering analysis of longitudinal data.(3)Based on the mixture growth curve model and Gibbs sampler method,we propose a longitudinal data clustering method based on random search.Compared with the whole subclustering method,this method is efficient and fast,and overcomes the disadvantage that the whole subclustering method can not be calculated when the number of individuals n is large.