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208    5 Neural Networks

















                          Figure 5.41.  An  MLP2:2:1 with  all  its weights plus biases. These are considered
                          chromosome genes.



                            For  this  MLP2:2:1  device  the  chromosome  can  be  expressed  by  an  ordered
                          sequence of the genes, the so-called genotype g, such as:




                            In  the  representation  of  g  we  used  a  semi-colon  to  separate  the  weights of
                          individual neurons.
                            Imagine that  we have a population,  G, of  a specified number of  such devices
                          (individuals) represented by their chromosomes. Each device is characterized by a
                          fitnessfactor,  which we  may relate in some way with the error achieved with the
                          device.  Using the sum of squares error E we can, for instance, define the fitness
                          factor for a device with genotype g, as:





                          where E,,,,  is any upper bound of the error.
                            Inspired  by  the Darwinian  theory of  biological evolution,  a genetic  algorithm
                          generates a new population by repetition of the following two-step process:

                          I. Select  the  fittest  individuals  of  the  population  (the  so-called  survival  of  the
                            fittest) for reproduction. A common selection process of parent individuals for
                            reproduction  is  based  on  the  assignment  of  a  probability  of  selection  to  an
                            individual, given by the individual's  relative fitness:




                            This selection rule is also known as roulette-wheel selection.

                          2. Generate a new population based on the selected parent individuals by  applying
                            the following operators to the parent chromosomes:
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