Monte Carlo - milidiu/inf2981/aulas/13_acceptance... A Agulha de Buffon Georges-Louis Leclerc –...
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Ruy Luiz Milidiú Monte Carlo
Transcript of Monte Carlo - milidiu/inf2981/aulas/13_acceptance... A Agulha de Buffon Georges-Louis Leclerc –...
Ruy Luiz Milidiú
Monte Carlo
Monte Carlo
Monte Carlo
Monte Carlo
A Agulha de Buffon
Georges-Louis Leclerc – Conde de Buffon, 1777
Elementos
uma agulha
um piso
Experimento
Lançar a agulha ao piso
A agulha
O piso
Lançamento
Lançamento
Lançamento
Lançamento
Contagens
Lançamentos 8
Cortes 5
Estimativa de
cortes
slançamento.2
Estimativa de
2.3
5
8.2
Corte
d
sin21
sin21d
Accept–reject
sin21d
0
0.1
0.2
0.3
0.4
0.5
0 0.5 1 1.5 2 2.5 3
Accept–reject
sin21d
0
0.1
0.2
0.3
0.4
0.5
0 0.5 1 1.5 2 2.5 3
accept
Accept–reject
sin21d
0
0.1
0.2
0.3
0.4
0.5
0 0.5 1 1.5 2 2.5 3
accept
Accept–reject
sin21d
0
0.1
0.2
0.3
0.4
0.5
0 0.5 1 1.5 2 2.5 3
accept
Accept–reject
sin21d
0
0.1
0.2
0.3
0.4
0.5
0 0.5 1 1.5 2 2.5 3
accept
21cossinaccept
21
21
021
21
021
d
P
Accept–reject
2
slançamento
cortes
2accept
P
cortes
slançamento.2
Accepted values
?accept P
0
0.1
0.2
0.3
0.4
0.5
0 0.5 1 1.5 2 2.5 3
Accepted values
cos1accept21 P
0
0.1
0.2
0.3
0.4
0.5
0 0.5 1 1.5 2 2.5 3
Accepted values
sinaccept21f
0
0.1
0.2
0.3
0.4
0.5
0 0.5 1 1.5 2 2.5 3
U(0,) D U(0,½)
cos1accept
sinaccept
accept
2accept
sinD,accept
sinDaccept
21
0 21
0
sin
0
0
sin
0
2
2
21
21
21
21
P
duuP
dudvP
dudvP
PP
PP
u
u
y = ½
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0 0.5 1 1.5 2 2.5 3
accept
y = 2.x.( - x)/
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0 0.5 1 1.5 2 2.5 3
accept
Accept–reject
032.2
021
0
2
021
32
2
2
cosaccept
.
sinaccept
P
d
dP
3accept P
ACEITA
MAIS
y = 2.x.( - x)/2
yyyg
yyyg
dvvv
.
.
3.
3
2
2
6
3
2
0
2
yyyyG 023 3213
G(y) = (3y2 – 2y3)/3
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.5 1 1.5 2 2.5 3
The Cubic formula
a
cr
a
adbcpq
a
bp
dcybyay
3
6
3
3
0
2
3
23
pprqqprqqy 3 3223 322
Inverse Transformation
032
023
323
321
1
3
uyy
yyyu
yGu
uGy
1221221
2
3 23 2 uuuuuuy
G D U(0,(/3).g())
0 21
0
sin
23
3
0 0
3
sin
23
sinaccept
3accept
,accept
sin
23
duuduugP
duugdvP
UPP
ug
u
g
ug
u
cos1accept21 P
sinaccept21DPP
G D U(0,1)
0 21
0
sin
23
3
0 0
3
sin
23
sinaccept
3accept
,accept
sin
23
duuduugP
duugdvP
UPP
ug
u
g
ug
u
cos1accept21 P
gUPP
sin
23accept
Acceptance – Rejection
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0 0.5 1 1.5 2 2.5 3
accept
y = g(x)
y = f(x)
X G D U(0,1)
00
1
0 0
1
1
accept
accept
,accept
1
duufduugMxXP
duugdvMxXP
UxXPxXP
ug
uf
M
M
Xg
Xf
M
ug
uf
M
xFxXP accept
Xg
Xf
MUxXPxXP 1accept
X G U U(0,1)
MM
xg
xf
M
Xg
Xf
M
dxxfP
dxxgP
dxxgxXUPP
11
1
1
accept
.accept
.accept
M
P 1accept
Xg
Xf
MUPP 1accept
Acceptance – Rejection
while ( TRUE )
gerar X com distribuição G
gerar U uniforme em [0,1]
if U ≤ f(X)/M.g(X)
return (X)
Want to generate X with density f(x)
The proposal density g(x) is easier to generate
X N(0,1)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
-6 -4 -2 0 2 4 6
X N(0,1)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
-6 -4 -2 0 2 4 6
M
212
21
2
21
2
21
2max2
2
2
max
max
eeM
eM
e
eM
xx
xx
x
x
eM 2
X N(0,1)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
-6 -4 -2 0 2 4 6
X meia N(0,1)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
-6 -4 -2 0 2 4 6
Acceptance – Rejection
while ( TRUE )
gerar X com distribuição G
gerar U uniforme em [0,1]
if U ≤ f(X)/M.g(X)
return (X)
f(x)/M.g(x)
212
21
2
21
.
. 2
2
xx
xe
x
exgM
xf
e
e
xgM
xf
221 1
.
xe
xgM
xf
X meia N(0,1)
while ( TRUE )
gerar X com distribuição G
gerar U uniforme em [0,1]
if U ≤ e-0.5(X-1)²
return (X)
X N(0,1)
while ( TRUE )
gerar X com distribuição exp(1)
gerar U uniforme em [0,1]
if U ≤ e-0.5(X-1)² break
gerar U uniforme em [0,1]
if U ≤ 0.5 then
return (X)
else return (–X)
