cis541 08 montecarlo - computer science and...
TRANSCRIPT
Mon
te-C
arlo
Tec
hniq
ues
Rog
er C
raw
fis
Aug
ust 1
2, 2
005
OSU
/CIS
541
2
Mon
te-C
arlo
Inte
grat
ion
•O
verv
iew
1.G
ener
atin
g Ps
uedo
-Ran
dom
Num
bers
2.M
ultid
imen
sion
al In
tegr
atio
na)
Han
dlin
g co
mpl
ex b
ound
arie
s.b)
Han
dlin
g co
mpl
ex in
tegr
ands
.
Aug
ust 1
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005
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3
Pseu
do-R
ando
m N
umbe
rs
•D
efin
ition
of r
ando
m fr
om M
erria
m-W
ebst
er:
•M
ain
Entry
: ran
dom
Func
tion:
adj
ectiv
eD
ate:
156
51
a:l
acki
ng a
def
inite
pla
n, p
urpo
se, o
r pat
tern
b:m
ade,
don
e, o
r ch
osen
at r
ando
m <
read
rand
ompa
ssag
es fr
om th
e bo
ok>
2 a
:rel
atin
g to
, hav
ing,
or b
eing
ele
men
ts o
r eve
nts w
ith d
efin
ite
prob
abili
ty o
f occ
urre
nce
<ran
dom
proc
esse
s> b
:bei
ng o
r rel
atin
g to
a
set o
r to
an e
lem
ent o
f a se
t eac
h of
who
se e
lem
ents
has
equ
alpr
obab
ility
of o
ccur
renc
e <a
rand
omsa
mpl
e>; a
lso
:cha
ract
eriz
ed b
y pr
oced
ures
des
igne
d to
obt
ain
such
sets
or e
lem
ents
<ra
ndom
sam
plin
g>
Aug
ust 1
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005
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4
Rand
om C
ompu
ter C
alcu
latio
ns?
•C
ompa
re th
is to
the
defin
ition
of a
n al
gorit
hm (d
ictio
nary
.com
):–
algo
rith
m•
n : a
pre
cise
rule
(or s
et o
f rul
es) s
peci
fyin
g ho
w to
so
lve
som
e pr
oble
m.
Aug
ust 1
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005
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/CIS
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5
Rand
om N
umbe
r
•W
hat i
s ran
dom
num
ber ?
Is 3
?–
Ther
e is
no
such
thin
g as
sing
le ra
ndom
num
ber
•R
ando
m n
umbe
r –
A se
t of n
umbe
rs th
at h
ave
noth
ing
to d
o w
ith th
e ot
her
num
bers
in th
e se
quen
ce
•In
a u
nifo
rm d
istri
butio
n of
rand
om n
umbe
rs in
th
e ra
nge
[0,1
] , e
very
num
ber h
as th
e sa
me
chan
ce o
f tur
ning
up.
–0.
0000
1 is
just
as l
ikel
y as
0.5
000
Aug
ust 1
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005
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6
Rand
om v
. Pse
udo-
rand
om
•R
ando
m n
umbe
rsha
ve n
o de
fined
sequ
ence
or
form
ulat
ion.
Thu
s, fo
r any
nra
ndom
num
bers
, ea
ch a
ppea
rs w
ith e
qual
pro
babi
lity.
•If
we
rest
rict o
urse
lves
to th
e se
t of 3
2-bi
t int
eger
s, th
en o
ur n
umbe
rs w
ill st
art t
o re
peat
afte
r som
e ve
ry la
rge
n. T
he n
umbe
rs th
us c
lum
p w
ithin
this
ra
nge
and
arou
nd th
ese
inte
gers
. •
Due
to th
is li
mita
tion,
com
pute
r alg
orith
ms a
re
rest
ricte
d to
gen
erat
ing
wha
t we
call
pseu
do-
rand
om n
umbe
rs.
Aug
ust 1
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Mon
te-C
arlo
Met
hods
•19
53, N
icol
ausM
etro
polis
•
Mon
te C
arlo
met
hod
refe
rs to
any
met
hod
that
mak
es u
se o
f ran
dom
num
bers
–Si
mul
atio
n of
nat
ural
phe
nom
ena
–Si
mul
atio
n of
exp
erim
enta
l app
arat
us
–N
umer
ical
ana
lysi
s
Aug
ust 1
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005
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/CIS
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How
to g
ener
ate
rand
om
num
bers
?
•U
se so
me
chao
tic sy
stem
(B
alls
in a
bar
rel –
Lotto
) •
Use
a p
roce
ss th
at is
inhe
rent
ly ra
ndom
–
Rad
ioac
tive
deca
y–
Ther
mal
noi
se–
Cos
mic
ray
arriv
al
•Ta
bles
of a
few
mill
ion
rand
om n
umbe
rs
•H
ooki
ng u
p a
rand
om m
achi
ne to
a c
ompu
ter.
Aug
ust 1
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005
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/CIS
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9
Pseu
do R
ando
m n
umbe
r ge
nera
tors
•Th
e cl
oses
t ran
dom
num
ber g
ener
ator
that
can
be
obta
ined
by
com
pute
r alg
orith
m.
•U
sual
ly a
uni
form
dis
tribu
tion
in th
e ra
nge
[0,1
] •
Mos
t pse
udo
rand
om n
umbe
r gen
erat
ors h
ave
two
thin
gs in
com
mon
–Th
e us
e of
larg
e pr
ime
num
bers
–
The
use
of m
odul
o ar
ithm
etic
•A
lgor
ithm
gen
erat
es in
tege
rs b
etw
een
0 an
d M
, map
to
zer
o an
d on
e.
MI
Xn
n/
=A
ugus
t 12,
200
5O
SU/C
IS 5
4110
An e
arly
exa
mpl
e (J
ohn
Von
Neu
man
n,19
46)
•To
gen
erat
e 10
dig
its o
f int
eger
–St
art w
ith o
ne o
f 10
digi
ts in
tege
rs
–Sq
uare
it a
nd ta
ke m
iddl
e 10
dig
its fr
om a
nsw
er–
Exam
ple:
5
7721
5664
92=
3331
7792
3805
9490
9291
•Th
e se
quen
ce a
ppea
rs to
be
rand
om, b
ut e
ach
num
ber i
s det
erm
ined
from
the
prev
ious
no
t ran
dom
.•
Serio
us p
robl
em :
Smal
l num
bers
(0 o
r 1) a
re lu
mpe
d to
geth
er, i
tcan
ge
t its
elf t
o a
shor
t loo
p. F
or e
xam
ple:
•61
002
= 37
2100
00•
2100
2=
0441
0000
•41
002
= 16
8100
00•
5100
2=
6561
0000
Aug
ust 1
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005
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/CIS
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11
Line
ar C
ongr
uent
ialM
etho
d
•Le
hmer
, 194
8•
Mos
t typ
ical
so-c
alle
dra
ndom
num
ber g
ener
ator
•A
lgor
ithm
:–
a,c
>=
0 , m
> I 0
,a,c
•
Adva
ntag
e :
–Ve
ry fa
st•
Prob
lem
: –
Poor
cho
ice
of th
e co
nsta
nts c
an le
ad to
ver
y po
or se
quen
ce–
The
rela
tions
hip
will
repe
at w
ith a
per
iod
no g
reat
er th
an m
(a
roun
d m
/4)
•C
com
plie
r RAN
D_M
AX :
m =
327
67
)m
od(
)(
1m
caI
In
n+
=+
Aug
ust 1
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005
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/CIS
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12
RAN
DU
Gen
erat
or
•19
60’s
IB
M•
Alg
orith
m
•Th
is g
ener
ator
was
late
r fou
nd to
hav
e a
serio
us p
robl
em
)2
mod
()
6553
9(
311
nn
II
×=
+
Aug
ust 1
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005
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13
1D a
nd 2
D D
istr
ibut
ion
of
RAN
DU
Aug
ust 1
2, 2
005
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/CIS
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14
Rand
om N
umbe
r Alg
orith
ms
•M
ost a
lgor
ithm
s ope
rate
by
calc
ulat
ing
som
e nu
mbe
r and
th
en ta
king
the
low
er-o
rder
bits
. For
exa
mpl
e: th
e cl
ass o
f m
ultip
licat
ive
cong
ruen
tialr
ando
m-n
umbe
r gen
erat
ors h
as
the
form
: . T
he c
hoic
e of
the
coef
ficie
nts i
s crit
ical
. Ex
ampl
e in
boo
k:
()
()
()
()
31
531
1
05
11
1031
102
2
1531
33
21
7m
od2
1
1 70.
0000
0782
6369
2594
2561
0890
3445
3541
5221
3e-6
7m
od2
17
0.13
1537
7881
4316
6242
2340
2060
6723
62
7m
od2
116
2265
0073
0.75
5605
3221
9503
3227
1843
3720
3942
4
nn n
n
lx l
l
l lx
lx
lx
+
=−
=−
= =⇒
=
=−
=⇒
=
=−
=⇒
=
()
()
531
44
5
7*1
6226
5007
3m
od2
198
4943
658
0.45
8650
1319
2344
9287
1553
8665
9854
74
0.53
2767
2374
1216
9220
5835
9217
8571
78
lx
x
=−
=⇒
=
=
Aug
ust 1
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005
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15
Use
of P
rim
e N
umbe
rs
•Th
e nu
mbe
r 231
–1
is a
prim
e nu
mbe
r, so
the
rem
aind
er w
hen
a nu
mbe
r is d
ivid
ed b
y a
prim
e is
ra
ther
, wel
l ran
dom
.•
Not
es o
n th
e pr
evio
us a
lgor
ithm
:–
The
l’s c
an re
ach
a m
axim
um v
alue
of t
he p
rime
num
ber.
–D
ivid
ing
by th
is n
umbe
r map
s the
inte
gers
into
real
sin
with
in th
e op
en in
terv
al (0
,1.0
).•
Why
ope
n in
terv
al?
–l 0
is c
alle
d th
e se
edof
the
rand
om p
roce
ss. W
e ca
n us
e an
ythi
ng h
ere.
Aug
ust 1
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005
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16
Oth
er A
lgor
ithm
s
•M
ultip
ly b
y a
larg
e pr
ime
and
take
the
low
er-o
rder
bits
. •
Her
e, w
e us
e hi
gher
-bi
t int
eger
s to
gene
rate
48
-bit
rand
om
num
bers
.•
Dra
nd48
()
()
481
0
1
2
3
4 5
2736
7316
3155
813
8m
od2 1
2736
7316
3169
621
6915
2289
5421
844
6648
5884
4294
1232
7642
4030
766
1624
1526
4731
678
2996
1701
4593
9051
8927
4149
3630
2517
1568
5692
926
3710
8576
9044
4616
3500
6476
2854
2
nn
xx
xx
x x x x
+=
+
==
= = = =
Aug
ust 1
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005
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17
Oth
er A
lgor
ithm
s
•M
any
mor
e su
ch a
lgor
ithm
s.
•So
me
do n
ot u
se in
tege
rs. I
nteg
ers w
ere
just
m
ore
effic
ient
on
old
com
pute
rs.
()5
1m
od1
nn
xx
π+=
+
()
18
3
2
nn
nn
q
ut
uu
x+=
−
=
t is a
ny la
rge
num
ber
Wha
t is t
his o
pera
tion?
Aug
ust 1
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005
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18
Oth
er A
lgor
ithm
s
•O
ne w
ay to
impr
ove
the
beha
vior
of
rand
om n
umbe
r gen
erat
or
)m
od(
)(
21
mI
bI
aI
nn
n−
−×
+×
=H
as tw
o in
itial
see
d an
d ca
n ha
ve a
per
iod
grea
ter
than
m
Aug
ust 1
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005
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19
The
RAN
MAR
gen
erat
or
•A
vaila
ble
in th
e C
ERN
Lib
rary
–
Req
uire
s 103
initi
al se
ed–
Perio
d : a
bout
1043
–Th
is se
ems t
o be
the
ultim
ate
rand
om n
umbe
r ge
nera
tor
Aug
ust 1
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005
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20
Prop
ertie
s of P
seud
o-Ra
ndom
N
umbe
rs
•Th
ree
key
prop
ertie
s tha
t you
shou
ld
rem
embe
r:1.
Thes
e al
gorit
hms g
ener
ate
perio
dic
sequ
ence
s (h
ence
not
rand
om).
To se
e th
is, c
onsi
der
wha
t hap
pens
whe
n a
rand
om n
umbe
r is
gene
rate
d th
at m
atch
es o
ur in
itial
seed
.
Aug
ust 1
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005
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21
Prop
ertie
s of P
seud
o-Ra
ndom
N
umbe
rs
2.Th
e re
stric
tion
to q
uant
ized
num
bers
(a fi
nite
-se
t), le
ads t
o pr
oble
ms i
n hi
gh-d
imen
sion
al
spac
e. M
any
poin
ts e
nd u
p to
be
co-p
lana
r. Fo
r ten
-dim
ensi
ons,
and
32-b
it ra
ndom
nu
mbe
rs, t
his l
eads
to o
nly
126
hype
r-pl
anes
in
10-
dim
ensi
onal
spac
e.
Aug
ust 1
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005
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22
3D D
istr
ibut
ion
from
RAN
DU
Prob
lem
s se
en w
hen
obse
rved
at t
he ri
ght
angl
e
Aug
ust 1
2, 2
005
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23
The
Mar
sagl
iaef
fect
•19
68, M
arsa
glia
•R
ando
nnu
mbe
rs fa
ll m
ainl
y in
the
plan
es
•Th
e re
plac
emen
t of t
he m
ultip
lier f
rom
65
539
to 6
9069
impr
oves
per
form
ance
si
gnifi
cant
ly
Aug
ust 1
2, 2
005
OSU
/CIS
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24
Prop
ertie
s of P
seud
o-Ra
ndom
N
umbe
rs
3.Th
e in
divi
dual
dig
its in
the
rand
om n
umbe
r m
ay n
ot b
e in
depe
nden
t. Th
ere
may
be
a hi
gher
pro
babi
lity
that
a 3
will
follo
w a
5.
Aug
ust 1
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005
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25
Avai
labl
e fu
nctio
ns
•St
anda
rd C
Lib
rary
–Ty
pe in
“m
an ra
nd”
on y
our C
IS U
nix
envi
ronm
ent.
•R
athe
r poo
r pse
udo-
rand
om n
umbe
r gen
erat
or.
•O
nly
resu
lts in
16-
bit i
nteg
ers.
•H
as a
per
iodi
city
of 2
**31
thou
gh.
–Ty
pe in
“m
an ra
ndom
” on
you
r CIS
Uni
x en
viro
nmen
t.•
Slig
htly
bet
ter p
seud
o-ra
ndom
num
ber g
ener
ator
.•
Res
ults
in 3
2-bi
t int
eger
s.•
Use
d ra
nd()
to b
uild
an
initi
al ta
ble.
•H
as a
per
iodi
city
of a
roun
d 2*
*69.
–#i
nclu
de <
stdl
ib.h
>
Aug
ust 1
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005
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26
Avai
labl
e fu
nctio
ns
•D
rand
48()
–re
turn
s a p
seud
o-ra
ndom
nu
mbe
r in
the
rang
e fr
om z
ero
to o
ne, u
sing
do
uble
pre
cisi
on.
–Pr
etty
goo
d ro
utin
e.–
May
not
be
as p
orta
ble.
Aug
ust 1
2, 2
005
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27
Initi
aliz
ing
with
See
ds
•M
ost o
f the
alg
orith
ms h
ave
som
e so
rt st
ate
that
can
be
initi
aliz
ed. M
any
times
this
is
the
last
gen
erat
ed n
umbe
r (no
t thr
ead
safe
).•
You
can
set t
his s
tate
usi
ng th
e ro
utin
es
initi
aliz
atio
n m
etho
ds (s
rand
, sra
ndom
or
sran
d48)
.–
Why
wou
ld y
ou w
ant t
o do
this
?
Aug
ust 1
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005
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28
Initi
aliz
ing
with
See
ds
•Tw
o re
ason
s to
initi
aliz
e th
e se
ed:
1.Th
e de
faul
t sta
te a
lway
s gen
erat
es th
e sa
me
sequ
ence
of r
ando
m n
umbe
rs. N
ot re
ally
ra
ndom
at a
ll, p
artic
ular
ly fo
r a sm
all s
et o
f ca
lls. S
olut
ion:
Cal
l the
seed
met
hod
with
the
low
er-o
rder
bits
of t
he sy
stem
clo
ck.
2.Y
ou n
eed
a de
term
inis
tic p
roce
ss th
at is
re
peat
able
.
Aug
ust 1
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005
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29
Initi
aliz
ing
with
See
ds
•W
e do
not
wan
t the
mou
ntai
n to
cha
nge
as
the
cam
era
mov
es.
Aug
ust 1
2, 2
005
OSU
/CIS
541
30
Map
ping
rand
om n
umbe
rs
•M
ost c
ompu
ter l
ibra
ry su
ppor
t for
rand
om
num
bers
onl
y pr
ovid
es ra
ndom
num
bers
ov
er a
fixe
d ra
nge.
•Y
ou n
eed
to m
ap th
is to
you
r des
ired
rang
e.•
Two
com
mon
cas
es:
–R
ando
m in
tege
rs fr
om z
ero
to so
me
max
imum
.–
Ran
dom
floa
ting-
poin
t or d
oubl
e-pr
ecis
ion
num
bers
map
ped
to th
e ra
nge
zero
to o
ne.
Aug
ust 1
2, 2
005
OSU
/CIS
541
31
Non
-rec
tang
ular
Are
as
•In
2D
, we
may
wan
t poi
ntsr
ando
mly
di
strib
uted
ove
r som
e re
gion
.–
Squa
re –
inde
pend
ently
det
erm
ine
xan
d y.
–R
ecta
ngle
-??
?–
Circ
le -
???
•W
rong
way
–in
depe
nden
tly d
eter
min
e r a
ndθ.
Aug
ust 1
2, 2
005
OSU
/CIS
541
32
Mon
te-C
arlo
Tec
hniq
ues
•Pr
oble
m: W
hat i
s the
pro
babi
lity
that
10
dice
thro
ws a
dd
up e
xact
ly to
32?
•E
xact
Way
. Cal
cula
te th
is e
xact
ly b
y co
untin
g al
l pos
sibl
e w
ays o
f mak
ing
32 fr
om 1
0 di
ce.
•A
ppro
xim
ate
(Laz
y) W
ay. S
imul
ate
thro
win
g th
e di
ce
(say
500
tim
es),
coun
t the
num
ber o
f tim
es th
e re
sults
add
up
to 3
2, a
nd d
ivid
e th
is b
y 50
0.
•L
azy
Way
can
get
qui
te c
lose
to th
e co
rrec
t ans
wer
qu
ite q
uick
ly.
Aug
ust 1
2, 2
005
OSU
/CIS
541
33
Mon
te-C
arlo
Tec
hniq
ues
•Sa
mpl
e A
pplic
atio
ns–
Inte
grat
ion
–Sy
stem
sim
ulat
ion
–C
ompu
ter g
raph
ics -
Ren
derin
g.–
Phys
ical
phe
nom
ena
-rad
iatio
n tra
nspo
rt –
Sim
ulat
ion
of B
ingo
gam
e–
Com
mun
icat
ions
-bi
t err
or ra
tes
–V
LSI d
esig
ns -
tole
ranc
e an
alys
is
Aug
ust 1
2, 2
005
OSU
/CIS
541
34
∫b adx
xp
)(
P(x)
ab
x
P(x)
ab
Sim
ple
Exam
ple:
.
•M
etho
d 1:
Ana
lytic
al In
tegr
atio
n•
Met
hod
2: Q
uadr
atur
e•
Met
hod
3: M
C --
rand
om sa
mpl
ing
the
area
enc
lose
d by
a<x
<b a
nd
0<y<
max
(p(x
))
⎟⎟ ⎠⎞⎜⎜ ⎝⎛
+−
≈∫
##
#)
))(
(m
ax(
)(
ab
xp
dxx
pb a
Aug
ust 1
2, 2
005
OSU
/CIS
541
35
Sim
ple
Exam
ple:
.
•In
tuiti
vely
:
⎟⎟ ⎠⎞⎜⎜ ⎝⎛
+−
≈∫
##
#)
))(
(m
ax(
)(
ab
xp
dxx
pb a
()
Prob
abili
tyBo
xAr
eay
fx
⇒•
≤
∫b adx
xp
)(
Aug
ust 1
2, 2
005
OSU
/CIS
541
36
Shap
e of
Hig
h D
imen
sion
al
Regi
on
•H
ighe
r dim
ensi
onal
shap
es c
an b
e co
mpl
ex.
•H
ow to
con
stru
ct w
eigh
ted
poin
ts in
a g
rid
that
cov
ers t
he re
gion
R ?
Prob
lem
: m
ean-
squa
re d
ista
nce
from
the
orig
in
∫∫∫∫
+>=
<dx
dy
dxdy
yx
r)
(2
22
Aug
ust 1
2, 2
005
OSU
/CIS
541
37
Inte
grat
ion
over
sim
ple
shap
e ?
0.5
0.5
22
20.
50.
50.
50.
5
0.5
0.5(
)(
,)
(,
)
dxdy
xy
sx
yr
dxdy
sx
y
++
−−
++
−−
+=∫
∫∫
∫
1
insi
de R
0 o
utsi
de R
s=⎧ ⎨ ⎩
Grid
mus
t be
fine
enou
gh !
Aug
ust 1
2, 2
005
OSU
/CIS
541
38
•In
tegr
ate
a fu
nctio
n ov
er a
co
mpl
icat
ed d
omai
n–
D: c
ompl
icat
ed d
omai
n.–
D’:
Sim
ple
dom
ain,
supe
rset
of D
.•
Pick
ran
dom
poi
nts o
ver
D’:
•C
ount
ing:
N: p
oint
s ove
r D
•N
’: po
ints
ove
r D
’
D
D’: rectangular
D
D’: circle
Mon
te-C
arlo
Inte
grat
ion
D D
Volu
me
NVo
lum
eN
′
≈′
Aug
ust 1
2, 2
005
OSU
/CIS
541
39
•T
he p
roba
bilit
y of
a r
ando
m p
oint
ly
ing
insi
de th
e un
it ci
rcle
:
•If
pic
k a
rand
om p
oint
Ntim
es a
nd
Mof
thos
e tim
es th
e po
int l
ies i
nsid
e th
e un
it ci
rcle
:
•If
Nbe
com
es v
ery
larg
e,
P=
P0→
N
M
(x,y)
Estim
atin
g π
usin
g M
onte
Car
lo
Aug
ust 1
2, 2
005
OSU
/CIS
541
40
Estim
atin
g π
usin
g M
onte
Car
lo
•R
esul
ts:
–N
=
10,0
00Pi
= 3.
1388
–N
=
100,
000
Pi=
3.14
52–
N =
1,
000,
000
Pi=
3.14
164
–N
= 1
0,00
0,00
0Pi
= 3.
1422
784
–…
Aug
ust 1
2, 2
005
OSU
/CIS
541
41
Estim
atin
g π
usin
g M
onte
Car
lo
doub
le x
, y, p
i;co
nst i
ntm
_nM
axSa
mpl
es=
1000
0000
0;in
tcou
nt=0
;fo
r (in
tk=0
; k<m
_nM
axSa
mpl
es; k
++)
x=2.
0*dr
and4
8() –
1.0;
// M
ap to
the
rang
e [-1
,1]
y=2.
0*dr
and4
8() –
1.0;
if (x
*x+y
*y<=
1.0)
cou
nt++
; pi
=4.0
* (d
oubl
e) s
/ (do
uble
)m
_nM
axSa
mpl
es;
Aug
ust 1
2, 2
005
OSU
/CIS
541
42
Stan
dard
Qua
drat
ure
•W
e ca
n fin
d nu
mer
ical
val
ue o
f a d
efin
ite
inte
gral
by
the
defin
ition
:
whe
re p
oint
s xia
re u
nifo
rmly
spac
ed.
1f(
)lim
f()
bN
ix
ia
xdx
xx
∆→∞
=
=∆
∑∫
Aug
ust 1
2, 2
005
OSU
/CIS
541
43
Erro
r in
Qua
drat
ure
•C
onsi
der i
nteg
ral i
n d
dim
ensi
ons:
•Th
e er
ror w
ith N
sam
plin
g po
ints
is
12
f()
f()
di
dX
dxdx
dxX
xΩ
≈∆
∑∫
L
1/f(
)f(
)|
|d
dX
dXX
xN
−−
∆∝
∑∫
Aug
ust 1
2, 2
005
OSU
/CIS
541
44
Mon
te C
arlo
Err
or
•Fr
om p
roba
bilit
y th
eory
one
can
show
that
th
e M
onte
Car
lo e
rror
dec
reas
es w
ith
sam
ple
size
Nas
inde
pend
ent o
f dim
ensi
on d
.
1 Nε∝
Aug
ust 1
2, 2
005
OSU
/CIS
541
45
Gen
eral
Mon
te C
arlo
•If
the
sam
ples
are
not
dra
wn
unifo
rmly
but
w
ith so
me
prob
abili
ty d
istri
butio
n P(
X), w
e ca
n co
mpu
te b
y M
onte
Car
lo:
1
1f(
)P(
)df(
)N
ii
XX
XX
N=
=∑
∫
Whe
re P
(X) i
s no
rmal
ized
, P(
)d1
XX
=∫