BBO生物地理学算法
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13
Benchmark Functions
14 standard benchmark functions were used to evaluate BBO relative to other optimizers.
S–1 species at time t, and 1 species immigrated S+1 species at time t, and 1 species emmigrated
7
Biogeography
Convert the difference equation into a differential equation
10
Biogeography
0 ⎤ … ⎡ −1 1/ n 0 ⎢n / n − 1 2 / n ⎥ ⎢ ⎥ ⎥ A = E⎢ ⎢ ⎥ 2 / n − 1 n / n⎥ ⎢ ⎢ … 0 1/ n −1 ⎥ ⎣ 0 ⎦ = EA' P = AP
So the population reaches equilibrium when P is equal to the eigenvector corresponding to the zero eigenvalue of A
Biogeography-Based Optimization
Dan Simon Cleveland State University Fall 2008
Outline
1. 2. 3. 4. 5. 6. Biogeography Optimization Other Population-Based Optimizers Benchmark Functions & Results Sensor Selection & Results Conclusion
12
Biogeography-Based Optimization
1. 2. 3. 4. 5. 6. 7. Initialize a set of solutions to a problem. Compute “fitness” (HSI) for each solution. Compute S, λ, and μ for each solution. Modify habitats (migration) based on λ, μ. Mutatation based on probability. Typically we implement elitism. Go to step 2 for the next iteration if needed.
− (λs + μ s ) Ps + μ s +1 Ps +1 ⎧ ⎪ Ps = ⎨− (λs + μ s ) Ps + λs −1 Ps −1 + μ s +1 Ps +1 ⎪ − (λs + μ s ) Ps + λs −1 Ps −1 ⎩ S =0 S = 1, , S max − 1 S = S max
11
Biogeography
v P (∞ ) = ∑ vi
v = [v1
v n +1 ]
T百度文库
n! ⎧ (i = 1,..., ceil((n + 1) / 2)) ⎪ vi = ⎨ (n − 1 − i )!(i − 1)! ⎪ v n + 2 −i (i = ceil((n + 1) / 2) + 1,..., n + 1) ⎩
⎤ ⎥ ⎥ ⎥ ⎥ − (λn −1 + μ n −1 ) μn ⎥ − (λn + μ n )⎥ λn −1 ⎦ … 0
9
Biogeography
Suppose E=I. Then μ k = Ek / n
λk = E (1 − k / n)
where k = species count, n = S max
2
Biogeography
The study of the geographic distribution of biological organisms
• Mauritius • 1600s
3
Biogeography
Species migrate between “islands” via flotsam, wind, flying, swimming, …
5
Biogeography
As habitat suitability improves:
– The species count increases – Emigration increases (more species exit the habitat) – Immigration decreases (fewer species come into the habitat)
6
Biogeography
Ps = probability that habitat contains S species
S species at time t, and no migration occurred
Ps (t + Δt ) = Ps (t )(1 − λs Δt − μ s Δt ) + Ps −1 (t )λs −1Δt + Ps +1 (t ) μ s +1Δt
(Smax+1) coupled differential equations that can be combined into a single matrix equation.
8
Biogeography
P = AP
μ1 0 ⎡− (λ0 + μ 0 ) ⎢ − (λ1 + μ1 ) μ 2 λ0 ⎢ A=⎢ ⎢ λn − 2 ⎢ ⎢ 0 … 0 ⎣
4
Biogeography
• Habitat Suitability Index (HSI): Some islands are more suitable for habitation than others • Suitability Index Variables (SIVs): Habitability is related to features such as rainfall, topography, diversity of vegetation, temperature, etc.
Benchmark Functions
14 standard benchmark functions were used to evaluate BBO relative to other optimizers.
S–1 species at time t, and 1 species immigrated S+1 species at time t, and 1 species emmigrated
7
Biogeography
Convert the difference equation into a differential equation
10
Biogeography
0 ⎤ … ⎡ −1 1/ n 0 ⎢n / n − 1 2 / n ⎥ ⎢ ⎥ ⎥ A = E⎢ ⎢ ⎥ 2 / n − 1 n / n⎥ ⎢ ⎢ … 0 1/ n −1 ⎥ ⎣ 0 ⎦ = EA' P = AP
So the population reaches equilibrium when P is equal to the eigenvector corresponding to the zero eigenvalue of A
Biogeography-Based Optimization
Dan Simon Cleveland State University Fall 2008
Outline
1. 2. 3. 4. 5. 6. Biogeography Optimization Other Population-Based Optimizers Benchmark Functions & Results Sensor Selection & Results Conclusion
12
Biogeography-Based Optimization
1. 2. 3. 4. 5. 6. 7. Initialize a set of solutions to a problem. Compute “fitness” (HSI) for each solution. Compute S, λ, and μ for each solution. Modify habitats (migration) based on λ, μ. Mutatation based on probability. Typically we implement elitism. Go to step 2 for the next iteration if needed.
− (λs + μ s ) Ps + μ s +1 Ps +1 ⎧ ⎪ Ps = ⎨− (λs + μ s ) Ps + λs −1 Ps −1 + μ s +1 Ps +1 ⎪ − (λs + μ s ) Ps + λs −1 Ps −1 ⎩ S =0 S = 1, , S max − 1 S = S max
11
Biogeography
v P (∞ ) = ∑ vi
v = [v1
v n +1 ]
T百度文库
n! ⎧ (i = 1,..., ceil((n + 1) / 2)) ⎪ vi = ⎨ (n − 1 − i )!(i − 1)! ⎪ v n + 2 −i (i = ceil((n + 1) / 2) + 1,..., n + 1) ⎩
⎤ ⎥ ⎥ ⎥ ⎥ − (λn −1 + μ n −1 ) μn ⎥ − (λn + μ n )⎥ λn −1 ⎦ … 0
9
Biogeography
Suppose E=I. Then μ k = Ek / n
λk = E (1 − k / n)
where k = species count, n = S max
2
Biogeography
The study of the geographic distribution of biological organisms
• Mauritius • 1600s
3
Biogeography
Species migrate between “islands” via flotsam, wind, flying, swimming, …
5
Biogeography
As habitat suitability improves:
– The species count increases – Emigration increases (more species exit the habitat) – Immigration decreases (fewer species come into the habitat)
6
Biogeography
Ps = probability that habitat contains S species
S species at time t, and no migration occurred
Ps (t + Δt ) = Ps (t )(1 − λs Δt − μ s Δt ) + Ps −1 (t )λs −1Δt + Ps +1 (t ) μ s +1Δt
(Smax+1) coupled differential equations that can be combined into a single matrix equation.
8
Biogeography
P = AP
μ1 0 ⎡− (λ0 + μ 0 ) ⎢ − (λ1 + μ1 ) μ 2 λ0 ⎢ A=⎢ ⎢ λn − 2 ⎢ ⎢ 0 … 0 ⎣
4
Biogeography
• Habitat Suitability Index (HSI): Some islands are more suitable for habitation than others • Suitability Index Variables (SIVs): Habitability is related to features such as rainfall, topography, diversity of vegetation, temperature, etc.