Evolution in a nutshell

an alternative outline on evoution

and some consequences concerning valuations

by

Gregor Kjellström

            References          

7 The simulation of normal adaptation

Thus far the theory only considers mean values of continuous distributions corresponding to an infinite number of individuals. In reality however, the number of individuals is always limited, which gives rise to an uncertainty in the estimation of m and M. And this may also affect the efficiency of the process. Unfortunately very little is known about this, at least theoretically.

The implementation of normal adaptation on a computer is a fairly simple task. The adaptation of m may be done by one sample (individual) at a time, for example

     m(i+1) = (1 – a) m(i) + ax

where x is a pass sample, and a < 1 a suitable constant so that the inverse of a represents the number of individuals in the population.

     M may in principle be updated after every step y  leading to a feasible point

     x = m + y   according to:

     M(i+1) = (1 – 2b) M(i) + 2byy^T,

where b << 1 is another suitable constant. In order to guarantee a suitable increase of average information, y should be normally distributed with moment matrix m²M, where the scalar m > 1 is used to increase information at a suitable rate. But M will never be used in the calculations. Instead we use the matrix W defined by WW^T = M.

 Thus, we have y = Wg, where g is normally distributed with the moment matrix mU, and U is the unit matrix. W and W^T may be updated by the formulas:

     W = (1 – b)W + byg^T   and   W^T = (1 – b)W^T + bgy^T

because multiplication gives

     M = (1 – 2b)M + 2byy^T,

where terms including b² have been neglected. Thus, M will be indirectly adapted with good approximation. In practice it will suffice to update W only 

     W(i+1)  = (1 – b)W(i)  + byg^T.

In fact, this is the formula used in our model of the brain satisfying the Hebbian rule of associative learning; see also section 2.8.