# Estimation Factor Numbers via Eigenvalue-Ratio Criterion 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. ```r TFM_FN(x, r = NULL, 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 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`. ## Details See Barigozzi et al. (2022) and Barigozzi et al. (2023) for details. ## Returns 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 $\hat r_1, \hat r_2, \cdots, \hat r_K$. ## 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,,,] } rank<-TFM_FN(X,r=NULL,method='PE') ```

TFM_FN function