Fall 200 Graduae Course on Dynamc Learnng Chaper 4: Parcle Flers Sepember 27, 200 Byoung-Tak Zhang School of Compuer Scence and Engneerng & Cognve Scence and Bran Scence Programs Seoul aonal Unversy hp://b.snu.ac.kr/~bzhang/
Overvew Flerng Problem Sequenal Bayesan Flerng Parcle Fler Mone Carlo (MC Approxmaon MC wh Imporance Samplng (IS Sequenal Imporance Samplng (SIS Samplng Imporance Resamplng (SIR 2
Flerng / Trackng We wan o rack he unknown sae x of a sysem as evolves over me based on he (nosy observaons y ha arrve sequenally. y - y y + sae Observaon p(y x x - p(x x - Transon x x +
Dynamcal Sysem x s sae vecor a me, y s observaons a me Sae equaon px ( x Observaon equaon py ( x oe: The forms of px ( x and py ( x depend on he sae ranson funcon f ( and observaon funcon f (. Sae equaon: x = f ( x, u f u X X X sae ranson funcon process nose wh known dsrbuon Observaon equaon: y = f ( x, v f Y u Y observaon funcon observaon nose wh known dsrbuon Y
Flerng Problem The objecve s o esmae unknown sae x, based on a sequence of observaons y, =0,, Fnd poseror dsrbuon p( x y 0: : By knowng poseror dsrbuon (of he saes a number of esmaes can be compued, e.g. he expeced value of some funcon f(. ha depends on he sae values: E [ f ( x ] = f ( x p ( x y dx 0: 0: 0: : 0:
Le: Formally Sae vecor x0: = ( x0,..., x Observaon vecor y : = ( y,...,, y Fnd: PDF px ( 0: y: poseror dsrbuon or px ( y: flerng dsrbuon Gven: px ( 0 pror dsrbuon (on sae p( x x ranson probably (e.g., moor model py ( x observaon probably (e.g., sensor model
px ( s gven. 0 = 0, observe y. 0 py ( 0 x0 Updae px ( 0 y0 = px ( 0 (Bayes heorem py ( 0 Predc p ( x y 0 = p ( x x 0 p ( x 0 y 0 dx 0 (Markovan =, observe y from x py ( x Updae px ( y = py ( px ( Predc px ( y = px ( x px ( y dx = 2, observe y from x 2 2 2 2 py ( x 2 2 Updae px ( 2 y:2 = px ( 2 y py ( 2 y Predc px ( y = px ( x px ( y dx 3 :2 3 2 2 :2 2 = 3, observe y3 from x3 py ( 3 x2 Updae px ( 3 y3 :3 = px ( 2 y2 :2 p( y y 3 :3 Predc px ( y = px ( x px ( y dx 4 :3 4 3 3 :3 3
Sequenal Bayesan Flerng Gven px ( y... pror (flerng dsrbuon (.e., before observng y. Predcon : px ( y = px ( x px ( y dx (Eqn. : : x snce px ( y: px (, x y: dx x = x = px ( x, y px ( y dx : : noe: pa ( = pabdb (, and pab (, c = p( a b, c p ( b c 2. Updae... px = b poseror dsrbuon (afer observng y py ( x px ( y : ( y: (Eqn. 2 py ( y: where py ( y = py ( x px ( y dx : : x y
observaon y - Graphcally y y +.5 Observe (lkelhood p(y x. Predc sae p(x y :- x - p(x x - 0.5 Transon x x + p(x - y :- p(x y : 0. Sar (pror 2. Updae (poseror
A Specal Case: Kalman Fler px ( y x ( m, P : = px ( y = x ( m, P : p ( x y = ( x m, P : m = Fm P = Q + FP F... T x = Fx + v, v (0, Q... lnear and Gaussan y = H x + n, n (0, R F H : ranson marx (known : observaon marx (known
Parcle Flers Parcle fler s a echnque for mplemenng recursve Bayesan fler by Mone Carlo samplng The dea s o represen he poseror densy by a se of random samples (parcles wh assocaed weghs. Compue esmaes based on hese samples and weghs. Many dfferen names. Sequenal Mone Carlo (SMC Condensaon mehod Survval of he fes (evoluonary compuaon?
Advanages of Parcle Flers Ably o represen arbrary denses Can deal wh non-lneares on-gaussan nose Parcle flers focus adapvely on probable regons of sae space In conras, HMM flers dscreze he sae space o fxed saes. Can be mplemened n O(s s: sample sze Easy o mplemen Easy o parallelze
Sample-Based PDF Represenaon Mone Carlo characerzaon of pdf Represen poseror densy by a se of random..d. samples (parcles from he pdf p(x 0: y : For large number of parcles equvalen o funconal descrpon of pdf For, Mone Carlo mehod approaches opmal Bayesan esmae.
Mone Carlo (MC Approxmaon E [ ( ] ( ( p f x = p x f x dx f x x p x = = σ 2 (, ( (0, Mone Carlo approach. Smulae random varables from p(x, e.g. ormal dsrbuon 2. Compue he average 2 x p ( x = (0, σ E [ f( x] f ( x, [ ( ] ( p = =
MC wh Imporance Samplng Ep[ f ( x ] = p ( x f ( x dx x px ( = qx ( f ( xdx qx ( x wf x = ( x qx qx w ( ( : proposal dsrbuon px ( = w : mporance wegh qx ( oe: q ( x s easer o sample from han p( x.
Imporance Samplng (IS E[ f( x ] = f( x p( x y dx 0: 0: 0: : 0: = 0: 0: : wf( x 0: x q( x y q( x: proposal dsrbuon w = px ( y 0: : 0: y: qx ( w : mporance wegh
Imporance Samplng: Procedure ( x0: x qx ( y. Draw samples from proposal dsrbuon b q(.. ( 0: 0: : 2. Compue mporance wegh px ( y ( 0: : wx ( 0: = qx0: y: ( 3. Esmae an arbrary funcon f(.: wx ( 0: 0: : 0: = = ( j wx0: j== E [ f ( x y ] f ( x w, w (
Sequenal Imporance Samplng (SIS: Recursve Esmaon Augmenng he samples qx ( y = qx ( y qx ( x, y 0: : 0: : 0: : x q( x x, y = qx ( y qx ( x, y 0: : ( (f (cf. non-sequenal lis IS: x q( x0: y: Wegh updae w py ( x px ( x w qx ( x, y
Sequenal Imporance Samplng: Idea Updae flerng densy usng Bayesan flerng Compue negrals usng mporance samplng px ( y The flerng densy : s represened usng parcles and her weghs ( { x, w } = Compue weghs usng: w p( x, y (, ( : = qx y:
Sequenal Imporance Samplng: Procedure. Parcle generaon x q( x x, y = p( x x ( ( ( = w = ( j w j= 2. Wegh compuaon w w p( y x Wh Wegh normalzaon 3. Esmaon compuaon E[ f( x w : = y ] = f( x w oe: Sep above assumes he proposal densy o be he pror. Ths does no use he nformaon from observaons. Alernavely, he proposal densy could be x q ( x x, y = p ( x x, y ha mnzes he varance of w (Douce el al., 999.
Resamplng SIS suffers from degeneracy problems,.e. a small number of parcles have bg weghs and he res have exremely small values. Remedy: SIR nroduces a selecon (resamplng sep o elmnae samples wh low mporance raos (weghs and mulply samples wh hgh mporance raos. Resamplng maps he weghed random measure on o he equally weghed random measure by samplng unformly wh replacemen from { x 0: } = ( wh probables { } : w = { x, } { x, w ( x } ( ( ( ( 0: = 0: 0: =
Samplng Imporance Resamplng (SIR = Sequenal Mone Carlo = Parcle Fler. Inalze 0 ( - For =,..., : sample x p( x0,. 2. Imporance samplng - For =,..., : sample x q( x x, y = p( x x Le x ( x, x 0: 0: ( - For =,..., : compue weghs w = p( y x - ormalze he weghs: 3. Resamplng ( j = j= w w w - Resample wh replacemen parcles x accordng o ( 0: x 0: = x0: = x 0: = he mporance weghs w, resulng n {, }. - ew parcle populaon { } { }. - Se + and go o sep 2.
References (Sldes Bolc, M., Theory and Implemenaon of Parcle Flers, Unversy of Oawa, slde fle, 2004. Copsey, K., Tuoral on Parcle Flers, DERA Malvern, slde fle, 200. Dellaer, F., Tuoral on Mone Carlo Mehods: Par 2 - Parcle Flers, Georga Ins. of Tech., slde fle 2007. Muehlch, M., Parcle Flers: An Overvew, Unversy of Frankfur, slde fle, 2003. Pfeffer, M., A Bref Inroducon o Parcle Flers, TU Graz, slde fle, 2004.