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)

Arguments

Details

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

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.

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

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

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