smile.base.rbf

Class RBF<T>

java.lang.Object

smile.base.rbf.RBF<T>

All Implemented Interfaces:

java.io.Serializable

public class RBF<T>
extends java.lang.Object
implements java.io.Serializable

A neuron in radial basis function network. A radial basis function network is an artificial neural network that uses radial basis functions as activation functions. It is a linear combination of radial basis functions.

In its basic form, radial basis function network is in the form y(x) = Σ wi φ(||x-ci||) where the approximating function y(x) is represented as a sum of N radial basis functions φ, each associated with a different center ci, and weighted by an appropriate coefficient wi. For distance, one usually chooses Euclidean distance. The weights wi can be estimated using the matrix methods of linear least squares, because the approximating function is linear in the weights.

The centers ci can be randomly selected from training data, or learned by some clustering method (e.g. k-means), or learned together with weight parameters undergo a supervised learning processing (e.g. error-correction learning).

See Also:

Serialized Form

Constructor Summary

Constructors

Constructor and Description
RBF(T center, smile.math.rbf.RadialBasisFunction rbf, smile.math.distance.Metric<T> distance) Constructor.

Method Summary

Modifier and Type	Method and Description
double	f(T x) The activation function.
static RBF<double[]>[]	fit(double[][] x, int k) Learns Gaussian RBF function and centers from data.
static RBF<double[]>[]	fit(double[][] x, int k, double r) Learns Gaussian RBF function and centers from data.
static RBF<double[]>[]	fit(double[][] x, int k, int p) Learns Gaussian RBF function and centers from data.
static <T> RBF<T>[]	fit(T[] x, smile.math.distance.Metric<T> distance, int k) Learns Gaussian RBF function and centers from data.
static <T> RBF<T>[]	fit(T[] x, smile.math.distance.Metric<T> distance, int k, double r) Learns Gaussian RBF function and centers from data.
static <T> RBF<T>[]	fit(T[] x, smile.math.distance.Metric<T> distance, int k, int p) Learns Gaussian RBF function and centers from data.
static <T> RBF<T>[]	of(T[] centers, smile.math.rbf.RadialBasisFunction[] basis, smile.math.distance.Metric<T> distance) Makes a set of RBF neurons.
static <T> RBF<T>[]	of(T[] centers, smile.math.rbf.RadialBasisFunction basis, smile.math.distance.Metric<T> distance) Makes a set of RBF neurons.

Methods inherited from class java.lang.Object

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

Constructor Detail

RBF

public RBF(T center,
           smile.math.rbf.RadialBasisFunction rbf,
           smile.math.distance.Metric<T> distance)

Constructor.

Parameters:

center - the center of neuron.

rbf - the radial basis functions.

distance - the distance metric functor.

Method Detail

f

public double f(T x)

The activation function.

of

public static <T> RBF<T>[] of(T[] centers,
                              smile.math.rbf.RadialBasisFunction basis,
                              smile.math.distance.Metric<T> distance)

Makes a set of RBF neurons.

of

public static <T> RBF<T>[] of(T[] centers,
                              smile.math.rbf.RadialBasisFunction[] basis,
                              smile.math.distance.Metric<T> distance)

Makes a set of RBF neurons.

fit

public static RBF<double[]>[] fit(double[][] x,
                                  int k)

Learns Gaussian RBF function and centers from data. The centers are chosen as the centroids of K-Means. Let d_max be the maximum distance between the chosen centers, the standard deviation (i.e. width) of Gaussian radial basis function is d_max / sqrt(2*k), where k is number of centers. This choice would be close to the optimal solution if the data were uniformly distributed in the input space, leading to a uniform distribution of centroids.

Parameters:

x - the training dataset.

k - the number of RBF neurons to learn.

Returns:

a Gaussian RBF function with parameter learned from data.

fit

public static RBF<double[]>[] fit(double[][] x,
                                  int k,
                                  int p)

Learns Gaussian RBF function and centers from data. The centers are chosen as the centroids of K-Means. The standard deviation (i.e. width) of Gaussian radial basis function is estimated by the p-nearest neighbors (among centers, not all samples) heuristic. A suggested value for p is 2.

Parameters:

x - the training dataset.

k - the number of RBF neurons to learn.

p - the number of nearest neighbors of centers to estimate the width of Gaussian RBF functions.

Returns:

Gaussian RBF functions with parameter learned from data.

fit

public static RBF<double[]>[] fit(double[][] x,
                                  int k,
                                  double r)

Learns Gaussian RBF function and centers from data. The centers are chosen as the centroids of K-Means. The standard deviation (i.e. width) of Gaussian radial basis function is estimated as the width of each cluster multiplied with a given scaling parameter r.

Parameters:

x - the training dataset.

k - the number of RBF neurons to learn.

r - the scaling parameter.

Returns:

Gaussian RBF functions with parameter learned from data.

fit

public static <T> RBF<T>[] fit(T[] x,
                               smile.math.distance.Metric<T> distance,
                               int k)

Learns Gaussian RBF function and centers from data. The centers are chosen as the medoids of CLARANS. Let d_max be the maximum distance between the chosen centers, the standard deviation (i.e. width) of Gaussian radial basis function is d_max / sqrt(2*k), where k is number of centers. In this way, the radial basis functions are not too peaked or too flat. This choice would be close to the optimal solution if the data were uniformly distributed in the input space, leading to a uniform distribution of medoids.

Parameters:

x - the training dataset.

k - the number of RBF neurons to learn.

distance - the distance functor.

Returns:

a Gaussian RBF function with parameter learned from data.

fit

public static <T> RBF<T>[] fit(T[] x,
                               smile.math.distance.Metric<T> distance,
                               int k,
                               int p)

Learns Gaussian RBF function and centers from data. The centers are chosen as the medoids of CLARANS. The standard deviation (i.e. width) of Gaussian radial basis function is estimated by the p-nearest neighbors (among centers, not all samples) heuristic. A suggested value for p is 2.

Parameters:

x - the training dataset.

k - the number of RBF neurons to learn.

distance - the distance functor.

p - the number of nearest neighbors of centers to estimate the width of Gaussian RBF functions.

Returns:

Gaussian RBF functions with parameter learned from data.

fit

public static <T> RBF<T>[] fit(T[] x,
                               smile.math.distance.Metric<T> distance,
                               int k,
                               double r)

Learns Gaussian RBF function and centers from data. The centers are chosen as the medoids of CLARANS. The standard deviation (i.e. width) of Gaussian radial basis function is estimated as the width of each cluster multiplied with a given scaling parameter r.

Parameters:

x - the training dataset.

k - the number of RBF neurons to learn.

distance - the distance functor.

r - the scaling parameter.

Returns:

Gaussian RBF functions with parameter learned from data.