smile.classification

Class FLD

java.lang.Object

smile.classification.FLD

All Implemented Interfaces:

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

public class FLD
extends java.lang.Object
implements Classifier<double[]>, Projection<double[]>

Fisher's linear discriminant. Fisher defined the separation between two distributions to be the ratio of the variance between the classes to the variance within the classes, which is, in some sense, a measure of the signal-to-noise ratio for the class labeling. FLD finds a linear combination of features which maximizes the separation after the projection. The resulting combination may be used for dimensionality reduction before later classification.

The terms Fisher's linear discriminant and LDA are often used interchangeably, although FLD actually describes a slightly different discriminant, which does not make some of the assumptions of LDA such as normally distributed classes or equal class covariances. When the assumptions of LDA are satisfied, FLD is equivalent to LDA.

FLD is also closely related to principal component analysis (PCA), which also looks for linear combinations of variables which best explain the data. As a supervised method, FLD explicitly attempts to model the difference between the classes of data. On the other hand, PCA is a unsupervised method and does not take into account any difference in class.

One complication in applying FLD (and LDA) to real data occurs when the number of variables/features does not exceed the number of samples. In this case, the covariance estimates do not have full rank, and so cannot be inverted. This is known as small sample size problem.

References

1. H. Li, K. Zhang, and T. Jiang. Robust and Accurate Cancer Classification with Gene Expression Profiling. CSB'05, pp 310-321.

See Also:

LDA, PCA, Serialized Form

Constructor Summary

Constructors

Constructor and Description
FLD(double[] mean, double[][] mu, smile.math.matrix.Matrix scaling) Constructor.
FLD(double[] mean, double[][] mu, smile.math.matrix.Matrix scaling, smile.util.IntSet labels) Constructor.

Method Summary

Modifier and Type	Method and Description
static FLD	fit(double[][] x, int[] y) Learn Fisher's linear discriminant.
static FLD	fit(double[][] x, int[] y, int L, double tol) Learn Fisher's linear discriminant.
static FLD	fit(smile.data.formula.Formula formula, smile.data.DataFrame data) Learn Fisher's linear discriminant.
static FLD	fit(smile.data.formula.Formula formula, smile.data.DataFrame data, java.util.Properties prop) Learn Fisher's linear discriminant.
smile.math.matrix.Matrix	getProjection() Returns the projection matrix W.
int	predict(double[] x) Predicts the class label of an instance.
double[]	project(double[] x) Project a data point to the feature space.
double[][]	project(double[][] x) Project a set of data to the feature space.

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

FLD

public FLD(double[] mean,
           double[][] mu,
           smile.math.matrix.Matrix scaling)

Constructor.

Parameters:

mean - the mean vector of all samples.

mu - the mean vectors of each class.

scaling - the projection matrix.

FLD

public FLD(double[] mean,
           double[][] mu,
           smile.math.matrix.Matrix scaling,
           smile.util.IntSet labels)

Constructor.

Parameters:

mean - the mean vector of all samples.

mu - the mean vectors of each class - mean.

scaling - the projection matrix.

labels - class labels

Method Detail

fit

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

Learn Fisher's linear discriminant.

Parameters:

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

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

fit

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

Learn Fisher's linear discriminant.

Parameters:

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

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

fit

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

Learn Fisher's linear discriminant.

Parameters:

x - training samples.

y - training labels.

fit

public static FLD fit(double[][] x,
                      int[] y,
                      int L,
                      double tol)

Learn Fisher's linear discriminant.

Parameters:

x - training samples.

y - training labels.

L - the dimensionality of mapped space.

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

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.

project

public double[] project(double[] x)

Description copied from interface: Projection

Project a data point to the feature space.

Specified by:

project in interface Projection<double[]>

project

public double[][] project(double[][] x)

Description copied from interface: Projection

Project a set of data to the feature space.

Specified by:

project in interface Projection<double[]>

getProjection

public smile.math.matrix.Matrix getProjection()

Returns the projection matrix W. The dimension reduced data can be obtained by y = W' * x.