DGP.CP {HDTSA}

Data generate process of matrix CP-factor model

Description

DGP.CP() function generate the matrix time series described in Chang et al. (2023):

{\bf{Y}}_t = {\bf A \bf X}_t{\bf B}^{'} + {\boldsymbol{\epsilon}}_t,

where {\bf X}_t = diag(x_{t,1},\ldots,x_{t,d}) is an d \times d latent process, {\bf A} and {\bf B} are , respectively, p \times d and q \times d unknown constant matrix, and {\boldsymbol{\epsilon}}_t is a p \times q matrix white noise process.

Usage

DGP.CP(n, p, q, d1, d2, d)

Arguments

n

Integer. Sample size of \bf Y_t, t=1,\ldots,n.

p

Integer. Number of rows of \bf Y_t.

q

Integer. Number of columns of \bf Y_t.

d1

Integer. Rank of \bf A.

d2

Integer. Rank of \bf B.

d

Integer. Number of columns of \bf A and \bf B.

Value

A list containing the following components:

Y

A n \times p \times q data array of \bf Y_t.

S

A n \times p \times q data array of \bf S_t = \bf A \bf X_t \bf B'.

A

A p \times d coefficient matrix.

B

A q \times d coefficient matrix.

X

A n \times d \times d data array of \bf X_t.

P

A p \times d_1 orthogonal matrix such that \bf A = \bf P \bf U.

Q

A q \times d_2 orthogonal matrix such that \bf B = \bf Q \bf V.

U

A d_1 \times d matrix such that \bf A = \bf P \bf U.

V

A d_2 \times d matrix such that \bf B = \bf Q \bf V.

W

A d_1 d_2 \times d matrix such that \bf W = (\bf v_1 \otimes \bf u_1,\ldots,\bf v_d \otimes \bf u_d).

Ws

A d_1 d_2 \times d matrix. An orthogonal basis of \bf W.

Xmat

A n \times d data matrix of diag(\bf X_t).

Smat

A n \times pq data matrix of vec(\bf S_t).

References

Chang, J., He, J., Yang, L. and Yao, Q.(2023). Modelling matrix time series via a tensor CP-decomposition. Journal of the Royal Statistical Society Series B: Statistical Methodology, Vol. 85(1), pp.127–148.

See Also

CP_MTS.

Examples

p = 10
q = 10
n = 400
d = d1 = d2 = 3
data <- DGP.CP(n,p,q,d,d1,d2)
Y = data$Y