TFM_est {RTFA}

Estimation of Factor Model for High-Dimensional Tensor Time Series

Description

This function is to estimate the tensor factor model via four different methods, namely the initial estimation without initial (IE), one-step projection estimation (PE), iterative projection estimation (iPE) and iterative weighted projection estimation by Huber loss (HUBER).

Usage

TFM_est(x, r, method = "PE", tol = 1e-04, maxiter = 100)

Arguments

x

T \times p_1 \times \cdots \times p_K tensor-valued time series.

r

input rank of the factor tensor.

method

character string, specifying the type of the estimation method to be used.

"IE",

Initial estimation, without projection.

"PE",

One-step projection estimation.

"iPE",

Iterative projection estimation.

"HUBER",

Iterative weighted projection estimation based on huber loss function.

tol

tolerance in terms of the Frobenius norm.

maxiter

maximum number of iterations if error stays above tol.

Details

See Barigozzi et al. (2022) and Barigozzi et al. (2023) for details.

Value

return a list containing the following:

Ft

estimated factor processes of dimension T \times r_1 \times r_2 \times \cdots \times r_K.

Ft.all

Summation of factor processes over time, of dimension r_1,r_2,\cdots,r_K.

Q

a list of estimated factor loading matrices Q_1,Q_2,\cdots,Q_K.

x.hat

fitted signal tensor, of dimension T \times p_1 \times p_2 \times \cdots \times p_K.

niter

number of iterations.

fnorm.resid

Frobenius norm of residuals, divide the Frobenius norm of the original tensor.

Author(s)

Matteo Barigozzi, Yong He, Lingxiao Li, Lorenzo Trapani.

References

Barigozzi M, He Y, Li L, Trapani L. Robust Estimation of Large Factor Models for Tensor-valued Time Series. <arXiv:2206.09800>

Barigozzi M, He Y, Li L, Trapani L. Statistical Inference for Large-dimensional Tensor Factor Model by Iterative Projection. <arXiv:2303.18163>

Examples

library(rTensor)
set.seed(1234)
p <- c(12,16,20)  # dimensions of tensor time series
r <- c(3,4,5)  # dimensions of factor series
A<-list()
Q<-list()
for(i in 1:3){
  A[[i]]<-matrix(rnorm(p[i]*r[i],0,1),p[i],r[i])
  Q[[i]]<-eigen(A[[i]]%*%t(A[[i]]))$vectors
}
T<-100
F<-array(NA,c(T,r))
E<-array(NA,c(T,p))
S<-array(NA,c(T,p))
X<-array(NA,c(T,p))
for(t in 1:T){
  F[t,,,]<-array(rnorm(prod(r),0,1),r)
  E[t,,,]<-array(rnorm(prod(p),0,1),p)
  S[t,,,]<-ttl(as.tensor(F[t,,,]),A,c(1,2,3))@data
  X[t,,,]<-S[t,,,]+E[t,,,]
}
result <- TFM_est(X,r,method='PE')  
Q.hat<-result$Q
Ft.hat <- result$Ft