smile.classification

Class QDA

java.lang.Object

smile.classification.QDA

All Implemented Interfaces:

Classifier<double[]>

public class QDA
extends Object
implements Classifier<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.

Author:

Haifeng Li

See Also:

LDA, RDA, NaiveBayes

Nested Class Summary

Nested Classes

Modifier and Type	Class and Description
static class	QDA.Trainer Trainer for quadratic discriminant analysis.

Constructor Summary

Constructors

Constructor and Description
QDA(double[][] x, int[] y) Learn quadratic discriminant analysis.
QDA(double[][] x, int[] y, double tol) Learn quadratic discriminant analysis.
QDA(double[][] x, int[] y, double[] priori) Learn quadratic discriminant analysis.
QDA(double[][] x, int[] y, double[] priori, double tol) Learn quadratic discriminant analysis.

Method Summary

Modifier and Type	Method and Description
double[]	getPriori() Returns a priori probabilities.
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.

Methods inherited from class java.lang.Object

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

Constructor Detail

QDA

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

Learn quadratic discriminant analysis.

Parameters:

x - training samples.

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

QDA

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

Learn 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.

QDA

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

Learn quadratic discriminant analysis.

Parameters:

x - training samples.

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

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

QDA

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

Learn 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.

Method Detail

getPriori

public double[] getPriori()

Returns a priori probabilities.

predict

public int predict(double[] x)

Description copied from interface: Classifier

Predicts the class label of an instance.

Specified by:

predict in interface Classifier<double[]>

Parameters:

x - the instance to be classified.

Returns:

the predicted class label

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.

Specified by:

predict in interface Classifier<double[]>

Parameters:

x - the instance to be classified.

posteriori - the array to store a posteriori probabilities on output.

Returns:

the predicted class label