Identifying Multiple Outliers in Multivariate Data

TLDR

Multivariate data are represented by an n × p matrix of observations on p variables. The authors propose a new procedure to detect multiple outliers in multivariate data. The method orders observations by a robust outlyingness measure, iteratively partitions the data into a basic subset of p + 1 (then p + 2) good points and a non‑basic subset, recomputes relative distances using the basic subset’s covariance, and repeats until a stopping criterion, declaring the final non‑basic subset as outliers; it is illustrated and compared with existing techniques. The procedure is simple, computationally inexpensive, automatable, implementable with standard software, and effectively mitigates masking and swamping while reliably identifying multivariate outliers.

Abstract

SUMMARY We propose a procedure for the detection of multiple outliers in multivariate data. Let X be an n × p data matrix representing n observations on p variates. We first order the n observations, using an appropriately chosen robust measure of outlyingness, then divide the data set into two initial subsets: A ‘basic’ subset which contains p +1 ‘good’ observations and a ‘non-basic’ subset which contains the remaining n - p - 1 observations. Second, we compute the relative distance from each point in the data set to the centre of the basic subset, relative to the (possibly singular) covariance matrix of the basic subset. Third, we rearrange the n observations in ascending order accordingly, then divide the data set into two subsets: A basic subset which contains the first p + 2 observations and a non-basic subset which contains the remaining n - p - 2 observations. This process is repeated until an appropriately chosen stopping criterion is met. The final non-basic subset of observations is declared an outlying subset. The procedure proposed is illustrated and compared with existing methods by using several data sets. The procedure is simple, computationally inexpensive, suitable for automation, computable with widely available software packages, effective in dealing with masking and swamping problems and, most importantly, successful in identifying multivariate outliers.