A splicing system based genetic algorithm is proposed to optimize dynamical radial basis function (RBF) neural network, which is used to extract valuable process information from input output data. The novel RBF network training technique includes the network structure into the set of function centers by compromising between the conflicting requirements of reducing prediction error and simultaneously decreasing model complexity. The effectiveness of the proposed method is illustrated through the development of dynamic models as a benchmark discrete example and a continuous stirred tank reactor by comparing with several different RBF network training methods.
Radial basis function (RBF) networks attracted
considerable interest in the past because of its several
advantages compared with other types of artificial
neural networks (ANNs), such as better approximation
capabilities, simpler network structures, and faster
learning algorithms[l]. However, the selection of appropriate
number of basis functions is a critical issue
for RBF networks[2]. The number of basis functions
controls the complexity of the structure , is., the generalization
capability of RBF networks. A IU3F network,
containing very few basis functions, yields poor
predictions on new data, i.e., poor generalization, as
the model has limited flexibility. The RBF network,
containing several basis functions, also yields poor
generalization, as it is too flexible and fits the noise in
the training data. The best generalization performance
is obtained via the compromise between the conflicting
requirements of simultaneously reducing the prediction
error and decreasing the complexity of the
model. This trade-off highlights the importance of
optimizing the complexity of RBF network to achieve
the best generalization.
More specifically, most of the standard RBF
training methods require the designer to fix the network
structure. These training procedures usually
proceed via two steps[3]: First, the centers of basis
function are determined using clustering method.
Second, the calculation of the final-layer weights is
reduced to solve a simple linear system using least
squares method. Therefore, the first stage is an unsupervised
method, and separated from the actual objective
to minimize the output prediction error. In this
study, the RBF networks are constructed using the
input data supervised by the output data.
The inclusion of the structure selection in the
formulation of the network optimization problem is
desirable, but it results in a rather difficult problem,
which cannot be easily solved using the standard optimization
methods. An interesting alternative forsolving this complicated problem is offered by the use
of the recently developed evolutionary computation
methods. Perhaps the most popular and successful
strategies are the so-called genetic algorithms (GAS),
which are stochastic methods based on the principles
of natural selection and evolution[4]. GAS have
proved to be successful in the structure selection of
several types of neural networks, such as BP neural
networks[5,6] and recurrent neural networks[7,8]. As
to the optimization of RBF networks, Vesin and
Gruter used GA to solve the complete optimization
problem, but the centers of the potential nodes were
restricted among the set of training data[9]. Esposito
et al. employed a GA based technique to determine the
widths of Gaussian functions in RBF networks[lO],
whereas Sarimveis et al. used GA approach to optimize
the parameters of RBF networks in terms of the
error minimization criterion[ 113.
In this study, the structure selection is included,
and the fitness of each chromosome is calculated on
the basis of the prediction error and the structure complexity
criterion. To simplify the optimization of RBF
network, the radial basis function is chosen as
thin-plate-spline function[ 121, where the detennination
of widths is not required. Therefore, the GA in
this study is used to determine the centers of basis
functions and the network structure. The final-layer
weights are derived using recursive least squares (RLS)
method with the same initial weight vector. The proposed
algorithm starts with a random population of
RBF networks, which are coded as chromosomes. As
all the function centers generated by stochastic chromosomes
are not feasible, two novel operators, i.e.,
elongation and deletion, enlightened by DNA splicing
system[13,14], are introduced in the GA approach.
This article presents a splicing system based GA
for the optimization of both structure and centers of
the RBF network model. This algorithm is based oninput-output data, and its objective function considers
both approximation capability and generalization performance
of the RBF network. In this manner. the
network simultaneously retains a reasonable size and
effectively describes thc whole system. Different
simulations of benchmark problem and typical
nonlinear dynamic CSTR system are performed to
illustrate the effectiveness of thc proposed method.
The results show that this method can produce highly
accurate prediction and keep a relatively simple network
structure.
