# Mass Volume Curves and Anomaly Ranking

## TL;DR

This paper introduces the Mass Volume (MV) curve, a novel unsupervised method for ranking multivariate observations by abnormality, using statistical estimation and optimization techniques to improve anomaly detection.

## Contribution

It formulates anomaly ranking as an M-estimation problem with the MV curve, providing estimation strategies, confidence regions, and bounds for the optimal scoring function.

## Key findings

- Proposes a new MV curve-based scoring function for multivariate anomalies.
- Develops a bootstrap method for confidence region estimation.
- Establishes generalization bounds for the empirical MV curve.

## Abstract

This paper aims at formulating the issue of ranking multivariate unlabeled observations depending on their degree of abnormality as an unsupervised statistical learning task. In the 1-d situation, this problem is usually tackled by means of tail estimation techniques: univariate observations are viewed as all the more `abnormal' as they are located far in the tail(s) of the underlying probability distribution. It would be desirable as well 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 by means of a novel functional performance criterion, referred to as the Mass Volume curve (MV curve in short), whose optimal elements are strictly increasing transforms of the density almost everywhere on the support of the density. We first study the statistical estimation of the MV curve of a given scoring function and we provide a strategy to build confidence regions using a smoothed bootstrap approach. Optimization of this functional criterion over the set of piecewise constant scoring functions is next tackled. This boils down to estimating a sequence of empirical minimum volume sets whose levels are chosen adaptively from the data, so as to adjust to the variations of the optimal MV curve, while controling the bias of its approximation by a stepwise curve. Generalization bounds are then established for the difference in sup norm between the MV curve of the empirical scoring function thus obtained and the optimal MV curve.