Package smile.classification

Class QDA

java.lang.Object

smile.classification.AbstractClassifier<double[]>

smile.classification.QDA

All Implemented Interfaces:

Serializable, ToDoubleFunction<double[]>, ToIntFunction<double[]>, Classifier<double[]>

Direct Known Subclasses:

RDA

public class QDA
extends AbstractClassifier<double[]>

Quadratic discriminant analysis. QDA is closely related to linear discriminant analysis (LDA). Like LDA, QDA models the conditional probability density functions as a Gaussian distribution, then uses the posterior distributions to estimate the class for a given test data. Unlike LDA, however, in QDA there is no assumption that the covariance of each of the classes is identical. Therefore, the resulting separating surface between the classes is quadratic.

The Gaussian parameters for each class can be estimated from training data with maximum likelihood (ML) estimation. However, when the number of training instances is small compared to the dimension of input space, the ML covariance estimation can be ill-posed. One approach to resolve the ill-posed estimation is to regularize the covariance estimation. One of these regularization methods is regularized discriminant analysis.

See Also:

• LDA
• RDA
• NaiveBayes
• Serialized Form

Nested Class Summary

Nested classes/interfaces inherited from interface smile.classification.Classifier

Classifier.Trainer<T,M extends Classifier<T>>

Field Summary

Fields inherited from class smile.classification.AbstractClassifier

classes

Constructor Summary

Constructors

Constructor	Description
QDA(double[] priori, double[][] mu, double[][] eigen, Matrix[] scaling)	Constructor.
QDA(double[] priori, double[][] mu, double[][] eigen, Matrix[] scaling, IntSet labels)	Constructor.

Method Summary

Modifier and Type	Method	Description
static QDA	fit(double[][] x, int[] y)	Fits quadratic discriminant analysis.
static QDA	fit(double[][] x, int[] y, double[] priori, double tol)	Fits quadratic discriminant analysis.
static QDA	fit(double[][] x, int[] y, Properties params)	Fits quadratic discriminant analysis.
int	predict(double[] x)	Predicts the class label of an instance.
int	predict(double[] x, double[] posteriori)	Predicts the class label of an instance and also calculate a posteriori probabilities.
double[]	priori()	Returns a priori probabilities.
boolean	soft()	Returns true if this is a soft classifier that can estimate the posteriori probabilities of classification.

Methods inherited from class smile.classification.AbstractClassifier

classes, numClasses

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Methods inherited from interface smile.classification.Classifier

applyAsDouble, applyAsInt, online, predict, predict, predict, predict, predict, predict, score, update, update, update

Constructor Details

QDA

public QDA(double[] priori,
 double[][] mu,
 double[][] eigen,
 Matrix[] scaling)

Constructor.

Parameters:

priori - a priori probabilities of each class.

mu - the mean vectors of each class.

eigen - the eigen values of each variance matrix.

scaling - the eigen vectors of each covariance matrix.

QDA

public QDA(double[] priori,
 double[][] mu,
 double[][] eigen,
 Matrix[] scaling,
 IntSet labels)

Constructor.

Parameters:

priori - a priori probabilities of each class.

mu - the mean vectors of each class.

eigen - the eigen values of each variance matrix.

scaling - the eigen vectors of each covariance matrix.

labels - the class label encoder.

Method Details

fit

public static QDA fit(double[][] x,
 int[] y)

Fits quadratic discriminant analysis.

Parameters:

x - training samples.

y - training labels in [0, k), where k is the number of classes.

Returns:

the model.

fit

public static QDA fit(double[][] x,
 int[] y,
 Properties params)

Fits quadratic discriminant analysis.

Parameters:

x - training samples.

y - training labels.

params - the hyper-parameters.

Returns:

the model.

fit

public static QDA fit(double[][] x,
 int[] y,
 double[] priori,
 double tol)

Fits quadratic discriminant analysis.

Parameters:

x - training samples.

y - training labels in [0, k), where k is the number of classes.

priori - the priori probability of each class. If null, it will be estimated from the training data.

tol - a tolerance to decide if a covariance matrix is singular; it will reject variables whose variance is less than tol².

Returns:

the model.

priori

public double[] priori()

Returns a priori probabilities.

Returns:

a priori probabilities.

predict

public int predict(double[] x)

Description copied from interface: Classifier

Predicts the class label of an instance.

Parameters:

x - the instance to be classified.

Returns:

the predicted class label.

soft

public boolean soft()

Description copied from interface: Classifier

Returns true if this is a soft classifier that can estimate the posteriori probabilities of classification.

Returns:

true if soft classifier.

predict

public int predict(double[] x,
 double[] posteriori)

Description copied from interface: Classifier

Predicts the class label of an instance and also calculate a posteriori probabilities. Classifiers may NOT support this method since not all classification algorithms are able to calculate such a posteriori probabilities.

Parameters:

x - an instance to be classified.

posteriori - a posteriori probabilities on output.

Returns:

the predicted class label