| 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 \boldsymbol{X}_{n} = (\mathbf{x}_{1}, \dots,
\mathbf{x}_n)^{\top}, a positive and decreasing weight function w,
and a tuning parameter \beta > 0, the pairwise one-step M-estimator
of scatter is defined as
\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
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
\mathbf{x}_{i} and \mathbf{x}_{j} based on the sample
covariance matrix \mathrm{COV}(\boldsymbol{X}_n). Here, the weight
function 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()