^{1}

^{2}

^{3}

^{1}

^{1}

^{*}

In this paper, we first study the linear regression model
and obtain a norm-minimized estimator of the parameter vector
by using the g-inverse and the singular value decomposition of matrix
*X*. We then investigate the growth curve model (GCM) and extend the GCM to a generalized GCM (GGCM) by using high order tensors. The parameter estimations in GGCMs are also achieved in this way.

Linear regression model, or called linear model (LM), is one of the most widely used models in statistics. There are many kinds of linear models including simple linear models, general linear models, generalized linear models, mixed effects linear models and some other extended forms of linear models [

In this paper, we first use the generalized inverse of matrices and the singular value decomposition to obtain the norm-minimized estimation of the parameters in the linear model. Then we introduce some basic knowledge about tensors before we employ tensors to express and extend the multivariate mixed effects linear models. The extended tensor form of the model can be also regarded as a generalization of the GCM.

Let us first begin with some basic linear regression models. We let y be a response variable and x 1 , ⋯ , x r be independent random variables for explaining y. The most general regression model between y and x 1 , ⋯ , x r is in form

y = f ( x 1 , ⋯ , x r ) + ε (1.1)

where ε is the error term, and f is an unknown regression function. In linear regression model, f is assumed to be a linear function, i.e.,

y i = β 0 + β 1 x 1 + β 2 x 2 + ⋯ + β r x r + ε (1.2)

where all β i are unknown parameters. Denote x = ( x 1 , ⋯ , x r ) which is called a random vector. Let P = ( y , x ) , an ( r + 1 ) -dimensional random vector, which is called an observable vector. Given N observations of P, say P i = ( y i , x i 1 , x i 2 , ⋯ , x i r ) , i = 1 , 2 , ⋯ , N . Here y i stands for the ith observation of the response variable y, and x i 1 , x i 2 , ⋯ , x i r are the corresponding explanatory observations. The sample model of Equation (1.2) turns out to be

y i = β 0 + β 1 x i 1 + β 2 x i 2 + ⋯ + β r x i r + ε (1.3)

or equivalently

y = X β + ε (1.4)

where y = ( y 1 , y 2 , ⋯ , y N ) T ∈ ℝ N (here and throughout the paper T stands for the transpose of a matrix/vector) is the sample vector of the response variable y, X = ( x i j ) ∈ ℝ N × ( r + 1 ) is the data matrix or the design matrix each of whose rows corresponding to an observation of x, β = ( β 0 , β 1 , ⋯ , β r ) ∈ ℝ r + 1 is the regression coefficient vector, which is to be estimated, and ε = ( ε 1 , ε 2 , ⋯ , ε N ) T is the random error vector. A general linear regression model, abbreviated GLM, is a LM (1.4) with the error terms ε i satisfying:

1) Zero-mean: Ε [ ε i ] = 0 , ∀ i ∈ [ N ] , i.e., the expected value of the error term is zero for all the observations.

2) Homoskedasticity: Var [ ε ] = σ 2 , i.e., all the error term are distributed with the same variance.

3) Uncorrelation: Cov ( ε i , ε j ) = 0 for all distinct i , j , i.e., distinct error terms are uncorrelated.

Equations (1)-(3) is called Gauss-Markov assumption [

Ε [ Y ] = X β , Cov ( ε ) = σ 2 I N (1.5)

where I N is the N × N identity matrix and σ 2 > 0 . In order to investigate the general linear model and extend the properties, we recall some known results concerning the linear combinations of some random variables. Suppose α ∈ ℝ n is a constant vector with the same length as that of y, the random vector under the investigation.

Let A ∈ ℝ m × n . The g-inverse of A, denoted A g , is a generalized inverse defined as an n × m matrix satisfying [

x = A g b + ( I − A g A ) ω , ∀ ω ∈ ℝ n . (1.6)

It is easy to verify that A g = A − 1 is unique when A is invertible. The g-inverse of a matrix (usually not unique) can be calculated by using singular value decomposition (SVD).

Lemma 1.1. Let X ∈ ℝ N × p with a SVD decomposition X = U D V T such that U ∈ ℝ N × N and V ∈ ℝ p × p are orthogonal matrices, and D ∈ ℝ N × p is in form D = d i a g ( σ 1 , σ 2 , ⋯ , σ r , 0 , ⋯ , 0 ) where r = r a n k ( X ) ≤ min ( N , p ) , and σ 1 ≥ σ 2 ≥ ⋯ ≥ σ r . Then

A g = V [ D r − 1 * * * ] U T ∈ ℝ n × m (1.7)

where * denotes any matrix of suitable size and D r = d i a g ( σ 1 , σ 2 , ⋯ , σ r ) .

The Gauss-Markov Theorem (e.g. Page 51 of [

Lemma 1.2. Suppose that model (1.4) satisfies the Gauss-Markov assumption and a ∈ ℝ N be a constant vector. Then z = a T y is estimable, and a T β ^ is the best (minimum variance) linear unbiased estimator (BLUE) of a T β , with β ^ = ( X T X ) g X T y .

Based on Lemma 1.2, we get

Proposition 1.3. If r a n k ( X ) = r < min ( N , p ) in Equation (1.4) and X satisfies condition in Lemma 1.2. Then the estimator of β with minimal 2-norm is in form

β ^ = V [ D r − 1 y ˜ 1 0 ] (1.8)

where y ˜ = U T y = [ y ˜ 1 T , y ˜ 2 T ] T with y ˜ 1 ∈ ℝ r , y ˜ 2 ∈ ℝ N − r .

Proposition 1.3 tells us that by taking D as a block upper triangle form in the decomposition

( X T X ) g = V [ G 11 G 12 0 G 22 ] V T .

We can reach a norm-minimised estimator of β . Now denote H : = ( X T X ) g X . By Gauss-Markov Theorem, we have

‖ β ⌢ ‖ 2 = 〈 H y , H y 〉 = y T H T H y = y T X ( X T X ) 2 g X T y

which implies that ‖ β ⌢ ‖ = ‖ ( X X T ) g y ‖ .

The generalized linear model (GLM) is a generalization of LM [

β ( m ) = ( X T W ( m − 1 ) X ) g X T W ( m − 1 ) z ( m − 1 ) , W = d i a g ( W 1 , W 2 , ⋯ , W N )

where W i = w i / { ϕ v ( μ i ) [ g T ( μ i ) ] 2 } with w i being a known priori weight,

ϕ the dispersion parameter, v ( ⋅ ) a variance function, g a link function, and z ∈ ℝ N the work dependent variable with z i = η i + ( y i − μ i ) g T ( μ i ) . The moment estimation of discrete parameters is

ϕ ^ = 1 N − k − 1 ∑ i = 1 N w i ( y i − μ ^ i ) 2 v ( μ ^ i ) . (1.9)

In order to extend the GLMS to more general case, we need some knowledge on tensors. In the next section, we will introduce some basic terminology and operations implemented on tensors, especially on low order tensors.

A tensor is an extension of a matrix in the case of high order, which is an important tool to study high-dimensional arrays. The origin of tensor can be traced back to early nineteenth century when Cayley studied linear transformation theory and invariant representation. Gauss and Riemann et al. promoted the development of tensor in mathematics. In 1915 Albert Einstein used tensor to describe his general relativity, leading tensor calculus more widely accepted. In the early twentieth century, Ricci and Levi-Civita further developed tensor analysis in absolute differential methods and explored their applications [

For our convenience we denote [ n ] : = { 1 , 2 , ⋯ , n } and use S ( m , n ) to denote the index set

S ( m , n ) : = { τ = ( i 1 , i 2 , ⋯ , i m ) : i k ∈ [ n ] , ∀ k ∈ [ m ] } .

Let I k ( k ∈ [ m ] ) be any positive integer (usually larger than 1). Sometimes we abuse I k as a set [ I k ] . Denote I : = I 1 × I 2 × ⋯ × I m . If I k stands for an index set, then I is a tensor product of I 1 , I 2 , ⋯ , I m . An m-order tensor A = ( A σ ) of size I is an m-array whose entries are denoted by A σ : = A i 1 i 2 ⋯ i m with σ = ( i 1 , i 2 , ⋯ , i m ) ∈ I . Note that a vector is a 1-order tensor and an m × n matrix is a 2-order or second order tensor. An m × n tensor is a tensor with I 1 = I 2 = ⋯ = I m = n . We denote by T m , n the set of all mth order n-dimensional real tensors . An m × n tensor A is called symmetric if A σ is constant under any permutation on its index.

An mth order n-dimensional real tensors A is always associated with an m-order homogeneous polynomial f A ( x ) which is defined by

f A ( x ) : = A x m = ∑ i 1 , i 2 , ⋯ , i m A i 1 , i 2 , ⋯ , i m x i 1 x i 2 ⋯ x i m . (2.10)

A is called positive definite or pd (positive semidefinite or psd) if

f A ( x ) : = A x m ≥ 0 , ∀ x ∈ ℝ n . (2.11)

A nonzero psd tensor must be of an even order. Let A be of size m × n × p . Given an r-order tensor A ∈ ℝ n 1 × n 2 × ⋯ × n r and a matrix U = ( u i j ) ∈ ℝ I k × J k where k ∈ [ r ] . The product of A with U along k-mode is defined as the r-order tensor A × k U defined as

( A × k U ) i 1 , ⋯ , i k − 1 , i k , i k + 1 , ⋯ , i m = ∑ i = 1 n A i 1 , ⋯ , i k − 1 , i k , i k + 1 , ⋯ , i m u i i k . (2.12)

Note that A × k U is compressed into an ( m − 1 ) -order tensor when U ∈ ℝ I k is a column vector ( J k = 1 ) . There are two kinds of tensor decomposition, i.e., the rank-1 decomposition, also called the CP decomposition, and the Tucker decomposition, or HOSVD. The former is the generalization of matrix rank-1 decomposition and the latter is the matrix singular value decomposition in the high order case. A zero tensor is a tensor with all entries being zero. A diagonal tensor is a tensor whose off-diagonal elements are all zero, i.e., A i 1 , i 2 , ⋯ , i m = 0 if i 1 , i 2 , ⋯ , i m are not identical. Thus an m × n tensor has n diagonal elements. By this way, we can define similarly (and analogous to matrix case) the identity tensor and a scalar tensor.

For any i ∈ [ n ] , an i-slice of an m-order tensor A = ( A i 1 i 2 ⋯ i m ) along mode k for any given k ∈ [ m ] is an ( m − 1 ) × n tensor B with

B i 1 , i 2 , ⋯ , i m − 1 = A i 1 , ⋯ , i k − 1 , i , i k + 1 , ⋯ , i m , i = 1 , 2 , ⋯ , n .

A slice of 3-order tensor A = ( A i j k ) ∈ ℝ m × n × p along mode-3 is an m × n matrix A ( : , : , k ) with k ∈ [ p ] , and a slice of a 4-order tensor is a 3-order tensor.

Let A ∈ ℝ m × n 1 × p , B ∈ ℝ n 1 × n × p be two tensors of 3-order. The slice-wise product of A , B , denoted by

where A ( : , : , k ) is the matrix consisting of n sample points of size m in class k and X ( : , : , k ) is the design matrix corresponding to the kth sample (there are n 1 observations in each class in this situation).

Let k be a positive integer. The k-moment of a random variable x is defined as the expectation of x, i.e., m x ( k ) ( x ) = Ε ( x k ) . The traditional extension of moments to a multivariate case is done by an iterative vectorization imposed on k. This technique is employed not only in the definition of moments but also in other definitions such as that of a characteristic function. By introducing the tensor form into these definitions, we find that the expressions will be much easier to handle than the classical ones. In the next section, we will introduce the tensor form of all these definitions.

Let x = ( x 1 , x 2 , ⋯ , x n ) T be a random vector. Denote by x m the (symmetric) rank-one m-order tensor with

x σ m = x i 1 x i 2 ⋯ x i m , ∀ σ : = ( i 1 , i 2 , ⋯ , i m ) ∈ S ( m , n ) .

x m is called a rank-1 tensor generated by x which is also symmetric. It is shown by Comon et al. [

where α i ( j ) ∈ ℝ I i for all j ∈ [ r ] , i ∈ [ m ] . The smallest positive integer r is called the rank of A , denoted by r a n k ( A ) . We note that Equation (2.14) can also be used to define the tensor product of two matrices, which will be used in our next work on the covariance of random matrices. Note that the tensor product of two rank-one matrices is

( α 1 × β 1 ) × ( α 2 × β 2 ) = α 1 × α 2 × β 1 × β 2 .

Now consider two matrices A ∈ ℝ m × n , B ∈ ℝ p × q . Then write A , B in a rank-1 decomposition, i.e.,

A = ∑ j = 1 R 1 α 1 ( j ) × β 1 ( j ) , B = ∑ k = 1 R 2 α 2 ( k ) × β 2 ( k ) .

Tucker decomposition decomposes the original tensor into a product of the core tensor and a number of unitary matrices in different directions [

A = S × 1 U 1 × 2 U 2 × 3 ⋯ × N U N (2.15)

where S is the core tensor, and U 1 , U 2 , ⋯ , U N are unitary matrices.

Example 2.1. Let

X ( : , : , 1 ) = [ 1 3 2 4 ] , X ( : , : , 2 ) = [ 5 7 6 8 ] .

Then the unfolded matrices along 1-mode, 2-mode and 3-mode are respectively

X 1 = [ 1 5 3 7 2 6 4 8 ] , X 2 = [ 1 3 5 7 2 4 6 8 ] , X 3 = [ 1 2 5 6 3 4 7 8 ] .

The growth curve model (GCM) is one of the GLMs introduced by Wishart in 1938 [

The analysis of a GCM focuses on the functional relationship among ordered responses. Conventional GCM methods apply to growth data and to other analogs such as dose-response data (indexed by dose), location-response data (indexed by distance), or response-surface data (indexed by two or more variables such as latitude and longitude). The GCM methods mainly focus on longitudinal observations on a one-dimensional characteristic even though they may also be used in multidimensional cases [

A general GCM can be indicated by

Y = X B T + E (3.16)

where Y ∈ ℝ N × p is the random response matrix whose rows are mutually independent and columns correspond to the response variables ordered according to d = [ d 1 , d 2 , ⋯ , d p ] T ; X ∈ ℝ N × q is the fixed design matrix with r : = r a n k ( X ) ≤ q ≤ N ; The matrix B ∈ ℝ q × m is a fixed parameter matrix whose entries are the regression coefficients; T ∈ ℝ m × p is a within-subject design matrix each of whose entries is a fixed function of d, and E ∈ ℝ N × p is a random error with matrix normal distribution E ∼ N N , p ( 0 , Σ , I N ) where Σ ∈ ℝ p × p is an unknown symmetric positive definite matrix. Suppose the samplings corresponding to each object are recorded at p different times (moments) d 1 , d 2 , ⋯ , d p . Consider an example of a pattern of the children’s weight. The plotting of the weights against the ages indicates a temporal pattern of growth. A univariate linear model for weight given age could be fitted with a design matrix T expressing the central tendency of the children’s weights as a linear or curvilinear function of age. Here T is an example of a within-subject design matrix. If N > 1 , a separate curve could be fitted for each subject to obtain a separate matrix of regression parameter estimators for each independent sampling units, β ^ i = Y i T ( T T T ) − 1 for i ∈ [ N ] , and a simple average of the N fitted curves is a proper (if not efficient) estimator of the population growth curve, that is,

B ^ = 1 N ∑ j = 1 N B ^ j .

The efficient estimator has the form

B ^ = [ ( X T X ) − 1 X T ] [ T ( T T T ) − 1 ] . (3.17)

If the subjects are grouped in a balanced way, i.e., The N observations are clustered into m groups, each containing the same number, say n, of observations. For simplicity, we may assume that first n each then X = l N , the all-one vector, is the appropriate choice for computing B ^ . The choice of T defines the functional form of the population growth curve by describing a function relationship between weight and age.

Example 3.1. We recorded the heights of n boys and n girls whose ages are 2, 3, 4 and 6 years. From the observations we make an assumption that the average height increases linearly with age. Since the observed data is partitioned into two groups (one is for the heights of n boys and another is for the height of n girls), each consisting of n objects, and p = 4 with age vector d = [ 2 , 3 , 4 , 6 ] T . Thus the model for the height vs. age shall be Y = X B T + E where

X = [ l n 0 0 l n ] , T = [ l p T d T ]

where l k ∈ ℝ k is an all-ones vector of dimension k.

Here β 11 , β 12 are respectively the intercept and the slope for girls and β 21 , β 22 are respectively the intercept and the slope for boys. We find that it is not so easy for us to investigate the relationship between the gender, height, weight, and age. In the following we employ tensor expression to deal with this issue.

Using the notation in tensor theory, we rewrite model (3.16) in form

Y = X × 2 B × 2 T + E

or equivalently

Y = B × 1 X T × 2 T + E (3.18)

where B is regarded as a second order tensor and X , T as two matrices. Actually, according to Equation (2.12), we have

( B × 1 X T ) i j = ∑ k = 1 q X i k B k j .

Similarly we can define B × 2 V . Note that

B × 1 U × 2 V = B × 2 V × 1 U .

Now we extend model (3.16) in a more general form as

A = B × 1 X 1 × 2 X 2 × 3 X 3 + E (3.19)

where A ∈ ℝ n 1 × n 2 × n 3 , B = ( B i j k ) ∈ ℝ m 1 × m 2 × m 3 is a 3-order tensor, which is usually an unknown constant parameter tensor or the kernel tensor, and X i ∈ ℝ n i × m i for i = 1 , 2 , 3 . Here the tensor-matrix multiplication is defined by Equation (2.12) according to the dimensional coherence along each mode.

The potential applications of Equation (3.21) are obvious. A HOSVD (high order singular value decomposition) of a 3-order tensor can be regarded as a good example for this model.

Example 3.2. A sequence of 1000 images extracted from a repository of face images of ten individuals, each with 100 face images. Suppose each face image is of size 256 × 256 . Then these images can be restored in an 256 × 256 × 1000 tensor A . Let A be decomposed as

A = B × 1 U 1 T × 2 U 2 T × 3 U 3 T (3.20)

where

B ∈ ℝ 16 × 16 × 50 , U 1 ∈ ℝ 256 × 16 , U 2 ∈ ℝ 256 × 16 , U 3 ∈ ℝ 1000 × 50 .

The decomposition Equation (3.20) yields a set of compressed images, each with size 16 × 16 . If each individual can be characterized by five images (this is called a balanced compression), then the kernel tensor B consists of 50 compressed images where each U i is a projection matrix along mode-i (i = 1, 2, 3). Specifically, U 1 and U 2 together play a role of compression of each image into an 16 × 16 image, while U 3 finds the representative elements (here is the 50 images) among a large set of images (the set of 1000 face images).

Analog to GCM, we let Y i j k be the measured value of Index I k in Class C i at time T j . A tensor Y = ( Y i j k ) ∈ ℝ m × n × p can be used to express m objects, say P 1 , ⋯ , P m , each having p indexes I 1 , ⋯ , I p measured respectively at times t 1 , ⋯ , t n . For each index I k , k ∈ [ p ] , we have GCM form:

Y k = B k × 1 X × 2 T + E k (3.21)

where Y k = ( y 1 ( k ) , ⋯ , y n ( k ) ) ∈ ℝ m × n . Suppose each row of Y k stands for a class of individuals, e.g., partitioned by ages. To make things more clear, we consider a concrete example.

Example 3.3. There are 30 persons under health test, each measured, at time T 1 , ⋯ , T 4 , 10 indexes such as the lower/higher blood pressures, heartbeat rate, urea, cholesterol, bilirubin, etc. We label these indexes respectively by I 1 , ⋯ , I 10 . Suppose that the 30 people are partitioned into three groups (denoted by C 1 , C 2 , C 3 ) with respect to their ages, consisting of 5, 10, 15 individuals respectively. Denote

X = [ l 5 0 0 0 l 10 0 0 0 l 15 ] , T = [ 1 1 1 1 t 1 t 2 t 3 t 4 t 1 2 t 2 2 t 3 2 t 4 2 ]

and

B k = [ β 11 k β 12 k β 1 p k β 21 k β 22 k β 2 p k β 31 k β 32 k β 3 p k ] .

Denote by Y i j k the measurement of Index I k in group C i at time T j . Set Y ( : , : , k ) = Y k , B ( : , : , k ) = B k for k = 1 , 2 , ⋯ , 10 . Then we have

Y = B × 1 X T × 2 T T + ε (3.22)

where Y , ε ∈ ℝ 30 × 4 × 10 , X ∈ ℝ 30 × 3 , T ∈ ℝ 4 × 3 and B ∈ ℝ 3 × 3 × 10 is an unknown constant parameter tensor to be estimated, where B ( : , : , k ) = B k is the parameter matrix corresponding to the k th index model. The model (3.22) can be further promoted to manipulate a balanced linear mixed model when multiple responses are measured for balanced clustered (i.e., there are same number of subjects in each cluster) subjects.

In the multivariate analysis, the correlations between the coordinates of a random vector x = ( x 1 , ⋯ , x n ) T are represented by the covariance matrix Σ : = Σ ( x ) , which is symmetric and positive semidefinite. When the variables are arrayed as a matrix, say X = ( X i j ) ∈ ℝ m × n , which is called a random matrix, the correlation between any pair of entries, say X i 1 j 1 and X i 2 j 2 of matrix X , is represented as an entry of a matrix Σ which is defined as the covariance matrix of the vector. A matrix normal distribution is defined. μ ∈ ℝ m × n , and Σ ∈ ℝ m × m , ϕ ∈ ℝ n × n are two positive definite matrices. A random matrix X ∈ ℝ m × n is said to obey a matrix normal distribution, denoted by X ∼ N m , n ( μ , Σ , ϕ ) , if it satisfies the following the conditions:

1)

2) Each row X i ⋅ of X obeys normal distribution X i ⋅ ∼ N p ( 0 , ϕ ) for i ∈ [ m ] .

3) Each column X ⋅ j obeys normal distribution X ⋅ j ∼ N q ( 0 , Σ ) .

It is easy to show that a matrix normal distribution X ∼ N p . q ( μ , Σ , ϕ ) is equivalent to vec ( X ) ∼ N p q ( vec ( μ ) , Σ ⊗ ϕ ) (see e.g. [

We now define the tensor normal distribution. Let A = ( A i 1 i 2 … i m ) ∈ T m be an m-order tensor of size N : = N 1 × N 2 × ⋯ × N m , each of whose entries is a random variable. Let μ = ( μ i 1 i 2 ⋯ i m ) be an m-order tensor of the same size as that of A , and Σ k ( k ∈ [ m ] ) be an N k × N k positive definite matrix. For convenience, we denote by I ( n ) the ( m − 1 ) -tuple ( i 1 , i 2 , ⋯ , i n − 1 , i n + 1 , ⋯ , i m ) with i k ∈ [ N k ] . We denote by A ( I ( n ) ) and μ ( I ( n ) ) , both in ℝ N n respectively the corresponding fibre (vector) of A and μ , indexed by I ( n ) , i.e.,

A ( I ( n ) ) : = A ( i 1 , i 2 , ⋯ , i n − 1 , : , i n + 1 , ⋯ , i m ) , ∀ i k ∈ N k , k ∈ [ m ] \ { n } .

A ( I ( n ) ) μ ( I ( n ) ) is called a fibre of A ( μ resp.) along mode-n indexed by I ( n ) . A is said to obey a tensor normal distribution with parameter matrices ( μ , Σ 1 , ⋯ , Σ m ) or denoted by

A ∼ N T ( μ , Σ 1 , ⋯ , Σ m )

if for any n ∈ [ m ] , we have

A ( i 1 , ⋯ , i n − 1 , : , i n + 1 , ⋯ , i m ) ∼ N N n ( μ ( I ( n ) ) Σ n ) .

A is said to follow a standard tensor normal distribution if all the Σ k ’s are identity matrices. A model (2.13) with a tensor normal distribution is called a general tensor normal (GTN) model.

To show the application of tensor normal distribution, we consider the 3-order tensor. For our convenience, we use ( i , j ) -value to denote the value related to the ith subject at jth measurement for any ( i , j ) ∈ [ m ] × [ n ] . For example, the kth response observation Y i j k at ( i , j ) represents the kth response value measured on the ith subject at time j. Now we let m , n , p be respectively the number of observed objects, number of measuring times for each subject, and the number of responses for each observation. Denote by Y the response tensor with Y i j k being the kth response at ( i , j ) , and by X the covariate tensor with X i j : ∈ ℝ r being the covariate vector at ( i , j ) for fixed effects, and by U i j : the covariate vector at ( i , j ) for random effects. Further, for each k ∈ [ p ] , we denote by B k ∈ ℝ r the coefficient vector related to the fixed effects corresponding to the kth response Y i j k at ( i , j ) for each pair ( i , j ) ∈ [ m ] × [ n ] , and similarly by C k ∈ ℝ q the coefficient vector related to the random effects. Now let B = [ B 1 , ⋯ , B p ] and γ = [ C 1 , ⋯ , C p ] . Then B ∈ ℝ r × p , γ ∈ ℝ q × p . We call X , U respectively the design matrix for fixed effects and the design matrix for random effects. Then we have

with Y = ( Y i j k ) ∈ ℝ m × n × p , X = ( X i j k ) ∈ ℝ m × n × r , B = ( β i j ) ∈ ℝ r × p , U = ( U i j k ) ∈ ℝ m × n × q , γ = ( γ i j ) ∈ ℝ q × p , ε = ( ε i j k ) ∈ ℝ m × n × p where ε i j k is the error term. Here the tensor-matrix multiplications X B and

( X B ) i j k = X ( i , j , : ) B k = ∑ k ′ = 1 r X i j k ′ β k ′ k , ∀ i ∈ [ m ] , j ∈ [ n ] , k ∈ [ p ]

Denote B = [ β 1 , ⋯ , β p ] , γ = [ γ 1 , ⋯ , γ p ] , and

E i ( 1 ) : = E ( i , : , : ) , E j ( 2 ) : = E ( : , j , : ) , E k ( 3 ) : = E ( : , : , k ) , ∀ i ∈ [ m ] , j ∈ [ n ] , k ∈ [ p ]

E j k ( 1 ) : = E ( : , j , k ) , E i k ( 2 ) : = E ( i , : , k ) , E i j ( 3 ) : = E ( i , j , : ) , ∀ i ∈ [ m ] , j ∈ [ n ] , k ∈ [ p ]

where each matrix E l ( s ) is called a slice on mode s, and each vector E l t [ s ] is called a fibre along mode-s. We also use E [ s ] to denote the set consisting of all fibres of ε along mode-s, and use notation E [ s ] ∼ P to express that each element of E [ s ] obeys distribution P where P is a distribution function. For example, E [ 1 ] ∼ N m ( 0 , I m ) means that each 1-mode fibre E j k [ 1 ] (there are n p 1-mode fibres) obeys a standard normal distribution, i.e., E j k [ 1 ] ∼ N m ( 0 , I m ) .

Now for convenience we let ( n 1 , n 2 , n 3 ) : = ( m , n , p ) . We assume that

1) γ obeys matrix normal distribution γ ∼ N q , p ( 0 , Σ , ϕ ) .

2) The random vectors in

3) For any

The model (4.23) with conditions (I, II, III) is called a 3-order general mixed tensor (GMT) model. We will generalise this model to a more general case. In the following we first define the standard normal 3-order tensor distribution:

Definition 4.1. Let

A 3-order random tensor satisfying TSN distribution has the following property:

Theorem 4.2. Let

Proof. □

Note that condition (III) is a generalization to the matrix normal distribution, and we denote it by

Note that

here

Now we unfold

where

The multivariate linear mixed model (4.23) or (4.25) can be transformed into a general linear model through the vectorization of matrices. Recall that the vectorization of a matrix

where

Lemma 4.3. Let

1)

2)

3)

The following property of the multiplication of a tensor with a matrix is shown by Kolda [

Lemma 4.4. Let

where

Proof. Let

From which the result Equation (4.27) is immediate. □

Note that our Formula (4.27) is different from that in Section 2.5 in [

We have the following result for the estimation of the parameter matrix

Theorem 4.5. Suppose

Proof. We first write Equation (4.25) in a matrix-vector form by vectorization by using the first item in Lemma 4.3,

where

By using (1) of Lemma 4.3 again (this time in the opposite direction), we get result (4.29). □

For any

Theorem 4.6. Suppose

where

Proof. By Theorem 4.5 and Equation (4.26), we have

It follows that

By employing Lemma 4.4, we get result (4.31). □

This research was partially supported by the Hong Kong Research Grant Council (No. PolyU 15301716) and the graduate innovation funding of USTS. We shall thank the anonymous referees for their patient and elaborate reading and their suggestions which improved the writing of the paper.

Lin, Z.R., Liu, D.Z., Liu, X.Y., He, L.L. and Xu, C.Q. (2018) High Order Tensor Forms of Growth Curve Models. Advances in Linear Algebra & Matrix Theory, 8, 18-32. https://doi.org/10.4236/alamt.2018.81003