covW {ICS}

One-step M-estimator

Description

Estimates the scatter matrix based on one-step M-estimator using mean and covariance matrix as starting point.

Usage

covW(X, na.action = na.fail, alpha = 1, cf = 1)

Arguments

X

numeric n \times p data matrix or dataframe.

na.action

a function which indicates what should happen when the data contain 'NA's. Default is to fail.

alpha

parameter of the one-step M-estimator. By default equals to 1.

cf

consistency factor of the one-step M-estimator. By default equals to 1.

Details

It is given for n \times p matrix X by

COV_{w}(X)=\frac{1}{n} {cf} \sum_{i=1}^n w(D^2(x_i)) (x_i - \bar{ x})^\top(x_i - \bar{ x}),

where \bar{x} is the mean vector, D^2(x_i) is the squared Mahalanobis distance, w(d)=d^\alpha is a non-negative and continuous weight function and {cf} is a consistency factor. Note that the consistency factor, which makes the estimator consistent at the multivariate normal distribution, is in most case unknown and therefore the default is to use simply cf = 1.

• If w(d)=1, we get the covariance matrix cov() (up to the factor 1/(n-1) instead of 1/n).

• If \alpha=-1, we get the covAxis().

• If \alpha=1, we get the cov4() with {cf} = \frac{1}{p+2}.

Value

A matrix containing the one-step M-scatter.

Author(s)

Aurore Archimbaud and Klaus Nordhausen

References

Archimbaud, A., Drmac, Z., Nordhausen, K., Radojicic, U. and Ruiz-Gazen, A. (2023). SIAM Journal on Mathematics of Data Science (SIMODS), Vol.5(1):97–121. doi:10.1137/22M1498759.

See Also

cov(), cov4(), covAxis()

Examples

data(iris)
X <- iris[,1:4]

# Equivalence with covAxis
covW(X, alpha = -1, cf = ncol(X))
covAxis(X)

# Equivalence with cov4
covW(X, alpha = 1, cf = 1/(ncol(X)+2))
cov4(X)

# covW with alpha = 0.5
covW(X, alpha = 0.5)