| ucov {ICSClust} | R Documentation |
Simple robust estimates of scatter
Description
Compute a one-step M-estimator of scatter with weights based on Mahalanobis distances, or a simple related estimator that is based on a transformation.
Usage
scov(x, beta = 0.2)
ucov(x, beta = 0.2)
Arguments
x |
a numeric matrix or data frame. |
beta |
a positive numeric value specifying the tuning parameter of the estimator (defaults to 0.2), 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 one-step M-estimator
of scatter is defined as
\mathrm{SCOV}_{\beta}(\boldsymbol{X}_{n}) =
\frac{\sum_{i=1}^{n}
w(\beta \, r^{2}(\mathbf{x}_{i}))
(\mathbf{x}_{i} - \mathbf{\bar{x}}_{n})
(\mathbf{x}_{i} - \mathbf{\bar{x}}_{n})^{\top}}{\sum_{i=1}^{n}
w(\beta \, r^{2}(\mathbf{x}_{i}))},
where
r^{2}(\mathbf{x}_{i}) =
(\mathbf{x}_{i} - \mathbf{\bar{x}}_{n})^{\top}
\mathrm{COV}(\boldsymbol{X}_n)^{-1}
(\mathbf{x}_{i} - \mathbf{\bar{x}}_{n})
denotes the squared Mahalanobis distance of observation \mathbf{x}_{i}
from the sample mean \mathbf{\bar{x}}_{n} based on the sample
covariance matrix \mathrm{COV}(\boldsymbol{X}_n). Here, the weight
function w(x) = \exp(-x/2) is used.
A simple robust estimator that is consistent under normality is obtained via the transformation
\mathrm{UCOV}_{\beta}(\boldsymbol{X}_{n}) =
(\mathrm{SCOV}_{\beta}(\boldsymbol{X}_{n})^{-1} -
\beta \, \mathrm{COV}(\boldsymbol{X}_{n})^{-1})^{-1}.
Value
A numeric matrix giving the estimate of the scatter matrix.
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.
Ruiz-Gazen, A. (1996) A Very Simple Robust Estimator of a Dispersion Matrix. Computational Statistics & Data Analysis, 21(2), 149-162. doi:10.1016/0167-9473(95)00009-7.
See Also
ICS_ucov(), tcov(), ICS_tcov()