Visualisation of Mahalanobis Distances for Trivariate JOINT Distributions

The Mahalanobis distance is a statistical measure used to quantify the distance between elliptic distributions with distinct locations and shared shapes, while accounting for the variables' covariance structure. It is applicable to both estimative and predictive estimation approaches, where variations are limited to location, and it assesses the similarity or dissimilarity between data and the mean (centroid) of a multivariate distribution, within the family of multivariate elliptic distributions. It is thus useful for outlier identification. The aim of the study is to provide, for the first time, a three-dimensional visualisation of the Mahalanobis distance when the underlying framework comprises three jointly connected variables (rather than the standard two variables presented in textbooks). Data with Mahalanobis distances exceeding a predefined threshold, determined using a  distribution, are considered outliers. This approach is analogous to identifying outliers for univariate distributions based on critical values derived from confidence levels. While the literature mainly discusses the Mahalanobis distance formulation for bivariate distributions, we extend the discussion to include one additional variable and provide a visualisation of the resulting Mahalanobis distance for a trivariate distribution. An empirical example is presented to illustrate a practical application of a trivariate Mahalanobis distance. Visualising outliers alongside other historical events within three-factor systems can offer valuable insights into the risk profile of the current environment and assess the probability of future extreme events.