# 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

$$d=\sqrt{(y-\mu ){\sum}^{-1}(y-\mu )\text{'}}.$$

This distance represents how far *y* is from the
mean in number of standard deviations.

`mahal`

returns the squared Mahalanobis distance *d*^{2} 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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