CP_MTS {HDTSA}

Estimation of matrix CP-factor model

Description

CP_MTS() deals with CP-decomposition for high-dimensional matrix time series proposed 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. This function aims to estimate the rank d and the coefficient matrices {\bf A} and {\bf B}.

Usage

CP_MTS(
  Y,
  xi = NULL,
  Rank = NULL,
  lag.k = 15,
  lag.ktilde = 10,
  method = c("CP.Direct", "CP.Refined", "CP.Unified")
)

Arguments

Y

A n \times p \times q data array, where n is the sample size and (p,q) is the dimension of {\bf Y}_t.

xi

A n \times 1 vector. If NULL (the default), then a PCA-based \xi_{t} is used [See Section 5.1 in Chang et al. (2023)] to calculate the sample auto-covariance matrix \widehat{\bf \Sigma}_{\bf Y, \xi}(k).

Rank

A list of the rank d,d_1 and d_2. Default to NULL.

lag.k

Integer. Time lag K is only used in CP.Refined and CP.Unified to calculate the nonnegative definte matrices \widehat{\mathbf{M}}_1 and \widehat{\mathbf{M}}_2:

\widehat{\mathbf{M}}_1\ =\ \sum_{k=1}^{K}\widehat{\mathbf{\Sigma}}_{\bf Y, \xi}(k)\widehat{\mathbf{\Sigma}}_{\bf Y, \xi}(k)',

,

\widehat{\mathbf{M}}_2\ =\ \sum_{k=1}^{K}\widehat{\mathbf{\Sigma}}_{\bf Y, \xi}(k)'\widehat{\mathbf{\Sigma}}_{\bf Y, \xi}(k),

where \widehat{\mathbf{\Sigma}}_{\bf Y, \xi}(k) is the sample auto-covariance of {\bf Y}_t and \xi_t at lag k.

lag.ktilde

Integer. Time lag \tilde K is only used in CP.Unified to calulate the nonnegative definte matrix \widehat{\mathbf{M}}:

\widehat{\mathbf{M}} \ =\ \sum_{k=1}^{\tilde K}\widehat{\mathbf{\Sigma}}_{\tilde{\bf Z}}(k)\widehat{\mathbf{\Sigma}}_{\tilde{\bf Z}}(k)'.

method

Method to use: CP.Direct and CP.Refined, Chang et al.(2023)'s direct and refined estimators; CP.Unified, Chang et al.(2024+)'s unified estimation procedure.

Value

An object of class "mtscp" is a list containing the following components:

A

The estimated p \times d left loading matrix \widehat{\bf A}.

B

The estimated q \times d right loading matrix \widehat{\bf B}.

f

The estimated latent process (\hat{x}_{1,t},\ldots,\hat{x}_{d,t}).

Rank

The estimated rank (\hat{d}_1,\hat{d}_2,\hat{d}) of the matrix CP-factor model.

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.

Chang, J., Du, Y., Huang, G. and Yao, Q.(2024+). On the Identification and Unified Estimation Procedure for the Matrix CP-factor Model, Working paper.

Examples

p = 10
q = 10
n = 400
d = d1 = d2 = 3
data <- DGP.CP(n,p,q,d,d1,d2)
Y = data$Y
res1 <- CP_MTS(Y,method = "CP.Direct")
res2 <- CP_MTS(Y,method = "CP.Refined")
res3 <- CP_MTS(Y,method = "CP.Unified")