# Estimation of Factor Model for High-Dimensional Tensor Time Series 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). ```r 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. ## Returns 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. ## References Barigozzi M, He Y, Li L, Trapani L. Robust Estimation of Large Factor Models for Tensor-valued Time Series. Barigozzi M, He Y, Li L, Trapani L. Statistical Inference for Large-dimensional Tensor Factor Model by Iterative Projection. ## Author(s) Matteo Barigozzi, Yong He, Lingxiao Li, Lorenzo Trapani. ## Examples ```r 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 ```

TFM_est function