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

## Description

Generate sample covariance matrix of residuals (includes diagnoal) described in Sun et al. (2016).

## Usage

 `1` ```covres(data, Omega.list, k = 1) ```

## Arguments

 `data` tensor object stored in a m1 * m2 * ... * mK * n array, where n is sample size and mk is dimension of the kth tensor mode. `Omega.list` list of precision matrices of tensor, i.e., `Omega.list[[k]]` is precision matrix for the kth tensor mode, 1 <= k <= K . `k` index of interested mode, default is 1.

## Details

This function computes sample covariance of residuals and is the basis for support recovery procedure in Sun et al. (2016). Note that output matrix includes diagnoal while bias corrected matrix (output of `biascor`) for inference is off-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 (includes diagnoal) is 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`, `biascor`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12``` ```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 rho ```