ClassificationDiscriminant class

Superclasses: CompactClassificationDiscriminant

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.

Construction

Create a ClassificationDiscriminant object by using fitcdiscr.

Properties

BetweenSigma	p-by-p matrix, the between-class covariance, where p is the number of predictors.
CategoricalPredictors	Categorical predictor indices, which is always empty ([]) .
ClassNames	List of the elements in the training data Y with duplicates removed. ClassNames can be a categorical array, cell array of character vectors, character array, logical vector, or a numeric vector. ClassNames has the same data type as the data in the argument Y. (The software treats string arrays as cell arrays of character vectors.)
Coeffs	k-by-k structure of coefficient matrices, where k is the number of classes. 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. If fitcdiscr had the FillCoeffs name-value pair set to 'off' when constructing the classifier, Coeffs is empty ([]).
Cost	Square matrix, where Cost(i,j) is the cost of classifying a point into class j if its true class is i (i.e., 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.
Delta	Value of the Delta threshold for a linear discriminant model, 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.
DeltaPredictor	Row vector of length equal to 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.
DiscrimType	Character vector specifying the discriminant type. One of: '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.
Gamma	Value of the Gamma regularization parameter, a scalar from 0 to 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').
HyperparameterOptimizationResults	Description of the cross-validation optimization of hyperparameters, stored 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 the determinant of the within-class covariance matrix. 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
MinGamma	Nonnegative scalar, the minimal value of the Gamma parameter so that the correlation matrix is invertible. If the correlation matrix is not singular, MinGamma is 0.
ModelParameters	Parameters used in training obj.
Mu	Class means, specified as a K-by-p matrix of scalar values class means of size. 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.
NumObservations	Number of observations in the training data, a numeric scalar. NumObservations can be less than the number of rows of input data X when there are missing values in X or response Y.
PredictorNames	Cell array of names for the predictor variables, in the order in which they appear in the training data X.
Prior	Numeric vector of prior probabilities for each class. 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.
ResponseName	Character vector describing the response variable Y.
ScoreTransform	Character vector representing a built-in transformation function, or 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
Sigma	Within-class covariance matrix or matrices. 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
W	Scaled weights, a vector with length n, the number of rows in X.
X	Matrix of predictor values. Each column of X represents one predictor (variable), and each row represents one observation.
Xcentered	X data with class means subtracted. If Y(i) is of class j, Xcentered(i,:) = X(i,:) – Mu(j,:), where Mu is the class mean property.
Y	A categorical array, cell array of character vectors, character array, logical vector, or a numeric vector with the same number of rows as X. Each row of Y represents the classification of the corresponding row of X.

Object Functions

compact	Compact discriminant analysis classifier
compareHoldout	Compare accuracies of two classification models using new data
crossval	Cross-validated discriminant analysis classifier
cvshrink	Cross-validate regularization of linear discriminant
edge	Classification edge
lime	Local interpretable model-agnostic explanations (LIME)
logp	Log unconditional probability density for discriminant analysis classifier
loss	Classification error
mahal	Mahalanobis distance to class means of discriminant analysis classifier
margin	Classification margins
nLinearCoeffs	Number of nonzero linear coefficients
partialDependence	Compute partial dependence
plotPartialDependence	Create partial dependence plot (PDP) and individual conditional expectation (ICE) plots
predict	Predict labels using discriminant analysis classification model
resubEdge	Classification edge by resubstitution
resubLoss	Classification error by resubstitution
resubMargin	Classification margins by resubstitution
resubPredict	Predict resubstitution labels of discriminant analysis classification model
shapley	Shapley values
testckfold	Compare accuracies of two classification models by repeated cross-validation

Copy Semantics

Value. To learn how value classes affect copy operations, see Copying Objects.

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]


  Properties, Methods

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