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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

Mdl = fitcdiscr(Tbl,ResponseVarName) returns a fitted discriminant analysis model based on the input variables (also known as predictors, features, or attributes) contained in the table Tbl and output (response or labels) contained in ResponseVarName.

Mdl = fitcdiscr(Tbl,formula) returns a fitted discriminant analysis model based on the predictor data and class labels in the table Tbl. formula is an explanatory model of the response and a subset of predictor variables in Tbl used to fit Mdl.

Mdl = fitcdiscr(Tbl,Y) returns a fitted discriminant analysis model based on the input variables contained in the table Tbl and response Y.

Mdl = fitcdiscr(X,Y) returns a discriminant analysis classifier based on the input variables X and response Y.

Mdl = fitcdiscr(___,Name,Value) fits a classifier with additional options specified by one or more name-value pair arguments, using any of the previous syntaxes. For example, you can specify the cost of misclassification, prior probabilities for each class, or observation weights.

Input Arguments

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Sample data used to train the model, specified as a table. Each row of Tbl corresponds to one observation, and each column corresponds to one predictor variable. Optionally, Tbl can contain one additional column for the response variable. Multi-column variables and cell arrays other than cell arrays of character vectors are not allowed.

If Tbl contains the response variable, and you want to use all remaining variables in Tbl as predictors, then specify the response variable using ResponseVarName.

If Tbl contains the response variable, and you want to use only a subset of the remaining variables in Tbl as predictors, then specify a formula using formula.

If Tbl does not contain the response variable, then specify a response variable using Y. The length of response variable and the number of rows of Tbl must be equal.

Data Types: table

Response variable name, specified as the name of a variable in Tbl.

You must specify ResponseVarName as a character vector. For example, if the response variable Y is stored as Tbl.Y, then specify it as 'Y'. Otherwise, the software treats all columns of Tbl, including Y, as predictors when training the model.

The response variable must be a categorical or character array, logical or numeric vector, or cell array of character vectors. If Y is a character array, then each element must correspond to one row of the array.

It is good practice to specify the order of the classes using the ClassNames name-value pair argument.

Data Types: char

Explanatory model of the response and a subset of the predictor variables, specified as a character vector in the form of 'Y~X1+X2+X3'. In this form, Y represents the response variable, and X1, X2, and X3 represent the predictor variables. The variables must be variable names in Tbl (Tbl.Properties.VariableNames).

To specify a subset of variables in Tbl as predictors for training the model, use a formula. If you specify a formula, then the software does not use any variables in Tbl that do not appear in formula.

Data Types: char

Class labels, specified as a categorical or character array, logical or numeric vector, or cell array of character vectors. Each row of Y represents the classification of the corresponding row of X.

The software considers NaN, '' (empty character vector), and <undefined> values in Y to be missing values. Consequently, the software does not train using observations with a missing response.

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

Predictor values, specified as a numeric matrix. Each column of X represents one variable, and each row represents one observation.

fitcdiscr considers NaN values in X as missing values. fitcdiscr does not use observations with missing values for X in the fit.

Data Types: single | double

Properties

BetweenSigma

p-by-p matrix, the between-class covariance, where p is the number of predictors.

CategoricalPredictors

List of categorical predictors, which is always empty ([]) for SVM and discriminant analysis classifiers.

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.

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

  • Class1ClassNames(i)

  • Class2ClassNames(j)

  • Const — A scalar

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

  • Quadraticp-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

Function handle for transforming scores, or character vector representing 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 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.

Methods

compactCompact discriminant analysis classifier
crossvalCross-validated discriminant analysis classifier
cvshrinkCross-validate regularization of linear discriminant
resubEdgeClassification edge by resubstitution
resubLossClassification error by resubstitution
resubMarginClassification margins by resubstitution
resubPredictPredict resubstitution response of classifier

Inherited Methods

compareHoldoutCompare accuracies of two classification models using new data
edgeClassification edge
logPLog unconditional probability density for discriminant analysis classifier
lossClassification error
mahalMahalanobis distance to class means
marginClassification margins
nLinearCoeffsNumber of nonzero linear coefficients
predictPredict 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:

y^=argminy=1,...,Kk=1KP^(k|x)C(y|k),

where

  • y^ is the predicted classification.

  • K is the number of classes.

  • P^(k|x) is the posterior probability of class k for observation x.

  • C(y|k) 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 X^ be the centered data (the data X minus the mean by class). Define

D=diag(X^T*X^).

The regularized covariance matrix Σ˜ is

Σ˜=(1γ)Σ+γD.

Whenever γ ≥ MinGamma, Σ˜ 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 C˜ be the regularized correlation matrix:

C˜=(1γ)C+γI,

where I is the identity matrix.

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

(xμ0)TΣ˜1(μkμ0)=[(xμ0)TD1/2][C˜1D1/2(μkμ0)].

The parameter δ enters into this equation as a threshold on the final term in square brackets. Each component of the vector [C˜1D1/2(μkμ0)] 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

|C˜1D1/2(μkμ0)|δ.

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

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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: [3×4 double]
                   Coeffs: [3×3 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 =

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

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.

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