A new procedure for finding the principal components of a data set is explored. The new procedure's goal is to find a few principal component directions in a high-dimensional setting, and to do so in a robust manner. This procedure, called elliptical principal component analysis (elliptical PCA), expands the current choices for performing robust principal component analysis (PCA). The necessity of the elliptical PCA method arose due to the limitations of current methods for obtaining estimates of principal components in high-dimensional data with possible rank deficiencies.

The core of the elliptical PCA method involves a new way to scale data to obtain a robust covariance matrix estimate. This will be referred to as the elliptical covariance matrix. A singular value decomposition (SVD) is then performed on the elliptical covariance matrix, yielding elliptical PCA estimates.

A motivational example from opthalmology is discussed, with a focus on drawbacks of traditional PCA. The talk outlines the development of elliptical PCA, including a discussion of estimation bias using this method, an a proposed solution. The two-dimensional elliptical PCA estimation problem is discussed, and an iterative estimation method for elliptical PCA is shown to converge to the target principal eigenvector. The result is then extended to arbitrary dimensions.

August 10 2001