# biascor: Bias Correction of Sample Covariance of Residuals In Tlasso: Non-Convex Optimization and Statistical Inference for Sparse Tensor Graphical Models

## Description

Generate a matrix of bias-corrected sample covariance of residuals (excludes diagnoal) described in Sun et al. (2016).

## Usage

 `1` ```biascor(rho, Omega.list, k = 1) ```

## Arguments

 `rho` matrix of sample covariance of residuals (includes diagnoal), e.g., output of `covres`. `Omega.list` list of precision matrices of tensor, i.e., `Omega.list[[k]]` is the precision matrix for the kth tensor mode, 1 <= k <= K . For example, output of `link{Tlasso.fit}`. `k` index of interested mode, default is 1.

## Details

This function computes bias-corrected sample covariance of residuals (excludes diagnoal, diagnoal is zero vector). Note that output matrix excludes diagnoal while sample covariance of residuals includes diagnoal, see Sun et al. (2016) for details. Elements in `Omega.list` are true precision matrices or estimation of the true ones, the latter can be output of `Tlasso.fit`.

## Value

A matrix whose (i,j) entry (excludes diagnoal; diagnoal is zero vector) is bias-corrected sample covariance of the ith and jth residuals in the kth mode. See Sun et al. (2016) for details.

## Author(s)

Will Wei Sun, Zhaoran Wang, Xiang Lyu, Han Liu, Guang Cheng.

`varcor`, `covres`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16``` ```m.vec = c(5,5,5) # dimensionality of a tensor n = 5 # sample size k=1 # index of interested mode lambda.thm = 20*c( sqrt(log(m.vec[1])/(n*prod(m.vec))), sqrt(log(m.vec[2])/(n*prod(m.vec))), sqrt(log(m.vec[3])/(n*prod(m.vec)))) DATA=Trnorm(n,m.vec,type='Chain') # obersavations from tensor normal distribution out.tlasso = Tlasso.fit(DATA,T=1,lambda.vec = lambda.thm) # output is a list of estimation of precision matrices rho=covres(DATA, out.tlasso, k = k) # sample covariance of residuals, including diagnoal bias_rho=biascor(rho,out.tlasso,k=k) bias_rho # bias-corrected sample covariance of residuals # diagnoal is zero vector ```