39
edits
Changes
Jump to navigation
Jump to search
Line 6:
Line 6: − + − + + + + + + + + + + + − In this case the rules of genetic variation such as crossover, inversion, transposition etcetera may be seen as random number generators for the phenotypes. So, in this sense GA may be seen as a genetic algorithm.
Gaussian adaptation (view source)
Revision as of 09:16, February 14, 2008
, 09:16, February 14, 2008→Gaussian adaptation as a model of evolution
== Gaussian adaptation as a model of evolution ==
== Gaussian adaptation as a model of evolution ==
It has also been compared to the natural evolution of populations of living organisms. In this case the region of acceptability is replaced by a probability function, s(x), where x is an array of phenotypes determining the organism. This is possible because the theorem of Gaussian adaptation is valid for any region of acceptability. Then it is fairly easily proved that the process is maximizing the mean fitness of a large population with respect to Gaussian distributed quantitative characters.
It has also been compared to the natural evolution of populations of living organisms. In this case the region of acceptability is preferably replaced by a probability function, s(x), where x is an array of quantitative characters (phenotypes) determining the organism. This is possible because the theorem of Gaussian adaptation is valid for any region of acceptability independent of structure and extension.
As long as the ontogenetic program my be seen as a stepwise modified recapitulation of the evolution of a particular individual organism, the central limit theorem states that ''the sum of contributions from many random steps tend to become Gaussian distributed.'' A necessary condition for the natural evolution to be able to fulfill the theorem of Gaussian adaptation is that it may push the centre of gravity of the Gaussian to the centre of gravity of the surviving individuals. The Hardy-Weinberg law may accomplish this.
Mean fitness may be defined as
P(m) = integral { s(x) N(m – x) dx }.
Here N is the Gaussian probability density function, p. d. f., of phenotypes and m is the centre of gravity of ditto. Because a Gaussian is the exponential of negative squared phenotypes, it is fairly easily proved that the process maximizes the mean fitness of a large population. The condition for maximal mean fitness is obtained by letting the derivative, dP(m)/dm, become equal to zero. Because the derivative of the exponential is equal to the exponential itself the result becomes proportional to P (m* - m) = 0, where m* is the centre of gravity of the phenotypes of the parents to offspring in the progeny. This leads to the theorem of Gaussian adaptation in its simplest form.
More generally, if m slightly deviates from its optimal position in any arbitrary direction, then mean fitness will be decreased, but may be recovered if determinant of the moment matrix is slightly decreased. According to the information theory due to [[Claude Shannon]] this means that the [[average information]] (entropy, disorder, diversity) is simultaneously maximal, also meaning that average information is as important to survival as mean fitness.
As long as the ontogenetic program my be seen as a stepwise modified recapitulation of the evolution of a particular individual organism, the [[central limit theorem]] states that the sum of contributions from many random steps tend to become Gaussian distributed. A necessary condition for the natural evolution to be able to fulfill the theorem of Gaussian adaptation is that it may push the centre of gravity of the Gaussian towards the centre of gravity of the surviving individuals. The [[Hardy-Weinberg]] law may accomplish this. In this case the rules of genetic variation such as crossover, inversion, transposition etcetera may be seen as random number generators for the phenotypes. So, in this sense GA may be seen as a genetic algorithm.
A remarcable thing seems to be that a Gaussian distribution is the most disordered distribution as compared to all other distributions having the same moment matrix, and is connected to some remarcable theorems such as the central limit theorem, the Hardy-Weinberg law, the entropy law and the theorem of gaussian adaptation. Those theorems make it possible to centre a gene pool in an arbitrary region of acceptability so as to maximize mean fitness, which is an extremely difficult problem in a high-dimensional space.
So, we may perhaps ask if this combination of chaos and mathematical order makes evolution a tool for [[intelligent design]], with the intelligence represented by the mathematical theorems?
== References ==
== References ==