# mahal

Mahalanobis distance to Gaussian mixture component

## Description

example

d2 = mahal(gm,X) returns the squared Mahalanobis distance of each observation in X to each Gaussian mixture component in gm.

## Examples

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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

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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.

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

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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 jth Gaussian mixture component.

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### 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

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

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

mahal returns the squared Mahalanobis distance d2 from an observation in X to a mixture component in gm.