ClassificationDiscriminant

Discriminant analysis classification

Description

A ClassificationDiscriminant object encapsulates a discriminant analysis classifier, which is a Gaussian mixture model for data generation. A ClassificationDiscriminant object can predict responses for new data using the predict method. The object contains the data used for training, so can compute resubstitution predictions.

Creation

Create a ClassificationDiscriminant object by using fitcdiscr.

Properties

BetweenSigma — Between-class covariance

square matrix

This property is read-only.

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

Data Types: double

CategoricalPredictors — Categorical predictor indices

[]

This property is read-only.

Categorical predictor indices, which is always empty ([]).

ClassNames — Class names in training data Y

categorical array | cell array of character vectors | character array | string array | logical vector | numeric vector

This property is read-only.

Class names in the training data Y with duplicates removed. ClassNames has the same data type as the data in the argument Y in the training data. ClassNames can have the following data types:

• Categorical array

• Cell array of character vectors

• Character array

• Logical vector

• Numeric vector

(The software treats string arrays as cell arrays of character vectors.)

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

Coeffs — Coefficient matrices

k-by-k structure | []

This property is read-only.

Coefficient matrices, specified as a k-by-k structure, where k is the number of classes. If fitcdiscr had the FillCoeffs name-value pair set to 'off' when constructing the classifier, Coeffs is empty ([]).

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

• DiscrimType

• Class1 — ClassNames(i)

• Class2 — ClassNames(j)

• Const — A scalar

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

• Quadratic — p-by-p matrix, exists for quadratic DiscrimType

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

Cost — Cost of classifying a point

square matrix

Cost of classifying a point, specified as a square matrix. Cost(i,j) is the cost of classifying a point into class j if its true class is i (the rows correspond to the true class and the columns correspond to the predicted class). The order of the rows and columns of Cost corresponds to the order of the classes in ClassNames. The number of rows and columns in Cost is the number of unique classes in the response.

Change a Cost matrix using dot notation: obj.Cost = costMatrix.

Data Types: double

Delta — Delta threshold for a linear discriminant model

nonnegative scalar

Value of the Delta threshold for a linear discriminant model, specified as a nonnegative scalar. If a coefficient of obj has magnitude smaller than Delta, obj sets this coefficient to 0, and so you can eliminate the corresponding predictor from the model. Set Delta to a higher value to eliminate more predictors.

Delta must be 0 for quadratic discriminant models.

Change Delta using dot notation: obj.Delta = newDelta.

Data Types: double

DeltaPredictor — Minimum value of Delta coefficient for predictor to be in model

row vector of length p

This property is read-only.

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

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

Data Types: double

DiscrimType — Discriminant type

character vector

Discriminant type, specified as a character vector or string. Available values:

• 'linear'

• 'quadratic'

• 'diagLinear'

• 'diagQuadratic'

• 'pseudoLinear'

• 'pseudoQuadratic'

Change DiscrimType using dot notation: obj.DiscrimType = newDiscrimType. You can change between linear types, or between quadratic types, but cannot change between linear and quadratic types.

Data Types: char | string

Gamma — Gamma regularization parameter

scalar from 0 through 1

Value of the Gamma regularization parameter, specified as a scalar from 0 through 1. Change Gamma using dot notation: obj.Gamma = newGamma.

• If you set 1 for linear discriminant, the discriminant sets its type to 'diagLinear'.

• If you set a value between MinGamma and 1 for linear discriminant, the discriminant sets its type to 'linear'.

• You cannot set values below the value of the MinGamma property.

• For quadratic discriminant, you can set either 0 (for DiscrimType 'quadratic') or 1 (for DiscrimType 'diagQuadratic').

Data Types: double

HyperparameterOptimizationResults — Description of cross-validation optimization of hyperparameters

BayesianOptimization object | table of hyperparameters and associated values

This property is read-only.

Description of the cross-validation optimization of hyperparameters, returned as a BayesianOptimization object or a table of hyperparameters and associated values. Nonempty when the OptimizeHyperparameters name-value pair is nonempty at creation. Value depends on the setting of the HyperparameterOptimizationOptions name-value pair at creation:

• 'bayesopt' (default) — Object of class BayesianOptimization

• 'gridsearch' or 'randomsearch' — Table of hyperparameters used, observed objective function values (cross-validation loss), and rank of observations from lowest (best) to highest (worst)

LogDetSigma — Logarithm of determinant of within-class covariance matrix

scalar | vector

This property is read-only.

Logarithm of the determinant of the within-class covariance matrix, returned as a scalar or vector. The type of LogDetSigma depends on the discriminant type:

• Scalar for linear discriminant analysis

• Vector of length K for quadratic discriminant analysis, where K is the number of classes

Data Types: double

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

nonnegative scalar

This property is read-only.

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

Data Types: double

ModelParameters — Parameters used in training the model

DiscriminantParams object

This property is read-only.

Parameters used in training the model, returned as a DiscriminantParams object. The returned parameters have the following properties.

Property	Value
DiscrimType	'linear' 'quadratic' 'diagLinear' 'diagQuadratic' 'pseudoLinear' 'pseudoQuadratic'
Gamma	scalar from 0 through 1
Delta	nonnegative scalar
FillCoeffs	logical scalar
SaveMemory	logical scalar
Version	scalar
Method	'Discriminant'
Type	'classification'

Mu — Class means

real K-by-p matrix

This property is read-only.

Class means, specified 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 in the ClassNames attribute.

Data Types: double

NumObservations — Number of observations in the training data

positive integer

This property is read-only.

Number of observations in the training data, returned as a positive integer. NumObservations can be less than the number of rows of input data when there are missing values in the input data or response data.

Data Types: double

PredictorNames — Names of predictor variables

cell array

This property is read-only.

Names of predictor variables, returned as a cell array. The names are in the order in which they appear in the training data X.

Data Types: cell

Prior — Prior probabilities for each class

numeric vector

Prior probabilities for each class, returned as a numeric vector. The order of the elements of Prior corresponds to the order of the classes in ClassNames.

Add or change a Prior vector using dot notation: obj.Prior = priorVector.

Data Types: double

ResponseName — Name of the response variable Y

character vector

This property is read-only.

Name of the response variable Y, returned as a character vector.

Data Types: char | string

RowsUsed — Rows of the original predictor data X used for fitting

logical vector

This property is read-only.

Rows of the original predictor data X used for fitting, returned as an n-element logical vector, where n is the number of rows of X. If the software uses all rows of X for constructing the object, then RowsUsed is an empty array ([]).

Data Types: logical

ScoreTransform — Score transformation function

name of a built-in function | function handle | 'none'

Score transformation function, specified as a character vector or string representing a built-in transformation function, or as a function handle for transforming scores. '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 fitcdiscr.

Implement dot notation to add or change a ScoreTransform function using one of the following:

• cobj.ScoreTransform = 'function'

• cobj.ScoreTransform = @function

Data Types: char | string | function_handle

Sigma — Within-class covariance

numeric array

This property is read-only.

Within-class covariance, returned as a numeric array. The dimensions depend on 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

W — Scaled observation weights

numeric vector of length n

This property is read-only.

Scaled observation weights, returned as a numeric vector of length n, where n is the number of rows in X.

Data Types: double

X — Predictor values

real matrix

This property is read-only.

Predictor values, returned as a real matrix. Each column of X represents one predictor (variable), and each row represents one observation.

Data Types: single | double

Xcentered — X data with class means subtracted

real matrix

This property is read-only.

X data with class means subtracted, returned as a real matrix. If Y(i) is of class j,

Xcentered(i,:) = X(i,:) – Mu(j,:),

where Mu is the class mean property.

Data Types: single | double

Y — Row classifications

classification variables

This property is read-only.

Row classifications, returned as a categorical array, cell array of character vectors, character array, logical vector, or numeric vector with the same number of rows as X. Each row of Y represents the classification of the corresponding row of X.

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

Object Functions

compact	Reduce size of discriminant analysis classifier
compareHoldout	Compare accuracies of two classification models using new data
crossval	Cross-validate machine learning model
cvshrink	Cross-validate regularization of linear discriminant
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
resubEdge	Resubstitution classification edge for discriminant analysis classifier
resubLoss	Resubstitution classification loss for discriminant analysis classifier
resubMargin	Resubstitution classification margins for discriminant analysis classifier
resubPredict	Classify observations in discriminant analysis classifier by resubstitution
shapley	Shapley values
testckfold	Compare accuracies of two classification models by repeated cross-validation

Examples

Train Discriminant Analysis Model

Load Fisher's iris data set.

load fisheriris

Train a discriminant analysis model using the entire data set.

Mdl = fitcdiscr(meas,species)

Mdl = 
  ClassificationDiscriminant
             ResponseName: 'Y'
    CategoricalPredictors: []
               ClassNames: {'setosa'  'versicolor'  'virginica'}
           ScoreTransform: 'none'
          NumObservations: 150
              DiscrimType: 'linear'
                       Mu: [3x4 double]
                   Coeffs: [3x3 struct]

Mdl is a ClassificationDiscriminant model. To access its properties, use dot notation. For example, display the group means for each predictor.

Mdl.Mu

ans = 3×4

    5.0060    3.4280    1.4620    0.2460
    5.9360    2.7700    4.2600    1.3260
    6.5880    2.9740    5.5520    2.0260

To predict labels for new observations, pass Mdl and predictor data to predict.

More About

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:

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

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. 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 $$\widehat{X}$$ be the centered data (the data X minus the mean by class). Define

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

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

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

Whenever 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.

References

[1] Guo, Y., T. Hastie, and R. Tibshirani. "Regularized linear discriminant analysis and its application in microarrays." Biostatistics, Vol. 8, No. 1, pp. 86–100, 2007.

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 pair argument cannot be an anonymous function.

For more information, see Introduction to Code Generation.

Version History

Introduced in R2011b

R2023b: Model stores observations with missing predictor values

Starting in R2023b, training observations with missing predictor values are included in the X, Xcentered, Y, and W data properties. The RowsUsed property indicates the training observations stored in the model, rather than those used for training. Observations with missing predictor values continue to be omitted from the model training process.

In previous releases, the software omitted training observations that contained missing predictor values from the data properties of the model.