Machine Learning for Signal Processing Prediction and Estimation, Part II
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- Reynold Webster
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1 Machine Learning fr Signal Prceing Predicin and Eimain, Par II Bhikha Raj Cla Nv Nv /8797
2 Adminirivia HW cre u Sme uden wh g really pr mark given chance upgrade Make i all he way he 50 percenile fr each prblem HW2 cre be u by nex week Pleae end u prjec updae 2 Nv /8797 2
3 Recap: An aumive example Deermine aumaically, by nly liening a running aumbile, if i i: Idling; r ravelling a cnan velciy; r Acceleraing; r Deceleraing Aume fr illurain ha we nly recrd energy level SPL in he und he SPL i meaured nce per ecnd 2 Nv /8797 3
4 he Mdel! 0.33 Px accel Px idle Idling ae 60 he ae-pace mdel Auming all raniin frm a ae are equally prbable 2 Nv / Acceleraing ae Deceleraing ae 0.33 Cruiing ae I A C D I A 0 /3 /3 /3 C 0 /3 /3 /3 D
5 Overall prcedure Predic he diribuin f he ae a =+ PS x 0:- = S S- PS - x 0:- PS S - PS x 0: = C. PS x 0:- Px S PREDIC Updae he diribuin f he ae a afer berving x UPDAE A =0 he prediced ae diribuin i he iniial ae prbabiliy A each ime, he curren eimae f he diribuin ver ae cnider all bervain x 0... x A naural ucme f he Markv naure f he mdel he predicin+updae i idenical he frward cmpuain fr HMM wihin a nrmalizing cnan 2 Nv /8797 5
6 Eimaing he ae =+ PS x 0:- = S S- PS - x 0:- PS S - PS x 0: = C. PS x 0:- Px S Predic he diribuin f he ae a Updae he diribuin f he ae a afer berving x EimaeS = argmax S PS x 0: EimaeS he ae i eimaed frm he updaed diribuin he updaed diribuin i prpagaed in ime, n he ae 2 Nv /8797 6
7 Predicing he nex bervain Predic he diribuin f he ae a =+ PS x 0:- = S S- PS - x 0:- PS S - PS x 0: = C. PS x 0:- Px S Updae he diribuin f he ae a afer berving x Predic Px x 0:- Predic x he prbabiliy diribuin fr he bervain a he nex ime i a mixure: Px x 0:- = S S Px S PS x 0:- he acual bervain can be prediced frm Px x 0:- 2 Nv /8797 7
8 Cninuu ae yem f g,, he ae i a cninuu valued parameer ha i n direcly een he ae i he piin f navlab r he ar he bervain are dependen n he ae and are he nly way f knwing abu he ae Senr reading fr navlab r recrded image fr he elecpe
9 Dicree v. Cninuu Sae Syem f g,, P Predicin a ime : O P 0:- O0:- P Updae afer O : P O : CP O0:- PO P O : CP O0:- PO 0 P O0 :- P O0:- P d 0
10 Special cae: Linear Gauian mdel A B P P 2 m m A linear ae dynamic equain Prbabiliy f ae driving erm i Gauian Smeime viewed a a driving erm m and addiive zer-mean nie A linear bervain equain Prbabiliy f bervain nie i Gauian A, B and Gauian parameer aumed knwn May vary wih ime 2 Nv / d d exp 0.5 exp 0.5 m m
11 he Linear Gauian mdel KF Ieraive predicin and updae 2 Nv /8797, ; 0 R Gauian P, ; m A Gauian P R Gauian P ˆ, ˆ ; : 0 R B B R B R B I R B B R B R B ˆ ˆ A A R R A ˆ ˆ m, ; B Gauian P R Gauian P, ; : 0 A B
12 he Kalman filer Predicin Updae 2 Nv / m ˆ A R K B I R ˆ A A R R ˆ B K ˆ B B R R B K A B
13 he Kalman filer Predicin Updae 2 Nv / m ˆ A R K B I R ˆ A R A R ˆ B K ˆ B B R R B K A B he prediced ae a ime i bained imply by prpagaing he eimaed ae a - hrugh he ae dynamic equain
14 he Kalman filer Predicin A ˆ m A B he Updae predicin i imperfec. he variance f he predicr K R= Bvariance B R B f + variance f A - Rˆ crrelaed wih R A R ˆ ˆ he w imply add becaue i n I K K B R 2 Nv / A B
15 he Kalman filer Predicin A ˆ m A B R A R ˆ We Updae can al predic he bervain frm he prediced K ae R B uing B R B he bervain equain ˆ K A B ˆ B Rˆ I K B R 2 Nv /8797 5
16 MAP Recap fr Gauian 2 Nv /8797 6, xy xx yx yy x xx yx y C C C C x C C N x y P m m If Px,y i Gauian:,, yy yx xy xx y x y x C C C C N P m m ˆ x xx yx y x C C y m m
17 MAP Recap: Fr Gauian 2 Nv /8797 7, xy xx yx yy x xx yx y C C C C x C C N x y P m m If Px,y i Gauian:,, yy yx xy xx y x x y C C C C N P m m ˆ x xx yx y x C C y m m Slpe f he line
18 he Kalman filer Predicin A ˆ m A Updae R A R ˆ A B K R B B R B hi i he lpe ˆ f he K MAP B eimar ha predic frm RB = C, BRB R + I K= CB R ˆ hi i al called he Kalman Gain 2 Nv /8797 8
19 he Kalman filer Predicin A ˆ m R A R ˆ A A B We Updae mu crrec he prediced value f he ae afer making an bervain K ˆ ˆ R B B R B K he crrecin i he difference beween R I K B R he acual bervain and he prediced bervain, caled by he Kalman Gain 2 Nv / B ˆ B
20 Predicin Updae: Updae he Kalman filer Rˆ I A ˆ K B m R A R ˆ he uncerainy in ae decreae if we berve he daa K Rand B make a crrecin he reducin i ˆ a muliplicaive K B hrinkage baed n Kalman gain and B R 2 Nv / A B R B A B ˆ B
21 he Kalman filer Predicin Updae: Updae 2 Nv / m ˆ A R K B I R ˆ A A R R ˆ B K ˆ B B R R B K A B
22 Linear Gauian Mdel P P - PO a priri raniin prb. Sae upu prb P 0 P A B P 0 O 0 C P 0 PO 0 0 O0 P 0 O0 P 0 P d P O 0: C P O 0 PO 0 2 O0: P O0: P 2 P d P 2 O 0:2 C P 2 O 0: PO All diribuin remain Gauian
23 Prblem f g,, f and/r g may n be nice linear funcin Cnveninal Kalman updae rule are n lnger valid and/r may n be Gauian Gauian baed updae rule n lnger valid 2 Nv /
24 Prblem f g,, f and/r g may n be nice linear funcin Cnveninal Kalman updae rule are n lnger valid and/r may n be Gauian Gauian baed updae rule n lnger valid 2 Nv /
25 he prblem wih nn-linear funcin f g,, P 0 :- P 0:- P d P : CP 0:- P 0 Eimain require knwledge f P Difficul eimae fr nnlinear g Even if i can be eimaed, may n be racable wih updae lp Eimain al require knwledge f P - Difficul fr nnlinear f May n be amenable cled frm inegrain 2 Nv /
26 he prblem wih nnlineariy he PDF may n have a cled frm Even if a cled frm exi iniially, i will ypically becme inracable very quickly 2 Nv / g g J P P, :,..., n n n n J g, g
27 Example: a imple nnlineariy lg exp P =? Aume i Gauian P = Gauian; m, 0 ; 2 Nv /
28 Example: a imple nnlineariy lg exp P =? 0 ; P Gauian ; m, P Gauian ; m lg exp, 2 Nv /
29 Example: A =0. lg exp ; =0 Aume iniial prbabiliy P i Gauian P 0 0 P Gauian ;, R Updae P 0 0 CP 0 0 P 0 P CGauian ; m lg exp 0, Gauian ;, R 2 Nv /
30 UPDAE: A =0. lg exp m 0; ; 0 R ; =0 P 0 P 0 0 CGauian ; m lg exp 0, Gauian ;, R m lg exp 0 lg exp C exp m R 0 = N Gauian 0 2 Nv /
31 Predicin fr = P Gauian ;0, rivial, linear ae raniin equain P Predicin Gauian ;, P 0 P 0 0 P 0 d m lg exp 0 lg exp 0 P 0 C exp m exp d R 0 = inracable 2 Nv /8797 3
32 Updae a = and laer Updae a = P : CP 0:- P 0 Inracable Predicin fr =2 P Inracable 0 :- P 0:- P d 2 Nv /
33 he Sae predicin Equain f, Similar prblem arie fr he ae predicin equain P - may n have a cled frm Even if i de, i may becme inracable wihin he predicin and updae equain Paricularly he predicin equain, which include an inegrain perain 2 Nv /
34 Simplifying he prblem: Linearize lg exp he angen a any pin i a gd lcal apprximain if he funcin i ufficienly mh 2 Nv /
35 Simplifying he prblem: Linearize lg exp he angen a any pin i a gd lcal apprximain if he funcin i ufficienly mh 2 Nv /
36 Simplifying he prblem: Linearize lg exp he angen a any pin i a gd lcal apprximain if he funcin i ufficienly mh 2 Nv /
37 Simplifying he prblem: Linearize he angen a any pin i a gd lcal apprximain if he funcin i ufficienly mh 2 Nv /
38 Linearizing he bervain funcin P Gauian, R g g J g Simple fir-rder aylr erie expanin J i he Jacbian marix Simply a deerminan fr calar ae Expanin arund a priri r prediced mean f he ae 2 Nv /
39 M prbabiliy i in he lw-errr regin P Gauian, R P i mall apprximain errr i large M f he prbabiliy ma f i in lw-errr regin 2 Nv /
40 Linearizing he bervain funcin P Gauian, R g g J g Obervain PDF i Gauian P Gauian ;0, P Gauian ; g J g, 2 Nv /
41 UPDAE. Gauian!! Ne: hi i acually nly an apprximain 2 Nv /8797 4, ;, ; R Gauian J g CGauian P g R J RJ J RJ I g RJ J RJ Gauian P g g g g g g g, ; J g g, ; J g Gauian P g P CP P, ; R Gauian P
42 Predicin? f P Gauian ;0, Again, direc ue f f can be diaru Sluin: Linearize P ; ˆ, ˆ 0: Gauian R f f ˆ ˆ ˆ J f Linearize arund he mean f he updaed diribuin f a - Which huld be Gauian 2 Nv /
43 Predicin f f ˆ ˆ ˆ J f P ; ˆ, ˆ 0: Gauian R P Gauian ;0, he ae raniin prbabiliy i nw: P ; ˆ ˆ ˆ Gauian f J f, he prediced ae prbabiliy i: P 0 :- P 0:- P d 2 Nv /
44 Predicin Gauian!! hi i acually nly an apprximain 2 Nv / , ˆ ˆ ˆ ; f J f Gauian P he prediced ae prbabiliy i: ˆ, ˆ ; 0: R Gauian P 0:- :- 0 d P P P :- 0, ˆ ˆ ˆ ; ˆ, ˆ ; f d J f Gauian R Gauian P f f J R J f Gauian P ˆ ˆ ˆ, ˆ ; 0:-
45 he linearized predicin/updae g f Given: w nn-linear funcin fr ae updae and bervain generain Ne: he equain are deerminiic nn-linear funcin f he ae variable hey are linear funcin f he nie! Nn-linear funcin f chaic nie are lighly mre cmplicaed handle 2 Nv /
46 Linearized Predicin and Updae Predicin fr ime 2 Nv / Updae a ime R Gauian P, ; :- 0 f f J R J R f ˆ ˆ ˆ ˆ R Gauian P ˆ, ˆ ; : 0 g g g g g g g R J R J J R J I R g R J J R J ˆ ˆ
47 Linearized Predicin and Updae Predicin fr ime 2 Nv / Updae a ime R Gauian P, ; :- 0 A A R R f ˆ ˆ R Gauian P ˆ, ˆ ; : 0 R B B R B R B I R g B R B R B ˆ ˆ ˆ g f J B J A
48 he Exended Kalman filer Predicin Updae 2 Nv / ˆ f R K B I R ˆ A A R R ˆ ˆ g K B B R R B K ˆ g f J B J A
49 he Kalman filer Predicin A ˆ m R A R ˆ A Updae K R B B R B ˆ K B Rˆ I K B R 2 Nv /
50 he Exended Kalman filer Predicin Updae ˆ f R A R ˆ A f g A B J f J ˆ g K R B B R B ˆ K g Rˆ I K B R 2 Nv /
51 he Exended Kalman filer Predicin f he prediced ae a ime i bained B J g imply by prpagaing he eimaed ae Updae a - hrugh he ae dynamic equain K ˆ f R A R A R B ˆ B R B f g A J ˆ ˆ K g Rˆ I K B R 2 Nv /8797 5
52 he Exended Kalman filer Predicin B J g Updae he predicin i imperfec. he variance f he predicr K R= Bvariance B R B f + variance f A - Rˆ ˆ f I K B R A R ˆ ˆ K A g A i bained by linearizing f R A f g ˆ J f 2 Nv /
53 he Exended Kalman filer Predicin ˆ f f g Updae R A R ˆ A B J g K R B B R B he Kalman gain ˆ i he K lpe g f he MAP RB = C, BRBR + I K= BC R eimar ha predic frm ˆ B i bained by linearizing g 2 Nv / g
54 he Exended Kalman filer Predicin ˆ f f g R A R ˆ We Updae can al predic he bervain frm he prediced ae uing he bervain K R B B R B equain ˆ K A g Rˆ I K B R g 2 Nv /
55 he Exended Kalman filer Predicin ˆ f f g R A R ˆ A We Updae mu crrec he prediced value f he ae afer making an bervain K ˆ R B B R B K g g he crrecin i he difference beween ˆ R I K B R he acual bervain and he prediced bervain, caled by he Kalman Gain 2 Nv /
56 he Exended Kalman filer Predicin ˆ f f g Updae R A R ˆ he uncerainy in ae decreae if we berve he daa K Rand B make a crrecin he reducin i ˆ a muliplicaive K g hrinkage baed n Kalman gain and B Rˆ I K B A B R B R B J g 2 Nv /
57 he Exended Kalman filer Predicin Updae ˆ f R A R ˆ A f g A B J f J ˆ g K R B B R B ˆ K g Rˆ I K B R 2 Nv /
58 EKF EKF are prbably he m cmmnly ued algrihm fr racking and predicin M yem are nn-linear Specifically, he relainhip beween ae and bervain i uually nnlinear he apprach can be exended include nn-linear funcin f nie a well he erm Kalman filer fen imply refer an exended Kalman filer in m cnex. Bu.. 2 Nv /
59 EKF have limiain If he nn-lineariy change quickly wih, he linear apprximain i invalid Unable he eimae i fen biaed he rue funcin lie enirely n ne ide f he apprximain Variu exenin have been prped: Invarian exended Kalman filer IEKF Uncened Kalman filer UKF 2 Nv /
60 A differen prblem: Nn-Gauian g PDF We have aumed far ha: P 0 i Gauian r can be apprximaed a Gauian P i Gauian P i Gauian f hi ha a happy cnequence: All diribuin remain Gauian Bu when any f hee are n Gauian, he reul are n happy 2 Nv /
61 Linear Gauian Mdel P P - PO a priri raniin prb. Sae upu prb P 0 P A B P 0 O 0 C P 0 PO 0 0 O0 P 0 O0 P 0 P d P O 0: C P O 0 PO 0 2 O0: P O0: P 2 P d P 2 O 0:2 C P 2 O 0: PO All diribuin remain Gauian
62 A differen prblem: Nn-Gauian PDF g f We have aumed far ha: P 0 i Gauian r can be apprximaed a Gauian P i Gauian P i Gauian hi ha a happy cnequence: All diribuin remain Gauian Bu when any f hee are n Gauian, he reul are n happy 2 Nv /
63 A imple cae P i a mixure f nly w Gauian i a linear funcin f Nn-linear funcin wuld be linearized anyway P i al a Gauian mixure! 2 Nv / B 0, ; i i i w i Gauian P m 0, ; i i i i B w Gauian B P P m P P
64 When diribuin are n Gauian P = P - = PO = a priri raniin prb. Sae upu prb P 0 P
65 When diribuin are n Gauian P = P - = PO = a priri raniin prb. Sae upu prb P 0 P P 0 O 0 C P 0 PO 0 0
66 When diribuin are n Gauian P = P - = PO = a priri raniin prb. Sae upu prb P 0 P P 0 O 0 C P 0 PO 0 0 O0 P 0 O0 P 0 P d 0
67 When diribuin are n Gauian P = P - = PO = a priri raniin prb. Sae upu prb P 0 P P 0 O 0 C P 0 PO 0 0 O0 P 0 O0 P 0 P d P O 0: C P O 0 PO 0 0
68 When diribuin are n Gauian P = P - = PO = a priri raniin prb. Sae upu prb P 0 P P 0 O 0 C P 0 PO 0 0 O0 P 0 O0 P 0 P d P O 0: C P O 0 PO 0 2 O0: P O0: P 2 P d 0
69 When diribuin are n Gauian P = P - = PO = a priri raniin prb. Sae upu prb P 0 P P 0 O 0 C P 0 PO 0 0 O0 P 0 O0 P 0 P d P O 0: C P O 0 PO 0 2 O0: P O0: P 2 P d P 2 O 0:2 C P 2 O 0: PO When PO ha mre han ne Gauian, afer nly a few ime ep
70 When diribuin are n Gauian P O0: We have many Gauian fr cmfr..
71 Relaed pic: Hw ample frm a Diribuin? Sampling frm a Diribuin Px; G wih parameer G Generae randm number uch ha he diribuin f a large number f generaed number i Px; G he parameer f he diribuin are G Many algrihm generae RV frm a variey f diribuin Generain frm a unifrm diribuin i well udied Unifrm RV ued ample frm mulinmial diribuin Oher diribuin: M cmmnly, ranfrm a unifrm RV he deired diribuin 2 Nv /8797 7
72 Sampling frm a mulinmial Given a mulinmial ver N ymbl, wih prbabiliy f i h ymbl = Pi Randmly generae ymbl frm hi diribuin Can be dne by ampling frm a unifrm diribuin 2 Nv /
73 Sampling a mulinmial.0 P P2 P3 PN Segmen a range 0, accrding he prbabiliie Pi he Pi erm will um.0 Randmly generae a number frm a unifrm diribuin Malab: rand. Generae a number beween 0 and wih unifrm prbabiliy If he number fall in he i h egmen, elec he i h ymbl 2 Nv /
74 Sampling a mulinmial P+P2.0 P+P2+P3 P P2 P3 PN Segmen a range 0, accrding he prbabiliie Pi he Pi erm will um.0 Randmly generae a number frm a unifrm diribuin Malab: rand. Generae a number beween 0 and wih unifrm prbabiliy If he number fall in he i h egmen, elec he i h ymbl 2 Nv /
75 Relaed pic: Sampling frm a Many algrihm Gauian Simple: add many ample frm a unifrm RV he um f 2 unifrm RV unifrm in 0, i apprximaely Gauian wih mean 6 and variance Fr calar Gauian, mean m, d dev : x i Malab : x = m + randn* randn draw frm a Gauian f mean=0, variance= 2 r i 6 2 Nv /
76 Relaed pic: Sampling frm a Gauian Mulivariae d-dimeninal Gauian wih mean m and cvariance Cmpue eigen value marix L and eigenvecr marix E fr = E L E Generae d 0-mean uni-variance number x..x d Arrange hem in a vecr: X = [x.. x d ] Muliply X by he quare r f L and E, add m Y = m + E qrl X 2 Nv /
77 Sampling frm a Gauian Mixure i w Gauian X; m, i Selec a Gauian by ampling he mulinmial diribuin f weigh: j ~ mulinmialw, w 2, Sample frm he eleced Gauian Gauian X; m, j j i i 2 Nv /
78 When diribuin are n Gauian P = P - = PO = a priri raniin prb. Sae upu prb P 0 P P 0 O 0 C P 0 PO 0 0 O0 P 0 O0 P 0 P d P O 0: C P O 0 PO 0 2 O0: P O0: P 2 P d P 2 O 0:2 C P 2 O 0: PO When PO ha mre han ne Gauian, afer nly a few ime ep
79 he prblem f he explding diribuin he cmplexiy f he diribuin increae expnenially wih ime hi i a cnequence f having a cninuu ae pace Only Gauian PDF prpagae wihu increae f cmplexiy Dicree-ae yem d n have hi prblem he number f ae in an HMM ay fixed Hwever, dicree ae pace are care Sluin: Cmbine he w cncep Dicreize he ae pace dynamically 2 Nv /
80 Dicree apprximain a diribuin A large-enugh cllecin f randmly-drawn ample frm a diribuin will apprximaely quanize he pace f he randm variable in equi-prbable regin We have mre randm ample frm high-prbabiliy regin and fewer ample frm lw-prbabiliy reign 2 Nv /
81 Dicree apprximain: Randm ampling A PDF can be apprximaed a a unifrm prbabiliy diribuin ver randmly drawn ample Since each ample repreen apprximaely he ame prbabiliy ma /M if here are M ample P x M M i0 x x i 2 Nv /8797 8
82 Ne: Prperie f a dicree diribuin he prduc f a dicree diribuin wih anher diribuin i imply a weighed dicree prbabiliy he inegral f he prduc i a mixure diribuin 2 Nv / M i x i x M x P 0 M i i x i x x y P x y P x P 0 M i i x i y w P dx x y P x P 0 M i i x i x w x P
83 Dicreizing he ae pace A each ime, dicreize he prediced ae pace P M 0 : i M i0 i are randmly drawn ample frm P 0: Prpagae he dicreized diribuin 2 Nv /
84 Paricle Filering P = P - = PO = a priri raniin prb. Sae upu prb predic P 0 P Auming ha we nly generae FOUR ample frm he prediced diribuin
85 Paricle Filering P = P - = PO = a priri raniin prb. Sae upu prb P 0 P ample Auming ha we nly generae FOUR ample frm he prediced diribuin
86 Paricle Filering P = P - = PO = a priri raniin prb. Sae upu prb P 0 P ample updae P 0 O 0 C P 0 PO 0 0 Auming ha we nly generae FOUR ample frm he prediced diribuin
87 Paricle Filering P = P - = PO = a priri raniin prb. Sae upu prb P 0 P ample predic P 0 O 0 C P 0 PO 0 0 O0 P 0 O0 P 0 P d 0 Auming ha we nly generae FOUR ample frm he prediced diribuin
88 Paricle Filering P = P - = PO = a priri raniin prb. Sae upu prb P 0 P predic P 0 O 0 C P 0 PO 0 0 O0 P 0 O0 P 0 P d 0 Auming ha we nly generae FOUR ample frm he prediced diribuin
89 Paricle Filering P = P - = PO = a priri raniin prb. Sae upu prb P 0 P ample updae P 0 O 0 C P 0 PO 0 0 O0 P 0 O0 P 0 P d P O 0: C P O 0 PO 0 0 Auming ha we nly generae FOUR ample frm he prediced diribuin
90 Paricle Filering P = P - = PO = a priri raniin prb. Sae upu prb P 0 P ample predic P 0 O 0 C P 0 PO 0 0 O0 P 0 O0 P 0 P d P O 0: C P O 0 PO 0 2 O0: P O0: P 2 P d 0 Auming ha we nly generae FOUR ample frm he prediced diribuin
91 Paricle Filering P = P - = PO = a priri raniin prb. Sae upu prb P 0 P ample predic P 0 O 0 C P 0 PO 0 0 O0 P 0 O0 P 0 P d P O 0: C P O 0 PO 0 2 O0: P O0: P 2 P d 0 Auming ha we nly generae FOUR ample frm he prediced diribuin
92 Paricle Filering P = P - = PO = ample a priri raniin prb. Sae upu prb updae Auming ha we nly generae FOUR ample frm he prediced diribuin P 0 P P 0 O 0 C P 0 PO 0 0 O0 P 0 O0 P 0 P d P O 0: C P O 0 PO 0 2 O0: P O0: P 2 P d P 2 O 0:2 C P 2 O 0: PO 2 2 0
93 Paricle Filering Dicreize ae pace a he predicin ep By ampling he cninuu prediced diribuin If apprpriaely ampled, all generaed ample may be cnidered be equally prbable Sampling reul in a dicree unifrm diribuin Updae ep updae he diribuin f he quanized ae pace Reul in a dicree nn-unifrm diribuin Prediced ae diribuin fr he nex ime inan will again be cninuu Mu be dicreized again by ampling A any ep, he curren ae diribuin will n have mre cmpnen han he number f ample generaed a he previu ampling ep he cmplexiy f diribuin remain cnan 2 Nv /
94 Paricle Filering P = P - = PO = ample a priri raniin prb. Sae upu prb predic updae Predicin a ime : O0 :- P O0:- P d Number f mixure cmpnen in prediced diribuin gverned by number f ample in dicree diribuin By deriving a mall number f ample a each ime inan, all diribuin are kep manageable P Updae a ime : P O : CP O0:- PO 0
95 Paricle Filering g P f P A = 0, ample he iniial ae diribuin P M P 0 0 i where i P M 0 i0 Updae he ae diribuin wih he bervain P M 0 : C P g i i i0 M 2 Nv / C i0 P g i
96 Paricle Filering Predic he ae diribuin a he nex ime Sample he prediced ae diribuin 2 Nv / f g P P 0 : 0 M i i i f P g P C P where 0: 0 : 0 i M i i P M P
97 Paricle Filering Predic he ae diribuin a Sample he prediced ae diribuin a Updae he ae diribuin a 2 Nv / f g P P 0 : 0 M i i i f P g P C P where 0: 0 : 0 i M i i P M P 0 : 0 M i i i g P C P 0 M i i g P C
98 Eimaing a ae he algrihm give u a dicree updaed diribuin ver ae: he acual ae can be eimaed a he mean f hi diribuin Alernaely, i can be he m likely ample 2 Nv / : 0 M i i i g P C P 0 ˆ M i i i g P C arg max : ˆ i i j g P j
99 Simulain wih a Linear Mdel x ha a Gauian diribuin wih 0 mean, knwn variance x ha a mixure Gauian diribuin wih knwn parameer Simulain: Generae ae equence frm mdel Generae equence f x frm mdel wih ne x erm fr every erm Generae bervain equence frm and x Aemp eimae frm
100 Simulain: Synheizing daa Generae ae equence accrding : i Gauian wih mean 0 and variance 0 2 Nv /
101 Simulain: Synheizing daa Generae ae equence accrding : i Gauian wih mean 0 and variance 0 Generae bervain equence frm ae equence accrding : x i mixure Gauian wih parameer: Mean = [-4, 0, 4, 8, 2, 6, 8, 20] Variance = [0, 0, 0, 0, 0, 0, 0, 0] Mixure weigh = [0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25] x 2 Nv /8797 0
102 Simulain: Synheizing daa Cmbined figure fr mre cmpac repreenain 2 Nv /
103 SIMULAION: IME = predic PREDICED SAE DISRIBUION A IME = 2 Nv /
104 SIMULAION: IME = predic ample SAMPLED VERSION OF PREDICED SAE DISRIBUION A IME = 2 Nv /
105 SIMULAION: IME = ample SAMPLED VERSION OF PREDICED SAE DISRIBUION A IME = 2 Nv /
106 SIMULAION: IME = ample updae UPDAED VERSION OF SAMPLED VERSION OF PREDICED SAE DISRIBUION A IME = AFER SEEING FIRS OBSERVAION 2 Nv /
107 SIMULAION: IME = updae updae, <= 2 Nv /
108 SIMULAION: IME = 2 updae predic updae, <= 2 Nv /
109 SIMULAION: IME = 2 predic updae, <= 2 Nv /
110 SIMULAION: IME = 2 predic ample updae, <= 2 Nv /8797 0
111 SIMULAION: IME = 2 ample updae, <= 2 Nv /8797
112 SIMULAION: IME = 2 ample updae updae, <= 2 Nv /8797 2
113 SIMULAION: IME = 2 updae updae, <= 2 2 Nv /8797 3
114 SIMULAION: IME = 3 updae predic updae, <= 2 2 Nv /8797 4
115 SIMULAION: IME = 3 predic updae, <= 2 2 Nv /8797 5
116 SIMULAION: IME = 3 predic ample updae, <= 2 2 Nv /8797 6
117 SIMULAION: IME = 3 ample updae, <= 2 2 Nv /8797 7
118 SIMULAION: IME = 3 ample updae updae, <= 2 2 Nv /8797 8
119 SIMULAION: IME = 3 updae he figure belw hw he cnur f he updaed ae prbabiliie fr all ime inan unil he curren inan updae, <= 3 2 Nv /8797 9
120 Simulain: Updaed Prb Unil =3 updae, <= 3 2 Nv /
121 Simulain: Updaed Prb Unil =00 updae, <= 00 2 Nv /8797 2
122 Simulain: Updaed Prb Unil =200 updae, <= Nv /
123 Simulain: Updaed Prb Unil =300 updae, <= Nv /
124 Simulain: Updaed Prb Unil =500 updae, <= Nv /
125 Simulain: Updaed Prb Unil =000 updae, <= Nv /
126 Updaed Prb Unil = 000 updae, <= Nv /
127 Updaed Prb Unil = 000 updae, <= Nv /
128 Updaed Prb: p View updae, <= Nv /
129 ESIMAED SAE 2 Nv /
130 Obervain, rue Sae, Eimae 2 Nv /
131 Paricle Filering Generally quie effecive in cenari where EKF/UKF may n be applicable Penial applicain include racking and edge deecin in image! N very cmmnly ued hwever Highly dependen n ampling A large number f ample required fr accurae repreenain Sample may n repreen mde f diribuin Sme diribuin are n amenable ampling Ue imprance ampling inead: Sample a Gauian and aign nnunifrm weigh ample 2 Nv /8797 3
132 Predicin filer HMM Cninuu ae yem Linear Gauian: Kalman Nnlinear Gauian: Exended Kalman Nn-Gauian: Paricle filering EKF are he m cmmnly ued kalman filer.. 2 Nv /
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