smile.projection

Class KPCA<T>

java.lang.Object

smile.projection.KPCA<T>

All Implemented Interfaces:

Projection<T>

public class KPCA<T>
extends Object
implements Projection<T>

Kernel principal component analysis. Kernel PCA is an extension of principal component analysis (PCA) using techniques of kernel methods. Using a kernel, the originally linear operations of PCA are done in a reproducing kernel Hilbert space with a non-linear mapping.

In practice, a large data set leads to a large Kernel/Gram matrix K, and storing K may become a problem. One way to deal with this is to perform clustering on your large dataset, and populate the kernel with the means of those clusters. Since even this method may yield a relatively large K, it is common to compute only the top P eigenvalues and eigenvectors of K.

Kernel PCA with an isotropic kernel function is closely related to metric MDS. Carrying out metric MDS on the kernel matrix K produces an equivalent configuration of points as the distance (2(1 - K(x_i, x_j)))^1/2 computed in feature space.

Kernel PCA also has close connections with Isomap, LLE, and Laplacian eigenmaps.

References

1. Bernhard Scholkopf, Alexander Smola, and Klaus-Robert Muller. Nonlinear Component Analysis as a Kernel Eigenvalue Problem. Neural Computation, 1998.

Author:

Haifeng Li

See Also:

MercerKernel, PCA, IsoMap, LLE, LaplacianEigenmap, SammonMapping

Constructor Summary

Constructors

Constructor and Description
KPCA(T[] data, MercerKernel<T> kernel, double threshold) Constructor.
KPCA(T[] data, MercerKernel<T> kernel, int k) Constructor.
KPCA(T[] data, MercerKernel<T> kernel, int k, double threshold) Constructor.

Method Summary

Modifier and Type	Method and Description
double[][]	getCoordinates() Returns the nonlinear principal component scores, i.e., the representation of learning data in the nonlinear principal component space.
double[][]	getProjection() Returns the projection matrix.
double[]	getVariances() Returns the eigenvalues of kernel principal components, ordered from largest to smallest.
double[]	project(T x) Project a data point to the feature space.
double[][]	project(T[] x) Project a set of data toe the feature space.

Methods inherited from class java.lang.Object

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

Constructor Detail

KPCA

public KPCA(T[] data,
            MercerKernel<T> kernel,
            double threshold)

Constructor. Learn kernel principal component analysis.

Parameters:

data - learning data.

kernel - Mercer kernel to compute kernel matrix.

threshold - only principal components with eigenvalues larger than the given threshold will be kept.

KPCA

public KPCA(T[] data,
            MercerKernel<T> kernel,
            int k)

Constructor. Learn kernel principal component analysis.

Parameters:

data - learning data.

kernel - Mercer kernel to compute kernel matrix.

k - choose upto k principal components (larger than 0.0001) used for projection.

KPCA

public KPCA(T[] data,
            MercerKernel<T> kernel,
            int k,
            double threshold)

Constructor. Constructor. Learn kernel principal component analysis.

Parameters:

data - learning data.

kernel - Mercer kernel to compute kernel matrix.

k - choose top k principal components used for projection.

threshold - only principal components with eigenvalues larger than the given threshold will be kept.

Method Detail

getVariances

public double[] getVariances()

Returns the eigenvalues of kernel principal components, ordered from largest to smallest.

getProjection

public double[][] getProjection()

Returns the projection matrix. The dimension reduced data can be obtained by y = W * K(x, ·).

getCoordinates

public double[][] getCoordinates()

Returns the nonlinear principal component scores, i.e., the representation of learning data in the nonlinear principal component space. Rows correspond to observations, columns to components.

project

public double[] project(T x)

Description copied from interface: Projection

Project a data point to the feature space.

Specified by:

project in interface Projection<T>

project

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

Description copied from interface: Projection

Project a set of data toe the feature space.

Specified by:

project in interface Projection<T>