Nonlinear Filtering using Particles and Outline Nonlinear - - PowerPoint PPT Presentation

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Nonlinear Filtering using Particles and Outline Nonlinear - - PowerPoint PPT Presentation

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Nonlinear Filtering Andreas Kl ockner Nonlinear Filtering using Particles and Outline Nonlinear Quadrature Filtering Monte Carlo Filters Particle Filters Importance Andreas Kl ockner Sampling Quadrature Filters Error Estimates

slide-1
SLIDE 1

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Nonlinear Filtering using Particles and Quadrature

Andreas Kl¨

  • ckner

May 8, 2007

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-2
SLIDE 2

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Outline

1 Nonlinear Filtering 2 Monte Carlo Filters

Particle Filters Importance Sampling

3 Quadrature Filters

Error Estimates

4 Continuous-Time Filtering

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-3
SLIDE 3

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Outline

1 Nonlinear Filtering 2 Monte Carlo Filters

Particle Filters Importance Sampling

3 Quadrature Filters

Error Estimates

4 Continuous-Time Filtering

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-4
SLIDE 4

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

The Nonlinear Filtering Problem

Have:

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-5
SLIDE 5

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

The Nonlinear Filtering Problem

Have: A hidden Markov process xt for t = 0, 1, 2, . . . with initial distribution p(x0) and transition probability p(xt+1|xt).

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-6
SLIDE 6

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

The Nonlinear Filtering Problem

Have: A hidden Markov process xt for t = 0, 1, 2, . . . with initial distribution p(x0) and transition probability p(xt+1|xt). Conditionally independent observations yt for t = 1, 2, 3, . . . characterized by p(yt|xt).

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-7
SLIDE 7

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

The Nonlinear Filtering Problem

Have: A hidden Markov process xt for t = 0, 1, 2, . . . with initial distribution p(x0) and transition probability p(xt+1|xt). Conditionally independent observations yt for t = 1, 2, 3, . . . characterized by p(yt|xt). Want:

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-8
SLIDE 8

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

The Nonlinear Filtering Problem

Have: A hidden Markov process xt for t = 0, 1, 2, . . . with initial distribution p(x0) and transition probability p(xt+1|xt). Conditionally independent observations yt for t = 1, 2, 3, . . . characterized by p(yt|xt). Want: (An estimate of) The posterior distribution p(xt|y1:t).

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-9
SLIDE 9

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Process and Observation, schematically

x0 x1 xt−2 xt−1 xt y1 yt−2 yt−1 yt

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-10
SLIDE 10

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Bayes’ Formula

Recall:

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-11
SLIDE 11

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Bayes’ Formula

Recall: p(X|Y )

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-12
SLIDE 12

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Bayes’ Formula

Recall: p(X|Y ) = p(Y |X)p(X) p(Y )

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-13
SLIDE 13

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Bayes’ Formula

Recall: p(X|Y ) = p(Y |X)p(X) p(Y ) = p(Y |X)p(X)

  • P(Y ∩ X)dX

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-14
SLIDE 14

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Bayes’ Formula

Recall: p(X|Y ) = p(Y |X)p(X) p(Y ) = p(Y |X)p(X)

  • P(Y ∩ X)dX =

p(Y |X)p(X)

  • p(Y |X)p(X)dX

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-15
SLIDE 15

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Bayes’ Formula

Recall: p(X|Y ) = p(Y |X)p(X) p(Y ) = p(Y |X)p(X)

  • P(Y ∩ X)dX =

p(Y |X)p(X)

  • p(Y |X)p(X)dX

So in our setting:

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-16
SLIDE 16

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Bayes’ Formula

Recall: p(X|Y ) = p(Y |X)p(X) p(Y ) = p(Y |X)p(X)

  • P(Y ∩ X)dX =

p(Y |X)p(X)

  • p(Y |X)p(X)dX

So in our setting: p(x0:t|y1:t) = p(y1:t|x0:t)p(x0:t)

  • p(y1:t|x0:t)p(x0:t)dx0:t

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-17
SLIDE 17

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Bayes’ Formula

Recall: p(X|Y ) = p(Y |X)p(X) p(Y ) = p(Y |X)p(X)

  • P(Y ∩ X)dX =

p(Y |X)p(X)

  • p(Y |X)p(X)dX

So in our setting: p(x0:t|y1:t) = p(y1:t|x0:t)p(x0:t)

  • p(y1:t|x0:t)p(x0:t)dx0:t

Not usable for on-line processing.

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-18
SLIDE 18

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Bayes’ Formula

Recall: p(X|Y ) = p(Y |X)p(X) p(Y ) = p(Y |X)p(X)

  • P(Y ∩ X)dX =

p(Y |X)p(X)

  • p(Y |X)p(X)dX

So in our setting: p(x0:t|y1:t) = p(y1:t|x0:t)p(x0:t)

  • p(y1:t|x0:t)p(x0:t)dx0:t

Not usable for on-line processing. Need a recursive algorithm.

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-19
SLIDE 19

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Joint Posterior Update

p(x0:t|y1:t) p(x0:t−1|y1:t−1)

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-20
SLIDE 20

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Joint Posterior Update

p(x0:t|y1:t) p(x0:t−1|y1:t−1) = p(y1:t|x0:t)p(x0:t) p(y1:t) · p(y1:t−1) p(y1:t−1|x0:t−1)p(x0:t−1)

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-21
SLIDE 21

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Joint Posterior Update

p(x0:t|y1:t) p(x0:t−1|y1:t−1) = p(y1:t|x0:t)p(x0:t) p(y1:t) · p(y1:t−1) p(y1:t−1|x0:t−1)p(x0:t−1)

CI

= p(yt|x0:t)p(x0:t) p(y1:t) · p(y1:t−1) p(x0:t−1)

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-22
SLIDE 22

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Joint Posterior Update

p(x0:t|y1:t) p(x0:t−1|y1:t−1) = p(y1:t|x0:t)p(x0:t) p(y1:t) · p(y1:t−1) p(y1:t−1|x0:t−1)p(x0:t−1)

CI

= p(yt|x0:t)p(x0:t) p(y1:t) · p(y1:t−1) p(x0:t−1)

Markov

= p(yt|x0:t)p(x0:t)p(xt|xt−1) p(y1:t) · p(y1:t−1) p(x0:t)

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-23
SLIDE 23

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Joint Posterior Update

p(x0:t|y1:t) p(x0:t−1|y1:t−1) = p(y1:t|x0:t)p(x0:t) p(y1:t) · p(y1:t−1) p(y1:t−1|x0:t−1)p(x0:t−1)

CI

= p(yt|x0:t)p(x0:t) p(y1:t) · p(y1:t−1) p(x0:t−1)

Markov

= p(yt|x0:t)p(x0:t)p(xt|xt−1) p(y1:t) · p(y1:t−1) p(x0:t) = p(yt|x0:t)p(xt|xt−1) p(y1:t) · p(y1:t−1)

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-24
SLIDE 24

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Joint Posterior Update

p(x0:t|y1:t) p(x0:t−1|y1:t−1) = p(y1:t|x0:t)p(x0:t) p(y1:t) · p(y1:t−1) p(y1:t−1|x0:t−1)p(x0:t−1)

CI

= p(yt|x0:t)p(x0:t) p(y1:t) · p(y1:t−1) p(x0:t−1)

Markov

= p(yt|x0:t)p(x0:t)p(xt|xt−1) p(y1:t) · p(y1:t−1) p(x0:t) = p(yt|x0:t)p(xt|xt−1) p(y1:t) · p(y1:t−1) = p(yt|xt)p(xt|xt−1) p(yt|y1:t−1)

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-25
SLIDE 25

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Joint Posterior Update

p(x0:t|y1:t) = p(yt|xt)p(xt|xt−1) p(yt|y1:t−1) p(x0:t−1|y1:t−1)

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-26
SLIDE 26

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Joint Posterior Update

p(x0:t|y1:t) = p(yt|xt)p(xt|xt−1) p(yt|y1:t−1) p(x0:t−1|y1:t−1) We can expand the denominator as

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-27
SLIDE 27

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Joint Posterior Update

p(x0:t|y1:t) = p(yt|xt)p(xt|xt−1) p(yt|y1:t−1) p(x0:t−1|y1:t−1) We can expand the denominator as p(yt|y1:t−1) =

  • p(yt|xt)p(xt|y1:t−1)dxt.

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-28
SLIDE 28

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Joint Posterior Update

p(x0:t|y1:t) = p(yt|xt)p(xt|xt−1) p(yt|y1:t−1) p(x0:t−1|y1:t−1) We can expand the denominator as p(yt|y1:t−1) =

  • p(yt|xt)p(xt|y1:t−1)dxt.

Two drawbacks:

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-29
SLIDE 29

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Joint Posterior Update

p(x0:t|y1:t) = p(yt|xt)p(xt|xt−1) p(yt|y1:t−1) p(x0:t−1|y1:t−1) We can expand the denominator as p(yt|y1:t−1) =

  • p(yt|xt)p(xt|y1:t−1)dxt.

Two drawbacks: Denominator not explicitly known

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-30
SLIDE 30

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Joint Posterior Update

p(x0:t|y1:t) = p(yt|xt)p(xt|xt−1) p(yt|y1:t−1) p(x0:t−1|y1:t−1) We can expand the denominator as p(yt|y1:t−1) =

  • p(yt|xt)p(xt|y1:t−1)dxt.

Two drawbacks: Denominator not explicitly known Don’t care about the joint posterior–only want p(xt|y1:t)

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-31
SLIDE 31

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Recursive Posterior Update

p(xt|y1:t)

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-32
SLIDE 32

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Recursive Posterior Update

p(xt|y1:t) =

  • p(x0:t|y1:t)dx0:t−1

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-33
SLIDE 33

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Recursive Posterior Update

p(xt|y1:t) =

  • p(x0:t|y1:t)dx0:t−1

JointUp

= p(yt|xt)p(xt|xt−1) p(yt|y1:t−1) p(x0:t−1|y1:t−1)dx0:t−1

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-34
SLIDE 34

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Recursive Posterior Update

p(xt|y1:t) =

  • p(x0:t|y1:t)dx0:t−1

JointUp

= p(yt|xt)p(xt|xt−1) p(yt|y1:t−1) p(x0:t−1|y1:t−1)dx0:t−1 = p(yt|xt) p(yt|y1:t−1)

  • p(xt|xt−1)p(x0:t−1|y1:t−1)dx0:t−1

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-35
SLIDE 35

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Recursive Posterior Update

p(xt|y1:t) =

  • p(x0:t|y1:t)dx0:t−1

JointUp

= p(yt|xt)p(xt|xt−1) p(yt|y1:t−1) p(x0:t−1|y1:t−1)dx0:t−1 = p(yt|xt) p(yt|y1:t−1)

  • p(xt|xt−1)p(x0:t−1|y1:t−1)dx0:t−1

Markov

= p(yt|xt) p(yt|y1:t−1)

  • p(xt|x0:t−1)p(x0:t−1|y1:t−1)dx0:t−1

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-36
SLIDE 36

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Recursive Posterior Update

p(xt|y1:t) =

  • p(x0:t|y1:t)dx0:t−1

JointUp

= p(yt|xt)p(xt|xt−1) p(yt|y1:t−1) p(x0:t−1|y1:t−1)dx0:t−1 = p(yt|xt) p(yt|y1:t−1)

  • p(xt|xt−1)p(x0:t−1|y1:t−1)dx0:t−1

Markov

= p(yt|xt) p(yt|y1:t−1)

  • p(xt|x0:t−1)p(x0:t−1|y1:t−1)dx0:t−1

= p(yt|xt) p(yt|y1:t−1)p(xt|y1:t−1)

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-37
SLIDE 37

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Recursive Posterior Update

p(xt|y1:t) =

  • p(x0:t|y1:t)dx0:t−1

JointUp

= p(yt|xt)p(xt|xt−1) p(yt|y1:t−1) p(x0:t−1|y1:t−1)dx0:t−1 = p(yt|xt) p(yt|y1:t−1)

  • p(xt|xt−1)p(x0:t−1|y1:t−1)dx0:t−1

Markov

= p(yt|xt) p(yt|y1:t−1)

  • p(xt|x0:t−1)p(x0:t−1|y1:t−1)dx0:t−1

= p(yt|xt) p(yt|y1:t−1)p(xt|y1:t−1) = p(yt|xt)p(xt|y1:t−1)

  • p(yt|xt)p(xt|y1:t−1)dxt

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-38
SLIDE 38

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

The Complete Recursion

Prediction t → Posterior t:

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-39
SLIDE 39

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

The Complete Recursion

Prediction t → Posterior t: p(xt|y1:t) = p(yt|xt)p(xt|y1:t−1)

  • p(yt|xt)p(xt|y1:t−1)dxt

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-40
SLIDE 40

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

The Complete Recursion

Prediction t → Posterior t: p(xt|y1:t) = p(yt|xt)p(xt|y1:t−1)

  • p(yt|xt)p(xt|y1:t−1)dxt

Posterior t − 1 → Prediction t:

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-41
SLIDE 41

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

The Complete Recursion

Prediction t → Posterior t: p(xt|y1:t) = p(yt|xt)p(xt|y1:t−1)

  • p(yt|xt)p(xt|y1:t−1)dxt

Posterior t − 1 → Prediction t: Start with

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-42
SLIDE 42

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

The Complete Recursion

Prediction t → Posterior t: p(xt|y1:t) = p(yt|xt)p(xt|y1:t−1)

  • p(yt|xt)p(xt|y1:t−1)dxt

Posterior t − 1 → Prediction t: Start with p(xt) =

  • p(xt|xt−1)p(xt−1)dxt−1,

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-43
SLIDE 43

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

The Complete Recursion

Prediction t → Posterior t: p(xt|y1:t) = p(yt|xt)p(xt|y1:t−1)

  • p(yt|xt)p(xt|y1:t−1)dxt

Posterior t − 1 → Prediction t: Start with p(xt) =

  • p(xt|xt−1)p(xt−1)dxt−1,

and obtain

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-44
SLIDE 44

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

The Complete Recursion

Prediction t → Posterior t: p(xt|y1:t) = p(yt|xt)p(xt|y1:t−1)

  • p(yt|xt)p(xt|y1:t−1)dxt

Posterior t − 1 → Prediction t: Start with p(xt) =

  • p(xt|xt−1)p(xt−1)dxt−1,

and obtain p(xt|y1:t−1) =

  • p(xt|xt−1)p(xt−1|y1:t−1)dxt−1.

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-45
SLIDE 45

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Outline

1 Nonlinear Filtering 2 Monte Carlo Filters

Particle Filters Importance Sampling

3 Quadrature Filters

Error Estimates

4 Continuous-Time Filtering

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-46
SLIDE 46

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Our Example Process

Throughout this presentation, we will stick to the following model:

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-47
SLIDE 47

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Our Example Process

Throughout this presentation, we will stick to the following model: xt = 1 2xt−1 + 25 xt−1 1 + x2

t−1

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-48
SLIDE 48

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Our Example Process

Throughout this presentation, we will stick to the following model: xt = 1 2xt−1 + 25 xt−1 1 + x2

t−1

xt−1 xt x → x xt

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-49
SLIDE 49

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Our Example Process

Throughout this presentation, we will stick to the following model: xt = 1 2xt−1 + 25 xt−1 1 + x2

t−1

+ 8cos(1.2t) + vt, xt−1 xt x → x xt

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-50
SLIDE 50

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Our Example Process

Throughout this presentation, we will stick to the following model: xt = 1 2xt−1 + 25 xt−1 1 + x2

t−1

+ 8cos(1.2t) + vt, yt = x2

t

20 + wt,

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-51
SLIDE 51

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Our Example Process

Throughout this presentation, we will stick to the following model: xt = 1 2xt−1 + 25 xt−1 1 + x2

t−1

+ 8cos(1.2t) + vt, yt = x2

t

20 + wt, where x0 ∼ N(0, σ2

1), vt ∼ N(0, σ2 v), wt ∼ N(0, σ2 w).

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-52
SLIDE 52

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Particle Approximations

Assume a discrete approximation of the posterior density ˆ p(dxt|y1:t) = 1 Nt

Nt

  • i=1

δx(i)

t (dxt). Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-53
SLIDE 53

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Particle Approximations

Assume a discrete approximation of the posterior density ˆ p(dxt|y1:t) = 1 Nt

Nt

  • i=1

δx(i)

t (dxt).

Then insert that into the prediction formula:

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-54
SLIDE 54

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Particle Approximations

Assume a discrete approximation of the posterior density ˆ p(dxt|y1:t) = 1 Nt

Nt

  • i=1

δx(i)

t (dxt).

Then insert that into the prediction formula: ˆ p(dxt+1|y1:t) = 1 Nt

Nt

  • i=1

p(dxt+1|x(i)

t )

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-55
SLIDE 55

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Particle Approximations

Assume a discrete approximation of the posterior density ˆ p(dxt|y1:t) = 1 Nt

Nt

  • i=1

δx(i)

t (dxt).

Then insert that into the prediction formula: ˆ p(dxt+1|y1:t) = 1 Nt

Nt

  • i=1

p(dxt+1|x(i)

t )

and the update formula:

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-56
SLIDE 56

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Particle Approximations

Assume a discrete approximation of the posterior density ˆ p(dxt|y1:t) = 1 Nt

Nt

  • i=1

δx(i)

t (dxt).

Then insert that into the prediction formula: ˆ p(dxt+1|y1:t) = 1 Nt

Nt

  • i=1

p(dxt+1|x(i)

t )

and the update formula: ˆ p(xt+1|y1:t+1) = 1 ˜ Ct+1 · Nt

Nt

  • i=1

p(yt+1|xt+1)p(xt+1|x(i)

t ).

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-57
SLIDE 57

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Particle Approximations

Assume a discrete approximation of the posterior density ˆ p(dxt|y1:t) = 1 Nt

Nt

  • i=1

δx(i)

t (dxt).

Then insert that into the prediction formula: ˆ p(dxt+1|y1:t) = 1 Nt

Nt

  • i=1

p(dxt+1|x(i)

t )

and the update formula: ˆ p(xt+1|y1:t+1) = 1 ˜ Ct+1 · Nt

Nt

  • i=1

p(yt+1|xt+1)p(xt+1|x(i)

t ).

Problem:

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-58
SLIDE 58

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Particle Approximations

Assume a discrete approximation of the posterior density ˆ p(dxt|y1:t) = 1 Nt

Nt

  • i=1

δx(i)

t (dxt).

Then insert that into the prediction formula: ˆ p(dxt+1|y1:t) = 1 Nt

Nt

  • i=1

p(dxt+1|x(i)

t )

and the update formula: ˆ p(xt+1|y1:t+1) = 1 ˜ Ct+1 · Nt

Nt

  • i=1

p(yt+1|xt+1)p(xt+1|x(i)

t ).

Problem: (Generally) Nontrivial to sample from!

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-59
SLIDE 59

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Importance Sampling

Importance Sampling lets you sample. . .

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-60
SLIDE 60

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Importance Sampling

Importance Sampling lets you sample. . . . . . from an (almost) arbitrary density. . .

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-61
SLIDE 61

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Importance Sampling

Importance Sampling lets you sample. . . . . . from an (almost) arbitrary density. . . . . . that may only be known up to a proportionality constant.

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-62
SLIDE 62

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Importance Sampling

Importance Sampling lets you sample. . . . . . from an (almost) arbitrary density. . . . . . that may only be known up to a proportionality constant. Recall our particle update formula:

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-63
SLIDE 63

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Importance Sampling

Importance Sampling lets you sample. . . . . . from an (almost) arbitrary density. . . . . . that may only be known up to a proportionality constant. Recall our particle update formula: ˆ p(xt+1|y1:t+1) = 1 ˜ Ct+1 · Nt

Nt

  • i=1

p(yt+1|xt+1)p(xt+1|x(i)

t ).

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-64
SLIDE 64

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Importance Sampling

Importance Sampling lets you sample. . . . . . from an (almost) arbitrary density. . . . . . that may only be known up to a proportionality constant. Recall our particle update formula: ˆ p(xt+1|y1:t+1) = 1 ˜ Ct+1 · Nt

Nt

  • i=1

p(yt+1|xt+1)p(xt+1|x(i)

t ).

That’s exactly our situation.

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-65
SLIDE 65

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Importance Sampling

Importance Sampling lets you sample. . . . . . from an (almost) arbitrary density. . . . . . that may only be known up to a proportionality constant. Recall our particle update formula: ˆ p(xt+1|y1:t+1) = 1 ˜ Ct+1 · Nt

Nt

  • i=1

p(yt+1|xt+1)p(xt+1|x(i)

t ).

That’s exactly our situation. Trick: Give each particle a weight in addition to its position.

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-66
SLIDE 66

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Importance Sampling in Detail

Suppose we have:

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-67
SLIDE 67

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Importance Sampling in Detail

Suppose we have: p(x) ∝ q(x) not easy to sample from, q known exactly

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-68
SLIDE 68

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Importance Sampling in Detail

Suppose we have: p(x) ∝ q(x) not easy to sample from, q known exactly π(x) a different density that is easy to sample from

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-69
SLIDE 69

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Importance Sampling in Detail

Suppose we have: p(x) ∝ q(x) not easy to sample from, q known exactly π(x) a different density that is easy to sample from –the importance density

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-70
SLIDE 70

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Importance Sampling in Detail

Suppose we have: p(x) ∝ q(x) not easy to sample from, q known exactly π(x) a different density that is easy to sample from –the importance density Then assign

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-71
SLIDE 71

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Importance Sampling in Detail

Suppose we have: p(x) ∝ q(x) not easy to sample from, q known exactly π(x) a different density that is easy to sample from –the importance density Then assign w(i) ∝ p(x(i)) π(x(i))

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-72
SLIDE 72

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Importance Sampling in Detail

Suppose we have: p(x) ∝ q(x) not easy to sample from, q known exactly π(x) a different density that is easy to sample from –the importance density Then assign and normalize the weights w(i) ∝ p(x(i)) π(x(i)) ˜ w(i) := w(i)

  • j w(j) .

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-73
SLIDE 73

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Importance Sampling in Detail

Suppose we have: p(x) ∝ q(x) not easy to sample from, q known exactly π(x) a different density that is easy to sample from –the importance density Then assign and normalize the weights w(i) ∝ p(x(i)) π(x(i)) ˜ w(i) := w(i)

  • j w(j) .

Any unkown proportionality constant in p or w goes away in the normalization step.

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-74
SLIDE 74

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Why and How Importance Sampling Works

Now p(x) ≈ ˆ p(x) :=

N

  • i=1

˜ w(i)δ(x − x(i)),

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-75
SLIDE 75

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Why and How Importance Sampling Works

Now p(x) ≈ ˆ p(x) :=

N

  • i=1

˜ w(i)δ(x − x(i)), assuming the particles x(i) have been drawn using π(x).

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-76
SLIDE 76

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Why and How Importance Sampling Works

Now p(x) ≈ ˆ p(x) :=

N

  • i=1

˜ w(i)δ(x − x(i)), assuming the particles x(i) have been drawn using π(x). Why?

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-77
SLIDE 77

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Why and How Importance Sampling Works

Now p(x) ≈ ˆ p(x) :=

N

  • i=1

˜ w(i)δ(x − x(i)), assuming the particles x(i) have been drawn using π(x). Why? A uniformly weighted distribution sampled from p(x) is (approximately) the same as a distribution weighted with p(x)/π(x) sampled from π(x).

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-78
SLIDE 78

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Why and How Importance Sampling Works

Now p(x) ≈ ˆ p(x) :=

N

  • i=1

˜ w(i)δ(x − x(i)), assuming the particles x(i) have been drawn using π(x). Why? A uniformly weighted distribution sampled from p(x) is (approximately) the same as a distribution weighted with p(x)/π(x) sampled from π(x).

  • f (x)p(x)

π(x)π(x)dx =

  • f (x)p(x)dx.

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-79
SLIDE 79

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Sequential Importance Sampling

Particle Framework + Importance Sampling = ?

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-80
SLIDE 80

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Sequential Importance Sampling

Particle Framework + Importance Sampling = ? To maintain the recursive update property, demand

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-81
SLIDE 81

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Sequential Importance Sampling

Particle Framework + Importance Sampling = ? To maintain the recursive update property, demand π(x0:t|y1:t) = π(x0:t−1|y1:t−1)π(xt|x0:t−1, y1:t).

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-82
SLIDE 82

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Sequential Importance Sampling

Particle Framework + Importance Sampling = ? To maintain the recursive update property, demand π(x0:t|y1:t) = π(x0:t−1|y1:t−1)π(xt|x0:t−1, y1:t). Recall the joint update equation

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-83
SLIDE 83

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Sequential Importance Sampling

Particle Framework + Importance Sampling = ? To maintain the recursive update property, demand π(x0:t|y1:t) = π(x0:t−1|y1:t−1)π(xt|x0:t−1, y1:t). Recall the joint update equation p(x0:t|y1:t) ∝ p(x0:t−1|y1:t−1)p(yt|xt)p(xt|xt−1).

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-84
SLIDE 84

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Sequential Importance Sampling

Particle Framework + Importance Sampling = ? To maintain the recursive update property, demand π(x0:t|y1:t) = π(x0:t−1|y1:t−1)π(xt|x0:t−1, y1:t). Recall the joint update equation p(x0:t|y1:t) ∝ p(x0:t−1|y1:t−1)p(yt|xt)p(xt|xt−1). This yields the weight update equation

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-85
SLIDE 85

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Sequential Importance Sampling

Particle Framework + Importance Sampling = ? To maintain the recursive update property, demand π(x0:t|y1:t) = π(x0:t−1|y1:t−1)π(xt|x0:t−1, y1:t). Recall the joint update equation p(x0:t|y1:t) ∝ p(x0:t−1|y1:t−1)p(yt|xt)p(xt|xt−1). This yields the weight update equation w(i)

t

∝ p(x(i)

0:t|y1:t)

π(x(i)

0:t|y1:t)

=

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-86
SLIDE 86

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Sequential Importance Sampling

Particle Framework + Importance Sampling = ? To maintain the recursive update property, demand π(x0:t|y1:t) = π(x0:t−1|y1:t−1)π(xt|x0:t−1, y1:t). Recall the joint update equation p(x0:t|y1:t) ∝ p(x0:t−1|y1:t−1)p(yt|xt)p(xt|xt−1). This yields the weight update equation w(i)

t

∝ p(x(i)

0:t|y1:t)

π(x(i)

0:t|y1:t)

= w(i)

t−1

p(yt|x(i)

t )p(x(i) t |x(i) t−1)

π(x(i)

t |x(i) 0:t−1, y1:t)

.

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-87
SLIDE 87

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Choice of Importance Density

Choosing a good π(x) is extremely important.

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-88
SLIDE 88

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Choice of Importance Density

Choosing a good π(x) is extremely important. It can direct particles to interesting parts of the state space–or far away from there.

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-89
SLIDE 89

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Choice of Importance Density

Choosing a good π(x) is extremely important. It can direct particles to interesting parts of the state space–or far away from there. Goal: Keep the weights as uniform as possible.

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-90
SLIDE 90

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Choice of Importance Density

Choosing a good π(x) is extremely important. It can direct particles to interesting parts of the state space–or far away from there. Goal: Keep the weights as uniform as possible. Optimal choice: πopt(xt|x(i)

t−1, yt) = p(xt|x(i) t−1yt).

Andreas Kl¨

  • ckner

Nonlinear Filtering

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SLIDE 91

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Choice of Importance Density

Choosing a good π(x) is extremely important. It can direct particles to interesting parts of the state space–or far away from there. Goal: Keep the weights as uniform as possible. Optimal choice: πopt(xt|x(i)

t−1, yt) = p(xt|x(i) t−1yt).

Not usually easy to sample from→back to original problem.

Andreas Kl¨

  • ckner

Nonlinear Filtering

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SLIDE 92

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Choice of Importance Density

Popular choice: πprior(xt|x(i)

t−1, yt) := p(xt|x(i) t−1)

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-93
SLIDE 93

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Choice of Importance Density

Popular choice: πprior(xt|x(i)

t−1, yt) := p(xt|x(i) t−1)

Readily available and often Gaussian→easy choice.

Andreas Kl¨

  • ckner

Nonlinear Filtering

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SLIDE 94

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Choice of Importance Density

Popular choice: πprior(xt|x(i)

t−1, yt) := p(xt|x(i) t−1)

Readily available and often Gaussian→easy choice. Ignores observations→not always a good choice.

Andreas Kl¨

  • ckner

Nonlinear Filtering

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SLIDE 95

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Choice of Importance Density

Popular choice: πprior(xt|x(i)

t−1, yt) := p(xt|x(i) t−1)

Readily available and often Gaussian→easy choice. Ignores observations→not always a good choice. Weight update for πprior: w(i)

t

∝ w(i)

t−1

p(yt|x(i)

t )p(x(i) t |x(i) t−1)

π(x(i)

t |x(i) t−1, yt)

= w(i)

t−1p(yt|x(i) t ).

Andreas Kl¨

  • ckner

Nonlinear Filtering

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SLIDE 96

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Choice of Importance Density

Popular choice: πprior(xt|x(i)

t−1, yt) := p(xt|x(i) t−1)

Readily available and often Gaussian→easy choice. Ignores observations→not always a good choice. Weight update for πprior: w(i)

t

∝ w(i)

t−1

p(yt|x(i)

t )p(x(i) t |x(i) t−1)

π(x(i)

t |x(i) t−1, yt)

= w(i)

t−1p(yt|x(i) t ).

→Actually implementable algorithm.

Andreas Kl¨

  • ckner

Nonlinear Filtering

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SLIDE 97

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

SIS Implementation

def s i s ( model , n=1000, max time =20): p a r t i c l e s = [ model . s a m p l e i n i t i a l () for i in range (n ) ] p a r t i c l e h i s t o r y = [ p a r t i c l e s ] weights = [1/ n for i in range (n ) ] w e i g h t h i s t o r y = [ weights ] x = model . s a m p l e i n i t i a l () x h i s t o r y = [ x ] for t in range (1 , max time ) : x = model . s a m p l e t r a n s i t i o n ( t , x ) y = model . sampl e observation ( t , x ) x h i s t o r y . append ( x ) e v o l v e d p a r t i c l e s = [ model . s a m p l e t r a n s i t i o n ( t , xp ) for xp weights = [ weight ∗model . o b s e r v a t i o n d e n s i t y ( t , y , xtp ) for weight , xtp in z i p ( weights , e v o l v e d p a r t i c l e s ) weight sum = sum( weights ) weights = [ weight / weight sum for weight in weights ] p a r t i c l e h i s t o r y . append ( p a r t i c l e s ) w e i g h t h i s t o r y . append ( weights )

Andreas Kl¨

  • ckner

Nonlinear Filtering

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SLIDE 98

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

SIS Results

2 4 6 8 10 12 14 16 18 20

  • 40
  • 30
  • 20
  • 10

10 20 30 40 0.2 0.4 0.6 0.8 1 Density SIS Method trajectory posterior Time x Density Andreas Kl¨

  • ckner

Nonlinear Filtering

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SLIDE 99

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Sequential Importance Resampling

Pure SIS degenerates→all weight on one particle.

Andreas Kl¨

  • ckner

Nonlinear Filtering

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SLIDE 100

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Sequential Importance Resampling

Pure SIS degenerates→all weight on one particle. Idea:

Andreas Kl¨

  • ckner

Nonlinear Filtering

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SLIDE 101

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Sequential Importance Resampling

Pure SIS degenerates→all weight on one particle. Idea: Kill off particles with small weight

Andreas Kl¨

  • ckner

Nonlinear Filtering

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SLIDE 102

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Sequential Importance Resampling

Pure SIS degenerates→all weight on one particle. Idea: Kill off particles with small weight Multiply the ones with high weight

Andreas Kl¨

  • ckner

Nonlinear Filtering

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SLIDE 103

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Sequential Importance Resampling

Pure SIS degenerates→all weight on one particle. Idea: Kill off particles with small weight Multiply the ones with high weight Rely on process noise to scatter them

Andreas Kl¨

  • ckner

Nonlinear Filtering

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SLIDE 104

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Sequential Importance Resampling

Pure SIS degenerates→all weight on one particle. Idea: Kill off particles with small weight Multiply the ones with high weight Rely on process noise to scatter them How?

Andreas Kl¨

  • ckner

Nonlinear Filtering

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SLIDE 105

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Sequential Importance Resampling

Pure SIS degenerates→all weight on one particle. Idea: Kill off particles with small weight Multiply the ones with high weight Rely on process noise to scatter them How? Sample N times from ˆ p(x) :=

N

  • i=1

˜ w(i)δ(x − x(i)).

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-106
SLIDE 106

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Sequential Importance Resampling

Pure SIS degenerates→all weight on one particle. Idea: Kill off particles with small weight Multiply the ones with high weight Rely on process noise to scatter them How? Sample N times from ˆ p(x) :=

N

  • i=1

˜ w(i)δ(x − x(i)). Particle descendant counts ξ satisfy ξ ∼ Multinomial(n, ˜ w(1), . . . , ˜ w(N)).

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-107
SLIDE 107

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Sequential Importance Resampling

Pure SIS degenerates→all weight on one particle. Idea: Kill off particles with small weight Multiply the ones with high weight Rely on process noise to scatter them How? Sample N times from ˆ p(x) :=

N

  • i=1

˜ w(i)δ(x − x(i)). Particle descendant counts ξ satisfy ξ ∼ Multinomial(n, ˜ w(1), . . . , ˜ w(N)). → Multinomial Resampling

Andreas Kl¨

  • ckner

Nonlinear Filtering

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SLIDE 108

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

SIR Implementation

def s i r ( model , n=1000, max time=20) : p a r t i c l e s = [ model . s a m p l e i n i t i a l () for i in range (n) ] p a r t i c l e h i s t o r y = [ p a r t i c l e s ] x = model . s a m p l e i n i t i a l () x h i s t o r y = [ x ] for t in range (1 , max time ) : x = model . s a m p l e t r a n s i t i o n ( t , x ) y = model . sampl e observation ( t , x ) x h i s t o r y . append ( x ) e v o l v e d p a r t i c l e s = [ model . s a m p l e t r a n s i t i o n ( t , xp ) for xp in p a r t i c l e s ] i w e i g h t s = [ model .

  • b s e r v a t i o n d e n s i t y ( t ,

y , xtp ) for xtp in e v o l v e d p a r t i c l e s ] p a r t i c l e s = s a m p l e d i s c r e t e ( e v o l v e d p a r t i c l e s , iweights , n) p a r t i c l e h i s t o r y . append ( p a r t i c l e s )

Andreas Kl¨

  • ckner

Nonlinear Filtering

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SLIDE 109

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

SIR Results

2 4 6 8 10 12 14 16 18 20

  • 50
  • 40
  • 30
  • 20
  • 10

10 20 30 40 0.2 0.4 0.6 0.8 1 Density SIR Method trajectory posterior Time x Density Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-110
SLIDE 110

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Binary-Tree-based Resampling

ξr

  • x(1)
  • ξm

ξm1 ξm2

  • x(N)

Construct random variables ξm ∈ {0, . . . , N} for each node m,

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-111
SLIDE 111

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Binary-Tree-based Resampling

ξr

  • x(1)
  • ξm

ξm1 ξm2

  • x(N)

Construct random variables ξm ∈ {0, . . . , N} for each node m, with ξm = ξm1 + ξm2

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-112
SLIDE 112

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Binary-Tree-based Resampling

ξr

  • x(1)
  • ξm

ξm1 ξm2

  • x(N)

Construct random variables ξm ∈ {0, . . . , N} for each node m, with ξm = ξm1 + ξm2 such that E[ξm] = wm and ξm = ⌊wm⌋ with probability 1 − wm − ⌊wm⌋, ⌊wm⌋ + 1 with probability wm − ⌊wm⌋.

Andreas Kl¨

  • ckner

Nonlinear Filtering

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SLIDE 113

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

A Convergence Result

Theorem (D. Crisan, 2001) Let the transition kernel p(xt|xt−1) have the Feller property, that is for any bounded and continuous f : ❘n → ❘, the map xt−1 →

  • f (x)p(dxt|xt−1)

must be bounded and continuous. Then as the number of particles N → ∞, the approximated posterior densities of the SIR method with multinomial resampling ˆ pN(xt|y0:t) → p(xt|y0:t) p-almost surely.

Andreas Kl¨

  • ckner

Nonlinear Filtering

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SLIDE 114

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Outline

1 Nonlinear Filtering 2 Monte Carlo Filters

Particle Filters Importance Sampling

3 Quadrature Filters

Error Estimates

4 Continuous-Time Filtering

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-115
SLIDE 115

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Gauss-Legendre Quadrature

Idea: Cleverly reinterpret Gaussian Quadrature as a particle approximation.

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-116
SLIDE 116

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Gauss-Legendre Quadrature

Idea: Cleverly reinterpret Gaussian Quadrature as a particle approximation. Recall:

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-117
SLIDE 117

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Gauss-Legendre Quadrature

Idea: Cleverly reinterpret Gaussian Quadrature as a particle approximation. Recall: B

A

f (x)dx ≈

N

  • i=1

f (x(i))w(i),

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-118
SLIDE 118

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Gauss-Legendre Quadrature

Idea: Cleverly reinterpret Gaussian Quadrature as a particle approximation. Recall: B

A

f (x)dx ≈

N

  • i=1

f (x(i))w(i), with some quadrature weights w(i) and points x(i).

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-119
SLIDE 119

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Gauss-Legendre Quadrature

Idea: Cleverly reinterpret Gaussian Quadrature as a particle approximation. Recall: B

A

f (x)dx ≈

N

  • i=1

f (x(i))w(i), with some quadrature weights w(i) and points x(i). Consider a probability density function p supported on (A, B).

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-120
SLIDE 120

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Gauss-Legendre Quadrature

Idea: Cleverly reinterpret Gaussian Quadrature as a particle approximation. Recall: B

A

f (x)dx ≈

N

  • i=1

f (x(i))w(i), with some quadrature weights w(i) and points x(i). Consider a probability density function p supported on (A, B). E[f (x)] = B

A

f (x)p(x)dx ≈

N

  • i=1

f (x(i))p(x(i))w(i).

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-121
SLIDE 121

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Gauss-Legendre Quadrature

Idea: Cleverly reinterpret Gaussian Quadrature as a particle approximation. Recall: B

A

f (x)dx ≈

N

  • i=1

f (x(i))w(i), with some quadrature weights w(i) and points x(i). Consider a probability density function p supported on (A, B). E[f (x)] = B

A

f (x)p(x)dx ≈

N

  • i=1

f (x(i))p(x(i))w(i). So view the quadrature points x(i) as the particles

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-122
SLIDE 122

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Gauss-Legendre Quadrature

Idea: Cleverly reinterpret Gaussian Quadrature as a particle approximation. Recall: B

A

f (x)dx ≈

N

  • i=1

f (x(i))w(i), with some quadrature weights w(i) and points x(i). Consider a probability density function p supported on (A, B). E[f (x)] = B

A

f (x)p(x)dx ≈

N

  • i=1

f (x(i))p(x(i))w(i). So view the quadrature points x(i) as the particles and ˜ p(x(i)) := p(x(i))w(i) as the weights.

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-123
SLIDE 123

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Construction of the Quadrature Filter

Recall the Update Formula p(xt|y1:t) = 1 Ct · p(yt|xt)p(xt|y1:t−1)

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-124
SLIDE 124

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Construction of the Quadrature Filter

Recall the Update Formula p(xt|y1:t) = 1 Ct · p(yt|xt)p(xt|y1:t−1) with Ct =

  • p(yt|xt)p(xt|y1:t−1)dxt.

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-125
SLIDE 125

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Construction of the Quadrature Filter

Recall the Update Formula p(xt|y1:t) = 1 Ct · p(yt|xt)p(xt|y1:t−1) with Ct =

  • p(yt|xt)p(xt|y1:t−1)dxt.

˜ p(x(i)

t |y1:t) := 1

ˆ Ct · p(yt|xt)w(i)

t

ˆ p(x(i)

t |y1:t−1)

  • prediction

.

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-126
SLIDE 126

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Construction of the Quadrature Filter

Recall the Update Formula p(xt|y1:t) = 1 Ct · p(yt|xt)p(xt|y1:t−1) with Ct =

  • p(yt|xt)p(xt|y1:t−1)dxt.

˜ p(x(i)

t |y1:t) := 1

ˆ Ct · p(yt|xt)w(i)

t

ˆ p(x(i)

t |y1:t−1)

  • prediction

. Compute this by finding q(i) := p(yt|xt)w(i)

t ˆ

p(x(i)

t |y1:t−1)

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-127
SLIDE 127

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Construction of the Quadrature Filter

Recall the Update Formula p(xt|y1:t) = 1 Ct · p(yt|xt)p(xt|y1:t−1) with Ct =

  • p(yt|xt)p(xt|y1:t−1)dxt.

˜ p(x(i)

t |y1:t) := 1

ˆ Ct · p(yt|xt)w(i)

t

ˆ p(x(i)

t |y1:t−1)

  • prediction

. Compute this by finding q(i) := p(yt|xt)w(i)

t ˆ

p(x(i)

t |y1:t−1)

ˆ Ct =

N

  • i=1

q(i),

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-128
SLIDE 128

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Construction of the Quadrature Filter

Recall the Update Formula p(xt|y1:t) = 1 Ct · p(yt|xt)p(xt|y1:t−1) with Ct =

  • p(yt|xt)p(xt|y1:t−1)dxt.

˜ p(x(i)

t |y1:t) := 1

ˆ Ct · p(yt|xt)w(i)

t

ˆ p(x(i)

t |y1:t−1)

  • prediction

. Compute this by finding q(i) := p(yt|xt)w(i)

t ˆ

p(x(i)

t |y1:t−1)

ˆ Ct =

N

  • i=1

q(i), ˜ p(x(i)

t |y1:t) = q(i)

ˆ Ct .

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-129
SLIDE 129

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Construction of the Filter: Prediction

Recall the original prediction formula: p(xt+1|y1:t) =

  • p(xt+1|xt)p(xt|y1:t)dxt

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-130
SLIDE 130

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Construction of the Filter: Prediction

Recall the original prediction formula: p(xt+1|y1:t) =

  • p(xt+1|xt)p(xt|y1:t)dxt

Approximate prediction formula: ˆ p(x(i)

t+1|y1:t−1) = N

  • j=1

p(x(i)

t+1|x(j) t )˜

p(x(j)

t |y1:t)

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-131
SLIDE 131

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Construction of the Filter: Prediction

Recall the original prediction formula: p(xt+1|y1:t) =

  • p(xt+1|xt)p(xt|y1:t)dxt

Approximate prediction formula: ˆ p(x(i)

t+1|y1:t−1) = N

  • j=1

p(x(i)

t+1|x(j) t )˜

p(x(j)

t |y1:t)

˜ p(x(j)

t |y1:t) already contains quadrature weights.

Andreas Kl¨

  • ckner

Nonlinear Filtering

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SLIDE 132

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Construction of the Filter: Prediction

Recall the original prediction formula: p(xt+1|y1:t) =

  • p(xt+1|xt)p(xt|y1:t)dxt

Approximate prediction formula: ˆ p(x(i)

t+1|y1:t−1) = N

  • j=1

p(x(i)

t+1|x(j) t )˜

p(x(j)

t |y1:t)

˜ p(x(j)

t |y1:t) already contains quadrature weights.

Note: O(N2) complexity.

Andreas Kl¨

  • ckner

Nonlinear Filtering

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SLIDE 133

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Quadrature Filter Results

2 4 6 8 10 12 14 16 18 20

  • 40
  • 30
  • 20
  • 10

10 20 30 40 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 Density Quadrature Method trajectory posterior Time x Density Andreas Kl¨

  • ckner

Nonlinear Filtering

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SLIDE 134

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Features of the Quadrature Filter

Deterministic, no sampling.

Andreas Kl¨

  • ckner

Nonlinear Filtering

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SLIDE 135

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Features of the Quadrature Filter

Deterministic, no sampling. Well-suited to comparing likelihoods p(y1:T) =

T

  • t=1

p(yt|y1:t−1) =

T

  • t=1

Ct

  • f different models.

Andreas Kl¨

  • ckner

Nonlinear Filtering

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SLIDE 136

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Features of the Quadrature Filter

Deterministic, no sampling. Well-suited to comparing likelihoods p(y1:T) =

T

  • t=1

p(yt|y1:t−1) =

T

  • t=1

Ct

  • f different models.

Suffers heavily with high-dimensional statespaces: O(N2), N = md. (SIS: O(N), SIR: O(N log N))

Andreas Kl¨

  • ckner

Nonlinear Filtering

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SLIDE 137

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Features of the Quadrature Filter

Deterministic, no sampling. Well-suited to comparing likelihoods p(y1:T) =

T

  • t=1

p(yt|y1:t−1) =

T

  • t=1

Ct

  • f different models.

Suffers heavily with high-dimensional statespaces: O(N2), N = md. (SIS: O(N), SIR: O(N log N)) Requires far fewer particles than SMC for comparable accuracy.

Andreas Kl¨

  • ckner

Nonlinear Filtering

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SLIDE 138

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Features of the Quadrature Filter

Deterministic, no sampling. Well-suited to comparing likelihoods p(y1:T) =

T

  • t=1

p(yt|y1:t−1) =

T

  • t=1

Ct

  • f different models.

Suffers heavily with high-dimensional statespaces: O(N2), N = md. (SIS: O(N), SIR: O(N log N)) Requires far fewer particles than SMC for comparable accuracy. Explicit error estimates available.

Andreas Kl¨

  • ckner

Nonlinear Filtering

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SLIDE 139

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Error Estimates: Setup

Error in the log-likelihood of the observation: δt := log{ˆ p(y1:t)} − log{p(y1:t)} =

t

  • s=1

log ˆ Cs −

t

  • s=1

log Cs.

Andreas Kl¨

  • ckner

Nonlinear Filtering

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SLIDE 140

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Error Estimates: Setup

Error in the log-likelihood of the observation: δt := log{ˆ p(y1:t)} − log{p(y1:t)} =

t

  • s=1

log ˆ Cs −

t

  • s=1

log Cs. Prediction error indicator: ε(xt+1|y1:t) := exp(δt)ˆ p(xt+1|y1:t) − p(xt+1|y1:t).

Andreas Kl¨

  • ckner

Nonlinear Filtering

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SLIDE 141

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Error Estimates: Setup

Error in the log-likelihood of the observation: δt := log{ˆ p(y1:t)} − log{p(y1:t)} =

t

  • s=1

log ˆ Cs −

t

  • s=1

log Cs. Prediction error indicator: ε(xt+1|y1:t) := exp(δt)ˆ p(xt+1|y1:t) − p(xt+1|y1:t). Indicator ignores error in normalization ˆ Ct

Andreas Kl¨

  • ckner

Nonlinear Filtering

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SLIDE 142

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Error Estimates: Setup

Error in the log-likelihood of the observation: δt := log{ˆ p(y1:t)} − log{p(y1:t)} =

t

  • s=1

log ˆ Cs −

t

  • s=1

log Cs. Prediction error indicator: ε(xt+1|y1:t) := exp(δt)ˆ p(xt+1|y1:t) − p(xt+1|y1:t). Indicator ignores error in normalization ˆ Ct→ exp(δt) necessary.

Andreas Kl¨

  • ckner

Nonlinear Filtering

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SLIDE 143

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Error Propagation Recursion

A nice bit of algebra (see notes) shows ε(xt+1|y1:t) = C −1

t

  • i

p(xt+1|x(i)

t )p(yt|x(i) t )w(i) t ε(xt|y1:t−1)

+

  • i

p(xt+1|x(i)

t )w(i) t p(x(i) t |y1:t) − p(xt+1|y1:t).

Andreas Kl¨

  • ckner

Nonlinear Filtering

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SLIDE 144

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Error Propagation Recursion

A nice bit of algebra (see notes) shows ε(xt+1|y1:t) = C −1

t

  • i

p(xt+1|x(i)

t )p(yt|x(i) t )w(i) t ε(xt|y1:t−1)

+

  • i

p(xt+1|x(i)

t )w(i) t p(x(i) t |y1:t) − p(xt+1|y1:t).

Split error into

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-145
SLIDE 145

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Error Propagation Recursion

A nice bit of algebra (see notes) shows ε(xt+1|y1:t) = C −1

t

  • i

p(xt+1|x(i)

t )p(yt|x(i) t )w(i) t ε(xt|y1:t−1)

+

  • i

p(xt+1|x(i)

t )w(i) t p(x(i) t |y1:t) − p(xt+1|y1:t).

Split error into Method error (First line)

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-146
SLIDE 146

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Error Propagation Recursion

A nice bit of algebra (see notes) shows ε(xt+1|y1:t) = C −1

t

  • i

p(xt+1|x(i)

t )p(yt|x(i) t )w(i) t ε(xt|y1:t−1)

+

  • i

p(xt+1|x(i)

t )w(i) t p(x(i) t |y1:t) − p(xt+1|y1:t).

Split error into Method error (First line) Quadrature error (Second line) on exact probabilities

Andreas Kl¨

  • ckner

Nonlinear Filtering

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SLIDE 147

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Error Estimates: Assumptions

Approximation Assumption: Assume we have enough particles so that

Andreas Kl¨

  • ckner

Nonlinear Filtering

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SLIDE 148

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Error Estimates: Assumptions

Approximation Assumption: Assume we have enough particles so that |

  • j

w(j)p(x(i)

t+1|x(j) t )p(x(j) t |y1:t) − p(x(i) t+1|y1:t)|

εp(x(i)

t+1|y1:t).

Andreas Kl¨

  • ckner

Nonlinear Filtering

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SLIDE 149

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Error Estimates: Assumptions

Approximation Assumption: Assume we have enough particles so that |

  • j

w(j)p(x(i)

t+1|x(j) t )p(x(j) t |y1:t) − p(x(i) t+1|y1:t)|

εp(x(i)

t+1|y1:t).

Note: Assumption is on quadrature of exact probabilities.

Andreas Kl¨

  • ckner

Nonlinear Filtering

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SLIDE 150

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Error Estimates: Assumptions

Approximation Assumption: Assume we have enough particles so that |

  • j

w(j)p(x(i)

t+1|x(j) t )p(x(j) t |y1:t) − p(x(i) t+1|y1:t)|

εp(x(i)

t+1|y1:t).

Note: Assumption is on quadrature of exact probabilities. Induction Assumption: rt+1 rt(1 + ε) + ε,

Andreas Kl¨

  • ckner

Nonlinear Filtering

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SLIDE 151

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Error Estimates: Assumptions

Approximation Assumption: Assume we have enough particles so that |

  • j

w(j)p(x(i)

t+1|x(j) t )p(x(j) t |y1:t) − p(x(i) t+1|y1:t)|

εp(x(i)

t+1|y1:t).

Note: Assumption is on quadrature of exact probabilities. Induction Assumption: rt+1 rt(1 + ε) + ε, where rt is the relative error at time t, i.e. the smallest positive real such that |ε(xt+1|y1:t)| rtp(xt+1|y1:t).

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-152
SLIDE 152

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Proof of Error Bound

ε(xt+1|y1:t)

  • |C −1

t

  • i

p(xt+1|x(i)

t )p(yt|x(i) t )w(i) t ε(xt|y1:t−1)|

+|

  • i

p(xt+1|x(i)

t )w(i) t p(x(i) t |y1:t) − p(xt+1|y1:t)|

Andreas Kl¨

  • ckner

Nonlinear Filtering

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SLIDE 153

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Proof of Error Bound

ε(xt+1|y1:t)

  • |C −1

t

  • i

p(xt+1|x(i)

t )p(yt|x(i) t )w(i) t ε(xt|y1:t−1)|

+|

  • i

p(xt+1|x(i)

t )w(i) t p(x(i) t |y1:t) − p(xt+1|y1:t)| AA

  • |C −1

t

  • i

p(xt+1|x(i)

t )p(yt|x(i) t )w(i) t ε(xt|y1:t−1)|

+εp(x(i)

t+1|y1:t)

Andreas Kl¨

  • ckner

Nonlinear Filtering

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SLIDE 154

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Proof of Error Bound

ε(xt+1|y1:t)

AA

  • |C −1

t

  • i

p(xt+1|x(i)

t )p(yt|x(i) t )w(i) t ε(xt|y1:t−1)|

+εp(x(i)

t+1|y1:t)

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-155
SLIDE 155

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Proof of Error Bound

ε(xt+1|y1:t)

AA

  • |C −1

t

  • i

p(xt+1|x(i)

t )p(yt|x(i) t )w(i) t ε(xt|y1:t−1)|

+εp(x(i)

t+1|y1:t) IA

  • |C −1

t

  • i

p(xt+1|x(i)

t )p(yt|x(i) t )w(i) t rt−1p(xt|y1:t−1)|

+εp(x(i)

t+1|y1:t)

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-156
SLIDE 156

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Proof of Error Bound

ε(xt+1|y1:t)

IA

  • |C −1

t

  • i

p(xt+1|x(i)

t )p(yt|x(i) t )w(i) t rt−1p(xt|y1:t−1)|

+εp(x(i)

t+1|y1:t)

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-157
SLIDE 157

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Proof of Error Bound

ε(xt+1|y1:t)

IA

  • |C −1

t

  • i

p(xt+1|x(i)

t )p(yt|x(i) t )w(i) t rt−1p(xt|y1:t−1)|

+εp(x(i)

t+1|y1:t)

= |rt−1

  • i

p(xt+1|x(i)

t )w(i) t p(x(i) t |y1:t)|

+εp(x(i)

t+1|y1:t)

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-158
SLIDE 158

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Proof of Error Bound

ε(xt+1|y1:t)

IA

  • |C −1

t

  • i

p(xt+1|x(i)

t )p(yt|x(i) t )w(i) t rt−1p(xt|y1:t−1)|

+εp(x(i)

t+1|y1:t)

= |rt−1

  • i

p(xt+1|x(i)

t )w(i) t p(x(i) t |y1:t)|

+εp(x(i)

t+1|y1:t) AA

  • rt−1|p(x(i)

t+1|y1:t)(1 + ε)| + εp(x(i) t+1|y1:t).

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-159
SLIDE 159

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Proof of Error Bound

ε(xt+1|y1:t)

IA

  • |C −1

t

  • i

p(xt+1|x(i)

t )p(yt|x(i) t )w(i) t rt−1p(xt|y1:t−1)|

+εp(x(i)

t+1|y1:t)

= |rt−1

  • i

p(xt+1|x(i)

t )w(i) t p(x(i) t |y1:t)|

+εp(x(i)

t+1|y1:t) AA

  • rt−1|p(x(i)

t+1|y1:t)(1 + ε)| + εp(x(i) t+1|y1:t).

⇒ rt rt−1(1 + ε) + ε.

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-160
SLIDE 160

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Error Estimates: Summary

We showed rt+1 rt(1 + ε) + ε.

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-161
SLIDE 161

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Error Estimates: Summary

We showed rt+1 rt(1 + ε) + ε. This implies rt (1 + ε)t − 1.

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-162
SLIDE 162

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Error Estimates: Summary

We showed rt+1 rt(1 + ε) + ε. This implies rt (1 + ε)t − 1. → Potentially exponential error growth.

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-163
SLIDE 163

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Error Estimates: Summary

We showed rt+1 rt(1 + ε) + ε. This implies rt (1 + ε)t − 1. → Potentially exponential error growth. A similar proof shows δt (t + 1) log(1 + ε′)

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-164
SLIDE 164

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Error Estimates: Summary

We showed rt+1 rt(1 + ε) + ε. This implies rt (1 + ε)t − 1. → Potentially exponential error growth. A similar proof shows δt (t + 1) log(1 + ε′) → Linear error growth in the log-likelihood.

Andreas Kl¨

  • ckner

Nonlinear Filtering

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SLIDE 165

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Outline

1 Nonlinear Filtering 2 Monte Carlo Filters

Particle Filters Importance Sampling

3 Quadrature Filters

Error Estimates

4 Continuous-Time Filtering

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-166
SLIDE 166

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Continuous-Time Nonlinear Filtering

Have: A Markov diffusion process dxi(t) = bi(x(t))dt + σij(x(t))dW j(t) where x = (x1, . . . , xd) and x(0) = x0.

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-167
SLIDE 167

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Continuous-Time Nonlinear Filtering

Have: A Markov diffusion process dxi(t) = bi(x(t))dt + σij(x(t))dW j(t) where x = (x1, . . . , xd) and x(0) = x0. An observation y(t) = t h(x(s))ds + V(t).

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-168
SLIDE 168

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Continuous-Time Nonlinear Filtering

Have: A Markov diffusion process dxi(t) = bi(x(t))dt + σij(x(t))dW j(t) where x = (x1, . . . , xd) and x(0) = x0. An observation y(t) = t h(x(s))ds + V(t). All coefficients are assumed smooth,

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-169
SLIDE 169

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Continuous-Time Nonlinear Filtering

Have: A Markov diffusion process dxi(t) = bi(x(t))dt + σij(x(t))dW j(t) where x = (x1, . . . , xd) and x(0) = x0. An observation y(t) = t h(x(s))ds + V(t). All coefficients are assumed smooth, and x0 and the Wiener processes W and V are assumed to be independent.

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-170
SLIDE 170

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

The Optimal Continuous-Time Filter

The optimal filter (= best mean-square estimate) for f (x(t)) is

❘ ❘

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-171
SLIDE 171

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

The Optimal Continuous-Time Filter

The optimal filter (= best mean-square estimate) for f (x(t)) is ˆ f (x(t)) =

  • ❘d f (x)u(t, x)dx
  • ❘d u(t, x)dx

,

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-172
SLIDE 172

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

The Optimal Continuous-Time Filter

The optimal filter (= best mean-square estimate) for f (x(t)) is ˆ f (x(t)) =

  • ❘d f (x)u(t, x)dx
  • ❘d u(t, x)dx

, where u is the unnormalized filtering density.

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-173
SLIDE 173

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

The Optimal Continuous-Time Filter

The optimal filter (= best mean-square estimate) for f (x(t)) is ˆ f (x(t)) =

  • ❘d f (x)u(t, x)dx
  • ❘d u(t, x)dx

, where u is the unnormalized filtering density. u obeys the Zakai SPDE du(t, x) = L∗u(t, x)dt + h(x)u(t, x)dy(t),

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-174
SLIDE 174

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

The Optimal Continuous-Time Filter

The optimal filter (= best mean-square estimate) for f (x(t)) is ˆ f (x(t)) =

  • ❘d f (x)u(t, x)dx
  • ❘d u(t, x)dx

, where u is the unnormalized filtering density. u obeys the Zakai SPDE du(t, x) = L∗u(t, x)dt + h(x)u(t, x)dy(t), where L∗u := 1 2 ∂2 ∂xixj ((σσ∗)iju) − ∂ ∂xi (biu).

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-175
SLIDE 175

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

One possible Solution Method

Sergey Lototsky’s spectral separating scheme expresses u as an expansion into Wick polynomials ξα:

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-176
SLIDE 176

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

One possible Solution Method

Sergey Lototsky’s spectral separating scheme expresses u as an expansion into Wick polynomials ξα: u(t, x) =

  • α

1 √ α! ϕα(t, x)ξα(y).

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-177
SLIDE 177

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

One possible Solution Method

Sergey Lototsky’s spectral separating scheme expresses u as an expansion into Wick polynomials ξα: u(t, x) =

  • α

1 √ α! ϕα(t, x)ξα(y). Coefficients ϕα satisfy a deterministic system of PDEs.

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-178
SLIDE 178

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

One possible Solution Method

Sergey Lototsky’s spectral separating scheme expresses u as an expansion into Wick polynomials ξα: u(t, x) =

  • α

1 √ α! ϕα(t, x)ξα(y). Coefficients ϕα satisfy a deterministic system of PDEs. The scheme separates

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-179
SLIDE 179

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

One possible Solution Method

Sergey Lototsky’s spectral separating scheme expresses u as an expansion into Wick polynomials ξα: u(t, x) =

  • α

1 √ α! ϕα(t, x)ξα(y). Coefficients ϕα satisfy a deterministic system of PDEs. The scheme separates

dependency on process parameters (ϕα) from

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-180
SLIDE 180

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

One possible Solution Method

Sergey Lototsky’s spectral separating scheme expresses u as an expansion into Wick polynomials ξα: u(t, x) =

  • α

1 √ α! ϕα(t, x)ξα(y). Coefficients ϕα satisfy a deterministic system of PDEs. The scheme separates

dependency on process parameters (ϕα) from dependency on observation (ξα).

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-181
SLIDE 181

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

One possible Solution Method

Sergey Lototsky’s spectral separating scheme expresses u as an expansion into Wick polynomials ξα: u(t, x) =

  • α

1 √ α! ϕα(t, x)ξα(y). Coefficients ϕα satisfy a deterministic system of PDEs. The scheme separates

dependency on process parameters (ϕα) from dependency on observation (ξα).

ϕα can be precomputed.

Andreas Kl¨

  • ckner

Nonlinear Filtering

slide-182
SLIDE 182

Nonlinear Filtering Andreas Kl¨

  • ckner

Outline Nonlinear Filtering Monte Carlo Filters

Particle Filters Importance Sampling

Quadrature Filters

Error Estimates

Continuous- Time Filtering

Questions?

?

Andreas Kl¨

  • ckner

Nonlinear Filtering