Pattern Recognition Techniques - Biomedical Signal Analysis: Contemporary Methods and Applications - page 187

Here, N is the number of neurons in the second hidden layer, x is the
n -dimensional input pattern vector, x

is the transformed input pattern

vector, m
i

is the center of a node, w i are the output weights, and y is
the m -dimensional output of the network. The n

× n covariance matrix

C i is of the form

C jk = σ jk

if m = n
0 t r ise

(6.32)

where σ jk is the standard deviation. Because the centers of the Gaussian
potential function units (GPFU) are defined in the feature space, they
will be subject to transformation by B as well. Therefore, the exponent
of a GPFU can be rewritten as

d ( x , m
i

m i ) T B T C − 1

)=( x − B ( x − m i ) (6.33)

i

and is in this form similar to equation (6.31).
For the moment, we will regard B as the identity matrix. The
network models the distribution of input vectors in the feature space
by the weighted summation of Gaussian normal distributions, which are
provided by the GPFU Ψ j . To measure the difference between these
distributions, we define the relevance ρ n for each feature x n :

PJ
p

m jn ) 2
2 σ jn

1 ( x pn −

ρ n = (6.34)

j

where P is the size of the training set and J is the number of the GPFUs.
If ρ n falls below the threshold ρ th , one will decide to discard feature x n .
This criterion will not identify every irrelevant feature. If two features are
correlated, one of them will be irrelevant, but this cannot be indicated
by the criterion.

Learning paradigm for the transformation radial-basis neural
network

We follow [151] for the implementation of the neuron allocation and
learning rules for the TRBNN. The network generation process starts
without any neuron.
The mutual dependency of correlated features can often be approx-
imated by a linear function, which means that a linear transformation