d = mahal(Y,X)
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.
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.