Mahalanobis Distance - covariance-adjusted distance accounting for inter-criterion correlations

Overview

d ≥ 0; d=0 iff a=b. Mahalanobis Distance is symmetric.

Output

distance, lower is better

Data

Crisp, complete numeric matrix

Size

2+ alternatives, 3-12 criteria works best

Used for

Similarity assessment, clustering

How it works

1. 1

   Compute the Mahalanobis Distance between vectors a and b.

   $$d_{Mah}(\mathbf{x},\mathbf{\mu })=\sqrt{(\mathbf{x}-\mathbf{\mu }{)}^{\top }{\Sigma }^{-1}(\mathbf{x}-\mathbf{\mu })}$$

   Mahalanobis 1936 (statistical distance; pending PDF page verification)

Edge cases and pitfalls

• Requires the covariance matrix Σ to be positive definite (invertible). With m < n or highly correlated criteria, Σ may be singular.

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