The Group Designs The Unknown Growth Curve Model

In statistics, the growth curve model is a special multivariate linear model, which is called the generalized multivariate variance analysis model(GMANOVA). The growth curve model is a generalization of the linear model, which has a wider range of applications and richer theoretical meaning, now it is widely concerned. Growth curve models are widely used in economic, biological, medical and epidemic areas, and also are the basic analysis tools for the longitudinal data of sequence correlation or repeated observation. But the parameter estimation and classification of growth curve model for group design matrix unknown has not been resolved.The aim of this paper is used to estimate the parameter and classify in design matrix unknown growth curve model.And the idea to sovle the problem is EM algorithm. In statistics, the EM algorithm is an iterative process to find the maximum likelihood function or the maximum a posteriori function. This algorithm can be used in the treatment of missing data, censored data, incomplete data and other so-called noise. Arthur proposed and analyzed EM algorithm in 1977. In the same year, Dempster proof the convergence of EM algorithm.Research contents and methods: Previous research in growth curve model mainly for group design matrix was known are not applicable for the growth curve model which design matrix is unknown.This paper mainly estimate the paratemer of the growth curve model which design matrix is unknown. In this paper, by analyzing the theoretical knowledge and application of EM algorithm in Gauss mixture model, we find that this method can solve the problem of parameter estimation of the growth curve model with unknown design matrix. If you want to use EM algorithm, you must calculate the growth curve model of the log likelihood function and its EM algorithm E-STEP and M-STEP, which E-STEP expect the missing date under the given information and parameters, and then M-STEP, solving for the unknown parameters by maximizing the likelihood function. Then we repeated iteration until convergence. Due to the computer precision, can’t randomly selected initial value, we design two methods to choose the initial value at last. First one, we can estimate original value of by assumptionunder least square method. Second, we can estimate original value of and by assumption a known design matrix. In this paper, we propose two methods to estimate the asymptotic variance of the parameters, and finally take the bootstrap method. This paper uses AIC criteria to select parameters of the model. Finally, the results show that the proposed method is effective in the actual data by computer simulation and real data.Generally speaking, this paper discuss the parameter estimation of the growth curve model with design matrix unknown and prove rationality of the EM algorithm in the growth curve model.This paper provides a way of thinking for parameter estimation and classification of the growth curve model with design matrix unknown. Above all, this paper provides a basic method and thinking path for the growth curve model with design matrix unknown. The method can be applied to the diagnosis and financial analysis in practice.