tcov {ICSClust}R Documentation

Pairwise one-step M-estimate of scatter

Description

Computes a pairwise one-step M-estimate of scatter with weights based on pairwise Mahalanobis distances. Note that it is based on pairwise differences and therefore does not require a location estimate.

Usage

tcov(x, beta = 2)

Arguments

x

a numeric matrix or data frame.

beta

a positive numeric value specifying the tuning parameter of the pairwise one-step M-estimator (defaults to 2), see ‘Details’.

Details

For a sample Xn=(x1,,xn)\boldsymbol{X}_{n} = (\mathbf{x}_{1}, \dots, \mathbf{x}_n)^{\top}, a positive and decreasing weight function ww, and a tuning parameter β>0\beta > 0, the pairwise one-step M-estimator of scatter is defined as

TCOVβ(Xn)=i=1n1j=i+1nw(βr2(xi,xj))(xixj)(xixj)i=1n1j=i+1nw(βr2(xi,xj)),\mathrm{TCOV}_{\beta}(\boldsymbol{X}_{n}) = \frac{\sum_{i=1}^{n-1} \sum_{j=i+1}^{n} w(\beta \, r^{2}(\mathbf{x}_{i}, \mathbf{x}_{j})) (\mathbf{x}_{i} - \mathbf{x}_{j}) (\mathbf{x}_{i} - \mathbf{x}_{j})^{\top}}{\sum_{i=1}^{n-1} \sum_{j=i+1}^{n} w(\beta \, r^{2}(\mathbf{x}_{i}, \mathbf{x}_{j}))},

where

r2(xi,xj)=(xixj)COV(Xn)1(xixj)r^{2}(\mathbf{x}_{i}, \mathbf{x}_{j}) = (\mathbf{x}_{i} - \mathbf{x}_{j})^{\top} \mathrm{COV}(\boldsymbol{X}_n)^{-1} (\mathbf{x}_{i} - \mathbf{x}_{j})

denotes the squared pairwise Mahalanobis distance between observations xi\mathbf{x}_{i} and xj\mathbf{x}_{j} based on the sample covariance matrix COV(Xn)\mathrm{COV}(\boldsymbol{X}_n). Here, the weight function w(x)=exp(x/2)w(x) = \exp(-x/2) is used.

Value

A numeric matrix giving the pairwise one-step M-estimate of scatter.

Author(s)

Andreas Alfons and Aurore Archimbaud

References

Caussinus, H. and Ruiz-Gazen, A. (1993) Projection Pursuit and Generalized Principal Component Analysis. In Morgenthaler, S., Ronchetti, E., Stahel, W.A. (eds.) New Directions in Statistical Data Analysis and Robustness, 35-46. Monte Verita, Proceedings of the Centro Stefano Franciscini Ascona Series. Springer-Verlag.

Caussinus, H. and Ruiz-Gazen, A. (1995) Metrics for Finding Typical Structures by Means of Principal Component Analysis. In Data Science and its Applications, 177-192. Academic Press.

See Also

ICS_tcov(), ucov(), ICS_ucov()


[Package ICSClust version 0.1.0 Index]