WeiszfeldCov {Gmedian}

WeiszfeldCov

Description

Estimation of the Geometric median covariation matrix with Weiszfeld's algorithm. Weights (such as sampling weights) for statistical units are allowed.

Usage

WeiszfeldCov(X, weights=NULL, scores=2, epsilon=1e-08, nitermax = 100) 

Arguments

X

Data matrix, with n (rows) observations in dimension d (columns).

weights

When NULL, all observations have the same weight, say 1/n. Else, the user can provide a size n vector of weights (such as sampling weights). These weights are used in the estimating equation (see details).

scores

An integer q, by default q=2. The function computes the eigenvectors of the median covariation matrix associated to the q largest eigenvalues and the corresponding principal component scores. No output if scores=0.

epsilon

Numerical tolerance. By defaut 1e-08.

nitermax

Maxium number of iterations of the algorithm. By default set to 100.

Details

This fast and accurate iterative algorithm can deal with moderate size datasets. For large datasets use preferably GmedianCov, if fast estimations are required. Weights can be given for statistical units, allowing to deal with data drawn from unequal probability sampling designs (see Lardin-Puech, Cardot and Goga, 2014). The principal components standard deviation is estimed robustly thanks to function scaleTau2 from package robustbase.

Value

median

Vector of the geometric median

covmedian

Median covariation matrix

vectors

The scores=q eigenvectors of the median covariation matrix associated to the q largest eigenvalues

scores

Principal component scores corresponding to the scores=q eigenvectors

sdev

The scores=q robust estimates of the standard deviation of the principal components scores

iterm

Number of iterations needed to estimate the median

itercov

Number of iterations needed to estimate the median covariation matrix.

References

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

Lardin-Puech, P., Cardot, H. and Goga, C. (2014). Analysing large datasets of functional data: a survey sampling point of view, Journal de la Soc. Fr. de Statis., 155(4), 70-94.

See Also

See also Weiszfeld and GmedianCov.

Examples

## Simulated data - Brownian paths
n <- 1e3
d <- 20
x <- matrix(rnorm(n*d,sd=1/sqrt(d)), n, d)
x <- t(apply(x,1,cumsum))

## Estimation
median.est <- WeiszfeldCov(x)

par(mfrow=c(1,2))
image(median.est$covmedian) ## median covariation function
plot(c(1:d)/d,median.est$vectors[,1]*sqrt(d),type="l",xlab="Time",
ylab="Eigenvectors",ylim=c(-1.4,1.4))
lines(c(1:d)/d,median.est$vectors[,2]*sqrt(d),lty=2)