Scoring anomalies : a M-estimation formulation.

Summary It is the purpose of this paper to formulate the issue of scoring multivariate observations depending on their degree of abnormality/novelty as an unsupervised learning task. Whereas in the 1-d situation, this problem can be dealt with by means of tail estimation techniques, observations being viewed as all the more "abnormal" as they are located far in the tail(s) of the underlying probability distribution. In a wide variety of applications, it is desirable to dispose of a scalar valued "scoring" function allowing for comparing the degree of abnormality of multivariate observations. Here we formulate the issue of scoring anomalies as a M-estimation problem. A (functional) performance criterion is proposed, whose optimal elements are, as expected, nondecreasing transforms of the density. The question of empirical estimation of this criterion is tackled and preliminary statistical results related to the accuracy of partition-based techniques for optimizing empirical estimates of the empirical performance measure are established.