Mahalanobisavstander

Mahalanobisavstander, or Mahalanobis distances, are a multivariate measure of the distance between a point and a distribution. They account for the correlations among variables and the scale of each variable, providing a geometry that reflects the data structure rather than mere coordinate differences. The concept is named after Indian statistician Prasanta Chandra Mahalanobis, who introduced it in 1936 to identify multivariate outliers and compare observations across different populations.

For a vector x in R^p, with mean vector μ and covariance matrix Σ, the squared Mahalanobis distance is D^2 = (x − μ)^T Σ^−1 (x − μ). The distance is D = sqrt(D^2). In practice, Σ can be the population covariance or, more commonly, the sample covariance S. If the variables are standardized and uncorrelated so that Σ is the identity matrix, the Mahalanobis distance reduces to the Euclidean distance.

Key properties include affine invariance under linear transformations and an interpretation in terms of ellipsoidal contours: points equidistant by this measure lie on ellipsoids aligned with the data’s covariance structure. Under a multivariate normal distribution, the squared distance D^2 follows a chi-squared distribution with p degrees of freedom, facilitating significance testing for outliers.

Applications span outlier detection, cluster analysis, classification, and finance, where understanding joint variability is important. Limitations include sensitivity to outliers in the covariance estimate and potential instability when the covariance matrix is singular or ill-conditioned, which motivates robust or regularized estimators.