Envelope Model In Statistics And Its Tensor Representation

In the twenty-first Century,the Internet technology was flourishing,and the rapid development of the Internet has led to the progress and rise of many industries.On the one hand,Internet technology makes the way that people access to information simple and quick.On the other hand,the ways of people obtaining resources and data become diverse,and it also makes the first-hand data resources large-scale,which analysts obtained.In summary,the data show the following characters: large scale,high dimensions and complex structure.In the perspective of data,some traditional ways of data processing are invalid.In this case,the tensor as a new method of data processing and storage has more and more scholars' attention.To be sure,tensor will become a powerful tool to deal with large-scale data in the future.From the perspective of statistical model,effects of data processing which use some traditional multiple regression models become not so ideal,and even gradually failed with the more and more complex data structure.Envelope model is an improved form of multiple statistical regression model.To a certain extent,it improves the original method and increases the estimation accuracy and efficiency.The thesis focus on two aspects: the linear mixed model and Envelope model.In the aspect of linear mixed model,the maximum likelihood function is given,which based on proper model assumptions.Then we get the maximum likelihood estimation of the parameter of fixed effects.Based on the maximum likelihood function and the method of matrix derivative,the estimation of covariance in the model is obtained by the iterative algorithm given in this paper.In addition,after a proper adjustment of the model hypothesis,we get the variance least square estimation(VLS)of the corresponding parameters in the linear mixed model,Which combined with the method of variance least square estimation(VLS)and an inversion algorithm of the special block matrix.In the research of Envelope model,the least square method and the maximum likelihood function method are used to get the estimation of parameters in multiple linear regression model.The objective function and corresponding estimation formula are introduced in detail,which were used to estimate the parameters of Envelope model.The multiple regression model is extended to a tensor model.And then the tensor model was optimized(the model is founded by the product of the coefficient tensor and index matrix).then we get the least square estimation of tensor model by the matricization of the corresponding tensor model,and also get the corresponding tensor representation of coefficient.At the same time,the definition of tensor response Envelope model was given in detail.Combined with the assumption of the separable covariance structure,based on the matricization of tensor response Envelope model,we get the objective function and its parameter estimation of tensor response Envelope model,which refers to the objective function and corresponding estimation formula of Envelope model under the multivariate conditions.Finally,comparing the estimation algorithm of the parameters of tensor response Envelope model,which Li and Zhang proposed,to the algorithm proposed in this thesis,we found that the two algorithms are based on the idea of turning the high dimensional case(tensor)into a low dimensional case(matrix).The difference of the two algorithms lies in:The algorithm which Li and Zhang proposed is based on the vectorized version of tensor model,while the algorithm in this thesis is based on the matricization of tensor model.From the angle of the calculation of tensor,the two methods are theoretically feasible.