Class RBFNetwork<T>
 Type Parameters:
T
 the data type of model input objects.
 All Implemented Interfaces:
Serializable
,ToDoubleFunction<T>
,ToIntFunction<T>
,Classifier<T>
In its basic form, radial basis function network is in the form
y(x) = Σ w_{i} φ(xc_{i})
where the approximating function y(x) is represented as a sum of N radial basis functions φ, each associated with a different center c_{i}, and weighted by an appropriate coefficient w_{i}. For distance, one usually chooses Euclidean distance. The weights w_{i} can be estimated using the matrix methods of linear least squares, because the approximating function is linear in the weights.
The centers c_{i} can be randomly selected from training data, or learned by some clustering method (e.g. kmeans), or learned together with weight parameters undergo a supervised learning processing (e.g. errorcorrection learning).
The popular choices for φ comprise the Gaussian function and the
socalled thin plate splines. The advantage of the thin plate splines is that
their conditioning is invariant under scalings. Gaussian, multiquadric
and inverse multiquadric are infinitely smooth and involve a scale
or shape parameter, r_{0} > 0
. Decreasing
r_{0} tends to flatten the basis function. For a
given function, the quality of approximation may strongly depend on this
parameter. In particular, increasing r_{0} has the
effect of better conditioning (the separation distance of the scaled points
increases).
A variant on RBF networks is normalized radial basis function (NRBF) networks, in which we require the sum of the basis functions to be unity. NRBF arises more naturally from a Bayesian statistical perspective. However, there is no evidence that either the NRBF method is consistently superior to the RBF method, or vice versa.
SVMs with Gaussian kernel have similar structure as RBF networks with Gaussian radial basis functions. However, the SVM approach "automatically" solves the network complexity problem since the size of the hidden layer is obtained as the result of the QP procedure. Hidden neurons and support vectors correspond to each other, so the center problems of the RBF network is also solved, as the support vectors serve as the basis function centers. It was reported that with similar number of support vectors/centers, SVM shows better generalization performance than RBF network when the training data size is relatively small. On the other hand, RBF network gives better generalization performance than SVM on large training data.
References
 Simon Haykin. Neural Networks: A Comprehensive Foundation (2nd edition). 1999.
 T. Poggio and F. Girosi. Networks for approximation and learning. Proc. IEEE 78(9):14841487, 1990.
 Nabil Benoudjit and Michel Verleysen. On the kernel widths in radialbasis function networks. Neural Process, 2003.
 See Also:

Nested Class Summary
Nested classes/interfaces inherited from interface smile.classification.Classifier
Classifier.Trainer<T,
M extends Classifier<T>> 
Field Summary
Fields inherited from class smile.classification.AbstractClassifier
classes

Constructor Summary
ConstructorDescriptionRBFNetwork
(int k, RBF<T>[] rbf, Matrix w, boolean normalized) Constructor.RBFNetwork
(int k, RBF<T>[] rbf, Matrix w, boolean normalized, IntSet labels) Constructor. 
Method Summary
Modifier and TypeMethodDescriptionstatic RBFNetwork
<double[]> fit
(double[][] x, int[] y, Properties params) Fits an RBF network.static <T> RBFNetwork
<T> Fits an RBF network.static <T> RBFNetwork
<T> Fits an RBF network.boolean
Returns true if the model is normalized.int
Predicts the class label of an instance.Methods inherited from class smile.classification.AbstractClassifier
classes, numClasses

Constructor Details

RBFNetwork
Constructor. Parameters:
k
 the number of classes.rbf
 the radial basis functions.w
 the weights of RBFs.normalized
 True if this is a normalized RBF network.

RBFNetwork
Constructor. Parameters:
k
 the number of classes.rbf
 the radial basis functions.w
 the weights of RBFs.normalized
 True if this is a normalized RBF network.labels
 the class label encoder.


Method Details

fit
Fits an RBF network. Type Parameters:
T
 the data type. Parameters:
x
 training samples.y
 training labels in [0, k), where k is the number of classes.rbf
 the radial basis functions. Returns:
 the model.

fit
Fits an RBF network. Type Parameters:
T
 the data type. Parameters:
x
 training samples.y
 training labels in [0, k), where k is the number of classes.rbf
 the radial basis functions.normalized
 true for the normalized RBF network. Returns:
 the model.

fit
Fits an RBF network. Parameters:
x
 training samples.y
 training labels.params
 the hyperparameters. Returns:
 the model.

isNormalized
public boolean isNormalized()Returns true if the model is normalized. Returns:
 true if the model is normalized.

predict
Description copied from interface:Classifier
Predicts the class label of an instance. Parameters:
x
 the instance to be classified. Returns:
 the predicted class label.
