Lecture on -0.05cm gMonte-Carlo Methodsg -0numscc/nums-hep2012/materials/... · 2012. 2. 20. ·...

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


(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


Inverse-transform



Basic concept

Metropolisalgorithm



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Monte-Carlo methods


Outline

Monte-Carlomethods

Basic concept


Inverse-transform



Basic concept

Metropolisalgorithm



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Basic concept of Monte-Carlo methods


Outline

Monte-Carlomethods

Basic concept


Inverse-transform



Basic concept

Metropolisalgorithm



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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]


Outline

Monte-Carlomethods

Basic concept


Inverse-transform



Basic concept

Metropolisalgorithm



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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


Outline

Monte-Carlomethods

Basic concept


Inverse-transform



Basic concept

Metropolisalgorithm



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Distribution of the dice points6 points, equal probability, 500 experiments, 6000 times

[DISW]


Outline

Monte-Carlomethods

Basic concept


Inverse-transform



Basic concept

Metropolisalgorithm



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Distribution of the dice points6 points, modified probabilities, 500 experiments, 6000 times

[DISW]


Outline

Monte-Carlomethods

Basic concept


Inverse-transform



Basic concept

Metropolisalgorithm



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[DISW]

1 2 3 4 5 6

1

2

3

4

√√√√√√

√√√ √√√√√√

××

×××××

××


Outline

Monte-Carlomethods

Basic concept


Inverse-transform



Basic concept

Metropolisalgorithm



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[DISW]

1 2 3 4 5 6

0.25

0.50

0.75

1.00


Outline

Monte-Carlomethods

Basic concept


Inverse-transform



Basic concept

Metropolisalgorithm



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Acceptance-rejection method


Outline

Monte-Carlomethods

Basic concept


Inverse-transform



Basic concept

Metropolisalgorithm



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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)


Outline

Monte-Carlomethods

Basic concept


Inverse-transform



Basic concept

Metropolisalgorithm



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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.


Outline

Monte-Carlomethods

Basic concept


Inverse-transform



Basic concept

Metropolisalgorithm



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Distribution of the generated eventsPoisson distribution, µ = 5, 500 events

σµ

[DISW]


Outline

Monte-Carlomethods

Basic concept


Inverse-transform



Basic concept

Metropolisalgorithm



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σµ

[DISW]


Outline

Monte-Carlomethods

Basic concept


Inverse-transform



Basic concept

Metropolisalgorithm



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σµ

[DISW]


Outline

Monte-Carlomethods

Basic concept


Inverse-transform



Basic concept

Metropolisalgorithm



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σµ

[DISW]


Outline

Monte-Carlomethods

Basic concept


Inverse-transform



Basic concept

Metropolisalgorithm



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Distribution of the generated eventsGaussian distribution, µ = 10, σµ = 5, 500 events

σµ

[DISW]


Outline

Monte-Carlomethods

Basic concept


Inverse-transform



Basic concept

Metropolisalgorithm



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σµ

[DISW]


Outline

Monte-Carlomethods

Basic concept


Inverse-transform



Basic concept

Metropolisalgorithm



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g





σµ

[DISW]


Outline

Monte-Carlomethods

Basic concept


Inverse-transform



Basic concept

Metropolisalgorithm



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σµ

[DISW]


Outline

Monte-Carlomethods

Basic concept


Inverse-transform



Basic concept

Metropolisalgorithm



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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 )


Outline

Monte-Carlomethods

Basic concept


Inverse-transform



Basic concept

Metropolisalgorithm



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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)


Outline

Monte-Carlomethods

Basic concept


Inverse-transform



Basic concept

Metropolisalgorithm



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Inverse-transform method


Outline

Monte-Carlomethods

Basic concept


Inverse-transform



Basic concept

Metropolisalgorithm



g


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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


Outline

Monte-Carlomethods

Basic concept


Inverse-transform



Basic concept

Metropolisalgorithm



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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


x(P;λ) = −1

λln (1− P) =⇒ x(P;λ) = −

1

λln P


Outline

Monte-Carlomethods

Basic concept


Inverse-transform



Basic concept

Metropolisalgorithm



g


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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


Outline

Monte-Carlomethods

Basic concept


Inverse-transform



Basic concept

Metropolisalgorithm



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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

)


x(P;α) = α tan

[(P−

1

2

)π

]


Outline

Monte-Carlomethods

Basic concept


Inverse-transform



Basic concept

Metropolisalgorithm



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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)


Outline

Monte-Carlomethods

Basic concept


Inverse-transform



Basic concept

Metropolisalgorithm



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Using auxiliary disrtribution functions


Outline

Monte-Carlomethods

Basic concept


Inverse-transform



Basic concept

Metropolisalgorithm



g


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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


Outline

Monte-Carlomethods

Basic concept


Inverse-transform



Basic concept

Metropolisalgorithm



g


g





σµ

[DISW]


Outline

Monte-Carlomethods

Basic concept


Inverse-transform



Basic concept

Metropolisalgorithm



g


g





σµ

[DISW]


Outline

Monte-Carlomethods

Basic concept


Inverse-transform



Basic concept

Metropolisalgorithm



g


g





σµ

[DISW]


Outline

Monte-Carlomethods

Basic concept


Inverse-transform



Basic concept

Metropolisalgorithm



g


g





σµ

[DISW]


Outline

Monte-Carlomethods

Basic concept


Inverse-transform



Basic concept

Metropolisalgorithm



g


g



Markov chain Monte-Carlo


Outline

Monte-Carlomethods

Basic concept


Inverse-transform



Basic concept

Metropolisalgorithm



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Basic concept of Markov processes


Outline

Monte-Carlomethods

Basic concept


Inverse-transform



Basic concept

Metropolisalgorithm



g


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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.


Outline

Monte-Carlomethods

Basic concept


Inverse-transform



Basic concept

Metropolisalgorithm



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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.


Outline

Monte-Carlomethods

Basic concept


Inverse-transform



Basic concept

Metropolisalgorithm



g


g





σµ

[DISW]


Outline

Monte-Carlomethods

Basic concept


Inverse-transform



Basic concept

Metropolisalgorithm



g


g





σµ

[DISW]


Outline

Monte-Carlomethods

Basic concept


Inverse-transform



Basic concept

Metropolisalgorithm



g


g





σµ

[DISW]


Outline

Monte-Carlomethods

Basic concept


Inverse-transform



Basic concept

Metropolisalgorithm



g


g





σµ

[DISW]


Outline

Monte-Carlomethods

Basic concept


Inverse-transform



Basic concept

Metropolisalgorithm



g


g





σµ

[DISW]


Outline

Monte-Carlomethods

Basic concept


Inverse-transform



Basic concept

Metropolisalgorithm



g


g





σµ

[DISW]


Outline

Monte-Carlomethods

Basic concept


Inverse-transform



Basic concept

Metropolisalgorithm



g


g





σµ

[DISW]


Outline

Monte-Carlomethods

Basic concept


Inverse-transform



Basic concept

Metropolisalgorithm



g


g





σµ

[DISW]


Outline

Monte-Carlomethods

Basic concept


Inverse-transform



Basic concept

Metropolisalgorithm



g


g



Applications of Monte-Carlo methods


Outline

Monte-Carlomethods

Basic concept


Inverse-transform



Basic concept

Metropolisalgorithm



g


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Area/volumn estimation


Outline

Monte-Carlomethods

Basic concept


Inverse-transform



Basic concept

Metropolisalgorithm



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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,


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Lecture on -0.05cm gMonte-Carlo Methodsg -0numscc/nums-hep2012/materials/... · 2012. 2. 20. ·...

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