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.
tcov(x, beta = 2)A numeric matrix giving the pairwise one-step M-estimate of scatter.
a numeric matrix or data frame.
a positive numeric value specifying the tuning parameter of the pairwise one-step M-estimator (defaults to 2), see ‘Details’.
Andreas Alfons and Aurore Archimbaud
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.
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.
ICS_tcov(), ucov(), ICS_ucov()