Mahalanobis Distance

What is Mahalanobis distance?

Most people know what Euclidean distance is…it is the shortest distance between any two points.  In other words, its what we typically think of when we think of distance – the distance we would measure with a ruler, and the one given by the Pythagorean formula.

Unlike Euclidean distance, Mahalanobis distance gauges the similarity of an unknown sample set to a known one. It differs from Euclidean distance in that it takes into account the correlations of the data set and is scale-invariant. In other words, it is a multivariate effect size.

Wikipedia defines Mahalanobis distance using the following intuition.

Consider the problem of estimating the probability that a test point in N-dimensional Euclidean space belongs to a set, where we are given sample points that definitely belong to that set. Our first step would be to find the average or center of mass of the sample points. Intuitively, the closer the point in question is to this center of mass, the more likely it is to belong to the set.

However, we also need to know if the set is spread out over a large range or a small range, so that we can decide whether a given distance from the center is noteworthy or not. The simplistic approach is to estimate the standard deviation of the distances of the sample points from the center of mass. If the distance between the test point and the center of mass is less than one standard deviation, then we might conclude that it is highly probable that the test point belongs to the set. The further away it is, the more likely that the test point should not be classified as belonging to the set.

This intuitive approach can be made quantitative by defining the normalized distance between the test point and the set to be  {x - \mu} \over \sigma . By plugging this into the normal distribution we can derive the probability of the test point belonging to the set.

The drawback of the above approach was that we assumed that the sample points are distributed about the center of mass in a spherical manner. Were the distribution to be decidedly non-spherical, for instance ellipsoidal, then we would expect the probability of the test point belonging to the set to depend not only on the distance from the center of mass, but also on the direction. In those directions where the ellipsoid has a short axis the test point must be closer, while in those where the axis is long the test point can be further away from the center.

Putting this on a mathematical basis, the ellipsoid that best represents the set’s probability distribution can be estimated by building the covariance matrix of the samples. The Mahalanobis distance is simply the distance of the test point from the center of mass divided by the width of the ellipsoid in the direction of the test point.

How does one calculate the Mahalanobis distance in practice? The Healthcare Economist has a simple example for you to examine here.


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