DEPARTMENT OF AVIATION COMPUTER INTEGRATED COMPLEXES
ARTIFICIAL NEURAL NETWORKS
REPORT ON LAB:
RADIAL BASIS FUNCTION
done by _________________________
Group 307
Teacher: Sineglazov V.M.
Kyiv 2011
1. 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. They are used in function approximation, time series prediction, and control. Radial basis function neural networks are essentially a combination of the ideas found in multi-level perceptrons and those described earlier in this Guide under the main heading ?deterministic surface interpolation?, subsection ?radial basis functions?. In addition to interpolation they can be used for function approximation, as explained further below.
2. Network architecture:
o Normalized
o Local linear models
3. Network architecture: Radial basis function (RBF) networks typically have three layers: an input layer, a hidden layer with a non-linear RBF activation function and a linear output layer. The output, , of the network is thus
where N is the number of neurons in the hidden layer, is the center vector for neuron i, and ai are the weights of the linear output neuron.
4.
Normalized architecture
In addition to the above unnormalized architecture, RBF networks can benormalized. Inthiscasethemappingis
where
is known as a "normalized radial basis function".
Local linear models
It is sometimes convenient to expand the architecture to include local linearmodels. Inthatcasethearchitecturesbecome, tofirstorder,
and
in the unnormalized and normalized cases, respectively. Here are weights to be determined. Higher order linear terms are also possible.
This result can be written
where
and
intheunnormalizedcaseand
in the normalized case.
Here δij is a Kronecker delta function defined as
.
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 ofpopulation dynamics and became the prototype for chaotic time series. Themap, inthefullychaoticregime, isgivenby
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. Thegoalistofindanestimate