CompactClassificationDiscriminant

R2026b

Compact discriminant analysis classification

Description

CompactClassificationDiscriminant is a compact version of a discriminant analysis classifier. The compact version does not include the data used to train the classifier. Therefore, you cannot perform some tasks, such as cross-validation, using the compact classifier. Use a compact classifier to make predictions (classifications) for new data.

Creation

You can create a CompactClassificationDiscriminant object in two ways:

• Create a compact model from a full ClassificationDiscriminant model object by using the compact object function.

• Create a compact model by using the makecdiscr function and specifying the class means Mu and covariance matrix Sigma.

Properties

Discriminant Analysis Properties

BetweenSigma — Between-class covariance

Read-only: square matrix

This property is read-only.

Between-class covariance, returned as a p-by-p matrix, where p is the number of predictors.

Data Types: double

Coeffs — Coefficient matrices

Read-only: k-by-k structure | []

This property is read-only.

Coefficient matrices, returned as a k-by-k structure, where k is the number of classes. If the value of the FillCoeffs name-value argument is "off" when you create the classifier object, Coeffs is empty ([]).

Coeffs(i,j) contains the coefficients of the linear or quadratic boundaries between classes i and j. The fields in Coeffs(i,j) are:

• DiscrimType — Character vector corresponding to the discriminant type

• Class1 — ClassNames(i) value

• Class2 — ClassNames(j) value

• Const — Numeric scalar

• Linear — Numeric vector with p components, where p is the number of columns in X

• Quadratic — p-by-p numeric matrix. This field exists only when the discriminant type is quadratic.

The equation of the boundary between class i and class j is

Const + Linear * x + x' * Quadratic * x = 0,

where x is a column vector of length p.

Data Types: struct

Delta — Delta threshold for linear discriminant model

nonnegative scalar

Delta threshold for a linear discriminant model, specified as a nonnegative scalar. If a coefficient of the classifier object has a magnitude smaller than Delta, the software sets the coefficient to 0, which means you can remove the corresponding predictor from the model. Set Delta to a higher value to remove more predictors.

Delta must be 0 for quadratic discriminant models.

To change the value of Delta to newDelta for a classifier object Mdl, you can use dot notation: Mdl.Delta = newDelta.

Data Types: double

DeltaPredictor — Minimum value of Delta coefficient

Read-only: row vector of length p

This property is read-only.

Minimum value of the Delta coefficient needed for the predictor to be included in the model, returned as a row vector of length p, where p is the number of predictors in the classifier object. If DeltaPredictor(i) < Delta, then coefficient i of the model is 0.

If the classifier is a quadratic discriminant model, all elements of DeltaPredictor are 0.

Data Types: double

DiscrimType — Discriminant type

character vector | string scalar

Discriminant type, specified as a character vector or string scalar corresponding to one of these values:

• "linear"

• "quadratic"

• "diagLinear"

• "diagQuadratic"

• "pseudoLinear"

• "pseudoQuadratic"

To change the value of DiscrimType to newDiscrimType for a classifier object Mdl, you can use dot notation: Mdl.DiscrimType = newDiscrimType. You can change one linear type to another, or one quadratic type to another, but cannot change a linear type to a quadratic type (or vice versa).

Data Types: char | string

Gamma — Gamma regularization parameter

scalar from 0 through 1

Gamma regularization parameter, specified as a scalar from 0 through 1. To change the Gamma value to newGamma for a classifier object Mdl, you can use dot notation: Mdl.Gamma = newGamma.

• If you set Gamma to 1 for a linear discriminant model, the software sets the DiscrimType property to 'diagLinear'.

• If you set Gamma to a value between MinGamma and 1 for a linear discriminant model, the software sets the DiscrimType property to 'linear'.

• You cannot set Gamma to a value below the value of the MinGamma property.

• For a quadratic discriminant model, you can set Gamma to 0 (when DiscrimType is 'quadratic') or 1 (when DiscrimType is 'diagQuadratic').

Data Types: double

LogDetSigma — Logarithm of determinant of within-class covariance matrix

Read-only: numeric scalar | numeric vector

This property is read-only.

Logarithm of the determinant of the within-class covariance matrix, returned as a numeric scalar or numeric vector.

• For linear discriminant analysis, LogDetSigma is a numeric scalar.

• For quadratic discriminant analysis, LogDetSigma is a numeric vector of length K, where K is the number of classes.

Data Types: double

MinGamma — Minimal value of Gamma parameter so that correlation matrix is invertible

Read-only: nonnegative scalar

This property is read-only.

Minimal value of the Gamma parameter so that the correlation matrix is invertible, returned as a nonnegative scalar. If the correlation matrix is not singular, MinGamma is 0.

Data Properties

CategoricalPredictors — Categorical predictor indices

Read-only: empty array []

This property is read-only.

Categorical predictor indices, returned as an empty array [].

ClassNames — Unique class names

Read-only: numeric vector | categorical vector | logical vector | character array | cell array of character vectors

This property is read-only.

Unique class names used in training, returned as a numeric vector, categorical vector, logical vector, character array, or cell array of character vectors. ClassNames has the same data type as the class labels Y. (The software treats string arrays as cell arrays of character vectors.) ClassNames also determines the class order.

Data Types: single | double | categorical | logical | char | cell

Cost — Misclassification costs

square numeric matrix

Misclassification costs, returned as a square numeric matrix. Cost has K rows and columns, where K is the number of classes.

Cost(i,j) is the cost of classifying a point into class j if its true class is i. The order of the rows and columns of Cost corresponds to the order of the classes in ClassNames.

To change the value of Cost to costMatrix for a classifier object Mdl, you can use dot notation: Mdl.Cost = costMatrix.

Data Types: double

Mu — Class means

Read-only: K-by-p matrix

This property is read-only.

Class means, returned as a K-by-p matrix of real values. K is the number of classes, and p is the number of predictors. Each row of Mu represents the mean of the multivariate normal distribution of the corresponding class. The class indices are stored in the ClassNames property.

Data Types: double

PredictorNames — Predictor variable names

Read-only: cell array of character vectors

This property is read-only.

Predictor variable names, returned as a cell array of character vectors. The order of the elements in PredictorNames corresponds to the order in which the predictor names appear in the training data.

Data Types: cell

Prior — Prior probabilities for each class

numeric vector

Prior probabilities for each class, returned as a K-element numeric vector, where K is the number of unique classes in the response. The order of the elements of Prior corresponds to the order of the classes in ClassNames.

To change the value of Prior to priorVector for a classifier object Mdl, you can use dot notation: Mdl.Prior = priorVector.

Data Types: double

ResponseName — Response variable name

Read-only: character vector

This property is read-only.

Response variable name, returned as a character vector.

Data Types: char

ScoreTransform — Function for transforming scores

function handle | name of a built-in transformation function | "none"

Function for transforming scores, specified as a function handle or the name of a built-in transformation function. "none" means no transformation; equivalently, "none" means @(x)x. For a list of built-in transformation functions and the syntax of custom transformation functions, see ScoreTransform.

To change the value of ScoreTransform for a classifier object Mdl, you can use dot notation:

Mdl.ScoreTransform = "function"
% or
Mdl.ScoreTransform = @function

Data Types: char | string | function_handle

Sigma — Within-class covariance

Read-only: numeric array

This property is read-only.

Within-class covariance, returned as a numeric array. The dimensions of the array depend on the value of DiscrimType:

• 'linear' (default) — Matrix of size p-by-p, where p is the number of predictors

• 'quadratic' — Array of size p-by-p-by-K, where K is the number of classes

• 'diagLinear' — Row vector of length p

• 'diagQuadratic' — Array of size 1-by-p-by-K

• 'pseudoLinear' — Matrix of size p-by-p

• 'pseudoQuadratic' — Array of size p-by-p-by-K

Data Types: double

Object Functions

compareHoldout	Compare accuracies of two classification models using new data
edge	Classification edge for discriminant analysis classifier
lime	Local interpretable model-agnostic explanations (LIME)
logp	Log unconditional probability density for discriminant analysis classifier
loss	Classification loss for discriminant analysis classifier
mahal	Mahalanobis distance to class means of discriminant analysis classifier
margin	Classification margins for discriminant analysis classifier
nLinearCoeffs	Number of nonzero linear coefficients in discriminant analysis classifier
partialDependence	Compute partial dependence
plotPartialDependence	Create partial dependence plot (PDP) and individual conditional expectation (ICE) plots
predict	Predict labels using discriminant analysis classifier
shapley	Shapley values

Examples

Reduce Size of Discriminant Analysis Classifier

Load Fisher's iris data set.

load fisheriris

Train a discriminant analysis classifier using the entire data set.

Mdl = fitcdiscr(meas,species);

Reduce the size of the discriminant analysis classifier.

CMdl = compact(Mdl)

CMdl = 
  CompactClassificationDiscriminant
             ResponseName: 'Y'
    CategoricalPredictors: []
               ClassNames: {'setosa'  'versicolor'  'virginica'}
           ScoreTransform: 'none'
              DiscrimType: 'linear'
                       Mu: [3×4 double]
                   Coeffs: [3×3 struct]


  Properties, Methods

CMdl is a trained CompactClassificationDiscriminant classifier.

Display the amount of memory used by each classifier.

whos("Mdl","CMdl")

  Name      Size            Bytes  Class                                                          Attributes

  CMdl      1x1             12328  classreg.learning.classif.CompactClassificationDiscriminant              
  Mdl       1x1             19139  ClassificationDiscriminant                                               

The compact classifier (CMdl) is smaller than the full classifier (Mdl).

Create Compact Discriminant Analysis Classifier Using Means and Covariances

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

load fisheriris

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;

compactMdl = makecdiscr(mu,sigstar, ...
    ClassNames=["setosa","versicolor","virginica"])

compactMdl = 
  CompactClassificationDiscriminant
           PredictorNames: {'x1'  'x2'  'x3'  'x4'}
             ResponseName: 'Y'
    CategoricalPredictors: []
               ClassNames: {'setosa'  'versicolor'  'virginica'}
           ScoreTransform: 'none'
              DiscrimType: 'linear'
                       Mu: [3×4 double]
                   Coeffs: [3×3 struct]


  Properties, Methods

More About

Discriminant Classification

In the model for discriminant analysis, 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 the means and the covariances of each class vary.

The predict function classifies observations in a way that minimizes the expected classification cost:

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

where

• $$\widehat{y}$$ is the predicted classification.

• K is the number of classes.

• $$\widehat{P}\left(k|x\right)$$ 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 Prediction Using Discriminant Analysis Models.

Regularization

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

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

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

The regularized covariance matrix $$\tilde{\Sigma }$$ is

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

When γ ≥ MinGamma, $$\tilde{\Sigma }$$ is nonsingular.

Let μ_k 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 $$\tilde{C}$$ be the regularized correlation matrix.

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

where I is the identity matrix.

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

$${\left(x-{\mu }_0\right)}^T{\tilde{\Sigma }}^{-1}\left({\mu }_k-{\mu }_0\right)=\left[{\left(x-{\mu }_0\right)}^T{D}^{-1/2}\right]\left[{\tilde{C}}^{-1}{D}^{-1/2}\left({\mu }_k-{\mu }_0\right)\right].$$

The parameter δ enters into this equation as a threshold on the final term in square brackets. Each component of the vector $$\left[{\tilde{C}}^{-1}{D}^{-1/2}\left({\mu }_k-{\mu }_0\right)\right]$$ 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

$$\left|{\tilde{C}}^{-1}{D}^{-1/2}\left({\mu }_k-{\mu }_0\right)\right|\le \delta .$$

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

Extended Capabilities

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

• The predict function supports code generation.

• When you train a discriminant analysis model by using fitcdiscr or create a compact discriminant analysis model by using makecdiscr, the value of the ScoreTransform name-value argument cannot be an anonymous function.

For more information, see Introduction to Code Generation for Statistics and Machine Learning Functions.

GPU Code Generation

Generate CUDA® code for NVIDIA® GPUs using GPU Coder™.

Refer to the usage notes and limitations in the C/C++ Code Generation section. The same usage notes and limitations apply to GPU code generation.

Version History

Introduced in R2011b