Computationally Easy Outlier Detection via Projection Pursuit with Finitely Many Directions

Outlier detection methods are fundamental to all of data analysis. They are desirably robust, affine invariant, and computationally easy in any dimension. The powerful projection pursuit approach yields the “projection outlyingness”, which is affine invariant and highly robust and does not impose ellipsoidal contours like the Mahalanobis distance approach. However, it is highly computationally intensive, being obtained by taking suprema of univariate scaled deviation outlyingness over all projections of the data onto lines. Here we introduce several outlyingness functions based on a vector of scaled deviations taken over only finitely many directions approximately uniform over the unit hypersphere. A preliminary transformation of the data to a strong invariant coordinate system makes such vectors affine invariant. We establish useful foundational theory for finite vectors of scaled deviations on projections. Also, using artificial and real data sets, we compare our affine invariant outlyingness functions with the usual projection outlyingness and with robust Mahalanobis distance outlyingness. AMS 2000 Subject Classification: Primary 62H99. Secondary 62G99