RTFA (version 0.1.0)

TFM_FN: Estimation Factor Numbers via Eigenvalue-Ratio Criterion

Description

This function is to estimate factor numbers via eigenvalue-ratio criterion corresponding to initial estimation without projection, one-step projection estimation, iterative projection estimation and iterative weighted projection estimation by Huber loss.

Usage

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

Value

return a list containing the following:

path

a \(K \times (\rm{niter}+1)\) matrix of the estimated Tucker rank of the factor process as a path of the maximum number of iteration (\(\rm{niter}\)) used. The \(i\)-th column is the estimated rank \(\hat r_1, \hat r_2, \cdots, \hat r_K\) at \((i-1)\)-th iteration.

factor.num

final solution of the estimated Tucker rank of the factor process \(\hat r_1, \hat r_2, \cdots, \hat r_K\).

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 factor 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.

Author

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

Details

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

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,,,]
}
rank<-TFM_FN(X,r=NULL,method='PE')