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onlbity `ial

Particle Filtering

Monte Carlo

zehiyl `ean

• Particle Filtering

zehiyl `ean

dxeva odilr zelgez aeyigle zeiebltzd beviil zehiy opid

Monte Carlo

zehiy

.idylkzixhnxtdgtynl zekiiy zeiebltzdd ikgipdl ilan,xnelk,zixhnxt-`

zeinpi zekxrna.zebltzddz`edylkote`abviindmb nz`ivni"rdyrpxa d

z``evnl e` edylk onfazkxrnd avnzebltzd z` zr l miaxmixwna mivex

aeyig zexyt`n

Particle Filtering

zehiy .ely divwpet ly zlgez e` ezlgez

Dynamic

zehiye zebltzd beviil

Monte Carlo

zehiy ly aeliy i"r zebltzdd

.(zeitvzaeliye)onfaavndjexriy z`m wlzpnlr

Programming

oexztyimip`iqe`bzitvzejildziyrxmrzix`pilzinpi zkxrnxearikepi`x

zehiyd .(

DP

i"r) dliri dxeva eze` `ven

Kalman Filter

de (ilnihte`) ihilp`

zeix`pild zegpd xy`k (

Kalman Filter

dn k" a miaeh) miaexiw ze`ven o"ldl

.zeni`znopi` zeip`iqe`bde

Monte Carlo

zehiyl `ean

:ze`adzeiradmrz enzdl ze`a

Mone Carlo

zehiy

.

p(x)

zebltzdjezn

{x m } M m=1 i.i.d.

mb nz`ivn.1

:

p(x)

zebltzdzgzdivwpetlyzlgez aeyig.2

Φ = E p [f (x)] =

Z

f (x)p(x)dx

z` aygl ozipy ixd

i.i.d.

mb n epi ia yie dpey`xd dirad z` epxzte d ina

i"r2diraadievxdzlgezd

Φ = ˆ 1 d

X

r

f (x r )

(2)

`ed

Φ ˆ

ly

variance

de(dhen izla nerdfxnelk)

E( ˆ Φ) = Φ

ikxexa

var( ˆ Φ) = var(f )

d = 1 d

Z

(f (x) − Φ) 2 p(x)dx

.ze`nbe dztqed mrzlgezdly oekpdjxrdaiaqfkxznjexrydxnelk

, `nle b

N

m` mby jk

x

ly

N

nina ielz epi` jexryd ly zeqpkzdd avw

ziira,z`fznerl.dievxdzlgezdlyaehjexrylzewitqnze eazenib yokzi

.dfdxwnaxzeiadywljetdldlelr dnib dz`ivn

.

p(x)

n meb l xwiyokzioii ryxazqn.

p(x)

z`

x

lklaygl mir eiep` ikgipp

zexazqddz`zaygn,

grid

lagxndz`zwlgnxy`dhiyaynzypygipp,lynl

mrelit`yixd

N = 1000

m`.zilepxazebltzdndiaewlkaznbe ediaewlkly

.

p(x)

ly miaeyig

2 1000

-l ww fpy ixd miixyt` mikxr ipy wx nin lka gwp

.( nind zllwkmizirl dre iefdira)

Importance sampling

d zhiy

.zixewndzebltzddn mb n`evnlilanzelgezdz`aygld repefdhiy

ep`y zxg` zebltzd

q(x)

idz .dpnn meb l mir ei eppi` j` dre i

p(x)

ik gipp

miyp .(

q(x) > 0

f`

p(x) > 0

m` xnelk)

supp(p) ⊆ supp(q)

edpnn meb l mir ei

ikal

E q



f (x) p(x) q(x)



= Z

f (x)q(x) p(x) q(x) dx =

Z

f (x)p(x)dx = E p [f (x)]

aeyige

q

n

{x m } M m=1

mb nzxivii"rzlgezdlyixitni`aexiwayglxyt`okl

i"raexiwd

Φ ˆ p (f ) = 1 M

M

X

m=1

f (x m ) p(x m ) q(x m )

ely

variance

de(dhen `l ner)

E p [ ˆ Φ] = Φ

ikze`xllwo`kmb

var( ˆ Φ) = 1

M E q

"



f (x) p(x) q(x)

 2 #

 E q



f (x) p(x) q(x)

 2 !

= 1

M E q

"



f (x) p(x) q(x)

 2 #

− (E p [f (x)]) 2

!

f (x) = 1

xear

var( ˆ Φ) = M 1 var  p(x) q(x)



.oekpdjxrdaiaqxzei fkexnjexryd

p

l aexw

q

ylkk,xnelk

(3)

rval ozipdf dxwna mb .(re i `l

α

e)

π(x) = αp(x)

wx mir ei ep`y gipp zrk

onqp.

importance sampling w(x) = π(x)

q(x)

yalmiyp

E q [w(x)] =

Z π(x)

q(x) q(x)dx = Z

π(x)dx = α

okl

Φ = E p (f (x) = Z

f (x)p(x)dx

= Z

f (x)q(x) p(x) q(x) dx

= 1

α Z

f (x)q(x) π(x) q(x) dx

= E q [f (x)w(x)]

α

= E q [f (x)w(x)]

E q [w(x)]

aexiwdaeyige

q

n

{x m } M m=1

mb nzxivii"r

Φ

lixitni`aexiwlawllkepyo`kn

i"r

Φ = ˆ P M

m=1 f (x m )w(x m ) P M

m=1 w(x m )

f`

M = 1

migwelmrz`f ze`xllw.dhen nerdf

E q ( ˆ Φ) = E q

 f (x 1 )w(x 1 ) w(x 1 )



= E q [f (x 1 )] 6= E p [f (x 1 )]

.

Φ

l qpkzn`ed

M → ∞

xearj`

dtixgn ef dirae

p

n dpey

q

y lkk ziziiral zkted

Imprtance sampling

d zhiy

rejection

lynl)

p

zebltzdd jezn meb l zeywrzn xy` zehiy yi .le b

N

yk

oia

tradeoff

miiwe dler dnib zxivi onf j` (

Metropolis method

-de ,

sampling

.ozeki`ezenib dxtqn

Particle Filtering

zehiyl `ean

z`xwpoklre)zeiaewxnze`xyxyl

Monte Carlo

zehiylydagxddpidefdhiy

avnd zebltzdy jkn raep

particle filtering

myd .(

Sequential Monte Carlo

mb

(4)

xy`(

particles

)miixyt`miavnzex qlylwyennmb ni"r(m ewink)zx`ezn

.(

filtering

)

dynamic programming

zxfra onfazen wzn

zinpi zkxrndpezp :diradzx bd

x k = f (x k−1 , ν k−1 ) y k = h(x k , µ k )

e` izlgzddavnd re i oke mire i(dn`zda)

i.i.d.

zitvze jildz iyrx

ν, µ

yk

mipezp.xnelk.dk rzeitvzozpdaavndzebltzdz``evnlepipevxa.ezebltzd

p(x 0 ) , p(x k |x k−1 ) , p(y k |x k )

zitvzdejildzd iyrxezix`pil dppi`zkxrndy oeeik .

p(x k |y 1:k )

z``evnl yie

zeiebltzdlyzixhnxtdgtynndidz

p(x k |y 1:k )

ydaiqlkoi`mip`iqe`bmpi`

.(dl`kodzepzendzeiebltzddm`mb)dre i

itxbddpandnmiraepdmiwexitayeniyjez

p(x 0:k |y 1:k )

xeardbiqpzgqepgztp

.(zeiaewxnd)

(zeiaewxne)

Bayes

llkitlr.

p(x 0:k−1 |y 1:k−1 )

z`ep`vnikgipp

p(x 0:k |y 1:k ) = p(y k |x 0:k , y 1:k−1 )p(x 0:k |y 1:k−1 )

p(y k |y 1:k−1 )

= p(y k |x 0:k , y 1:k−1 )p(x k |x 0:k−1 , y 1:k−1 )p(x 0:k−1 |y 1:k−1 ) p(y k |y 1:k−1 )

= p(y k |x k )p(x k |x k−1 )p(x 0:k−1 |y 1:k−1 ) p(y k |y 1:k−1 )

∝ p(y k |x k )p(x k |x k−1 )p(x 0:k−1 |y 1:k−1 ) (1)

yjk

q

zebltzdxgap

q(x 0:k |y 1:k ) = q(x k |x 0:k−1 , y 1:k )q(x 0:k−1 |y 1:k−1 ) (2)

xi bp

w i = p(x i 0:k |y 1:k ) q(x i 0:k |y 1:k )

lawpe

w i

lydx bda2e1ze`eeyn z`alyp

˜

w i = p(y k |x i k )p(x i k |x i k−1 )p(x i 0:k−1 |y 1:k−1 ) q(x k |x 0:k−1 , y 1:k )q(x 0:k−1 |y 1:k−1 )

= p(y k |x i k )p(x i k |x i k−1 )

q(x k |x 0:k−1 , y 1:k ) w i k−1 (3)

mr giay

{x i 0:k } M i=1

miwiwlgepizeyxayi

k

d rvayeplaiw.

w i = P w ˜

i

j

w ˜

j xi bpe

oexg`davndlrzebltzdd .zex qd lrzebltzddz`mibviinmdlyzelewynd

(5)

p(x k |y 1:k ) = lim

M→∞

M

X

i=1

w i k δ(x k − x i k )

f` iaewxn

p

m` .jexryd aiharixkn jxr yi

p

l aexwy

q

zxigal ,xn`py itk

dxiga ef

p(x k |x k−1 )

z` mir ei ep`y oeeik .

q(x k |x k−1 , y k )

dxevdn

q

`evnl i

-ynly mevnvlawznefdxigaxear.(dpnnmeb lozipe d ina)

q

xearzixletet

:l3d`ee

˜

w i = p(y k |x i k )w k−1 i

zelewyndoeeipziira

zepeepznzelewyndmi rvxtqnxg`l:d`addiraadwell"pddhiydlyyenin

dyrnl.0lx`yde1ls`ey(z eazlewynelit`ile`)zelewyndnohwwlgyjk

q

y lkk dtixgn dirad . rv lka l b zelewynd ly

variance

d ik gikedl ozip

i"r llweynd mb nd jezn mb n zxivi i"r `id z`f repnl zg` jx .

p

n dpey

y g mb n zlawl ,(zexfgd mr) mb nd jezn (zelewynd t"r) zllweyn dnib

m`zdaminlrpwlge miltkeyn

particles

dnwlg,dyrnl)ze ig` zelewyn lra

Sequential Importance Resamplling

z`xwpefdhiy.(zelewynl

:(y gn dnib dllek)x`ezymzixebl`do"ldl

Algorithm 1 Particle filter (SIR) [{x i k , w i k } M i=1 ] = P F ({x i k−1 , w i k−1 } M i=1 , y k )

• for i=1:M

– draw x i k ∼ q(x k |x i k−1 , y k ) – assign ˜ w i k = p(y q(x

k

|x

ik

)p(x

ik

|x

ik−1

)

k

|x

0:k−1

,y

1:k

) w k−1 i ,

• for each i, w i = P w ˜

i

j

w ˜

j

• optionally, [{¯ x i , w ¯ i } M i=1 ] = RESAM P LE({x i , w i } M i=1 )

(6)

Algorithm 2 Resample

[{¯ x i , w ¯ i } M i=1 ] = RESAM P LE({x i , w i } M i=1 )

• for each i in {0, . . . , M }, c i = P i j=1 w i

• for each i in {1, . . . , M }

– u i = unif orm random in [0, 1]

– x ¯ i = x j such that c j−1 ≤ u i < c j

– w ¯ i = M 1

:d`ad zkxrndxear

E(x k )

z`aygl epilr:`nbe ldira

x k = x k−1

2 + 25x k−1

1 + x 2 k−1 + 8cos(1.2k) + ν k−1

y k = x 2 k 20 + µ k

ν ∼ N (0, 10) µ ∼ N (0, 1)

lydnbe oldl.

SIR

mzixebl`aynzype

N (x i k−1 , 10)

zeidl

q(x i k |x i k−1 )

z`xi bp

:mzixebl`dzvix

mipekpdmikxrd

0 10 20 30 40 50

−3

−2

−1 0 1 2 3

Measurement Noise

Time

0 10 20 30 40 50

−6

−4

−2 0 2 4 6 8

Process Noise

Time

0 10 20 30 40 50

−25

−20

−15

−10

−5 0 5 10 15 20

State x

Time

0 10 20 30 40 50

−5 0 5 10 15 20 25

Observation y

Time

(7)

0 5 10 15 20 25 30 35 40 45 50

−25

−20

−15

−10

−5 0 5 10 15 20

State x

Time

0 5 10 15 20 25 30 35 40 45 50

−20

−15

−10

−5 0 5 10 15 20 25 30

Output y

Time

0 5 10 15 20 25 30 35 40 45

−40

−30

−20

−10 0 10 20 30

Time

Sequential state estimate

−300 −20 −10 0 10 20 30

0.005 0.01 0.015 0.02 0.025 0.03

Likelihood function

Hidden state support Posterior mean estimate

True value

particles and their weights smoothed likelyhood posterior mean posterior max (MAP)

xzeiadaxezetitvyavndzxigalavndzlgezzxigaoiad`eeyde,dvixdmekiq

divwi xtk

0 5 10 15 20 25 30 35 40 45 50

−25

−20

−15

−10

−5 0 5 10 15 20

Time

State estimate

True value Posterior mean estimate MAP estimate

−25 −20 −15 −10 −5 0 5 10 15 20 25

−25

−20

−15

−10

−5 0 5 10 15 20 25

True state

Posterior mean estimate

−25 −20 −15 −10 −5 0 5 10 15 20 25

−25

−20

−15

−10

−5 0 5 10 15 20 25

True state

MAP estimate

−20

−10 0

10 20

0 10 20 30 40 50

0 100 200 300 400 500

Sample space Time

Posterior density

References

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