RobVar: RobVar

Description

Robust estimate of the variance

Usage

RobVar(X, c=2, alpha=0.75, model='Gaussian', methodMCM='Weiszfeld',
                methodMC='Robbins' , mc_sample_size=1000, init=rep(0, ncol(X)),
                init_cov=diag(ncol(X)),
                epsilon=10^(-8), w=2, df=3, niterMC=50,
                cgrad=2, niterWeisz=50, epsWeisz=10^-8, alphaMedian=0.75, cmedian=2)

Value

An object of class list with the following outputs:

median: The median of X.
variance: The robust variance of X.
median: The Median Covariation Matrix of X.

Arguments

X: A numeric matrix of whose rows correspond to observations.
c: A positive scalar giving the constant in the stepsequence of the Robbins-Monro or Gradient method if methodMC='RobbinsMC' or methodMC='GradMC'. Default is 2.
alpha: A scalar between 1/2 and 1 giving the power in the stepsequence for the Robbins-Monro algorithm is methodMC='RobbinsMC'. Default is 0.75.
model: A string character specifying the model: can be 'Gaussian' (default), 'Student' or 'Laplace'.
methodMCM: A string character specifying the method to estimate the Median Covariation Matrix. Can be 'Gmedian' or 'Weiszfeld' (defualt).
methodMC: A string character specifying the method to estimate robustly the variance. Can be 'Robbins' (default), 'Fix' or 'Grad'.
mc_sample_size: A positive integer giving the number of data simulated for the Monte-Carlo method. Default is 1000.
init: A numeric vector giving the initialization for estimating the median.
init_cov: A numeric matrix giving an initialization for estimating the Median Covariation Matrix.
epsilon: A positive scalar giving a stoping condition for algorithm.
w: A positive integer specifying the power for the weighted averaged Robbins-Monro algorithm if methodMC='RobbinsMC'.
df: An integer larger (or equal) than 3 specifying the degrees of freedom for the Student law if model='Student'. See also Gen_MM. Default is 3.
niterMC: An integer giving the number of iterations for iterative algorithms if the selected method is 'Grad' or 'Fix'. Default is 50.
cgrad: A numeric vector with positive values giving the stepsequence of the gradient algorithm for estimating the variance if methodMC='Grad'. Its length has to be equal to niter.
niterWeisz: A positive integer giving the maximum number of iterations for the Weiszfeld algorithms if methodMCM='Weiszfeld'. Default is 50.
epsWeisz: A stopping factor for the Weiszfeld algorithm.
alphaMedian: A scalar betwwen 1/2 and 1 giving the power of the stepsequence of the gradient algorithm for estimating the Median Covariation Matrix if methodMCM='Gmedian'. Default is 0.75.
cmedian: A positive scalar giving the constant in the stepsequence of the gradient algorithm for estimating the Median Covariation Matrix if methodMCM='Gmedian'. Default is 2.

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.

Examples

Run this code


n <- 2000
d <- 5
Sigma <-diag(1:d)
mean <- rep(0,d)
X <- mvtnorm::rmvnorm(n,mean,Sigma)
RVar=RobVar(X)