Foundaons of Sae Esmaon Par II Tocs: Hdden Markov Models Parcle Flers Addonal readng: L.R. Rabner, A uoral on hdden Markov models," Proceedngs of he IEEE, vol. 77,. 57-86, 989. Sequenal Mone Carlo Mehods n Pracce. A. Douce, N. de Freas, N. Gordon eds. Srnger-Verlag, 00. Radford M. Neal, 993. Probablsc Inference Usng Markov Chan Mone Carlo Mehods. Unversy of Torono CS Tech Reor. Robus Mone Carlo Localzaon for Moble Robos. S. Thrun, D. Fo, W. Burgard and F. Dellaer. Arfcal Inellgence. 8:-, 99-4 00. Hdden Markov Models Acons a Observable Hdden Belefs Observaons Oz b Z T a, b Z Saes Dscree saes, acons and observaons f,,, h, can now be wren as ables
Somewha Useful for Localzaon n Toologcal Mas :,a.9 3 : 3,a.05 4 : 4,a.05 Observaons can be feaures such as corrdor feaures, uncon feaures, ec. Belef Trackng Esmang s now easy Afer each acon a and observaon z, X, udae : z a, ' X ' Ths algorhm s quadrac n X. Recall ha Kalman Fler s quadrac n number of sae feaures. Connuous X means nfne number of saes.
The Three Basc Problems for HMMs Gven he hsory Oa,z,a,z,...,a T,z T, and a model λa,b,π, how do we effcenly comue POλ, he robably of he hsory, gven he model? Gven he hsory Oa,z,a,z,...,a T,z T and a model λ, how do we choose a corresondng sae sequence X,,..., T whch s omal n some meanngful sense.e., bes elans he observaons? 3 How do we adus he model arameers λa,b,π o mamze POλ? HMM Basc Problem Probably of hsory O gven λ s sum over all sae sequences Q,, 3,..., T,: P O λ all Q all q, q P O Q, λ P Q λ π,... z Summng over all sae sequences s T X T Insead, buld lace of saes forward n me, comung robables of each ossble raecory as lace s bul Forward algorhm s X T, a z 3, a...
HMM Basc Problem π α, X + + α α X α λ 3 Termnaon 4 Back rackng HMM Basc Problem algorhm, wh an era erm Inalzaon Inducon 0 ψ π δ [ ] [ ],, ma X X δ ψ δ δ [ ] [ ] ma * P T T T δ δ * * + + ψ. Inalzaon. Inducon: Reea for :T 3. Termnaon: z z a O Verb Decodng: Same rncle as forward z ma arg a z a arg ma X X Imlemenaon of he comuaon of n erms of a lace of observaons, and saes. Observaon, Sae 3 T N α
HMM Basc Problem 3 Gven labelled daa sequence, D{,a,z,,a,z,..., T,a T,z T }, esmang z, and,a k s us counng Gven unlabelled daa sequence, D{a,z,a, z,...,a T,z T }, esmang z, and,a k s equvalen o smulaneous localzaon and mang ne lecure Parcle Flers
Mone Carlo Localzaon: The Parcle Fler Sae Sace Samle arcles randomly from dsrbuon Carry around arcles, raher han full dsrbuon Samlng from unform dsrbuons s easy Samlng from Gaussans and oher arameerc dsrbuons s a lle harder Wha abou arbrary dsrbuons? Many algorhms Reecon samlng Imorance samlng Gbbs samlng Merools samlng. How o samle Wan o samle from arbrary Don know Do know for any secfc Do know how o samle from q Samle from q Comare q o Adus samles accordngly
Reecon Samlng Sae Sace Samle from an easy funcon Sae Sace Comue reecon rao: α /cq Sae Sace Kee arcles wh robably α, reec wh robably -α Samle Imorance Resamlng Sae Sace Samle from an easy funcon Sae Sace Comue morance weghs Sae Sace Resamle arcles from arcle se, accordng o morance weghs
Robo Localzaon usng SIR I. Samle { } from, y, θ II. Ierae: Samle Predcon from moon model accordng o acon a, o ge roosal dsrbuon q Comue morance weghs w q Measuremen 3 Resamle from { } accordng o {w } Samlng from Moon Model A common moon model: Decomose moon no roaon, ranslaon, roaon Roaon: µ θ, σ θ α d+ α θ Translaon: µ d, σ d α 3 d+ α 4 θ + θ Roaon: µ θ,σ θ α d+ α θ Comue roaon, ranslaon, roaon from odomery For each arcle, samle a new moon rle by from Gaussans descrbed above Use geomery o generae oseror arcle oson
Sensor Model 0.5 Aromaed Measured Probably y, 0. Eeced dsance 0.075 0.05 0.05 0 00 00 300 400 500 Measured dsance y [cm] Sensor Model Laser model bul from colleced daa Laser model fed o measured daa, usng aromae geomerc dsrbuon
Problem How o comue eeced dsance for any gven, y, θ? Ray-racng Cached eeced dsances for all, y, θ. Aromaon: Assume a symmerc sensor model deendng only on d: absolue dfference beween eeced and measured ranges Comue eeced dsance only for, y Much faser o comue hs sensor model Only useful for hghly-accurae range sensors e.g., laser range sensors, bu no sonar Comung Imorance Weghs Aromae Mehod Off-lne, for each emy grd-cell, y Comue d, y he dsance o neares flled cell from, y Sore hs eeced dsance ma A run-me, for a arcle, y and observaon z r, θ Comue end-on, y +rcosθ,y+rsnθ Rereve d, y, error n measuremen Comue robably of error, d, from Gaussan sensor model of secfc σ
Bayes Flers Kalman fler Unmodal Gaussan HMM Lnear-Gaussan moon and sensor models Daa assocaon roblem Quadrac n number of sae feaures Dscree mulmodal dsrbuon Arbrary moon and sensor models Quadrac n number of saes Parcle Flers Arbrary dsrbuons Arbrary moon and sensor models Eonenal n number of sae feaures Wha you should know Kalman Mulhyohess rackng Grd HMM Toology Parcle Belef Unmodal Mulmodal Dscree Dscree Non-aramerc or dscree Accuracy + + 0 - + Robusness 0 + + + + Sensor varey - - + 0 + Effcency + 0-0 0 Imlemenaon 0-0 0 +
Wha you should know Wha a Hdden Markov Model s The Forward algorhm The Verb algorhm How o mlemen arcle flerng Pros and cons of arcle flers How o mlemen robo localzaon usng arcle flers