Robust_Variance {RobRegression}

Robust_Variance

Description

The aim is to provide a robust estimation of the variance for Guassian models with reduction dimension. More precisely we considering a q dimensional random vector whose variance can be written as \Sigma = C + \sigma I where C is a matrix of rank d, with d possibly much smaller than q, sigma is a positive scalar, and I is the identity matrix.

Usage

Robust_Variance(X,K=ncol(X),par=TRUE,alphaRM=0.75,
                c='default',w=2,mc_sample_size='default',
                methodMC='Robbins',niterMC=50,method_MCM='Weiszfeld',
                eps_vp=10^(-6))

Arguments

X

A matrix whose raws are the vector we want to estimate the variance.

K

A vector containing the possible values of d. The 'good' d is chosen with the help of a penatly criterion if the length of K is larger than 10. Default is ncol(X).

par

Is equal to TRUE if the parallelization of the algorithm is allowed.

mc_sample_size

The number of data generated for the Monte-Carlo method for estimating robustly the eigenvalues of the variance.

methodMC

The method chosen to estimate robustly the variance. Can be 'Robbins', 'Grad' or 'Fix'. Default is 'Robbins'.

niterMC

The number of iterations for estimating robustly the variance of each class if methodMC='Fix' or methodMC='Grad'.

method_MCM

The method chosen to estimate Median Covariation Matrix. Can be 'Weiszfeld' or 'ASGD'.

alphaRM

A scalar between 1/2 and 1 used in the stepsequence for the Robbins-Monro method if methodMC='Robbins'.

c

The constant in the stepsequence if methodMC='Robbins'.

w

The power for the weighted averaged Robbins-Monro algorithm if methodMC='Robbins'. Default is 2.

eps_vp

The minimum values for the estimates of the eigenvalues of the Variance can take. Default is 10^-6.

Value

A list with:

Sigma

The robust estimation of the variance.

invSigma

The robuste estimation of the inverse of the variance.

MCM

The Median Covariation Matrix.

eigenvalues

A vector containing the estimation of the d+1 main eigenvalues of the variance, where d+1 is the optimal choice belong K.

MCM_eigenvalues

A vector containing the estimation of the d+1 main eigenvalues of the Median Covariation Matrix, where d+1 is the optimal choice belong K.

cap

The result given for capushe for selecting d if the length of K is larger than 10.

reduction_results

A list containing the results for all possible K.

References

Cardot, H., Cenac, P. and Zitt, P-A. (2013). Efficient and fast estimation of the geometric median in Hilbert spaces with an averaged stochastic gradient algorithm. Bernoulli, 19, 18-43.

Cardot, H. and Godichon-Baggioni, A. (2017). Fast Estimation of the Median Covariation Matrix with Application to Online Robust Principal Components Analysis. Test, 26(3), 461-480

Vardi, Y. and Zhang, C.-H. (2000). The multivariate L1-median and associated data depth. Proc. Natl. Acad. Sci. USA, 97(4):1423-1426.

See Also

See also Robust_Mahalanobis_regression, Robust_regression and RobRegression-package.

Examples

q<-100
d<-10
n<-2000
Sigma<- diag(c(d:1,rep(0,q-d)))+ diag(rep(0.1,q))
X=mvtnorm::rmvnorm(n=n,sigma=Sigma)
RobVar = Robust_Variance(X,K=q)
sum((RobVar$Sigma-Sigma)^2)/q