Robustness and multivariate analysis

Invariant coordinate selection (ICS) is a method for �nding structures in multivariate data using the eigenvalue-eigenvector decomposition of two different scatter matrices. The performance of the ICS depends on the structure of the data and the choice of the scatter matrices.
The main goal of this thesis is to understand how ICS works in some situations, and does not in other. In particular, we look at ICS under three different structures: two-group mixtures, long-tailed distributions, and parallel line structure.
Under two-group mixtures, we explore ICS based on the fourth-order moment matrix, ^K , and the covariance matrix S. We find the explicit form of ^K , and the ICS criterion under this model. We also explore the projection pursuit (PP) method, a variant of ICS, based on the univariate kurtosis. A comparison is made between PP, based on kurtosis, and ICS, based on ^K and S, through a simulation study. The results show that PP is more accurate than ICS. The asymptotic distributions of the ICS and PP estimates of the groups separation direction are derived.
We explore ICS and PP based on two robust measures of spread, under twogroup mixtures. The use of common location measures, and pairwise differencing of the data in robust ICS and PP are investigated using simulations. The simulation results suggest that using a common location measure can be sometimes useful.
The second structure considered in this thesis, the long-tailed distribution, is modelled by two dimensional errors-in-variables model, where the signal can have a non-normal distribution. ICS based on ^K and S is explored. We gain insight into how ICS �nds the signal direction in the errors in variables problem. We also compare the accuracy of the ICS estimate of the signal direction and Geary's
fourth-order cumulant-based estimates through simulations. The results suggest that some of the cumulant-based estimates are more accurate than ICS, but ICS has the advantage of affine equivariance.
The third structure considered is the parallel lines structure. We explore ICS based on the W-estimate based on the pairwise di�erencing of the data, ^ V , and S. We give a detailed analysis of the e�ect of the separation between points, overall and conditional on the horizontal separation, on the power of ICS based on ^ V and S.