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David C. Silverman |
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General Regression Artificial Neural Network The general regression neural network was originally proposed for system modeling and prediction (D. F. Specht, "A General Regression Neural Network", IEEE Transactions on Neural Networks, 2, p. 568, 1991). It has been used to learn the same problems as the back-propagation network, the radial basis function network, the probabilistic neural network, and the modular neural network. The network has a relationship to the probabilistic neural network and has sometimes been used in place of it for classification problems.This network has certain characteristics:
(8)and 
(9)In equations 8 and 9, N is the number of input nodes, Ntrain is total number of training sets, σ is the width parameter subject to certain constraints, and dj is the Euclidean distance between an input and the mean of those inputs. The statistically most likely value for the random variable y (output) given 
(the vector of random input variables) is estimated substituting equations (8) and
(9) into
(10)The algorithm uses each of the training vectors as the center of a Gaussian function defined as  .
The larger the number of training samples, the larger is the amount of memory
required and the longer is the time for the calculation. Methodologies exist
to decrease the required time and memory allocation.
The following example is used for illustration. It is identical to the one used for the back-propagation network , the radial basis function network , the probabilistic neural network , and the modular neural network . The example is provided to show that a general regression neural network can sometimes be used in place of the others.
br> Five hundred training sets were created about evenly divided between those that round to 1 and those that round to 0. The network constructed is shown in this figure ) .
For this case, the hidden layer contained 500 nodes (1 epoch). Either 1 or
2 output nodes could have been used. If one output, the value was 1 or
0 depending on rounding to 1 or 0. If two outputs, the output corresponding to rounding to 0
was 1 if the value was rounded to 0 and 0 if the value was rounded to 1. The other
output was 1 if the value was rounded to 1 and 0 if the value was rounded to 0.
Both were trained. A different set of 100 randomly generated input-output
combinations were used to test the trained networks. The calculated outputs
were between about -.1 and +1.1. The strategy used to assess error was to
assume that if the value is less than 0.5, the prediction would have been zero
and if the value is greater than or equal to 0.5, the prediction would have been 1.
These values were compared to the actual outputs to determine error. No attempt
was made to compare actual values because that information
did not enter into the decision. This figure
) .
shows correct and incorrect responses for the neural network with one output.
This figure
) .
shows correct and incorrect responses for the neural network with two outputs.
Only two points were in error and both were at the boundary. They were the same for both
networks.
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David C. Silverman, Ph.D. - Primary Consultant |