Quantifying Uncertainty

Transcription

1 Partcle Flters Quantfyng Uncertanty Sa Ravela M. I. T Last Updated: Sprng Quantfyng Uncertanty

2 Partcle Flters Partcle Flters Appled to Sequental flterng problems Can also be appled to smoothng problems Soluton va Recursve Bayesan Estmaton Approxmate Soluton Can work wth non-gaussan dstrbutons/non-lnear dynamcs Applcable to many other problems e.g. Spatal Inference 2 Quantfyng Uncertanty

3 Partcle Flters Notaton x t, X k : Models states n contnuous and dscrete space-tme respectvely. xt T : True system state y t,y k : Contnous and Dscrete measurements, respectvely. Xk n : n th sample of dscrete vector at step k. M: model, P: probablty mass functon. Q: Proposal Dstrbuton, δ : kronecker or drac delta functon. We follow Arulampalam et al. s paper. Non-Gaussanty Samplng SIS Kernel SIR RPF 3 Quantfyng Uncertanty

4 Partcle Flters Sequental Flterng Recall: Ensemble Kalman flter & Smoother y 1 y 2 Observatons x 0 x 1 x 2 Model States We are nterested n studyng the evoluton of y t f (x T ), observed t system, usng a model wth state x t. 4 Quantfyng Uncertanty

5 Partcle Flters Ths means (n dscrete tme, dscretzed space): P(X k Y 1:k ) Can be solved recursvely step P(X k, Y 1:k ) P(X k Y 1:k ) = P(Y 1:k ) 5 Quantfyng Uncertanty

6 Partcle Flters Sequental Flterng va Recursve Bayesan Estmaton Y 1:k s a collecton of varables Y 1... Y k So: P(X k Y 1:k ) = P(X k, Y 1:k ) P(Y 1:k ) = P(Y k X k )P(X k Y k ) P(Y 1:k 1) P(Y k Y 1:k 1 ) P(Y 1:k 1) = P(Y k X k )P(X k Y 1:k 1 ) P(Y k Y 1:k 1 ) 6 Quantfyng Uncertanty

7 Partcle Flters Contd. P(Y k X k ) P(X k X k 1 )P(X k 1 Y 1:k 1 ) _ X 2 k 1 _ 1 P(X k Y 1:k ) = P(Yk X k )P(X k X k 1 )P(X k 1 Y k 1 ) X k X k 1 _ 1. From the Chapman-Kolmogorov equaton 2. The measurement model/observaton equaton 3. Normalzaton Constant When can ths recursve master equaton be solved? 3 7 Quantfyng Uncertanty

8 Partcle Flters Let s say X k = F k X k 1 + V k Z k = H k X k + η k v k = N(, P k k ) η k = N(0, R) Lnear Gaussan Kalman Flter 8 Quantfyng Uncertanty

9 Partcle Flters For non lnear problems Extended Kalman Flter, va lnearzaton Ensemble Kalman flter No lnearzaton Gaussan assumpton Ensemble members are partcles that moved around n state space They represent the moments of uncertanty 9 Quantfyng Uncertanty

10 Partcle Flters How may we relax the Gaussan assumpton? If P(X k X k 1 ) and P(Y k X k ) are non-gaussan; How do we represent them, let alone perform these ntegratons n (2) & (3)? 10 Quantfyng Uncertanty

11 Partcle Flters Partcle Representaton Genercally N P(X) = w δ(x X ) =1 pmf/pdf defned as a weghted sum Recall from Samplng lecture Response Surface Modelng lecture 11 Quantfyng Uncertanty

12 Partcle Flters Contd. w 1 w 2 w 10 X 1 X 2 X 10 Even so, Whlst P(X) can be evaluated samplng from t may be dffcult. 12 Quantfyng Uncertanty

13 Partcle Flters Importance Samplng Suppose we wsh to evaluate x x f (x)p(x)dx (e.g. moment calculaton) P(x) f (x) Q(x)dx, X Q(x) Q(x) N 1 P(x = X ) = f (x = X )w, w = N Q(x = X ) =1 13 Quantfyng Uncertanty

14 Partcle Flters So: Sample from Q Proposal dstrbuton Evaluate from P the densty P(X ) Apply mportance weght = w = Q(X ) Now let s consder P(x) = Pˆ(x) P Q (x) = P(x)dx Q Z p Q(x) = Qˆ (x) Q(x) Q = QQ(x)dx Z q 14 Quantfyng Uncertanty

15 Partcle Flters So: N 1 Z q f (x = X )wq N Z p =1 where QP(x = X ) wq = These are un-normalzed mere potentals Q(x Q = X ) It turns out: NZ p = ŵ Z q N w =1 f (x = X ) ˆ f (x)p(x)dx = N j ŵ j 15 Quantfyng Uncertanty

16 Partcle Flters f (X )ŵ j ŵ j s just a weghted sum Where a proposal dstrbuton was used to get around samplng dffcultes and the mportance weghts manage all the normalzaton. It s mportant to select a good proposal dstrbuton. Not one that focus on a small part of the state space and perhaps better than an unnformatve pror. 16 Quantfyng Uncertanty

17 Partcle Flters Applcaton of Importance Samplng to Bayesan Recursve Estmaton Partcle Flter ŵ δ(x X ) P(X ) = = w δ(x X ) j ŵ j w s normalzed. 17 Quantfyng Uncertanty

18 Partcle Flters Let s consder agan: X k = f (X k 1 ) + V k Y k = h(x k ) + η k A relatonshp between the observaton and the state (measurement) Addtve nose, but can be generalzed 18 Quantfyng Uncertanty

19 Partcle Flters Let s consder the jont dstrbuton P(X 0:k Y 1:K ) Y 1 Y k X 0 X 1 X k IC We may factor ths dstrbuton usng partcles 19 Quantfyng Uncertanty

20 Partcle Flters Chan Rule wth Weghts And let s factor P(X 0:k Y 1:k ) as N P(X 0:k Y 1:k ) = w δ(x 0:k X 0 :k ) =1 P(X 0 :k Y 1:k ) w Q(X0 :k Y 1:k ) P(Y k X 0:k, Y 1:k 1 )P(X 0:k Y 1:k 1 ) P(X 0:k Y 1:k ) = P(Y k Y 1:k 1 ) P(Y k X k )P(X k X k 1 )P(X k 1 Y 1:k 1 ) = P(Y k Y 1:k ) 20 Quantfyng Uncertanty

21 Partcle Flters Proposal Dstrbuton Propertes Suppose we pck 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 ).e. there s some knd of recurson on the proposal dstrbuton. Further, f we approxmate.e. there s a Markov property. Q(X k X 0:k 1, Y 1:k ) = Q(X k X k 1, Y k ) 21 Quantfyng Uncertanty

22 Partcle Flters Recursve Weght Updates Then we may have found an update equaton for the weghts. P(X 0:k Y 1:k ) P(Y k X k )P(X k X k 1 )P(X 0:k 1, Y 1:k 1 ) = Q(X 0:k Y 1:k ) P(Y k Y 1:k 1 )Q(X k X k 1, Y k )Q(X 0:k 1 Y 1:k 1 ) P(Y k X k )P(X k X w k 1 = ) P(X 0:k 1, Y 1:k 1 ) k Q(X k X k 1, Y k )P(Y k Y 1:k 1 ) Q(X 0:k 1, Y 1:k 1 ) P(Y k X k )P(X k Xk 1 = ) Q(Xk X k 1, Y k )P(Y k Y 1:k 1 ) w k 1 P(Y k X k )P(X k Xk 1 ) w Q(X k X k 1, Y k ) 22 k 1 Quantfyng Uncertanty

23 Partcle Flters The Partcle Flter In the flterng problem P(X k Y 1:k ) P(Y k X k )P(X k X k 1 ) w k w k 1 Q(X k X k 1, Y k ) N (So) P(X k Y 1:k ) = w k δ(x k X Where the x k Q(X k X k 1, Y k ) The method essentally draws partcles from a proposal dstrbuton and recursvely update ts weghts. No gaussan assumpton Neat 23 =1 k ) Quantfyng Uncertanty

24 Partcle Flters Algorthm Sequental Importance Samplng end Input: {X k 1, w k 1 }, Y k = 1... N for: = 1... N Draw: X k Q(X k X k 1, Y k ) P(Y k X k )P(X k X k 1 ) w k w k 1 Q(X k X k 1,Y k ) 24 Quantfyng Uncertanty

25 Partcle Flters BUT The Problem X k 1 Q X k In a few ntervals one partcle wll have a non neglgble weght; all but one wll have neglgble weghts! 1 QN eff = N =1 (w k ) 2 25 Quantfyng Uncertanty

26 Partcle Flters Contd. QN eff Effectve Sample sze When N Q eff << N Degenaracy sets n Resamplng s a way to address ths problem Man dea weghts Resample weghts are reset You can sample unformly and set weghts to obtan a representaton. You can sample pdf to get partcles and reset ther weghts. 26 Quantfyng Uncertanty

27 Partcle Flters Resamplng algorthm w 4 w 5 w 3 w 6 w 2 w 7 w 1 Cdf(w) X 1 X 2 X 3 X 4 X 5 X 6 X 7 w 1 w 7 w 2 w 6 w 3 w 5 w 4 Unform weghts Many ponts Resamplng Sample more probable states more 27 Quantfyng Uncertanty

28 Partcle Flters Algorthm Input {X k, w k } 1. Construct cdf for = 2 : N C C 1 + w k (sorted) 2. Seed u 1 U[0, N 1 ] 3. for j = 1 : N 1 u j = u 1 + N (j 1) fnd(c u j ) Xˆ j = X w j = 1 k k k N Set Parent of j 28 Quantfyng Uncertanty

29 Partcle Flters Contd. So the resamplng method can avod degeneracy because t produces more samples for hgher probablty ponts But Sample mpovershment may result; Too many samples too close mpovershment or loss of dversty MCMC may be a way out 29 Quantfyng Uncertanty

30 Partcle Flters Generc Partcle flter end η = w k w k w k /η Input: {X k 1, w k 1 }, Y k for = 1 : N X k Q(X k X k 1, Y k ) w k w P(Y k X k )P(X k X k 1) k 1 Q(X k X k 1,Y k ) If N Q eff < N T {X k, w k } Resample {X k, w k } 1 QN eff = N =1 (w k ) 2 30 Quantfyng Uncertanty

31 Partcle Flters What s the optmal Q functon? N we try to mnmze =1 (w k ) 2 Then: Q (X k X k 1, Y k ) = P(X k X k 1, Y k ) P(Y k X k, X k 1 )P(X k X = P(Y k X k 1 ) k 1 ) w k w P(Y k X k )P(X k X k 1 ) P(Y k X P(Y k X k )P(X k X k 1 ) k 1 k 1) w k 1 P(Y k X k )P(X k X k 1 )dx k X _ k Not easy to do! 31 Quantfyng Uncertanty

32 Partcle Flters Asymptotcally: Q Q P(X k X k 1 ) Common choce Q P(X k X k 1 ) Sometmes feasble to use proposal from process nose Then w K w k 1 P(Y k X k ) If resamplng s done at every step: w k p(y k X k ) 1 (w k 1 N ) 32 Quantfyng Uncertanty

33 Partcle Flters SIR -Samplng Importance Resamplng Input {X k 1, w k 1 }, Y k for = 1 : N Xk P(X k X w = P(Y k X k 1 ) k k ) end η = w k w k = w k /η {xk, w k } Resample [{X k, w k }] 33 Quantfyng Uncertanty

34 Partcle Flters Example Y k = η k N(0, R) v k 1 N(0, Q k 1 ) X k 1 25X k 1 X k = cos(1.2k) + v k X 2 k 1 X 2 k + η k w 34 Quantfyng Uncertanty

35 MIT OpenCourseWare 12.S990 Quantfyng Uncertanty Fall 2012 For nformaton about ctng these materals or our Terms of Use, vst: