Robust Singular Value Decompositions: A New Approach to Projection Pursuit

Abstract

Robust location and covariance estimators are developed via general M estimation for covariance matrix eigenvectors and eigenvalues. The solution to this GM estimation problem is obtained by transforming it into a series of robust regression problems based on a new algorithm for the singular value decomposition. It is shown here that the singular value decomposition can be represented as an iteration of two steps: a least squares regression fit of the data matrix followed by a rotation to the regression hyperplanes. An algorithm to obtain the solution to this GM estimation problem is presented, along with results of a Monte Carlo study and examples of its application. In addition, it is shown how the output of this algorithm can be used to numerically search for multivariate outliers, which is especially useful in exploratory data analysis with high-dimensional data and large sample sizes, where standard graphical techniques are difficult to implement. Because the algorithm computes robust estimates of the eigenvectors and eigenvalues of the covariance matrix, it can be used as a basis for other multivariate methods such as errors-in-variables regression, discriminant analysis, and principal components.

Key Words:

• Robust covariance estimation
• Robust principal components