Mahalanobis distance between two vectors in MATLAB

Question

I have the following two vectors, and trying to find the Mahalanobis distance between them. The two vectors are as follows:

A=[2,4,5,7];
B=[6,3,8,1];

For calculating the Mahalanobis distance, I did the following:

> mahal(A(:),B(:))

For that, I got the following results:

0.6466
0.0259
0.0259
0.6466

But, how can I get one value, as when you calculate Euclidean distance for instance?

Thanks.

Answer 1

The Mahalanobis distance is actually the distance from the mean of a distribution. So if there is no distribution it becomes similar (not equal) to the Euclidean distance.

According to MATLAB:

mahal(Y,X) computes the Mahalanobis distance (in squared units) of each observation in Y from the reference sample in matrix X. If Y is n-by-m, where n is the number of observations and m is the dimension of the data, d is n-by-1. X and Y must have the same number of columns, but can have different numbers of rows. X must have more rows than columns.

so you will have something like this, you can compare the Mahalanobis with Euclidean distances:

X = mvnrnd([0;0],[1 .9;.9 1],100);
Y = [1 1;1 -1;-1 1;-1 -1];

d1 = mahal(Y,X) % Mahalanobis (**it still gives one value**)
d1 =
    1.3592
   21.1013
   23.8086
    1.4727

d2 = sum((Y-repmat(mean(X),4,1)).^2, 2) % Squared Euclidean
d2 =
    1.9310
    1.8821
    2.1228
    2.0739
% if you check the figure it will be easier to understand
scatter(X(:,1),X(:,2))
hold on
scatter(Y(:,1),Y(:,2),100,d1,'*','LineWidth',2)
hb = colorbar;
ylabel(hb,'Mahalanobis Distance')
legend('X','Y','Location','NW')

Mahalanobis distance (or "generalized squared interpoint distance" for its squared value) can also be defined as a dissimilarity measure between two random vectors x and y of the same distribution with the covariance matrix S:

If the covariance matrix is the identity matrix, the Mahalanobis distance reduces to the Euclidean distance. If the covariance matrix is diagonal, then the resulting distance measure is called a normalized Euclidean distance:

where Si is the standard deviation of the Xi and Yi over the sample set.