Mahalanobis Distance

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It is also called quadratic distance. It measures the separation of two groups of objects. Suppose we have two groups with means and , Mahalanobis distance is given by the following

Formula

The data of the two groups must have the same number of variables (the same number of columns) but not necessarily to have the same data (each group may have different number of rows).

For example: Suppose we have two groups of data, each of group consists of two variables (x, y). The scattered plot of data is shown below.

First, we center the data on the arithmetic mean of each variable.

Covariance matrix of group is computed using centered data matrix

It produces covariance matrices for group 1 and 2 as follow

The pooled covariance matrix of the two groups is computed as weighted average of the covariance matrices. The weighted average takes this form

.

The pooled covariance is computed using weighted average (10/15)*Covariance group 1 + (5/15)*Covariance group 2 yields

The Mahalanobis distance is simply quadratic multiplication of mean difference and inverse of pooled covariance matrix.

The final result of Mahalanobis distance is

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