# Documentation

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# CompactClassificationDiscriminant class

Compact discriminant analysis class

## Description

A CompactClassificationDiscriminant object is a compact version of a discriminant analysis classifier. The compact version does not include the data for training the classifier. Therefore, you cannot perform some tasks with a compact classifier, such as cross validation. Use a compact classifier for making predictions (classifications) of new data.

## Construction

cobj = compact(obj) constructs a compact classifier from a full classifier.

cobj = makecdiscr(Mu,Sigma) constructs a compact discriminant analysis classifier from the class means Mu and covariance matrix Sigma. For syntax details, see makecdiscr.

### Input Arguments

 obj Discriminant analysis classifier, created using fitcdiscr.

## Methods

 compareHoldout Compare accuracies of two classification models using new data edge Classification edge logP Log unconditional probability density for discriminant analysis classifier loss Classification error mahal Mahalanobis distance to class means margin Classification margins nLinearCoeffs Number of nonzero linear coefficients predict Predict labels using discriminant analysis classification model

## Definitions

### Discriminant Classification

The model for discriminant analysis is:

• Each class (Y) generates data (X) using a multivariate normal distribution. That is, the model assumes X has a Gaussian mixture distribution (gmdistribution).

• For linear discriminant analysis, the model has the same covariance matrix for each class, only the means vary.

• For quadratic discriminant analysis, both means and covariances of each class vary.

predict classifies so as to minimize the expected classification cost:

$\stackrel{^}{y}=\underset{y=1,...,K}{\mathrm{arg}\mathrm{min}}\sum _{k=1}^{K}\stackrel{^}{P}\left(k|x\right)C\left(y|k\right),$

where

• $\stackrel{^}{y}$ is the predicted classification.

• K is the number of classes.

• $\stackrel{^}{P}\left(k|x\right)$ is the posterior probability of class k for observation x.

• $C\left(y|k\right)$ is the cost of classifying an observation as y when its true class is k.

For details, see How the predict Method Classifies.

### Regularization

Regularization is the process of finding a small set of predictors that yield an effective predictive model. For linear discriminant analysis, there are two parameters, γ and δ, that control regularization as follows. cvshrink helps you select appropriate values of the parameters.

Let Σ represent the covariance matrix of the data X, and let $\stackrel{^}{X}$ be the centered data (the data X minus the mean by class). Define

$D=\text{diag}\left({\stackrel{^}{X}}^{T}*\stackrel{^}{X}\right).$

The regularized covariance matrix $\stackrel{˜}{\Sigma }$ is

$\stackrel{˜}{\Sigma }=\left(1-\gamma \right)\Sigma +\gamma D.$

Whenever γ ≥ MinGamma, $\stackrel{˜}{\Sigma }$ is nonsingular.

Let μk be the mean vector for those elements of X in class k, and let μ0 be the global mean vector (the mean of the rows of X). Let C be the correlation matrix of the data X, and let $\stackrel{˜}{C}$ be the regularized correlation matrix:

$\stackrel{˜}{C}=\left(1-\gamma \right)C+\gamma I,$

where I is the identity matrix.

The linear term in the regularized discriminant analysis classifier for a data point x is

${\left(x-{\mu }_{0}\right)}^{T}{\stackrel{˜}{\Sigma }}^{-1}\left({\mu }_{k}-{\mu }_{0}\right)=\left[{\left(x-{\mu }_{0}\right)}^{T}{D}^{-1/2}\right]\left[{\stackrel{˜}{C}}^{-1}{D}^{-1/2}\left({\mu }_{k}-{\mu }_{0}\right)\right].$

The parameter δ enters into this equation as a threshold on the final term in square brackets. Each component of the vector $\left[{\stackrel{˜}{C}}^{-1}{D}^{-1/2}\left({\mu }_{k}-{\mu }_{0}\right)\right]$ is set to zero if it is smaller in magnitude than the threshold δ. Therefore, for class k, if component j is thresholded to zero, component j of x does not enter into the evaluation of the posterior probability.

The DeltaPredictor property is a vector related to this threshold. When δ ≥ DeltaPredictor(i), all classes k have

$|{\stackrel{˜}{C}}^{-1}{D}^{-1/2}\left({\mu }_{k}-{\mu }_{0}\right)|\le \delta .$

Therefore, when δ ≥ DeltaPredictor(i), the regularized classifier does not use predictor i.

## Copy Semantics

Value. To learn how value classes affect copy operations, see Copying Objects in the MATLAB® documentation.

## Examples

expand all

Construct a discriminant analysis classifier for the sample data.

fullobj = fitcdiscr(meas,species);

Construct a compact discriminant analysis classifier, and compare its size to that of the full classifier.

cobj = compact(fullobj);
b = whos('fullobj'); % b.bytes = size of fullobj
c = whos('cobj'); % c.bytes = size of cobj
[b.bytes c.bytes] % shows cobj uses 60% of the memory
ans =
18578       11498

The compact classifier is smaller than the full classifier.

Construct a compact discriminant analysis classifier from the means and covariances of the Fisher iris data.

mu(1,:) = mean(meas(1:50,:));
mu(2,:) = mean(meas(51:100,:));
mu(3,:) = mean(meas(101:150,:));

mm1 = repmat(mu(1,:),50,1);
mm2 = repmat(mu(2,:),50,1);
mm3 = repmat(mu(3,:),50,1);
cc = meas;
cc(1:50,:) = cc(1:50,:) - mm1;
cc(51:100,:) = cc(51:100,:) - mm2;
cc(101:150,:) = cc(101:150,:) - mm3;
sigstar = cc' * cc / 147;
cpct = makecdiscr(mu,sigstar,...
'ClassNames',{'setosa','versicolor','virginica'});