### Description

Adjusts tensor with respect to covariates to achieve a more accurate performance. Tensor depends on the covariates through a linear regression model. The function returns the coefficients of covariates in regression and adjusted tensor list for further classifier modeling. It estimates coefficients based on training data, and then adjusts training tensor. When testing data is provided, the function will automatically adjust testing data by learned coefficients as well.

### Usage

adjten(x, z, y, testx = NULL, testz = NULL, is.centered = FALSE)


### Arguments

 x Input tensor or matrix list of length N, where N is the number of observations. Each element of the list is a tensor or matrix. The order of tensor can be any integer not less than 2. z Input covariate matrix of dimension N*q, where q

### Details

The model CATCH assumes the linear relationship bewteen covariates and tensor as

\mathbf{X}=\boldsymbol{μ}_k+\boldsymbol{α}\overline{\times}_{M+1}\mathbf{Z}+\mathbf{E},

where \boldsymbol{α} is the matrix of estimated coefficient of covariates. The function removes the effects of covariates on response variable through tensor and obtain \mathbf{X}-\boldsymbol{α}\overline{\times}_{M+1}\mathbf{Z} as adjusted tensor to fit tensor discriminant analysis model.

In estimating \boldsymbol{α}, which is the alpha in the package, adjten first centers both tensor and covariates within their individual classes, then performs tensor response regression which regresses {\mathbf{X}} on {\mathbf{Z}}.

### Value

 gamma The estimated coefficients of covariates to plug in classifier. gamma is the \boldsymbol{γ}_k defined function catch of dimension q*(K-1), where q is the size of covariates and K is the number of classes. xres Adjusted training tensor list \mathbf{X}-\boldsymbol{α}\overline{\times}_{M+1}\mathbf{Z} after adjusting for covariates. The effect of the covariate is removed. testxres Adjusted testing tensor list \mathbf{X}-\boldsymbol{α}\overline{\times}_{M+1}\mathbf{Z} after adjusting for covariates. The effect of the covariate is removed.

### Author(s)

Yuqing Pan, Qing Mai, Xin Zhang

### References

Pan, Y., Mai, Q., and Zhang, X. (2018) Covariate-Adjusted Tensor Classification in High-Dimensions, arXiv:1805.04421.

catch

### Examples

n <- 20
p <- 4
k <- 2
nvars <- p*p*p
x <- array(list(),n)
vec_x <- matrix(rnorm(n*nvars),nrow=n,ncol=nvars)
vec_x[1:10,] <- vec_x[1:10,]+2
z <- matrix(rnorm(n*2),nrow=n,ncol=2)
z[1:10,] <- z[1:10,]+0.5
y <- c(rep(1,10),rep(2,10))
for (i in 1:n){
x[[i]] <- array(vec_x[i,],dim=c(p,p,p))
}