mahal
Mahalanobis distance to reference samples
Syntax
Description
returns the squared Mahalanobis
distance of each observation in d2
= mahal(Y
,X
)Y
to the reference
samples in X
.
Examples
Compare Mahalanobis and Squared Euclidean Distances
Generate a correlated bivariate sample data set.
rng('default') % For reproducibility X = mvnrnd([0;0],[1 .9;.9 1],1000);
Specify four observations that are equidistant from the mean of X
in Euclidean distance.
Y = [1 1;1 -1;-1 1;-1 -1];
Compute the Mahalanobis distance of each observation in Y
to the reference samples in X
.
d2_mahal = mahal(Y,X)
d2_mahal = 4×1
1.1095
20.3632
19.5939
1.0137
Compute the squared Euclidean distance of each observation in Y
from the mean of X
.
d2_Euclidean = sum((Y-mean(X)).^2,2)
d2_Euclidean = 4×1
2.0931
2.0399
1.9625
1.9094
Plot X
and Y
by using scatter
and use marker color to visualize the Mahalanobis distance of Y
to the reference samples in X
.
scatter(X(:,1),X(:,2),10,'.') % Scatter plot with points of size 10 hold on scatter(Y(:,1),Y(:,2),100,d2_mahal,'o','filled') hb = colorbar; ylabel(hb,'Mahalanobis Distance') legend('X','Y','Location','best')
All observations in Y
([1,1]
, [-1,-1,]
, [1,-1]
, and [-1,1]
) are equidistant from the mean of X
in Euclidean distance. However, [1,1]
and [-1,-1]
are much closer to X than [1,-1]
and [-1,1]
in Mahalanobis distance. Because Mahalanobis distance considers the covariance of the data and the scales of the different variables, it is useful for detecting outliers.
Input Arguments
Y
— Data
n-by-m numeric matrix
Data, specified as an n-by-m numeric matrix, where n is the number of observations and m is the number of variables in each observation.
X
and Y
must have the same
number of columns, but can have different numbers of rows.
Data Types: single
| double
X
— Reference samples
p-by-m numeric matrix
Reference samples, specified as a p-by-m numeric matrix, where p is the number of samples and m is the number of variables in each sample.
X
and Y
must have the same
number of columns, but can have different numbers of rows.
X
must have more rows than columns.
Data Types: single
| double
Output Arguments
d2
— Squared Mahalanobis distance
n-by-1 numeric vector
Squared Mahalanobis distance of each observation in
Y
to the reference samples in
X
, returned as an n-by-1 numeric
vector, where n is the number of observations in
X
.
More About
Mahalanobis Distance
The Mahalanobis distance is a measure between a sample point and a distribution.
The Mahalanobis distance from a vector y to a distribution with mean μ and covariance Σ is
This distance represents how far y is from the mean in number of standard deviations.
mahal
returns the squared Mahalanobis distance d2 from an observation in Y
to the reference
samples in X
. In the mahal
function,
μ and Σ are the sample mean and covariance
of the reference samples, respectively.
Version History
Introduced before R2006a
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