notes on the distinction of gaussian and cauchy mutations
DESCRIPTION
Notes on the Distinction of Gaussian and Cauchy Mutations. Speaker ： Kuo-Torng, Lan. Ph. D. Takming Univ. of Science and Technology. I. Introduction II. Analyses of Two Mutations III. Simulation Results IV. Conclusions. I. Introduction. Rank or Roulette-wheel selection? - PowerPoint PPT PresentationTRANSCRIPT
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Notes on the Distinction of Gaussian and Cauchy
Mutations
Speaker： Kuo-Torng, Lan. Ph. D.
Takming Univ. of Science and Technology
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I. Introduction
II. Analyses of Two Mutations
III. Simulation Results
IV. Conclusions
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I. Introduction
• Rank or Roulette-wheel selection?
• Gaussian or Cauchy mutation?
• Population size? Mutation step size? …
• escaping local optima & converging to the global optimum
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I. Introduction
Local optimumLocal optimum
Global optimum
Individuals: walk randomly
Population: go toward the local(global) optimum
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II. Analyses of Two Mutations
• Assume the dimension of the individual is 1.
• Assume the mutation step size is
• The mutation is
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II. Analyses of Two Mutations
• And X is a random variable with the Gaussian distribution. Its pdf is
• And X is a random variable with the Cauchy distribution. Its pdf is
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II. Analyses of Two Mutations
• Condition 1: Local Escape on Valley landscape
g e n e r a t i o n t + 1
t g e n e r a t i o n δ
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II. Analyses of Two Mutations
• Condition 1: Local Escape on Valley landscape
For GMO:
For CMO:
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II. Analyses of Two Mutations
• Condition 2: Local Convergence on hill landscape
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II. Analyses of Two Mutations
• Condition 2: Local Convergence on hill landscape
For GMO:
For CMO:
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II. Analyses of Two Mutations
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III. Simulation Results
• Benchmark function 1: Ackey function
• Benchmark function 2: modified Schaffer function
• DC motor control (2005)
• 2D fractal pattern Design (2006)
• 3D fractal pattern Design (2008)
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III. Simulation Results
• Benchmark function 1: Ackey function
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III. Simulation Results
• Benchmark function 1: Ackey function
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III. Simulation Results • Benchmark function 1: Ackey function - by Gaussian mutation
0
5
10
15
20
25
30
35
40
45
1 101 201 301 401 501 601 701 801 901
Generations
Fitn
ess
valu
e
step size=0.4step size=0.6step size=0.8
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III. Simulation Results • Benchmark function 1: Ackey function - by Cauchy mutation
0
5
10
15
20
25
30
35
40
45
1 101 201 301 401 501 601 701 801 901
Generations
Fitn
ess
valu
e
step size=0.4
step size=0.6
step size=0.8
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III. Simulation Results
• Benchmark function 2: modified Schaffer function
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III. Simulation Results
• Benchmark function 2: modified Schaffer function
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III. Simulation Results
• Benchmark function 2: modified Schaffer function
0
1
2
3
4
5
6
7
1 101 201 301 401 501 601 701 801 901
Generations
Fitn
ess
valu
e
Gaussian
Cauchy
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III. Simulation Results
• Benchmark function 2: modified Schaffer function
0
1
2
3
4
5
6
7
1 101 201 301 401 501 601 701 801 901
Generations
Fitn
ess
valu
e
Gaussian
Cauchy
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III. Simulation Results
• DC motor control: (K. T. Lan, “Design a rule-based controller for DC servo-motor Control by evolutionary computation,” TAAI 2005, in Chinese.)
G ss s s s
( )( )( . )( . )( . )
1
1 1 0 2 1 0 05 1 0 01
G(s)+
Vin Vout
Rule-Based Controller
e
ddt
VK
c
p
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III. Simulation Results
• DC motor control: (K. T. Lan, “Design a rule-based controller ...)
The chromosome (i.e. control table)
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III. Simulation Results • DC motor control: (K. T. Lan, “Design a rule-based controller …,” )
0
0.5
1
1.5
2
2.5
0.1 0.6 1.1 1.6 2.1 2.6 3.1
mutation step size
Ris
eTim
e(R
2) EA(Gauss, avg)
EA(Gauss, best)
EA(Cauchy, avg)
EA(Cauchy, best)
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III. Simulation Results
• 2D fractal pattern Design : (K. T. Lan, et al., “Design a 2D f
ractal pattern by using the evolutionary computation,” TAAI 2006, in Chinese.)
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III. Simulation Results
• 2D fractal pattern Design : (K. T. Lan, et al., “Design a ...)
The chromosome (i.e. 2D pattern)
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III. Simulation Results
• 2D fractal pattern Design : (K. T. Lan, et al., “Design a ...)
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III. Simulation Results
• 3D fractal pattern Design : (K. T. Lan, et al., “The problems for design a 3D fractal pattern by using the evolutionary computation,” TAAI 2008, in Chinese.)
The Cauchy mutation is predominant to Gaussian.
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III. Simulation Results • 3D fractal pattern Design : (K. T. Lan, et al., “The problems
for design a 3D fractal pattern by using the evolutionary computation,” TAAI 2008, in Chinese.)
Searching space: 10x10x10No. of Reef: 60Near optimal design: FD= 2.3843
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III. Simulation Results • 3D fractal pattern Design : (K. T. Lan, et al., “The problems
for design a 3D fractal pattern by using the evolutionary computation,” TAAI 2008, in Chinese.)
Searching space: 12x12x12No. of Reef: 94Near optimal design: FD=2.4055
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IV. Conclusions
• A larger mutation step size can lead population to escape local optima and tend towards the global optimum
• A smaller mutation step size can finely tune the population
• Cauchy mutation possesses more power in escaping local optima
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IV. Conclusions
• For local convergence, the Cauchy technique is nearly equal to the Gaussian after evolving more generations.
• Therefore, Cauchy mutation is suggested to avoid the dilemma problem and achieve the acceptable performance for evolutionary computation.
Thanks for your kindly attention.