Computationally easy outlier detection via projection pursuit with finitely many directions

Outlier detection is fundamental to data analysis. Desirable properties are affine invariance, robustness, low computational burden, and nonimposition of elliptical contours. However, leading methods fail to possess all of these features. The Mahalanobis distance outlyingness (MD) imposes elliptical contours. The projection outlyingness, powerfully involving projections of the data onto all univariate directions, is highly computationally intensive. Computationally easy variants using projection pursuit with but finitely many directions have been introduced, but these fail to capture at once the other desired properties. Here, we develop a 'robust Mahalanobis spatial outlyingness on projections' (RMSP) function, which indeed satisfies all the four desired properties. Pre-transformation to a strong invariant coordinate system yields affine invariance, 'spatial trimming' yields robustness, and 'spatial Mahalanobis outlyingness' is used to obtain computational ease and smooth, unconstrained contours. From empirical study using artificial and actual data, our findings are that SUP is outclassed by MD and RMSP, that MD and RMSP are competitive, and that RMSP is especially advantageous in describing the intermediate outlyingness structure when elliptical contours are not assumed.