Mahalanobis Distance

Mahalanobis distance is a Euclidean distance that is transformed through a change of basis to normalize variance.

Definition

Euclidean distance is typically described as x^Tx, but an equivalent formulation is x^TI^TIx. In a two-dimensional graph, plotting the points with a Euclidean distance of 1 around the origin results in a unit circle.

The distance can be transformed to a different basis by swapping the identity matrix with some other A: x^TA^TAx. The two-dimensional graph will now appear as an ellipsoid. This ellipsoid is axis-aligned (i.e. appears to be stretched along the x or y axes) if the A is diagonal.

Of course distance can be calculated from any arbitrary point, not just the origin. Subtract the difference between the origin and the true reference point, leading to (x-m)^T(x-m) or (x-m)^TA^TA(x-m).

Application

For computing the variance-normalized distance between two testable measurements, instead of using a simple Euclidean distance (i.e. x^Tx), use a Mahalanobis distance with the respective means and the covariance matrix (usually notated as Σ).

The measurement must be normalized to the mean: (x-μ).

Given the normalized measurement, the covariance matrix describes how the unit variance was transformed into some other variances. Therefore the inverse of the covariance matrix (Σ^-1) describes the inverse transformation. Specifically, A is substituted with Σ^-0.5. A covariance matrix is always positive semi-definite so it can always be inverted and can always have the square root taken. A^TA then evaluates to Σ^-1.

The Mahalanobis distance is thus implemented as (x-μ)^TΣ^-1(x-μ).

CategoryRicottone