DGP.CP: 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)

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 $vec(\bf S_t)$.

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 $\bf B$.

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.

Examples

Run this code

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