Lecture ongMonte-Carlo Methodsg
Chung-Lin Shan
Institute of Physics, Academia Sinica
Second Taiwan Winter Camp onNumerical Simulation in High Energy Physics
(NumS-HEP 2012)
February 13, 2012
Outline
Monte-Carlomethods
Basic concept
Acceptance-rejection
Inverse-transform
Auxiliarydisrtributions
Markov chainMonte-Carlo
Basic concept
Metropolisalgorithm
Applications ofMC methods
Area/volumnestimation
g
Lecture on Monte-Carlo Methods
g
Second Taiwan Winter Camp onNumerical Simulation in High Energy Physics
(NumS-HEP 2012)February 13, 2012
Monte-Carlo methodsBasic concept of Monte-Carlo methodsAcceptance-rejection methodInverse-transform methodUsing auxiliary disrtribution functions
Markov chain Monte-CarloBasic concept of Markov processesMetropolis algorithm
Applications of Monte-Carlo methodsArea/volumn estimation
C.-L. Shan, AS IoP p. 1 / 49
Outline
Monte-Carlomethods
Basic concept
Acceptance-rejection
Inverse-transform
Auxiliarydisrtributions
Markov chainMonte-Carlo
Basic concept
Metropolisalgorithm
Applications ofMC methods
Area/volumnestimation
g
Lecture on Monte-Carlo Methods
g
Second Taiwan Winter Camp onNumerical Simulation in High Energy Physics
(NumS-HEP 2012)February 13, 2012
Monte-Carlo methods
C.-L. Shan, AS IoP p. 2 / 49
Outline
Monte-Carlomethods
Basic concept
Acceptance-rejection
Inverse-transform
Auxiliarydisrtributions
Markov chainMonte-Carlo
Basic concept
Metropolisalgorithm
Applications ofMC methods
Area/volumnestimation
g
Lecture on Monte-Carlo Methods
g
Second Taiwan Winter Camp onNumerical Simulation in High Energy Physics
(NumS-HEP 2012)February 13, 2012
Basic concept of Monte-Carlo methods
C.-L. Shan, AS IoP p. 3 / 49
Outline
Monte-Carlomethods
Basic concept
Acceptance-rejection
Inverse-transform
Auxiliarydisrtributions
Markov chainMonte-Carlo
Basic concept
Metropolisalgorithm
Applications ofMC methods
Area/volumnestimation
g
Lecture on Monte-Carlo Methods
g
Second Taiwan Winter Camp onNumerical Simulation in High Energy Physics
(NumS-HEP 2012)February 13, 2012
Introduction
p
In the 1930s Enrico Fermi made some numerical experimentsthat would now be called Monte Carlo calculations.
[Monte Carlo Methods, M. H. Kalos and P. A. Whitlock, Chap. 1, p. 3]
The name Monte Carlo was applied to a class ofmathematical methods first by scientists working on thedevelopment of nuclear weapons in Los Alamos in the 1940s.
[Monte Carlo Methods, M. H. Kalos and P. A. Whitlock, Chap. 1, p. 1]
Relationship between theory,experiment, and numericalsimulation: each is distinct,but each is strongly connectedto the other two.
-��� J
JJJJ]JJJJJ
Theory
Experiment Simulation
[A Guide to M. C. Simu. in Stat. Phys., D. P. Landau and K. Binder, Sec. 1.5, p. 5]
C.-L. Shan, AS IoP p. 4 / 49
Outline
Monte-Carlomethods
Basic concept
Acceptance-rejection
Inverse-transform
Auxiliarydisrtributions
Markov chainMonte-Carlo
Basic concept
Metropolisalgorithm
Applications ofMC methods
Area/volumnestimation
g
Lecture on Monte-Carlo Methods
g
Second Taiwan Winter Camp onNumerical Simulation in High Energy Physics
(NumS-HEP 2012)February 13, 2012
Basic concept of Monte-Carlo methods
Start with the dice game
Equal probability for all six points
P(1) : P(2) : P(3) : P(4) : P(5) : P(6) = 1 : 1 : 1 : 1 : 1 : 1
Probability distribution function (probability density)
P(n) = 1 for all n = 1, 2, · · · , 6
Modified probabilities
P(1) : P(2) : P(3) : P(4) : P(5) : P(6) = 2 : 4 : 2 : 1 : 2 : 4
Modified probability distribution function
P(n) =
1.00 for n = 2, 6
0.50 for n = 1, 3, 5
0.25 for n = 4
C.-L. Shan, AS IoP p. 5 / 49
Outline
Monte-Carlomethods
Basic concept
Acceptance-rejection
Inverse-transform
Auxiliarydisrtributions
Markov chainMonte-Carlo
Basic concept
Metropolisalgorithm
Applications ofMC methods
Area/volumnestimation
g
Lecture on Monte-Carlo Methods
g
Second Taiwan Winter Camp onNumerical Simulation in High Energy Physics
(NumS-HEP 2012)February 13, 2012
Basic concept of Monte-Carlo methods
Distribution of the dice points6 points, equal probability, 500 experiments, 6000 times
[DISW]
C.-L. Shan, AS IoP p. 6 / 49
Outline
Monte-Carlomethods
Basic concept
Acceptance-rejection
Inverse-transform
Auxiliarydisrtributions
Markov chainMonte-Carlo
Basic concept
Metropolisalgorithm
Applications ofMC methods
Area/volumnestimation
g
Lecture on Monte-Carlo Methods
g
Second Taiwan Winter Camp onNumerical Simulation in High Energy Physics
(NumS-HEP 2012)February 13, 2012
Basic concept of Monte-Carlo methods
Distribution of the dice points6 points, modified probabilities, 500 experiments, 6000 times
[DISW]
C.-L. Shan, AS IoP p. 7 / 49
Outline
Monte-Carlomethods
Basic concept
Acceptance-rejection
Inverse-transform
Auxiliarydisrtributions
Markov chainMonte-Carlo
Basic concept
Metropolisalgorithm
Applications ofMC methods
Area/volumnestimation
g
Lecture on Monte-Carlo Methods
g
Second Taiwan Winter Camp onNumerical Simulation in High Energy Physics
(NumS-HEP 2012)February 13, 2012
Basic concept of Monte-Carlo methods
Distribution of the dice points6 points, modified probabilities, 500 experiments, 6000 times
[DISW]
1 2 3 4 5 6
1
2
3
4
√√√√√√
√√√ √√√√√√
××
×××××
××
C.-L. Shan, AS IoP p. 8 / 49
Outline
Monte-Carlomethods
Basic concept
Acceptance-rejection
Inverse-transform
Auxiliarydisrtributions
Markov chainMonte-Carlo
Basic concept
Metropolisalgorithm
Applications ofMC methods
Area/volumnestimation
g
Lecture on Monte-Carlo Methods
g
Second Taiwan Winter Camp onNumerical Simulation in High Energy Physics
(NumS-HEP 2012)February 13, 2012
Basic concept of Monte-Carlo methods
Distribution of the dice points6 points, modified probabilities, 500 experiments, 6000 times
[DISW]
1 2 3 4 5 6
0.25
0.50
0.75
1.00
C.-L. Shan, AS IoP p. 9 / 49
Outline
Monte-Carlomethods
Basic concept
Acceptance-rejection
Inverse-transform
Auxiliarydisrtributions
Markov chainMonte-Carlo
Basic concept
Metropolisalgorithm
Applications ofMC methods
Area/volumnestimation
g
Lecture on Monte-Carlo Methods
g
Second Taiwan Winter Camp onNumerical Simulation in High Energy Physics
(NumS-HEP 2012)February 13, 2012
Acceptance-rejection method
C.-L. Shan, AS IoP p. 10 / 49
Outline
Monte-Carlomethods
Basic concept
Acceptance-rejection
Inverse-transform
Auxiliarydisrtributions
Markov chainMonte-Carlo
Basic concept
Metropolisalgorithm
Applications ofMC methods
Area/volumnestimation
g
Lecture on Monte-Carlo Methods
g
Second Taiwan Winter Camp onNumerical Simulation in High Energy Physics
(NumS-HEP 2012)February 13, 2012
Acceptance-rejection method
Poisson distribution
Poi(n;µ) =µne−µ
n!(n ∈ N ; µ > 0)
Gaussian/normal distribution
Gau(x;µ, σµ) =1
√2π σµ
e−(x−µ)2/2σ2µ (x ∈ R; µ ∈ R, σµ > 0)
Circular distribution
Cir(x;µ) =2
πµ2
√2µx− x2 (x ∈ [0, 2µ]; µ > 0)
C.-L. Shan, AS IoP p. 11 / 49
Outline
Monte-Carlomethods
Basic concept
Acceptance-rejection
Inverse-transform
Auxiliarydisrtributions
Markov chainMonte-Carlo
Basic concept
Metropolisalgorithm
Applications ofMC methods
Area/volumnestimation
g
Lecture on Monte-Carlo Methods
g
Second Taiwan Winter Camp onNumerical Simulation in High Energy Physics
(NumS-HEP 2012)February 13, 2012
Acceptance-rejection method
Basic process:
1 Generate n ∈ {0, 1, 2, · · · , nmax} randomly.
2 Generate Pcheck ∈ [0, 1] randomly.
3 Check whether Pcheck ≤ Poi(n;µ) / Poi(µ;µ).
4 “Yes” =⇒ “Valid” point =⇒ Take this!“No” =⇒ “Invalid” point =⇒ Throw it away!
5 Repeat Nevent times.
6 Count N(n), n ∈ {0, 1, 2, · · · , nmax}.nmax∑n=0
N(n) = Nevent
7 Repeat Nexpt times.
8 Draw the distribution (histogram) of N(n) (normalized byNevent) with Poi(n;µ) together.
C.-L. Shan, AS IoP p. 12 / 49
Outline
Monte-Carlomethods
Basic concept
Acceptance-rejection
Inverse-transform
Auxiliarydisrtributions
Markov chainMonte-Carlo
Basic concept
Metropolisalgorithm
Applications ofMC methods
Area/volumnestimation
g
Lecture on Monte-Carlo Methods
g
Second Taiwan Winter Camp onNumerical Simulation in High Energy Physics
(NumS-HEP 2012)February 13, 2012
Acceptance-rejection method
Distribution of the generated eventsPoisson distribution, µ = 5, 500 events
σµ
[DISW]
C.-L. Shan, AS IoP p. 13 / 49
Outline
Monte-Carlomethods
Basic concept
Acceptance-rejection
Inverse-transform
Auxiliarydisrtributions
Markov chainMonte-Carlo
Basic concept
Metropolisalgorithm
Applications ofMC methods
Area/volumnestimation
g
Lecture on Monte-Carlo Methods
g
Second Taiwan Winter Camp onNumerical Simulation in High Energy Physics
(NumS-HEP 2012)February 13, 2012
Acceptance-rejection method
Distribution of the generated eventsPoisson distribution, µ = 5, 500 events
σµ
[DISW]
C.-L. Shan, AS IoP p. 14 / 49
Outline
Monte-Carlomethods
Basic concept
Acceptance-rejection
Inverse-transform
Auxiliarydisrtributions
Markov chainMonte-Carlo
Basic concept
Metropolisalgorithm
Applications ofMC methods
Area/volumnestimation
g
Lecture on Monte-Carlo Methods
g
Second Taiwan Winter Camp onNumerical Simulation in High Energy Physics
(NumS-HEP 2012)February 13, 2012
Acceptance-rejection method
Distribution of the generated eventsPoisson distribution, µ = 5, 5000 events
σµ
[DISW]
C.-L. Shan, AS IoP p. 15 / 49
Outline
Monte-Carlomethods
Basic concept
Acceptance-rejection
Inverse-transform
Auxiliarydisrtributions
Markov chainMonte-Carlo
Basic concept
Metropolisalgorithm
Applications ofMC methods
Area/volumnestimation
g
Lecture on Monte-Carlo Methods
g
Second Taiwan Winter Camp onNumerical Simulation in High Energy Physics
(NumS-HEP 2012)February 13, 2012
Acceptance-rejection method
Distribution of the generated eventsPoisson distribution, µ = 5, 5000 events
σµ
[DISW]
C.-L. Shan, AS IoP p. 16 / 49
Outline
Monte-Carlomethods
Basic concept
Acceptance-rejection
Inverse-transform
Auxiliarydisrtributions
Markov chainMonte-Carlo
Basic concept
Metropolisalgorithm
Applications ofMC methods
Area/volumnestimation
g
Lecture on Monte-Carlo Methods
g
Second Taiwan Winter Camp onNumerical Simulation in High Energy Physics
(NumS-HEP 2012)February 13, 2012
Acceptance-rejection method
Distribution of the generated eventsGaussian distribution, µ = 10, σµ = 5, 500 events
σµ
[DISW]
C.-L. Shan, AS IoP p. 17 / 49
Outline
Monte-Carlomethods
Basic concept
Acceptance-rejection
Inverse-transform
Auxiliarydisrtributions
Markov chainMonte-Carlo
Basic concept
Metropolisalgorithm
Applications ofMC methods
Area/volumnestimation
g
Lecture on Monte-Carlo Methods
g
Second Taiwan Winter Camp onNumerical Simulation in High Energy Physics
(NumS-HEP 2012)February 13, 2012
Acceptance-rejection method
Distribution of the generated eventsGaussian distribution, µ = 10, σµ = 5, 500 events
σµ
[DISW]
C.-L. Shan, AS IoP p. 18 / 49
Outline
Monte-Carlomethods
Basic concept
Acceptance-rejection
Inverse-transform
Auxiliarydisrtributions
Markov chainMonte-Carlo
Basic concept
Metropolisalgorithm
Applications ofMC methods
Area/volumnestimation
g
Lecture on Monte-Carlo Methods
g
Second Taiwan Winter Camp onNumerical Simulation in High Energy Physics
(NumS-HEP 2012)February 13, 2012
Acceptance-rejection method
Distribution of the generated eventsGaussian distribution, µ = 10, σµ = 5, 5000 events
σµ
[DISW]
C.-L. Shan, AS IoP p. 19 / 49
Outline
Monte-Carlomethods
Basic concept
Acceptance-rejection
Inverse-transform
Auxiliarydisrtributions
Markov chainMonte-Carlo
Basic concept
Metropolisalgorithm
Applications ofMC methods
Area/volumnestimation
g
Lecture on Monte-Carlo Methods
g
Second Taiwan Winter Camp onNumerical Simulation in High Energy Physics
(NumS-HEP 2012)February 13, 2012
Acceptance-rejection method
Distribution of the generated eventsGaussian distribution, µ = 10, σµ = 5, 5000 events
σµ
[DISW]
C.-L. Shan, AS IoP p. 20 / 49
Outline
Monte-Carlomethods
Basic concept
Acceptance-rejection
Inverse-transform
Auxiliarydisrtributions
Markov chainMonte-Carlo
Basic concept
Metropolisalgorithm
Applications ofMC methods
Area/volumnestimation
g
Lecture on Monte-Carlo Methods
g
Second Taiwan Winter Camp onNumerical Simulation in High Energy Physics
(NumS-HEP 2012)February 13, 2012
Acceptance-rejection method
Bernoulli distribution
Ber(x; p) = px(1− p)1−x (x ∈ {0, 1}; 0 ≤ p ≤ 1)
Binomial distribution
Bin(x, n; p) =
(nx
)px(1− p)n−x
(x ∈ {0, 1, 2, · · · , n}, n ∈ N ; 0 ≤ p ≤ 1)
Geometric distribution
Geo(n; p) = p(1− p)n−1 (n ∈ N ; 0 ≤ p ≤ 1)
Discret uniform distribution
DUni(x, n) =1
n(x ∈ {1, 2, · · · , n}, n ∈ N )
C.-L. Shan, AS IoP p. 21 / 49
Outline
Monte-Carlomethods
Basic concept
Acceptance-rejection
Inverse-transform
Auxiliarydisrtributions
Markov chainMonte-Carlo
Basic concept
Metropolisalgorithm
Applications ofMC methods
Area/volumnestimation
g
Lecture on Monte-Carlo Methods
g
Second Taiwan Winter Camp onNumerical Simulation in High Energy Physics
(NumS-HEP 2012)February 13, 2012
Acceptance-rejection method
Uniform distribution
Uni(x;α, β) =1
β − α(x ∈ [α, β]; α < β)
Exponential distribution
Exp(x;λ) = λ e−λx (x ∈ R+; λ > 0)
Gamma distribution
Gamma(x;λ, α) =λαe−λxxα−1
Γ(α)(x ∈ R+; λ, α > 0)
Beta distribution
Beta(x;α, β) =Γ(α+ β)
Γ(α) Γ(β)xα−1(1− x)β−1 (x ∈ [0, 1]; α, β > 0)
C.-L. Shan, AS IoP p. 22 / 49
Outline
Monte-Carlomethods
Basic concept
Acceptance-rejection
Inverse-transform
Auxiliarydisrtributions
Markov chainMonte-Carlo
Basic concept
Metropolisalgorithm
Applications ofMC methods
Area/volumnestimation
g
Lecture on Monte-Carlo Methods
g
Second Taiwan Winter Camp onNumerical Simulation in High Energy Physics
(NumS-HEP 2012)February 13, 2012
Inverse-transform method
C.-L. Shan, AS IoP p. 23 / 49
Outline
Monte-Carlomethods
Basic concept
Acceptance-rejection
Inverse-transform
Auxiliarydisrtributions
Markov chainMonte-Carlo
Basic concept
Metropolisalgorithm
Applications ofMC methods
Area/volumnestimation
g
Lecture on Monte-Carlo Methods
g
Second Taiwan Winter Camp onNumerical Simulation in High Energy Physics
(NumS-HEP 2012)February 13, 2012
Inverse-transform method
p
Probability distribution function (probability density)
f(x) ≥ 0 ∀ x ∈ (−∞,∞)∫ ∞−∞
f(x) dx = 1∫ xmax
xmin
f(x) dx = P(xmin, xmax) 0 ≤ P(xmin, xmax) ≤ 1
Cumulative distribution function∫ x
−∞f(x′) dx′ = P(−∞, x) = P(x′ ≤ x) ≡ F (x)
F (x) increases monotonically with x:
F (x1) ≤ F (x2) ∀ x1 ≤ x2F (x) is strictly monotonic:
F (x1) < F (x2) once ∃ x1 ≤ x ≤ x2 where f(x) > 0
C.-L. Shan, AS IoP p. 24 / 49
Outline
Monte-Carlomethods
Basic concept
Acceptance-rejection
Inverse-transform
Auxiliarydisrtributions
Markov chainMonte-Carlo
Basic concept
Metropolisalgorithm
Applications ofMC methods
Area/volumnestimation
g
Lecture on Monte-Carlo Methods
g
Second Taiwan Winter Camp onNumerical Simulation in High Energy Physics
(NumS-HEP 2012)February 13, 2012
Inverse-transform method
p
Cumulative uniform distribution
PUni(x;α, β) =
0 for x ≤ αx− αβ − α
for α ≤ x ≤ β
1 for x ≥ β
Generate P ∈ [0, 1].
x(P;α, β) = α+ (β − α) P
Cumulative exponential distribution
PExp(x;λ) = 1− e−λx
Generate P ∈ [0, 1].
x(P;λ) = −1
λln (1− P) =⇒ x(P;λ) = −
1
λln P
C.-L. Shan, AS IoP p. 25 / 49
Outline
Monte-Carlomethods
Basic concept
Acceptance-rejection
Inverse-transform
Auxiliarydisrtributions
Markov chainMonte-Carlo
Basic concept
Metropolisalgorithm
Applications ofMC methods
Area/volumnestimation
g
Lecture on Monte-Carlo Methods
g
Second Taiwan Winter Camp onNumerical Simulation in High Energy Physics
(NumS-HEP 2012)February 13, 2012
Inverse-transform method
p
Cumulative Gaussian/normal distribution
PGau(x;µ, σµ) =1
2
[1 + erf
(x− µ√
2σµ
)]Generate P ∈ [0, 1].
x(P;µ, σµ) = µ+√
2σµ erf−1 (2P− 1)
Inverse error function
erf−1(x) ≈ sgn(x)
( 2
πaerf+
ln(1− x2
)2
)2
−ln(1− x2
)aerf
1/2
−(
2
πaerf+
ln(1− x2
)2
)}1/2
aerf =8 (π − 3)
3π (4− π)≈ 0.140 or aerf ≈ 0.147
C.-L. Shan, AS IoP p. 26 / 49
Outline
Monte-Carlomethods
Basic concept
Acceptance-rejection
Inverse-transform
Auxiliarydisrtributions
Markov chainMonte-Carlo
Basic concept
Metropolisalgorithm
Applications ofMC methods
Area/volumnestimation
g
Lecture on Monte-Carlo Methods
g
Second Taiwan Winter Camp onNumerical Simulation in High Energy Physics
(NumS-HEP 2012)February 13, 2012
Inverse-transform method
p
Cauchy distribution
Cau(x;α) =α
π
1
(α2 + x2)(x ∈ R; α > 0)
Cumulative Cauchy distribution
PCau(x;α) =1
2+
1
πtan−1
( xα
)
Generate P ∈ [0, 1].
x(P;α) = α tan
[(P−
1
2
)π
]
C.-L. Shan, AS IoP p. 27 / 49
Outline
Monte-Carlomethods
Basic concept
Acceptance-rejection
Inverse-transform
Auxiliarydisrtributions
Markov chainMonte-Carlo
Basic concept
Metropolisalgorithm
Applications ofMC methods
Area/volumnestimation
g
Lecture on Monte-Carlo Methods
g
Second Taiwan Winter Camp onNumerical Simulation in High Energy Physics
(NumS-HEP 2012)February 13, 2012
Inverse-transform method
p
2-D Gaussian distribution
Gau2D(x, y;µ = 0, σµ) =1(√
2π σµ)2 e−(x2+y2)/2σ2
µ
(x, y ∈ R; σµ > 0)
Radial Gaussian distribution
Gau2D(r, θ;σµ) =r
σ2µ
e−r2/2σ2
µ ·1
2π(r ∈ R+, θ ∈ [0, 2π])
Cumulative radial Gaussian distribution
PGau2D(r;σµ) = 1− e−r
2/2σ2µ
Generate Pr,Pθ ∈ [0, 1].
r(Pr;σµ) = σµ√−2 ln (1− Pr)
=⇒x(Pr,Pθ;σµ) = σµ
√−2 ln Pr cos (2πPθ)
y(Pr,Pθ;σµ) = σµ√−2 ln Pr sin (2πPθ)
C.-L. Shan, AS IoP p. 28 / 49
Outline
Monte-Carlomethods
Basic concept
Acceptance-rejection
Inverse-transform
Auxiliarydisrtributions
Markov chainMonte-Carlo
Basic concept
Metropolisalgorithm
Applications ofMC methods
Area/volumnestimation
g
Lecture on Monte-Carlo Methods
g
Second Taiwan Winter Camp onNumerical Simulation in High Energy Physics
(NumS-HEP 2012)February 13, 2012
Using auxiliary disrtribution functions
C.-L. Shan, AS IoP p. 29 / 49
Outline
Monte-Carlomethods
Basic concept
Acceptance-rejection
Inverse-transform
Auxiliarydisrtributions
Markov chainMonte-Carlo
Basic concept
Metropolisalgorithm
Applications ofMC methods
Area/volumnestimation
g
Lecture on Monte-Carlo Methods
g
Second Taiwan Winter Camp onNumerical Simulation in High Energy Physics
(NumS-HEP 2012)February 13, 2012
Using auxiliary disrtribution functions
Consider the Gaussian distribution
Gau(x;µ = 0, σµ = 1) =1√
2πe−x
2/2 (x ∈ R+)
Use the exponential distribution.
Exp(x;λ = 1) = e−x
Require C · Exp(x;λ = 1) ≥ Gau(x;µ = 0, σµ = 1) ∀ x.
=⇒ C ≥√
e
2π
=⇒ AuxGau(x;µ = 0, σµ = 1) =
√e
2πe−x
=⇒ AuxGau(x;µ, σµ) =
√e
√2π σµ
e−|x−µ|/σµ
C.-L. Shan, AS IoP p. 30 / 49
Outline
Monte-Carlomethods
Basic concept
Acceptance-rejection
Inverse-transform
Auxiliarydisrtributions
Markov chainMonte-Carlo
Basic concept
Metropolisalgorithm
Applications ofMC methods
Area/volumnestimation
g
Lecture on Monte-Carlo Methods
g
Second Taiwan Winter Camp onNumerical Simulation in High Energy Physics
(NumS-HEP 2012)February 13, 2012
Using auxiliary disrtribution functions
Distribution of the generated eventsGaussian distribution, µ = 10, σµ = 5, 500 events
σµ
[DISW]
C.-L. Shan, AS IoP p. 31 / 49
Outline
Monte-Carlomethods
Basic concept
Acceptance-rejection
Inverse-transform
Auxiliarydisrtributions
Markov chainMonte-Carlo
Basic concept
Metropolisalgorithm
Applications ofMC methods
Area/volumnestimation
g
Lecture on Monte-Carlo Methods
g
Second Taiwan Winter Camp onNumerical Simulation in High Energy Physics
(NumS-HEP 2012)February 13, 2012
Using auxiliary disrtribution functions
Distribution of the generated eventsGaussian distribution, µ = 10, σµ = 5, 500 events
σµ
[DISW]
C.-L. Shan, AS IoP p. 32 / 49
Outline
Monte-Carlomethods
Basic concept
Acceptance-rejection
Inverse-transform
Auxiliarydisrtributions
Markov chainMonte-Carlo
Basic concept
Metropolisalgorithm
Applications ofMC methods
Area/volumnestimation
g
Lecture on Monte-Carlo Methods
g
Second Taiwan Winter Camp onNumerical Simulation in High Energy Physics
(NumS-HEP 2012)February 13, 2012
Using auxiliary disrtribution functions
Distribution of the generated eventsGaussian distribution, µ = 10, σµ = 5, 5000 events
σµ
[DISW]
C.-L. Shan, AS IoP p. 33 / 49
Outline
Monte-Carlomethods
Basic concept
Acceptance-rejection
Inverse-transform
Auxiliarydisrtributions
Markov chainMonte-Carlo
Basic concept
Metropolisalgorithm
Applications ofMC methods
Area/volumnestimation
g
Lecture on Monte-Carlo Methods
g
Second Taiwan Winter Camp onNumerical Simulation in High Energy Physics
(NumS-HEP 2012)February 13, 2012
Using auxiliary disrtribution functions
Distribution of the generated eventsGaussian distribution, µ = 10, σµ = 5, 5000 events
σµ
[DISW]
C.-L. Shan, AS IoP p. 34 / 49
Outline
Monte-Carlomethods
Basic concept
Acceptance-rejection
Inverse-transform
Auxiliarydisrtributions
Markov chainMonte-Carlo
Basic concept
Metropolisalgorithm
Applications ofMC methods
Area/volumnestimation
g
Lecture on Monte-Carlo Methods
g
Second Taiwan Winter Camp onNumerical Simulation in High Energy Physics
(NumS-HEP 2012)February 13, 2012
Markov chain Monte-Carlo
C.-L. Shan, AS IoP p. 35 / 49
Outline
Monte-Carlomethods
Basic concept
Acceptance-rejection
Inverse-transform
Auxiliarydisrtributions
Markov chainMonte-Carlo
Basic concept
Metropolisalgorithm
Applications ofMC methods
Area/volumnestimation
g
Lecture on Monte-Carlo Methods
g
Second Taiwan Winter Camp onNumerical Simulation in High Energy Physics
(NumS-HEP 2012)February 13, 2012
Basic concept of Markov processes
C.-L. Shan, AS IoP p. 36 / 49
Outline
Monte-Carlomethods
Basic concept
Acceptance-rejection
Inverse-transform
Auxiliarydisrtributions
Markov chainMonte-Carlo
Basic concept
Metropolisalgorithm
Applications ofMC methods
Area/volumnestimation
g
Lecture on Monte-Carlo Methods
g
Second Taiwan Winter Camp onNumerical Simulation in High Energy Physics
(NumS-HEP 2012)February 13, 2012
Basic concept of Markov processes
A Markov process
generates the (n+ 1)-th event from the n-th event.
without information about the (n− 1)-th, the (n− 2)-th, ...events.
approaches the desired probability distribution functionasymptotically (with a large event number Nevent).
requires only specification of the target probabilitydistribution function.
Generated events
are (strongly) correlated.
has consequences in uncertainty estimation/error analysis.
C.-L. Shan, AS IoP p. 37 / 49
Outline
Monte-Carlomethods
Basic concept
Acceptance-rejection
Inverse-transform
Auxiliarydisrtributions
Markov chainMonte-Carlo
Basic concept
Metropolisalgorithm
Applications ofMC methods
Area/volumnestimation
g
Lecture on Monte-Carlo Methods
g
Second Taiwan Winter Camp onNumerical Simulation in High Energy Physics
(NumS-HEP 2012)February 13, 2012
Metropolis algorithm
Basic process:
1 Give/Generate a valid starting point n1 ∈ {0, 1, 2, · · · , nmax}.2 Generate n′2 ∈ {0, 1, 2, · · · , nmax} randomly.
3 Check whether n′2 ∈ [n1, µ] (n1 ≤ µ) or n′2 ∈ [µ, n1] (n1 ≥ µ).
4 “Yes” =⇒ “n2 = n′2”.
5 “No” =⇒ Generate Pcheck ∈ [0, 1] randomly.
6 Check whether Pcheck ≤ Poi(n′2;µ) / Poi(n1;µ).
7 “Yes” =⇒ “n2 = n′2”;“No” =⇒ “n2 = n1”.
8 Repeat Nevent times.
9 Draw the distribution (histogram) of N(n) (normalized byNevent) with Poi(n;µ) together.
C.-L. Shan, AS IoP p. 38 / 49
Outline
Monte-Carlomethods
Basic concept
Acceptance-rejection
Inverse-transform
Auxiliarydisrtributions
Markov chainMonte-Carlo
Basic concept
Metropolisalgorithm
Applications ofMC methods
Area/volumnestimation
g
Lecture on Monte-Carlo Methods
g
Second Taiwan Winter Camp onNumerical Simulation in High Energy Physics
(NumS-HEP 2012)February 13, 2012
Metropolis algorithm
Distribution of the generated eventsPoisson distribution, µ = 5, 500 events
σµ
[DISW]
C.-L. Shan, AS IoP p. 39 / 49
Outline
Monte-Carlomethods
Basic concept
Acceptance-rejection
Inverse-transform
Auxiliarydisrtributions
Markov chainMonte-Carlo
Basic concept
Metropolisalgorithm
Applications ofMC methods
Area/volumnestimation
g
Lecture on Monte-Carlo Methods
g
Second Taiwan Winter Camp onNumerical Simulation in High Energy Physics
(NumS-HEP 2012)February 13, 2012
Metropolis algorithm
Distribution of the generated eventsPoisson distribution, µ = 5, 500 events
σµ
[DISW]
C.-L. Shan, AS IoP p. 40 / 49
Outline
Monte-Carlomethods
Basic concept
Acceptance-rejection
Inverse-transform
Auxiliarydisrtributions
Markov chainMonte-Carlo
Basic concept
Metropolisalgorithm
Applications ofMC methods
Area/volumnestimation
g
Lecture on Monte-Carlo Methods
g
Second Taiwan Winter Camp onNumerical Simulation in High Energy Physics
(NumS-HEP 2012)February 13, 2012
Metropolis algorithm
Distribution of the generated eventsPoisson distribution, µ = 5, 5000 events
σµ
[DISW]
C.-L. Shan, AS IoP p. 41 / 49
Outline
Monte-Carlomethods
Basic concept
Acceptance-rejection
Inverse-transform
Auxiliarydisrtributions
Markov chainMonte-Carlo
Basic concept
Metropolisalgorithm
Applications ofMC methods
Area/volumnestimation
g
Lecture on Monte-Carlo Methods
g
Second Taiwan Winter Camp onNumerical Simulation in High Energy Physics
(NumS-HEP 2012)February 13, 2012
Metropolis algorithm
Distribution of the generated eventsPoisson distribution, µ = 5, 5000 events
σµ
[DISW]
C.-L. Shan, AS IoP p. 42 / 49
Outline
Monte-Carlomethods
Basic concept
Acceptance-rejection
Inverse-transform
Auxiliarydisrtributions
Markov chainMonte-Carlo
Basic concept
Metropolisalgorithm
Applications ofMC methods
Area/volumnestimation
g
Lecture on Monte-Carlo Methods
g
Second Taiwan Winter Camp onNumerical Simulation in High Energy Physics
(NumS-HEP 2012)February 13, 2012
Metropolis algorithm
Distribution of the generated eventsGaussian distribution, µ = 10, σµ = 5, 500 events
σµ
[DISW]
C.-L. Shan, AS IoP p. 43 / 49
Outline
Monte-Carlomethods
Basic concept
Acceptance-rejection
Inverse-transform
Auxiliarydisrtributions
Markov chainMonte-Carlo
Basic concept
Metropolisalgorithm
Applications ofMC methods
Area/volumnestimation
g
Lecture on Monte-Carlo Methods
g
Second Taiwan Winter Camp onNumerical Simulation in High Energy Physics
(NumS-HEP 2012)February 13, 2012
Metropolis algorithm
Distribution of the generated eventsGaussian distribution, µ = 10, σµ = 5, 500 events
σµ
[DISW]
C.-L. Shan, AS IoP p. 44 / 49
Outline
Monte-Carlomethods
Basic concept
Acceptance-rejection
Inverse-transform
Auxiliarydisrtributions
Markov chainMonte-Carlo
Basic concept
Metropolisalgorithm
Applications ofMC methods
Area/volumnestimation
g
Lecture on Monte-Carlo Methods
g
Second Taiwan Winter Camp onNumerical Simulation in High Energy Physics
(NumS-HEP 2012)February 13, 2012
Metropolis algorithm
Distribution of the generated eventsGaussian distribution, µ = 10, σµ = 5, 5000 events
σµ
[DISW]
C.-L. Shan, AS IoP p. 45 / 49
Outline
Monte-Carlomethods
Basic concept
Acceptance-rejection
Inverse-transform
Auxiliarydisrtributions
Markov chainMonte-Carlo
Basic concept
Metropolisalgorithm
Applications ofMC methods
Area/volumnestimation
g
Lecture on Monte-Carlo Methods
g
Second Taiwan Winter Camp onNumerical Simulation in High Energy Physics
(NumS-HEP 2012)February 13, 2012
Metropolis algorithm
Distribution of the generated eventsGaussian distribution, µ = 10, σµ = 5, 5000 events
σµ
[DISW]
C.-L. Shan, AS IoP p. 46 / 49
Outline
Monte-Carlomethods
Basic concept
Acceptance-rejection
Inverse-transform
Auxiliarydisrtributions
Markov chainMonte-Carlo
Basic concept
Metropolisalgorithm
Applications ofMC methods
Area/volumnestimation
g
Lecture on Monte-Carlo Methods
g
Second Taiwan Winter Camp onNumerical Simulation in High Energy Physics
(NumS-HEP 2012)February 13, 2012
Applications of Monte-Carlo methods
C.-L. Shan, AS IoP p. 47 / 49
Outline
Monte-Carlomethods
Basic concept
Acceptance-rejection
Inverse-transform
Auxiliarydisrtributions
Markov chainMonte-Carlo
Basic concept
Metropolisalgorithm
Applications ofMC methods
Area/volumnestimation
g
Lecture on Monte-Carlo Methods
g
Second Taiwan Winter Camp onNumerical Simulation in High Energy Physics
(NumS-HEP 2012)February 13, 2012
Area/volumn estimation
C.-L. Shan, AS IoP p. 48 / 49
Outline
Monte-Carlomethods
Basic concept
Acceptance-rejection
Inverse-transform
Auxiliarydisrtributions
Markov chainMonte-Carlo
Basic concept
Metropolisalgorithm
Applications ofMC methods
Area/volumnestimation
g
Lecture on Monte-Carlo Methods
g
Second Taiwan Winter Camp onNumerical Simulation in High Energy Physics
(NumS-HEP 2012)February 13, 2012
Area/volumn estimation
p
Count the total number of the valid points.nmax∑n=0
Nvalid(n) = Nevent, valid < Nevent
Estimate the area under the distribution.
A ≈Nevent, valid
Nevent×[(xmax − xmin) · fmax
]Repeat Nexpt times.
Determine the (1σ lower and upper bounds of the)mean and median values.
Compare the distribution (histogram) of Ai withAmean+ Gaussian distribution andAmedian+ double-Gaussian distribution,
C.-L. Shan, AS IoP p. 49 / 49