Biomedical Engineering Reference
In-Depth Information
Fig. 11 A typical artificial neural network structure, indicating layers of neurons; only a
selection of network weights is illustrated to reduce the complexity of the figure
A typical non-linear transformation of MLPs is a sigmoidal function, although
other non-linear functions can also be applied. The algorithm for adjusting the
weighted connections is typically based on the 'back-propagation' of the error
between the actual output and the network estimation.
In RBFs the data in the input space are non-linearly transformed from the input
layer space to the hidden layer space, typically using a form of a Gaussian
function, and then the transformation from the hidden layer space to the output
layer space is linear. This type of mapping allows a non-linearly separable clas-
sification problem to be transformed into a linearly separable one, which is easier
to solve. RBF networks are thus capable of faster learning and are less sensitive to
the order in which the training data are presented to the model. However, a very
large number of RBF may be required to span the input space adequately.
An added benefit of using an RBF network is the indication of the reliability of
the network estimations, based on the data density estimation. During model
development, the number of radial basis units is specified and the centre of each
function is found by k-means clustering, where the data matrix is arranged into a
number of clusters, the average positions of the data points in each cluster are
taken to be the cluster centres, and the data included within each cluster are
optimised so that the distance between the data points and the centre of the cluster
is minimised. The final cluster centres represent the centres of the RBF of the
trained network. Once these centres have been found, the width of each function is
defined based on a p-nearest-neighbour heuristic [
57
]. The number of nearest
neighbours (NN) is user-specified, and the distance between each of the function
centres is calculated. The width of each function is adjusted so that it overlaps with
the centres of NN functions around it. This structure allows an indication of the
reliability of the model prediction, as it indicates when new data points are pre-
sented to the network outside the data space used for the model development.
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