Statistics Toolbox™

mahal - Mahalanobis distance

Syntax

d = mahal(Y,X)

Description

d = 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.

For observation I, the Mahalanobis distance is defined by d(I) = (Y(I,:)-mu)*inv(SIGMA)*(Y(I,:)-mu)', where mu and SIGMA are the sample mean and covariance of the data in X. mahal performs an equivalent, but more efficient, computation.

Example

Generate some correlated bivariate data in X and compare the Mahalanobis and squared Euclidean distances of observations in Y:

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

d1 = mahal(Y,X) % Mahalanobis
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

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')

The observations in Y with equal coordinate values are much closer to X in Mahalanobis distance than observations with opposite coordinate values, even though all observations are approximately equidistant from the mean of X in Euclidean distance. The Mahalanobis distance, by considering the covariance of the data and the scales of the different variables, is useful for detecting outliers in such cases.

See Also

pdist, mahal

mad

mahal