Robust_regression {RobRegression}

Robust_regression

Description

This function gives robust estimates of the paramter of the Multivariate Linear regression with the help of the euclidean distance, or with the help of the Mahalanobis distance for some matrice Sigma. More precisely, the aim is to minimize

G(\hat{\beta}) = \mathbb{E}[ \| Y-X\hat{\beta} \|_{\Sigma}] + \lambda \| \hat{\beta}\|^{\text{ridge}} .

Usage

Robust_regression(X,Y, Mat_Mahalanobis=diag(rep(1,ncol(Y))),
                  niter=50,lambda=0,c='default',method='Offline',
                  alpha=0.66,w=2,ridge=1,nlambda=50,
                  init=matrix(runif(ncol(X)*ncol(Y))-0.5,nrow=ncol(X),ncol=ncol(Y)),
                  epsilon=10^(-8), Mahalanobis_distance = FALSE,
                  par=TRUE,scale='none',tol=10^(-3))

Arguments

X

A (n,p)-matrix whose raws are the explaining data.

Y

A (n,q)-matrix whose raws are the variables to be explained.

method

The method used for estimating the parameter. Should be method='Offline' if the fix point algorithm is used, and 'Online' if the (weighted) averaged stochastic gradient algorithm is used. Default is 'Offline'.

Mat_Mahalanobis

A (q,q)-matrix giving \Sigma for the Mahalanobis distance. Default is identity.

Mahalanobis_distance

A logical telling if the Mahalanobis distance is used. Default is FALSE.

scale

If a scaling is used. scale='robust' should be used if a robust scaling of Y is desired. Default is 'none'

niter

The maximum number of iteration if method='Offline'.

init

A (p,q)-matrix which gives the initialization of the algorithm.

ridge

The power of the penalty: i.e should be 2 if the squared norm is considered or 1 if the norm is considered.

lambda

A vector giving the different studied penalizations. If lambda='default', would be a vector of preselected penalizations.

nlambda

The number of tested penalizations if lambda='default'.

par

Is equal to TRUE if the parallelization of the algorithm for estimating robustly the variance of the noise is allowed.

c

The constant in the stepsequence if the averaged stochastic gradient algorithm, i.e if method='Online'.

alpha

A scalar between 1/2 and 1 used in the stepsequence for stochastic gradient algorithm if method='Online'.

w

The power for the weighted averaged Robbins-Monro algorithm if method='Online'.

epsilon

Stoping condition for the fix point algorithm if method='Offline'.

tol

A scalar that avoid numerical problems if method='Offline'. Default is 10^(-3).

Value

A list with:

beta

A (p,q)-matrix giving the estimation of the parameters.

criterion

A vector giving the loss for the different chosen lambda. If sale='robust', it is calculated on the scaled data.

all_beta

A list containing the different estimation of the parameters (with respect to the different coices of lambda).

lambda_opt

A scalar giving the selected lambda.

References

Godichon-Baggioni, A., Robin, S. and Sansonnet, L. (2023): A robust multivariate linear regression based on the Mahalanobis distance

See Also

See also Robust_Variance, Robust_Mahalanobis_regression and RobRegression-package.

Examples

p=5
q=10
n=2000
mu=rep(0,q)
epsilon=mvtnorm::rmvnorm(n = n,mean = mu)
X=mvtnorm::rmvnorm(n=n,mean=rep(0,p))
beta=matrix(rnorm(p*q),ncol=q)
Y=X %*% beta+epsilon
Res_reg=Robust_regression(X,Y)
sum((Res_reg$beta-beta)^2)