# mahal

Mahalanobis distance to Gaussian mixture component

## Syntax

## Description

## Examples

### Measure Mahalanobis Distance

Generate random variates that follow a mixture of two bivariate Gaussian distributions by using the `mvnrnd`

function. Fit a Gaussian mixture model (GMM) to the generated data by using the `fitgmdist`

function, and then compute Mahalanobis distances between the generated data and the mixture components of the fitted GMM.

Define the distribution parameters (means and covariances) of two bivariate Gaussian mixture components.

rng('default') % For reproducibility mu1 = [1 2]; % Mean of the 1st component sigma1 = [2 0; 0 .5]; % Covariance of the 1st component mu2 = [-3 -5]; % Mean of the 2nd component sigma2 = [1 0; 0 1]; % Covariance of the 2nd component

Generate an equal number of random variates from each component, and combine the two sets of random variates.

r1 = mvnrnd(mu1,sigma1,1000); r2 = mvnrnd(mu2,sigma2,1000); X = [r1; r2];

The combined data set `X`

contains random variates following a mixture of two bivariate Gaussian distributions.

Fit a two-component GMM to `X`

.

gm = fitgmdist(X,2)

gm = Gaussian mixture distribution with 2 components in 2 dimensions Component 1: Mixing proportion: 0.500000 Mean: -2.9617 -4.9727 Component 2: Mixing proportion: 0.500000 Mean: 0.9539 2.0261

`fitgmdist`

fits a GMM to `X`

using two mixture components. The means of `Component`

`1`

and `Component`

`2`

are `[-2.9617,-4.9727]`

and `[0.9539,2.0261]`

, which are close to `mu2`

and `mu1`

, respectively.

Compute the Mahalanobis distance of each point in `X`

to each component of `gm`

.

d2 = mahal(gm,X);

Plot `X`

by using `scatter`

and use marker color to visualize the Mahalanobis distance to `Component`

`1`

.

scatter(X(:,1),X(:,2),10,d2(:,1),'.') % Scatter plot with points of size 10 c = colorbar; ylabel(c,'Mahalanobis Distance to Component 1')

## Input Arguments

`gm`

— Gaussian mixture distribution

`gmdistribution`

object

Gaussian mixture distribution, also called Gaussian mixture model (GMM), specified as a `gmdistribution`

object.

You can create a `gmdistribution`

object using `gmdistribution`

or `fitgmdist`

. Use the `gmdistribution`

function to create a
`gmdistribution`

object by specifying the distribution parameters.
Use the `fitgmdist`

function to fit a `gmdistribution`

model to data given a fixed number of components.

`X`

— 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.

If a row of `X`

contains `NaNs`

, then
`mahal`

excludes the row from the computation.
The corresponding value in `d2`

is
`NaN`

.

**Data Types: **`single`

| `double`

## Output Arguments

`d2`

— Squared Mahalanobis distance

*n*-by-*k* numeric matrix

Squared Mahalanobis distance of each observation in `X`

to each Gaussian
mixture component in `gm`

, returned as an
*n*-by-*k* numeric matrix, where
*n* is the number of observations in `X`

and
*k* is the number of mixture components in
`gm`

.

`d2(i,j)`

is the squared distance of observation `i`

to the
`j`

th Gaussian mixture component.

## More About

### Mahalanobis Distance

The Mahalanobis distance is a measure between a sample point and a distribution.

The Mahalanobis distance from a vector *x* to a distribution with
mean *μ* and covariance *Σ* is

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

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

`mahal`

returns the squared Mahalanobis distance *d*^{2} from an observation in `X`

to a mixture
component in `gm`

.

## Version History

**Introduced in R2007b**

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