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.
Examples
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 (gmdistribution) |  |
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