Package smile.feature.extraction

Class KernelPCA

java.lang.Object

smile.feature.extraction.Projection

smile.feature.extraction.KernelPCA

All Implemented Interfaces:

Serializable, Function<smile.data.Tuple,smile.data.Tuple>, smile.data.transform.Transform

public class KernelPCA
extends Projection

Kernel PCA transform. 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.

See Also:

• PCA
• MercerKernel
• ] *
• Serialized Form

Field Summary

Fields

Modifier and Type	Field	Description
final KPCA<double[]>	kpca	Kernel PCA.

Fields inherited from class smile.feature.extraction.Projection

columns, projection, schema

Constructor Summary

Constructors

Constructor	Description
KernelPCA(KPCA<double[]> kpca, String... columns)	Constructor.

Method Summary

Modifier and Type	Method	Description
double[]	apply(double[] x)	Project a data point to the feature space.
static KernelPCA	fit(smile.data.DataFrame data, smile.math.kernel.MercerKernel<double[]> kernel, int k, double threshold, String... columns)	Fits kernel principal component analysis.
static KernelPCA	fit(smile.data.DataFrame data, smile.math.kernel.MercerKernel<double[]> kernel, int k, String... columns)	Fits kernel principal component analysis.

Methods inherited from class smile.feature.extraction.Projection

apply, apply, apply, postprocess, preprocess

Methods inherited from class java.lang.Object

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

Methods inherited from interface java.util.function.Function

andThen, compose

Methods inherited from interface smile.data.transform.Transform

andThen, compose

Field Details

kpca

public final KPCA<double[]> kpca

Kernel PCA.

Constructor Details

KernelPCA

public KernelPCA(KPCA<double[]> kpca,
 String... columns)

Constructor.

Parameters:

kpca - kernel PCA object.

columns - the columns to fit kernel PCA. If empty, all columns will be used.

Method Details

fit

public static KernelPCA fit(smile.data.DataFrame data,
 smile.math.kernel.MercerKernel<double[]> kernel,
 int k,
 String... columns)

Fits kernel principal component analysis.

Parameters:

data - training data.

kernel - Mercer kernel.

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

columns - the columns to fit kernel PCA. If empty, all columns will be used.

Returns:

the model.

fit

public static KernelPCA fit(smile.data.DataFrame data,
 smile.math.kernel.MercerKernel<double[]> kernel,
 int k,
 double threshold,
 String... columns)

Fits kernel principal component analysis.

Parameters:

data - training data.

kernel - Mercer kernel.

k - choose top k principal components used for projection.

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

columns - the columns to fit kernel PCA. If empty, all columns will be used.

Returns:

the model.

apply

public double[] apply(double[] x)

Description copied from class: Projection

Project a data point to the feature space.

Overrides:

apply in class Projection

Parameters:

x - the data point.

Returns:

the projection in the feature space.