文章将线性混合效应模型(LME)推广至线性混合张量模型. 首先建立单变量线性混合张量模型,并推广至多变量线性混合张量模型. 利用矩阵向量化方法,结合矩阵函数的导数运算和矩阵的Kronecker积, 得出LMEM模型的参数估计特别是方差参数矩阵的估计式,最后给出线性混合张量模型的VLS参数估计式.

In this paper, we initialize the linear mixed tensor model (LMTM) after the introduction of the linear mixed effects model (LMEM), which is further generalized to a multivariate LMTM (MMTM). The parameter estimation of an MMTM is also obtained based upon the approaches of vectorization, matricization, Kronecker product and the derivative of matrices. Finally we obtain the VLS (variance least square) estimation formula for both the variance and the covariance matrix.