Locally centred Mahalanobis distance: A new distance measure with salient features towards outlier detection

• A new measure of distance is proposed.
• Two different kinds of outliers are discussed.
• Two new indices for outlier detection are defined.
• A new plot is proposed as outlier detection tool.

Outlier detection is a prerequisite to identify the presence of aberrant samples in a given set of data. The identification of such diverse data samples is significant particularly for multivariate data analysis where increasing data dimensionality can easily hinder the data exploration and such outliers often go undetected. This paper is aimed to introduce a novel Mahalanobis distance measure (namely, a pseudo-distance) termed as locally centred Mahalanobis distance, derived by centering the covariance matrix at each data sample rather than at the data centroid as in the classical covariance matrix. Two parameters, called as Remoteness and Isolation degree, were derived from the resulting pairwise distance matrix and their salient features facilitated a better identification of atypical samples isolated from the rest of the data, thus reflecting their potential application towards outlier detection. The Isolation degree demonstrated to be able to detect a new kind of outliers, that is, isolated samples within the data domain, thus resulting in a useful diagnostic tool to evaluate the reliability of predictions obtained by local models (e.g. k-NN models).To better understand the role of Remoteness and Isolation degree in identification of such aberrant data samples, some simulated and published data sets from literature were considered as case studies and the results were compared with those obtained by using Euclidean distance and classical Mahalanobis distance.

The main topic of the paper is a new measure of Mahalanobis distance, centred on each sample and not on the data centroid. This new distance matrix gives interesting information for outlier detection and a new graphic tool – also useful for exploratory data analysis – is proposed.Figure optionsDownload as PowerPoint slide