Contents
In what units is the Mahalanobis distance?
The Mahalanobis distance is defined as the distance between a (multidimensional) point and a distribution. It is the multivariate form of the distance measured in units of standard deviation and is named after the famous Indian statistician R.P. Mahalanobis (1893 – 1972).
Is Mahalanobis distance always positive?
For Mahalanobis distance to be a valid distance, Σ must be a positive definite matrix. This stems directly from the definition of a positive definite matrix, and the non-negativity axiom of distance. (Whether or not Σ has negative entires is not important here; what is important is its eigenvalues.)
What does measuring distance between clusters mean in the case of a complete linkage?
In complete-linkage clustering, the link between two clusters contains all element pairs, and the distance between clusters equals the distance between those two elements (one in each cluster) that are farthest away from each other.
How does Mahalanobis distance take correlation into account?
The Mahalanobis distance takes correlation into account; the covariance matrix contains this information. However, it’s difficult to look at the Mahalanobis equation and gain an intuitive understanding as to how it actually does this. We can gain some insight into it, though, by taking a different approach.
How to calculate Mahalanobis distance between two vectors?
With that in mind, below is the general equation for the Mahalanobis distance between two vectors, x and y, where S is the covariance matrix. (Side note: As you might expect, the probability density function for a multivariate Gaussian distribution uses the Mahalanobis distance instead of the Euclidean. See the equation here .)
How to calculate the distance between two clusters?
If you want a distance of two clusters, the following two approaches stand out: the weighted average distance of each object to the other cluster, using the other clusters Mahalanobis distance. You could approximate this by using the distance of the centroid only. Maybe use the maximum of the two clusters to resolve asymmetry.