The Tensor Expression Of Linear Mixed Effects Model

As we all know,matrix is a two-dimensional representation of data,which brings great convenience to data collecting and processing.For the multi-factors sample(referring to the number of components of each sample point is greater than 2)situation,the traditional processing method is through "data folding" or data dimensionality reducing to achieve matrix representation,which can further get the estimations of parameters or problem's solutions by linear equations or LS(least square)method.But the "data folding" can destroy the original structure of data,and data dimensionality reducing may cause unnecessary loss of useful information.Such that the final results may be meaningless.The tensor expression can overcome these two drawbacks.In this paper we first introduce the Tucker Decomposition of a tensor,which is a generalization of SVD(Singular Value Decomposition)of a matrix in the higher order case.As the SVD method in the matrix case,a three-order tensor can always be decomposed into a product of a kernel tensor(analog to the standard form of a matrix)with three orthogonal matrices along three different directions.This lead us to extract simultaneously the principal components of the data from all directions.The package dedicated to the Tucker Decomposition in some software such as MATLAB is also available.The focus of this paper is that testing the independent variables effectiveness by two-stage estimation of Linear Mixed Effects Model and further improving test based on the SVD.The process is that transforming the Linear Mixed Effects Model into a traditional Linear Regression Model,in order to eliminate the influence of random variables.Then testing the effectiveness of the model variables;It is the first time to introduce the concept of tensor into the Linear Mixed Effects Model.Constructing the new Linear Mixed Effects Model by tensor definition.Then estimating fixed effects parameter based on Tucker Decomposition,When the collinearity problem exists between the independent variables,compared with the matrix singular value decomposition,tensor Tucker Decomposition tap the data from more dimensions.On the one hand to cut out the more irrelevant information,get more useful information.On the other hand realized to further compress the data,reduce the storage of data.