RBF Neural Networks
A Radial Basis Function (RBF) neural network has an input layer, a hidden
layer and an output layer. The neurons in the hidden layer contain Gaussian
transfer functions whose outputs are inversely proportional to the distance
from the center of the neuron.
RBF networks are similar to KMeans clustering
and PNN/GRNN networks. The main difference is
that PNN/GRNN networks have one neuron for each point in the training file,
whereas RBF networks have a variable number of neurons that is usually much
less than the number of training points. For problems with small to medium
size training sets, PNN/GRNN networks are usually more accurate than RBF
networks, but PNN/GRNN networks are impractical for large training sets.
How RBF networks work
Although the implementation is very different, RBF neural networks are
conceptually similar to KNearest Neighbor (kNN) models. The basic idea
is that a predicted target value of an item is likely to be about the same
as other items that have close values of the predictor variables. Consider
this figure:
Assume that each case in the training set has two predictor variables, x and
y. The cases are plotted using their x,y coordinates as shown in the
figure. Also assume that the target variable has two categories, positive
which is denoted by a square and negative which is denoted by a dash. Now,
suppose we are trying to predict the value of a new case represented by the
triangle with predictor values x=6, y=5.1.
Should we predict the target as positive or negative?
Notice that the triangle is position almost exactly on top of a dash
representing a negative value. But that dash is in a fairly unusual
position compared to the other dashes which are clustered below the squares
and left of center. So it could be that the underlying negative value is an
odd case.
The nearest neighbor classification performed for this example depends on
how many neighboring points are considered. If 1NN is used and only the
closest point is considered, then clearly the new point should be classified
as negative since it is on top of a known negative point. On the other
hand, if 9NN classification is used and the closest 9 points are
considered, then the effect of the surrounding 8 positive points may
overbalance the close negative point.
An RBF network positions one or more RBF neurons in the space described by
the predictor variables (x,y in this example). This space has as many
dimensions as there are predictor variables. The Euclidean distance is
computed from the point being evaluated (e.g., the triangle in this figure)
to the center of each neuron, and a radial basis function (RBF) (also called
a kernel function) is applied to the distance to compute the weight
(influence) for each neuron. The radial basis function is so named because
the radius distance is the argument to the function.
Weight = RBF(distance)
The further a neuron is from the point being evaluated, the less influence
it has.
Radial Basis Function
Different types of radial basis functions could be used, but the most common
is the Gaussian function:
If there is more than one predictor variable, then the RBF function has as
many dimensions as there are variables. The following picture illustrates
three neurons in a space with two predictor variables, X and Y.
Z is the value coming out of the RBF functions:
The best predicted value for the new point is found by summing the output
values of the RBF functions multiplied by weights computed for each neuron.
The radial basis function for a neuron has a center and a radius (also
called a spread). The radius may be different for each neuron, and, in RBF
networks generated by DTREG, the radius may be different in each dimension.
With larger spread, neurons at a distance from a point have a greater
influence.
RBF Network Architecture
RBF networks have three layers:
 Input layer – There is one neuron in
the input layer for each predictor variable. In the case of categorical
variables, N1 neurons are used where N is the number of categories. The
input neurons (or processing before the input layer) standardizes the range
of the values by subtracting the median and dividing by the interquartile
range. The input neurons then feed the values to each of the neurons in the
hidden layer.
 Hidden layer – This layer has a variable number of neurons (the optimal
number is determined by the training process). Each neuron consists of a
radial basis function centered on a point with as many dimensions as there
are predictor variables. The spread (radius) of the RBF function may be
different for each dimension. The centers and spreads are determined by the
training process. When presented with the x vector of input values from the
input layer, a hidden neuron computes the Euclidean distance of the test
case from the neuron’s center point and then applies the RBF kernel function
to this distance using the spread values. The resulting value is passed to
the the summation layer.
 Summation layer – The value coming out of a neuron in the hidden layer
is multiplied by a weight associated with the neuron (W1, W2, ...,Wn in this
figure) and passed to the summation which adds up the weighted values and
presents this sum as the output of the network. Not shown in this figure is
a bias value of 1.0 that is multiplied by a weight W0 and fed into the
summation layer. For classification problems, there is one output (and a
separate set of weights and summation unit) for each target category. The
value output for a category is the probability that the case being evaluated
has that category.
Training RBF Networks
The following parameters are determined by the training process:
 The number of neurons in the hidden layer.
 The coordinates of the center of each hiddenlayer RBF function.
 The radius (spread) of each RBF function in each dimension.
 The weights applied to the RBF function outputs as they are passed to
the summation layer.
Various methods have been used to train RBF networks. One approach first
uses Kmeans clustering to find cluster centers which are then used as the
centers for the RBF functions. However, Kmeans clustering is a
computationally intensive procedure, and it often does not generate the
optimal number of centers. Another approach is to use a random subset of
the training points as the centers.
DTREG uses a training algorithm developed by Sheng Chen, Xia Hong and Chris
J. Harris. This algorithm uses an evolutionary
approach to determine the optimal center points and spreads for each neuron.
It also determines when to stop adding neurons to the network by monitoring
the estimated leaveoneout (LOO) error and terminating when the LOO error
beings to increase due to overfitting.
The computation of the optimal weights between the neurons in the hidden
layer and the summation layer is done using ridge regression. An iterative
procedure developed by Mark Orr (Orr, 1966) is used to compute the optimal
regularization Lambda parameter that minimizes generalized crossvalidation
(GCV) error.
The RBF Network Property Page
Controls for RBF network analyses are provided on a screen in DTREG
that has the following image:
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