tenFM.est  R Documentation 
Estimation function for Tucker structure factor models of tensorvalued time series.
Two unfolding methods of the autocovariance tensor, Time series OuterProduct Unfolding Procedure (TOPUP), Time series InnerProduct Unfolding Procedure (TIPUP),
are included, as determined by the value of method
.
tenFM.est(x,r,h0=1,method='TIPUP',iter=TRUE,tol=1e4,maxiter=100)
x 

r 
input rank of factor tensor. 
h0 
the number of lags used in autocovariance tensor. If h0=0, covariance tensor is used. 
method 
character string, specifying the type of the estimation method to be used.

iter 
boolean, specifying using an iterative approach or an noniterative approach. 
tol 
tolerance in terms of the Frobenius norm. 
maxiter 
maximum number of iterations if error stays above 
Tensor factor model with Tucker structure has the following form,
X_t = F_t \times_{1} A_1 \times_{2} \cdots \times_{K} A_k + E_t,
where A_k
is the deterministic loading matrix of size d_k \times r_k
and r_k \ll d_k
,
the core tensor F_t
itself is a latent tensor factor process of dimension r_1 \times \cdots \times r_K
,
and the idiosyncratic noise tensor E_t
is uncorrelated (white) across time. Two estimation approaches, named TOPUP and TIPUP, are studied.
Time series OuterProduct Unfolding Procedure (TOPUP) are based on
{\rm{TOPUP}}_{k}(X_{1:T}) = \left(\sum_{t=h+1}^T \frac{{\rm{mat}}_{k}( X_{th}) \otimes {\rm{mat}}_k(X_t)} {Th}, \ h=1,...,h_0 \right),
where h_0
is a predetermined positive integer, \otimes
is tensor product. Note that
{\rm{TOPUP}}_k(\cdot)
is a function mapping a tensor time series to an order5 tensor.
Time series InnerProduct Unfolding Procedure (TIPUP) replaces the tensor product in TOPUP with the inner product:
{\rm{TIPUP}}_k(X_{1:T})={\rm{mat}}_1\left(\sum_{t=h+1}^T \frac{{\rm{mat}}_k(X_{th}) {\rm{mat}}_k^\top(X_t)} {Th}, \ h=1,...,h_0 \right).
returns 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 d_1 \times d_2 \times \cdots \times d_k
.
niter
number of iterations.
fnorm.resid
Frobenius norm of residuals, divide the Frobenius norm of the original tensor.
Chen, Rong, Dan Yang, and CunHui Zhang. "Factor models for highdimensional tensor time series." Journal of the American Statistical Association (2021): 159.
Han, Yuefeng, Rong Chen, Dan Yang, and CunHui Zhang. "Tensor factor model estimation by iterative projection." arXiv preprint arXiv:2006.02611 (2020).
set.seed(333)
dims < c(16,18,20) # dimensions of tensor time series
r < c(3,3,3) # dimensions of factor series
Ft < tenAR.sim(t=100, dim=r, R=1, P=1, rho=0.9, cov='iid')
lambda < sqrt(prod(dims))
x < tenFM.sim(Ft,dims=dims,lambda=lambda,A=NULL,cov='iid') # generate t*dims tensor time series
result < tenFM.est(x,r,h0=1,iter=TRUE,method='TIPUP') # Estimation
Ft < result$Ft
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