How to deal with Acontamenated & less than Full rank data in amultivariate data set

Abstract

When faced with high-dimensional data, one often uses principal component analysis (PCA) for dimension reduction. Classical PCA constructs aset of uncorrelated variables, which correspond to eigenvectors for the sample covariance matrix. However, it is well-known that this covariance matrix is strongly affected by anomalous observations. It is therefore necessary to apply robust method that are resistant to possible outliers. Li & Chen (1985) proposed asdution based on Projection Pursuit (PP). The idea is to search for the direction in which the projected observations have the largest robust scal. In subsequnt steps each new direction is constrained to be orthogonal to all previous direction. This method is very well suited for high dimensional data even when the number of variables (P) is higher than the number of observations (n) (Less than full rank data set or redused data). Our gool is to stady the redused data set spicially when this set contain an outliers observations and finaly we used arobust methodes (S,M,R, MVE, MCD,…) and find aroubest estimators that we can depent on it to make a good dissisions about our problems.