smile.classification

Class QDA

java.lang.Object

smile.classification.QDA

All Implemented Interfaces:

java.io.Serializable, java.util.function.ToDoubleFunction<double[]>, java.util.function.ToIntFunction<double[]>, Classifier<double[]>, SoftClassifier<double[]>

Direct Known Subclasses:

RDA

public class QDA
extends java.lang.Object
implements SoftClassifier<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

Constructor Summary

Constructors

Constructor and Description
QDA(double[] priori, double[][] mu, double[][] eigen, smile.math.matrix.DenseMatrix[] scaling) Constructor.
QDA(double[] priori, double[][] mu, double[][] eigen, smile.math.matrix.DenseMatrix[] scaling, smile.util.IntSet labels) Constructor.

Method Summary

Modifier and Type	Method and Description
static QDA	fit(double[][] x, int[] y) Learn quadratic discriminant analysis.
static QDA	fit(double[][] x, int[] y, double[] priori, double tol) Learn quadratic discriminant analysis.
static QDA	fit(double[][] x, int[] y, java.util.Properties prop) Learns quadratic discriminant analysis.
static QDA	fit(smile.data.formula.Formula formula, smile.data.DataFrame data) Learns quadratic discriminant analysis.
static QDA	fit(smile.data.formula.Formula formula, smile.data.DataFrame data, java.util.Properties prop) Learns 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.

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, f, predict

Constructor Detail

QDA

public QDA(double[] priori,
           double[][] mu,
           double[][] eigen,
           smile.math.matrix.DenseMatrix[] 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,
           smile.math.matrix.DenseMatrix[] scaling,
           smile.util.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 - class labels

Method Detail

fit

public static QDA fit(smile.data.formula.Formula formula,
                      smile.data.DataFrame data)

Learns quadratic discriminant analysis.

Parameters:

formula - a symbolic description of the model to be fitted.

data - the data frame of the explanatory and response variables.

fit

public static QDA fit(smile.data.formula.Formula formula,
                      smile.data.DataFrame data,
                      java.util.Properties prop)

Learns quadratic discriminant analysis.

Parameters:

formula - a symbolic description of the model to be fitted.

data - the data frame of the explanatory and response variables.

fit

public static QDA fit(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.

fit

public static QDA fit(double[][] x,
                      int[] y,
                      java.util.Properties prop)

Learns quadratic discriminant analysis.

Parameters:

x - training samples.

y - training labels.

fit

public static QDA fit(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.

priori

public double[] priori()

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: SoftClassifier

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 SoftClassifier<double[]>

Parameters:

x - an instance to be classified.

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

Returns:

the predicted class label