m = 41 members n = 27 (nonfounders), f = 14 (founders) 8 markers from chromosome 19

Transcription

1 Sequenial Imporance Sampling (SIS) AKA Paricle Filering, Sequenial Impuaion (Kong, Liu, Wong, 994) For many problems, sampling direcly from he arge disribuion is difficul or impossible. One reason possible reason for his is he size of he space ha needs o be drawn from Examples: ) Linkage Analysis (Irwin, Cox, & Kong, 994) m = 4 members n = 27 (nonfounders), f = 4 (founders) 8 markers from chromosome 9

2 #alleles ranges from 6 o 8 4 members in op 2 generaion have no marker daa Wan o sample oin haploypes for all pedigree members condiional on he observed marker and disease daa Assume ha marker has n possible alleles and he disease locus has wo alleles. Then he number of possible haploypes for each person is 2 h = 4 n and he maximum number of oin haploypes possible is H = h m If n = 8 for all markers, h = 67 H = and Noe ha no all possible oin haploypes included in H have posiive probabiliy since hey won be consisen wih Mendelian segregraion. 2

3 In addiion he observed daa will also reduce he number of possible haploypes wih posiive probabiliy. 2) Targe racking (Irwin, Cressie, & Johannesson, 2002) Movemen Model: Posiion: x = x + v x, y = y + v y, Velociy: v v = v + δ x, x, x, = v + δ y, y, y, v v v + v = = v + δ 2 2 v + v 2 2 x, x, x, x, x, y, y, y, = = vy, + δy, where δ x, and δ y, are he average acceleraions in he x and y direcions from ime o ime. 3

4 This gives x = x + v + x, 2 x, y = y + v + δ δ y, 2 x, This can be wrien in marix forma X = GX + Hδ where T X = x y vx, v y, T δ = δx, δx, G = ; H = 0 0 Assume ha he model for he average acceleraions is 2 ( ) δ ~ N 0, Λ 4

5 Measuremen model: Two radars rack he arges posiion wih error where = + ; ε ~ N ( 0, Σ ) Z FX ε F = The probabiliy srucure can be described by he following graph X 0 X X 2 X 3 Z Z 2 Z 3 5

6 The model is an example of he hidden Markov model. The sae variables X are described by a coninuous sae Markov Chain bu are unobserved (hidden). All ha is observed are he Z, he observed arge posiions Problem: Wan o know he disribuion of X Z : for each ( Z: {,, = Z Z} ). Since his is a linear dynamic model, i can easily be solved by he Kalman filer (KF) (Kalman, 960). In his case, X Z : is Gaussian and he means and variances can be deermined by he following simple updae formulas. µ = E X Z: ; µ = E X Z : = Var ( : ); P ( X Z: ) P X Z Var = 6

7 Assuming E [ δ ] = [ ] calculaions are E ε = 0, he KF where µ = G µ T T P = GPG + HΛH T K = P F FP F Σ + ( ) µ = µ + K Z Fµ P = P K FP T K is known as he Kalman gain. Noe ha he Kalman filer calculaions here reduce o Normal condiional disribuion calculaions. For example ( ( )) K = Cov( X, Z ) Var Z, exacly wha you need o calculae for a mulivariae regression of Z on X. P reduces o a sandard condiional variance calculaion. 7

8 300 Example Pah Truh Radar Radar 2 Firs 20 observaions

9 The above daa was generaed under he model described above wih Λ = ; and X 0 = [ 0 0 ] T Λ = Σ = However if he movemen or measuremen models are non linear or conain non-normal random componens, and he Kalman filer or is modificaions, such as he Exended Kalman filer (EKF) can give poor answers. The EKF linearizes he sysem hrough Taylor series approximaions and hen runs he sandard Kalman filer on his linear sysem. 9

10 An example wih a nonlinear componen is given wih he movemen model log s x y = log s + δ s, θ = θ + δ θ, = x + = y + s s cosθ + s 2 sinθ + s 2 cosθ sinθ In his seing, simulaing realizaions of X will give a beer approximaion o µ = E X Z: and P = Var ( X Z: ), (or any oher funcional of X ). In his case he disribuion of X can be exremely difficul o deal wih direcly, bu is fairly easy o deal wih condiional of he earlier pars of he pah (drawing X given X and Z is racable) For boh examples (linkage analysis and arge racking), sequenial imporance sampling is a useful echnique for sampling from he desired poserior disribuions. 0

11 =, 2,, k be some decomposiion of he random variable you wish o sample from and Y = { Y, Y2,, Y k } be he corresponding decomposiion of he daa you wish o condiion on. Le X { X X X } Wan o sample from ( ) p X Y = ( ) p( Y X) p X ( ) py which is assumed o be difficul o do. Wan o find a disribuion q( X Y ) ha is easy o sample from and use imporance sampling. SIS ) Sample ~ ( ) X q X Y and calculae w ( X ) 2) Then for = 2,, k = ( ) ( X Y ) p X Y q X q X Y X and calculae Sample ( ) ~ :, :

12 ( : ) = ( : ) w X w X The facor p( X ) : Y: ( ) ( ) :, : : : q X Y X p X Y p( X: Y: ) ( :, : ) ( : : ) q X Y X p X Y is ofen easy o calculae. The resuling sample X = X:k is a weighed sample from p( X Y ) wih unnormalized imporance sampling weigh where w k ( X ) : k = ( : k Y: k) ( : k Y: k) p X q X k ( : k : k ) ( ) ( :, : ) q X Y = q X Y q X Y X = 2 The componens of he proposal need o be chosen so ha ha hey are easy o sample from. 2

13 Two popular choices are and (, : : ) (, = : : ) q X Y X p X Y X * ( :, : ) = ( : ) q X Y X p X X ' The firs choice is opimal in ha is minimizes he variance of he imporance sampling weighs (which will increase he ESS). The second choice is ofen easy, such as wih he arge racking example. However by ignoring he daa, i can significanly increase he imporance sampling weigh variance. Opimal proposal properies For he opimal proposal k ( ) ( ) ( ) : k :, : w X = p Y p Y Y X = 2 which implies (Kong e al, 994, Irwin e al, 994) 3

14 or so ( : k : k) * q X Y ( ) w X = = : k * ( : k) ( : k : k) py p X Y ( ) w X : k ( : k) ( : k : k) q ( X: k Y: k) py p X Y * ( : ) ( ) ( : : ) : py ( : k) p( X: k Y: k) = w( X) E w X = w X q X Y dx q k k k k = = ( ) ( ) ( ) ( ) : k : k : k : k : k : k w X py p X Y dx py dx Thus he likelihood of he daa can be esimaed wih he average of he unnormalized imporance sampling weighs. : k 4

15 Implemening SIS for he arge racking example. Since he movemen is described by a Markov chain and he observaions are assumed o be independen ( ) ( ) :, : = :, : q X Y X p X Y X * = ( ), X p X Y ( ) ( ) p X X p Y X So he opimal proposal is racable here. In fac, X, ~ (, X Y N θ ) θ = GX Γ = HΛ T Γ where ( ) ( ) + HΛ H F Σ + FHΛ H F Y FGX H T T T T ( ) T T T T T HΛ H F Σ + FHΛ H F FHΛ H 5

16 In addiion, he muliplier for he weigh is ( ) ( ) :, : = py Y X py X which is he densiy of a T T N FGX, Σ + FHΛ H F random variable. ( ) Sequenial Impuaion Truh

17 8 6 4 Truh Sequenial Impuaion Kalman Filer Firs 20 observaions Sequenial Impuaion Kalman Filer Firs 20 observaions

18 x 0-3 Variance of Normalized Weighs Time Maximum Normalized Weigh Time 8

19 One poenial problem wih SIS is ha he variance of he imporance sampling weighs increases over ime, which implies ha ESS decreases as he sampler proceeds. Thus he esimaes of he mean are less precise, he furher ino he sampler we go. Soluion: Resampling. 9

20 Targe racking example. Sep ( =,, k): i ) Sample X, ~ (, X Y N θ ) θ = GX Γ = HΛ T Γ where ( ) ( ) + HΛ H F Σ + FHΛ H F Y FGX H T T T T ( ) T T T T T HΛ H F Σ + FHΛ H F FHΛ H This is goen by plugging he appropriae marices ino θ = E X X Γ = ( X Y X ) ( Y X ) ( Y E Y X ) ( X X ) ( X Y X ) ( Y X ) ( Y X X ) + Cov, Var Var Cov, Var Cov, 20

21 ii ) Updae he weigh since ( ) ( ) ( ) ( ) : = : w X w X p Y X ( ) ( ) :, : = py Y X py X py X is a normal densiy wih Var µ = FGX = E Y X ( ) T T =Σ + FH Λ H F = Var Y X Wha o do wih more complicaed models, such as log s x y = log s + δ s, θ = θ + δ θ, = x + s s = y + cosθ + s 2 sinθ + s 2 cosθ sinθ 2

22 The opimal proposal disribuion sill has he form ( ) ( ) :, : = :, : q X Y X p X Y X * However ( ) = ( ), X p X Y ( ) ( ) p X X p Y X p X X is no longer normal, hough i is based on normal, assuming he random changes in speed and direcion are normal. One approach is o approximae i wih a normal maching he mean and variance (approximaely). Then he combinaion of he wo pieces is approximaely normal. The normal approximaion may be deermined by Taylor series approximaion (Dela rule) Numerical quadraure (Scaled unscened ransformaion)??? 22

23 For he nonlinear model above, i can also be deal wih by seing he sae vecor o * X = [ logs θ] and having he measuremen * * model for Z depend on X,, X in a nonlinear fashion. The models are exacly he same, us paramerized differenly. However he differen paramerizaions lead o a differen normal approximaions, and in fac for his example he nonlinear measuremen model works beer (lower CV for he imporance sampling weighs and smaller sandard errors for he filered arge locaions. Daa decomposiions = { } and Y = { Y Y } X X,, Xk,, k Efficiency of SIS depends on how his decomposiion is made. In some problems here may be many ways of doing his decomposiion. 23

24 For he arge racking example here isn. The only decomposiion ha makes sense is o mach i wih ime. (Physical consrains of daa collecion force his.) Here are a couple where i does make a difference Example : Mulivariae normal daa wih missing values (Kong e al, 994) Bayesian analysis using he Jeffreys noninformaive prior. 269 observaions of a 6 componen vecor 88 observaion complee 8 observaions had a leas componen missing wih some missing up o 4 24

25 They performed simulaion in he order given above ) complee daa 2) componen missing 3) 2 componens missing ec Anoher approach would be o deal wih he daa in he daa collecion order (which probably was random) The imporance sampling weighs in his second approach will be more variable and hus more impuaions will be needed o reach he same precision. Noe ha in he analysis in his paper, i was based on simulaed daa. However i was based on he srucure of a real daa se from he social sciences. One poenial problem wih SIS is ha he variance of he imporance sampling weighs increases over ime, which implies ha ESS decreases as he sampler proceeds. 25

26 Example 2: Linkage Analysis Similar o he example presened las ime, bu on a differen daa se (Irwin e al, 994) Wan o esimae he disease locaion of a puaive gene for a form of diabees locaed on 20q wih 8 markers. Two approaches: ) Process all marker daa firs hen he disease daa 2) Process marker RM292 firs, hen he disease daa, hen he oher 7 markers In boh approaches he disease was processed in he middle of he marker inerval of ineres and he likelihood for oher poins in he inerval were deermined by reweighing he sample 26

27 CEPH Disances: 27

28 MCEM Disances 28

29 In all cases processing he disease early gave more precise esimaes, in some cases by a facor over 30. When possible, you wan include as much informaion in Y. Wan o sample wih rial disribuion based on insead of ( ) = ( ) ( 2, ) p( Xk X: k, Y ) p X Y p X Y p X X Y ( ) = ( ) ( 2, :2) q( Xk X: k, Y: k) q X Y q X Y q X X Y The firs case will have imporance sampling weighs = (assuming ha you don need o use imporance sampling for any of he p X, X: Y ). componens ( ) Thus careful hough can help alleviae he problem I alked abou las ime, he increasing variance of he imporance sampling weighs as he sampler progresses. 29

30 For example, suppose you have a process ha you wan o model wih he following hierarchical srucure Process level : [ Y ] Process level 2: XY Daa: Zx, Zy X, Y = Zx X Zy Y Wan o sample X and Y from XYZ, x, Z y. One possible scheme is o use he following SIS scheme ) Sample X from XZx w X. weighs ( ) x by SIS giving 2) Sample Y from Y X, Zx, Z y. Given he probabiliy srucure above Y X, Zx, Z y Y X, Z = y If i is possible o sample direcly from Y X Z, y, w ( X Y ) = w ( X) xy,, x 30

31 i.e. he simulaion of Y in his case won increase he variance of he imporance sampling weighs. I have been able o do his wih some geneics example, where X are he haploypes, and Y is he inheriance vecor. However his idea won work in he arge racking example. Les look a how he normalized imporance sampling weighs can evolve over ime in he arge racking example. 3

32 x 0-3 Normalized Weighs - Time 2 x 0-3 Normalized Weighs - Time Impuaion # Impuaion # 0.02 Normalized Weighs - Time 3 0. Normalized Weighs - Time Impuaion # Impuaion # 0.2 Normalized Weighs - Time Normalized Weighs - Time Impuaion # Impuaion # 32

33 One way o hink abou imporance sampling weighs is in erms of how many samples would you expec o see if you sampled from he arge disribuion insead of he rial disribuion you acually sampled from (if weighs normalize o have mean ). So if w( X ) = 2, you would expec o see abou wice as many copies of X if you sampled direcly from he arge disribuion. w( X ) = 0.5 implies you would expec half as many (From: van der Merwe e al, 2000, The Unscened Paricle Filer) 33

34 Resampling: n Sample realizaions from he se { X:,, X: } wih probabiliies proporional o he weighs n w X,, w X. ( : ) ( : ) Trea his new sample as an equally weighed from he arge disribuion. Sequenial Impuaion wih Resampling For i =,, n i ) Sample ( i X ~ ) q X Y:, X: 2) Updae weigh i i ( : ) = ( : ) w X w X ( i p X ) : Y: ( i i ) ( i ) :, : : : q X Y X p X Y 3) If appropriae, resample n realizaions n from{ X:,, X: } wih probabiliies n w X,, w X. proporional o ( : ) ( : ) i Rese weighs w ( X: ) = n 34

35 Noe ha he resampling sep does no have o be done hrough each pass. Two approaches are ) resample every m imes hrough ( = m, 2m, ) 2) monior he weighs and resample when he behaviour sars o ge poor (e.g. when CV > C) 35

36 Wih resampling some realizaions will ge replicaed and some will drop ou. There are a number of ways of doing he sampling. Le i ( ) w X be he normalized weighs i ( ) ( ) w X = w X ) Mulinomial sampling (Gordon, 994) Sample ( ) ( { }),, ~Muli, l ln n w X where l is he number of copies of he new sample X in 36

37 This is equivalen o Draw U ~Unif( 0,) i, i =,, n Se i = X if X l l ( ) i < ( ) w X U w X l= l= 2) Residual sampling (Higuchi (997), Liu and Chen (998)) 37

38 3) Minimum variance sampling (Kiagawa (996), Crisan (200)) Sample U ~Unif( 0, ) n Le U = U + for = 2,, n n Se i if X = X U < n n l l ( ) i < ( ) w X U w X l= l= This procedure has he propery ha X will occur eiher nw ( X ) nw X + imes in he new sample. or ( ) This implies ha samples wih high weighs mus be included in he new sample and ha lowly weighed samples can ge in very ofen. This will minimize he variances on { l } 38