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Wikipedia / Radial basis function network

In the field of mathematical modeling, a radial basis function network is an artificial neural network that uses radial basis functions as activation functions. The output of the network is a linear combination of radial basis functions of the inputs and neuron parameters. Radial basis function networks have many uses, including function approximation, time series prediction, classification, and system control. They were first formulated in a 1988 paper by Broomhead and Lowe, both researchers at the Royal Signals and Radar Establishment.

Network architecture

, and is given by

in the linear output neuron. Functions that depend only on the distance from a center vector are radially symmetric about that vector, hence the name radial basis function. In the basic form, all inputs are connected to each hidden neuron. The norm is typically taken to be the Euclidean distance (although the Mahalanobis distance appears to perform better with pattern recognition) and the radial basis function is commonly taken to be Gaussian

  1. .

The Gaussian basis functions are local to the center vector in the sense that

i.e. changing parameters of one neuron has only a small effect for input values that are far away from the center of that neuron.

Given certain mild conditions on the shape of the activation function, RBF networks are universal approximators on a compact. This means that an RBF network with enough hidden neurons can approximate any continuous function on a closed, bounded set with arbitrary precision.

and the data.

Normalization

Normalized architecture

In addition to the above unnormalized architecture, RBF networks can be normalized. In this case the mapping is

where

is known as a normalized radial basis function.

Theoretical motivation for normalization

There is theoretical justification for this architecture in the case of stochastic data flow. Assume a stochastic kernel approximation for the joint probability density

are exemplars from the data and we require the kernels to be normalized

and

  1. .

The probability densities in the input and output spaces are

and

is

where

. The conditional probability is related to the joint probability through Bayes' theorem

which yields

  1. .

This becomes

when the integrations are performed.

Local linear models

It is sometimes convenient to expand the architecture to include local linear models. In that case the architectures become, to first order,

and

are weights to be determined. Higher order linear terms are also possible.

This result can be written

where

and

is a Kronecker delta function defined as

  1. .

Training

by a two-step algorithm.

of the RBF functions in the hidden layer are chosen. This step can be performed in several ways; centers can be randomly sampled from some set of examples, or they can be determined using k-means clustering. Note that this step is unsupervised.

to the hidden layer's outputs with respect to some objective function. A common objective function, at least for regression/function estimation, is the least squares function:

where

  1. .

We have explicitly included the dependence on the weights. Minimization of the least squares objective function by optimal choice of weights optimizes accuracy of fit.

There are occasions in which multiple objectives, such as smoothness as well as accuracy, must be optimized. In that case it is useful to optimize a regularized objective function such as

where

and

is known as a regularization parameter.

A third optional backpropagation step can be performed to fine-tune all of the RBF net's parameters.

Interpolation

the weights can be solved from the equation

can be solved by simple linear algebra:

.

Function approximation

If the purpose is not to perform strict interpolation but instead more general function approximation or classification the optimization is somewhat more complex because there is no obvious choice for the centers. The training is typically done in two phases first fixing the width and centers and then the weights. This can be justified by considering the different nature of the non-linear hidden neurons versus the linear output neuron.

Training the basis function centers

Basis function centers can be randomly sampled among the input instances or obtained by Orthogonal Least Square Learning Algorithm or found by clustering the samples and choosing the cluster means as the centers.

The RBF widths are usually all fixed to same value which is proportional to the maximum distance between the chosen centers.

Pseudoinverse solution for the linear weights

have been fixed, the weights that minimize the error at the output can be computed with a linear pseudoinverse solution:

  1. ,

where the entries of G.

The existence of this linear solution means that unlike multi-layer perceptron (MLP) networks, RBF networks have an explicit minimizer (when the centers are fixed).

Gradient descent training of the linear weights

Another possible training algorithm is gradient descent. In gradient descent training, the weights are adjusted at each time step by moving them in a direction opposite from the gradient of the objective function (thus allowing the minimum of the objective function to be found),

is a "learning parameter."

, the algorithm becomes

in the unnormalized case and

in the normalized case.

For local-linear-architectures gradient-descent training is

Projection operator training of the linear weights

, the algorithm becomes

in the unnormalized case and

in the normalized case and

in the local-linear case.

For one basis function, projection operator training reduces to Newton's method.

Examples

Logistic map

The basic properties of radial basis functions can be illustrated with a simple mathematical map, the logistic map, which maps the unit interval onto itself. It can be used to generate a convenient prototype data stream. The logistic map can be used to explore function approximation, time series prediction, and control theory. The map originated from the field of population dynamics and became the prototype for chaotic time series. The map, in the fully chaotic regime, is given by

where t is a time index. The value of x at time t+1 is a parabolic function of x at time t. This equation represents the underlying geometry of the chaotic time series generated by the logistic map.

Generation of the time series from this equation is the forward problem. The examples here illustrate the inverse problem; identification of the underlying dynamics, or fundamental equation, of the logistic map from exemplars of the time series. The goal is to find an estimate

for f.

Function approximation

Unnormalized radial basis functions

The architecture is

where

  1. .

Since the input is a scalar rather than a vector are trained with projection operator training:

where the learning rate is taken to be 0.3. The training is performed with one pass through the 100 training points. The rms error is 0.15.

Normalized radial basis functions

The normalized RBF architecture is

where

  1. .

Again:

  1. .

are trained with projection operator training:

where the learning rate is again taken to be 0.3. The training is performed with one pass through the 100 training points. The rms error on a test set of 100 exemplars is 0.084, smaller than the unnormalized error. Normalization yields accuracy improvement. Typically accuracy with normalized basis functions increases even more over unnormalized functions as input dimensionality increases.

Time series prediction

Once the underlying geometry of the time series is estimated as in the previous examples, a prediction for the time series can be made by iteration:

  1. .

A comparison of the actual and estimated time series is displayed in the figure. The estimated times series starts out at time zero with an exact knowledge of x(0). It then uses the estimate of the dynamics to update the time series estimate for several time steps.

Note that the estimate is accurate for only a few time steps. This is a general characteristic of chaotic time series. This is a property of the sensitive dependence on initial conditions common to chaotic time series. A small initial error is amplified with time. A measure of the divergence of time series with nearly identical initial conditions is known as the Lyapunov exponent.

Control of a chaotic time series

such that

  1. .

. This can be done if we choose the control parameter to be

where

is an approximation to the underlying natural dynamics of the system.

The learning algorithm is given by

where

  1. .

See also

References

Further reading