MastersWork.NumericalNotes History
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||!Branins's (12) || All perform equally ||||||||
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||!Branins's (12) || All similar |||||| Somewhat better ||
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* An exception is #12, it is fairly smooth, and they all do similarly. However: the function for it should be checked. The results don't seem to be in the expected range.
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* NB: the base 10 coding simply picks a new value over it's range at the moment. Try a new random value for real perhaps?
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* real coding performs better than the others where the area is easy to find, but the minima hard to pin down
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* real coding performs better than the others where the area is easy to find, but the minima hard to pin down, i.e. a fairly wide, shallow space
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||!Hyper-ellipsoid (3) || All similar |||||| Notably better* ||
||!Rosenbrock's (4) || All similar |||||| Notably better, right from start* ||
||!Rosenbrock's (4) || All similar |||||| Notably better, right from start
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||!Hyper-ellipsoid (3) || All similar |||||| Notably better ||
||!Rosenbrock's (4) || All similar |||||| Notably better, right from start ||
||!Rosenbrock's (4) || All similar |||||| Notably better, right from start ||
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||!Schwefel's (6) || All similar |||||| Notably worse ||
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||!Schwefel's (6) || All similar |||||| Notably worse* ||
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||!Branins's (12) || All perform equally ||||||||
[=*=] This shows an interesting pattern: the real coding is very marginally better for a while, and then all of a sudden it stops increasing as fast.
[=*=] This shows an interesting pattern: the real coding is very marginally better for a while, and then all of a sudden it stops increasing as fast.
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The functions are [[http://www.systemtechnik.tu-ilmenau.de/~pohlheim/GA_Toolbox/fcnfun1.html|De Jong's]], [[http://www.systemtechnik.tu-ilmenau.de/~pohlheim/GA_Toolbox/fcnfun2.html|Rosenbrock's valley]], [[http://www.systemtechnik.tu-ilmenau.de/~pohlheim/GA_Toolbox/fcnfun6.html|Rastrigin's function]], [[http://www.systemtechnik.tu-ilmenau.de/~pohlheim/GA_Toolbox/fcnfun7.html|Schwefel's function]], [[http://www.systemtechnik.tu-ilmenau.de/~pohlheim/GA_Toolbox/fcnfun8.html|Griewangk's function]], and [[http://www.systemtechnik.tu-ilmenau.de/~pohlheim/GA_Toolbox/fcnfun10.html|Ackley's Path function]].
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The functions are [[http://www.systemtechnik.tu-ilmenau.de/~pohlheim/GA_Toolbox/fcnfun1.html|De Jong's]], [[http://www.systemtechnik.tu-ilmenau.de/~pohlheim/GA_Toolbox/fcnfun2.html|Rosenbrock's valley]],
[[http://www.systemtechnik.tu-ilmenau.de/~pohlheim/GA_Toolbox/fcnfun1b.html|Rotated hyper-ellipsoid function]], [[http://www.systemtechnik.tu-ilmenau.de/~pohlheim/GA_Toolbox/fcnfun6.html|Rastrigin's function]], [[http://www.systemtechnik.tu-ilmenau.de/~pohlheim/GA_Toolbox/fcnfun7.html|Schwefel's function]], [[http://www.systemtechnik.tu-ilmenau.de/~pohlheim/GA_Toolbox/fcnfun8.html|Griewangk's function]], and [[http://www.systemtechnik.tu-ilmenau.de/~pohlheim/GA_Toolbox/fcnfun10.html|Ackley's Path function]].
[[http://www.systemtechnik.tu-ilmenau.de/~pohlheim/GA_Toolbox/fcnfun1b.html|Rotated hyper-ellipsoid function]], [[http://www.systemtechnik.tu-ilmenau.de/~pohlheim/GA_Toolbox/fcnfun6.html|Rastrigin's function]], [[http://www.systemtechnik.tu-ilmenau.de/~pohlheim/GA_Toolbox/fcnfun7.html|Schwefel's function]], [[http://www.systemtechnik.tu-ilmenau.de/~pohlheim/GA_Toolbox/fcnfun8.html|Griewangk's function]], and [[http://www.systemtechnik.tu-ilmenau.de/~pohlheim/GA_Toolbox/fcnfun10.html|Ackley's Path function]].
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||!Rosenbrock's (4) || All similar |||||| Notably better, right from start ||
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||!Hyper-ellipsoid (3) || All similar |||||| Notably better* ||
||!Rosenbrock's (4) || All similar |||||| Notably better, right from start* ||
||!Rosenbrock's (4) || All similar |||||| Notably better, right from start* ||
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[=*=] These show an unexpected behaviour that may lead to the results being invalidated. There seems to be some bias in the production of the initial population that affects these problems specifically.
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* Changing the mutation method of real numbers from adjusting by [-0.1:0.1], to a Gaussian distribution with standard deviation 0.25 seems to have no significant effect.
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||!Ackley's (9) || All perform equally |||||| Notably worse ||
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||!Ackley's (9) || All similar |||||| Notably worse ||
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** the larger the local minima, the worse the real coding performance seems to be, compared to the others
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Thoughts about the why:
* '''Why it does bad:''' all the encodings other than real have the ability to change a specific coordinate value by crossover, as crossover can operate in the middle of a part of the phenotype, whereas the real representation is much closer to the phenotype, and can't alter a phenotype unit (i.e. coordinate) without relying on mutation. This puts it at a disadvantage, as if the specific value of the minima isn't in the population, it must be created by mutation. This becomes more evident when mutation is switched off totally, as then real does worse in all but Rosenbrock's valley. (''not verified, but stands up to the evidence that I have so far'')
* '''Why it does good:''' no certain ideas yet
* '''Why it does bad:''' all the encodings other than real have the ability to change a specific coordinate value by crossover, as crossover can operate in the middle of a part of the phenotype, whereas the real representation is much closer to the phenotype, and can't alter a phenotype unit (i.e. coordinate) without relying on mutation. This puts it at a disadvantage, as if the specific value of the minima isn't in the population, it must be created by mutation. This becomes more evident when mutation is switched off totally, as then real does worse in all but Rosenbrock's valley. (''not verified, but stands up to the evidence that I have so far'')
* '''Why it does good:''' no certain ideas yet
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* check to see the effect of increasing/decreasing the mutation
to:
* check to see the effect of increasing/decreasing the mutation
* double-check to ensure the genetic operators for the real numbers are working correctly.
* see why it is that base 10 and base 2 especially don't perform well when real performs well.
* double-check to ensure the genetic operators for the real numbers are working correctly.
* see why it is that base 10 and base 2 especially don't perform well when real performs well.
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||!Ackley's (9) || All perform equally |||||| Notably worse ||
to:
||!Ackley's (9) || All perform equally |||||| Notably worse ||
Preliminary conclusions:
* base 10, base 2 and Gray all perform very similarly
* real coding performs the same as the others where the problem is quite simple
* real coding performs better than the others where the area is easy to find, but the minima hard to pin down
* real coding performs worse than the others when the search space is deceptive, i.e there are many local minima
Things to look at around this:
* change the mutation amount of the real coding (and probably the same for the base10). Currently, it can be changed up to 0.1 (working within [-1,1] space). Perhaps make this change to be a Gaussian or similar distribution over the entire range.
** NB: the base 10 coding simply picks a new value over it's range at the moment. Try a new random value for real perhaps?
* check to see the effect of increasing/decreasing the mutation
Preliminary conclusions:
* base 10, base 2 and Gray all perform very similarly
* real coding performs the same as the others where the problem is quite simple
* real coding performs better than the others where the area is easy to find, but the minima hard to pin down
* real coding performs worse than the others when the search space is deceptive, i.e there are many local minima
Things to look at around this:
* change the mutation amount of the real coding (and probably the same for the base10). Currently, it can be changed up to 0.1 (working within [-1,1] space). Perhaps make this change to be a Gaussian or similar distribution over the entire range.
** NB: the base 10 coding simply picks a new value over it's range at the moment. Try a new random value for real perhaps?
* check to see the effect of increasing/decreasing the mutation
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||!De Jong's (1)|| All perform equally ||||||||
||!Rosenbrock's (4)|| All similar |||||| Notably better, right from start ||
||!Rastringin's (5)|| All similar |||||| Very slightly worse ||
||!Schwefel's (6)|| All similar |||||| Notably worse ||
||!Griewangk's (7)|| All perform equally ||||||||
||!Ackley's (9)|| All perform equally |||||| Notably worse ||
||!Rosenbrock's (4)|| All similar |||||| Notably better, right from start ||
||!Rastringin's (5)|| All similar |||||| Very slightly worse ||
||!Schwefel's (6)|| All similar |||||| Notably worse ||
||!Griewangk's (7)|| All perform equally ||||||||
||!Ackley's (9)|| All perform equally |||||| Notably worse ||
to:
||!De Jong's (1) || All perform equally ||||||||
||!Rosenbrock's (4) || All similar |||||| Notably better, right from start ||
||!Rastringin's (5) || All similar |||||| Very slightly worse ||
||!Schwefel's (6) || All similar |||||| Notably worse ||
||!Griewangk's (7) || All perform equally ||||||||
||!Ackley's (9) || All perform equally |||||| Notably worse ||
||!Rosenbrock's (4) || All similar |||||| Notably better, right from start ||
||!Rastringin's (5) || All similar |||||| Very slightly worse ||
||!Schwefel's (6) || All similar |||||| Notably worse ||
||!Griewangk's (7) || All perform equally ||||||||
||!Ackley's (9) || All perform equally |||||| Notably worse ||
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||!De Jong's|| All perform equally ||||||||
||!Rosenbrock's|| All similar |||||| Notably better, right from start ||
||!Rastringin's|| All similar |||||| Very slightly worse ||
||!Schwefel's|| All similar |||||| Notably worse ||
||!Griewangk's|| All perform equally ||||||||
||!Ackley's|| All perform equally |||||| Notably worse ||
||!Rosenbrock's
||!Rastringin's
||!Schwefel's
||!Griewangk's
||!Ackley's|| All perform equally |||||| Notably worse ||
to:
||!De Jong's (1)|| All perform equally ||||||||
||!Rosenbrock's (4)|| All similar |||||| Notably better, right from start ||
||!Rastringin's (5)|| All similar |||||| Very slightly worse ||
||!Schwefel's (6)|| All similar |||||| Notably worse ||
||!Griewangk's (7)|| All perform equally ||||||||
||!Ackley's (9)|| All perform equally |||||| Notably worse ||
||!Rosenbrock's (4)|| All similar |||||| Notably better, right from start ||
||!Rastringin's (5)|| All similar |||||| Very slightly worse ||
||!Schwefel's (6)|| All similar |||||| Notably worse ||
||!Griewangk's (7)|| All perform equally ||||||||
||!Ackley's (9)|| All perform equally |||||| Notably worse ||
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The functions are [[http://www.systemtechnik.tu-ilmenau.de/~pohlheim/GA_Toolbox/fcnfun1.html|De Jong's]], [[http://www.systemtechnik.tu-ilmenau.de/~pohlheim/GA_Toolbox/fcnfun2.html|Rosenbrock's valley]], [[http://www.systemtechnik.tu-ilmenau.de/~pohlheim/GA_Toolbox/fcnfun6.html|Rastrigin's function]], [[http://www.systemtechnik.tu-ilmenau.de/~pohlheim/GA_Toolbox/fcnfun7.html|Schwefel's function]], [[http://www.systemtechnik.tu-ilmenau.de/~pohlheim/GA_Toolbox/fcnfun8.html|Griewangk's function]], and [[http://www.systemtechnik.tu-ilmenau.de/~pohlheim/GA_Toolbox/fcnfun10.html|ckley's Path function]].
to:
The functions are [[http://www.systemtechnik.tu-ilmenau.de/~pohlheim/GA_Toolbox/fcnfun1.html|De Jong's]], [[http://www.systemtechnik.tu-ilmenau.de/~pohlheim/GA_Toolbox/fcnfun2.html|Rosenbrock's valley]], [[http://www.systemtechnik.tu-ilmenau.de/~pohlheim/GA_Toolbox/fcnfun6.html|Rastrigin's function]], [[http://www.systemtechnik.tu-ilmenau.de/~pohlheim/GA_Toolbox/fcnfun7.html|Schwefel's function]], [[http://www.systemtechnik.tu-ilmenau.de/~pohlheim/GA_Toolbox/fcnfun8.html|Griewangk's function]], and [[http://www.systemtechnik.tu-ilmenau.de/~pohlheim/GA_Toolbox/fcnfun10.html|Ackley's Path function]].
|| border=1
|| ||!base 10||!base 2||!Gray||!real||
||!De Jong's|| All perform equally ||||||||
||!Rosenbrock's|| All similar |||||| Notably better, right from start ||
||!Rastringin's|| All similar |||||| Very slightly worse ||
||!Schwefel's|| All similar |||||| Notably worse ||
||!Griewangk's|| All perform equally ||||||||
||!Ackley's|| All perform equally |||||| Notably worse ||
|| border=1
|| ||!base 10||!base 2||!Gray||!real||
||!De Jong's|| All perform equally ||||||||
||!Rosenbrock's|| All similar |||||| Notably better, right from start ||
||!Rastringin's|| All similar |||||| Very slightly worse ||
||!Schwefel's|| All similar |||||| Notably worse ||
||!Griewangk's|| All perform equally ||||||||
||!Ackley's|| All perform equally |||||| Notably worse ||
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These are points found in the testing of numerical problems. The representations used are: [[NumBase10|base 10]], [[NumBase2|base 2]], [[NumGray|Gray coding]], [[NumReal|real coding]].
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These are points found in the testing of numerical problems.
The representations used are: [[NumBase10|base 10]], [[NumBase2|base 2]], [[NumGray|Gray coding]], [[NumReal|real coding]].
The representations used are: [[NumBase10|base 10]], [[NumBase2|base 2]], [[NumGray|Gray coding]], [[NumReal|real coding]].
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These are points found in the testing of numerical problems. The representations used are: [[NumBase10|base 10]], [[NumBase2|base 2]], [[NumGray|Gray coding]], [[NumReal|real coding]].
The functions are [[http://www.systemtechnik.tu-ilmenau.de/~pohlheim/GA_Toolbox/fcnfun1.html|De Jong's]], [[http://www.systemtechnik.tu-ilmenau.de/~pohlheim/GA_Toolbox/fcnfun2.html|Rosenbrock's valley]], [[http://www.systemtechnik.tu-ilmenau.de/~pohlheim/GA_Toolbox/fcnfun6.html|Rastrigin's function]], [[http://www.systemtechnik.tu-ilmenau.de/~pohlheim/GA_Toolbox/fcnfun7.html|Schwefel's function]], [[http://www.systemtechnik.tu-ilmenau.de/~pohlheim/GA_Toolbox/fcnfun8.html|Griewangk's function]], and [[http://www.systemtechnik.tu-ilmenau.de/~pohlheim/GA_Toolbox/fcnfun10.html|ckley's Path function]].
The functions are [[http://www.systemtechnik.tu-ilmenau.de/~pohlheim/GA_Toolbox/fcnfun1.html|De Jong's]], [[http://www.systemtechnik.tu-ilmenau.de/~pohlheim/GA_Toolbox/fcnfun2.html|Rosenbrock's valley]], [[http://www.systemtechnik.tu-ilmenau.de/~pohlheim/GA_Toolbox/fcnfun6.html|Rastrigin's function]], [[http://www.systemtechnik.tu-ilmenau.de/~pohlheim/GA_Toolbox/fcnfun7.html|Schwefel's function]], [[http://www.systemtechnik.tu-ilmenau.de/~pohlheim/GA_Toolbox/fcnfun8.html|Griewangk's function]], and [[http://www.systemtechnik.tu-ilmenau.de/~pohlheim/GA_Toolbox/fcnfun10.html|ckley's Path function]].
Page last modified on January 12, 2006, at 07:07 PM