sis sequential importance sampling advanced methods in simulation 096320 winter 2009 presented by:...

SISSequential Importance

Sampling

Advanced Methods In Simulation 096320

Winter 2009

Presented by: Chen Bukay, Ella Pemov, Amit Dvash

Talk Layout

SIS – Overview and algorithm Random walk – SIS simulation Nonlinear Filtering – Overview & Added

value Nonlinear Filtering – Simulation

Importance Sampling - General Overview

Importance Sampling – The most fundamental variance reduction

technique Leads to a dramatic variance reduction –

particularly when estimating rare event probabilities

Target – Expected performance of-

Likelihood Ratio Estimator -)(

)()(

1

1

^

k

kN

kk xg

xfxH

Nl

the sample performance

importance sampling density

probability density of X

dxxfxHxHEl )()()]([

SIS - Overview

Sequential Importance Sampling Also known as “Dynamic Importance Sampling”. Simply means importance sampling that carried out

in sequential manner.Why Sequential?

Problematic to sample from multi-dimensional vector Dependency between the variables It is difficult to sample from

),...,( 1 nxxx

f

SIS - Overview

Assumptions –

X is decomposable can present g(x) –

Easy to sample from g(x) sequentially

),...,|()...|()()( 1112211 nnn xxxgxxgxgxg

SIS – Overview (cont’)

It is easy to generate sequentially from Generate from Generate from …..We get –

Due to the product rule of probability we can write -

The Likelihood function -

),...,|()...|()(

),...,|()...|()()(

1112211

11121

nnn

nn

xxxgxxgxg

xxxfxxfxfxW

1x

2x )|( 122 xxg

)(xg)( 11 xg

),...,|()...|()()( 11121 nn xxxfxxfxfxf)(

),()|(

BP

BAPBAP

),...,( 1 nxxx

SIS – Overview (cont’)

)(

)(

)|(

)|(

1

1

12

12122 xg

xf

xxg

xxfwuw

)();( 1;111 tttt xwutxw

Likelihood till time t

1)0;( 10 xw)(

)(

1

1011 xg

xfwuw

Likelihood till time t-1

)(

),()|(

BP

BAPBAP

)|()(

)(

)|(

)|(

1:11:1

:1

1:1

1:1

tttt

t

ttt

ttt xxgxf

xf

xxg

xxfu

SIS – Overview

In order to update the likelihood ratio recursively, we need to know how to calculate

We know In order to calculate it requires integrating

over

There options to solve this – Use auxiliary pdfs that can be easy evaluated and each

is a good approximation to

);( 1 nxf);( 1 txf

)( ;1 txf

)(xfnt xx ,...,1

nfff ,...,, 21 )( ;1 tt xf)( ;1 txf

nt xx

t dxnxfxf...

1;1

1

);()( considered hard integral

Easy to calculate

Where 1)( 0;10 xf)()(

)(...

)(

)(

1

)()( ;1

1:11

:1

11

2:1211nn

nn

nn xfxf

xf

xf

xfxfxf

)|()(

)(

1:11:11

:1

ttttt

ttt xxgxf

xfu

SIS – Algorithm

SIS algorithm (Sequential)1. For each finite t = 1,…,n, Sample from2. Compute where and

3. Repeat N times and estimate via

SIS algorithm (Dynamic) 1. At time t, arrival of tth sample2. Sample xt N times according to

3. Calculate

4. estimate according to the existing samples (1,…,t)

tx )|( 1:1 ttt xxg

1 ttt wuw 10 w

l^

l

1 ttt wuw

^

l

t = 1,…,n)|()(

)(

1:11:11

:1

ttttt

ttt xxgxf

xfu

)|( 1:1 ttt xxg

Parallel computing

)|()(

)(

1:11:11

:1

ttttt

ttt xxgxf

xfu

SIS Algorithm - Sequential

1st sample:

2nd sample:. .. .. .Nth sample:

2 1( ,..., )nx xx

1 1( ,..., )nx xx

1( ,..., )N nx xx

Calculate )(1 xw

Calculate

Calculate )(xwN

)(2 xw

Estimate by Computing )()(1

1

^

k

N

kk xWxH

Nl

l

SIS Algorithm - Dynamic

1st sample:

2nd sample:.

..

..

.Nth sample:

),...,( 1 nxx1x

At Timet =1

)( 1x2x

)( 1x1x

),...,( 1 nxxNx

)(1 xw

)(xwN

)(2 xw

)()(1

1

^

kk xx WHN

lN

k

l

),( 21 xx1x

)( 1xNx ),( 21 xxNx

),...,( 1 nxx2x ),( 21 xx2x

Calculaterecalculate

recalculateCalculate

Calculaterecalculate

At time t =2

At time t=n

Estimate by Computing

With the existing samples)](),..([ 1 xwxwW t

Nt

t

Random Walk

Problem statement

Reminder - Go forward Probability p Go backward probability q

p < q (has drift to - ) Goal – estimating the rare event probability of reaching state K (large

number) before 0 (zero) starting at k.l

p

0 K1 2 … k

q

p

qqq

p p pstart

Random Walk – Simulation Result

0.6

0.4

p

q


0.8

0.2

p

q

SIS Application:

Non Linear Filtering

State Space Models

( )1 kk kx x V T w

( ) k kky x v

x

Tx1 x2 x3

y2 y3y1

State Space Models

k k ky h x v

Dynamic Model

Measurement Equation

State Equation

Observation Equation

1k k kx f x w HMM

State Space Models cont’

k k ky h x v

1k k kx f x w Known pdf - Pw

Known pdf - Pv

Markov Property1 |

|k k w

k k v

x x P

y x P

Linear Models

Kalman Filter Linear Dynamic models Linear Measurement Equations v, w, x0 – Gaussian & independent

Kalman Filter is the optimal estimator (MSE)

Assuming models Motion models - Linear/Non-Linear State

Dynamic Linear/Non-Linear Measurement Equations

v, u, x0 – independent, not necessarily Gaussian

General Models

1

k k k

k k k

x f x w

y h x v

Problem Description

θa

θb

θc

LOP – Line Of Position

Observers – Known exact location

(xa,,ya)

(xb,yb)

(xc,,yc)

Target – Unknown location

(xe,,ye)

θa

θb

θc

Bearing Only Measurements

(xe,,ye)

(xa,,ya)

(xb,yb)

(xc,,yc)

tan e

e

c

c

y yc

xx

Bearing Only Measurements

2

tan1

kk k

k

xy v

x

1k k kx x w

1 | ,

2| tan ,

1

k k k

kk k

k

x x N x Q

xy x N R

x

Non-Linear Filtering

Motivation Non linear dynamic/measurement equations Noise distribution not Gauss

Kalman Filter: No longer the optimal estimator (MSE)

EKF – Linearization of the state space Equations Suboptimal estimator Convergence is not guaranteed

The Bootstrap Filter

Represent the pdf as a set of rv (and not as a function)

The Bootstrap Filter – Recursive algorithm for propagating and updating these rv samples

Samples are naturally concentrated in regions of high probability

“Novel Approach to nonlinear/non Gaussian Bayesian state estimation” N.J. Gordon, D.J. Salmond & A.F.M Smith

Motivation For having P(X(k)|Y(1:k))

1:1

ˆ |N

i iN k k k k k

i

p x Y q x x

ˆ: |MSE x E x y

MSE

ˆ: arg max |x

ML x f y x

ML

The Bootstrap FilterRecursive Calculation of P(X(k)|Y(1:k))

1: 1

1: 11:

11

:

| || |

|| t t t t

tt t t tt t

t

P y x P x yP y x P x y

P yx y

yP

1: 1 1 1 1: 1 1| | |t t t t t t tP x y P x x P x y dx

1 1|t t w t tP x x P x f x

|t t v t tP y x P y h x 1 1: 1|t tP x y

Assume we know

Bayes & yt|xt independent of y1:t-1

The importance Sampling

1: 1: 1| | |

t t tg xf x w

t t t t t tP x y P y x P x y


Algorithm1. Initialization: k = 0, Generate x0

i ~Px0 i = 1…N

2. Measurement Update:Given yk calculate likelihood for each current sample

*

* *

**

*11

2tan(

| 1 = i = 1...N

2| tan(1

k i

v kk k i k ii

k NN

k ik k j

v kjj k i

xP y

P y x xq

xP y x P yx

* *| ,k k k k k ky h x v y x N h x R

Algorithm (cont’)

3. Re-Sampling - Sample N samples from {xk

*i}i=1:N, with replacement, where the probability to choose the i-th particle is qk

i at stage k

4. Prediction: Pass the new samples through the system Equation

5. Set k = k+1 and return to 2


*

1 ,k k kk i k ix f x w x w

-50 0 50 100 150 200 2500

50

100

150Problem Geometry

X-Axis [km]

Y-Ax

is [k

m]

Target True Poistion (141,141)Initial Estimation Of Target Poistion (100,120)Observer Course

Vx= -0.1 [km/sec]

Vy= 0.01 [km/sec]

dt = 300 [sec]

Measurement Variance ~ 1o

(141,,141) [km]

(100,120) [km]

1 2 3 4 5 6 7 8 9 10 110

500

1000

1500

2000

Measurement index

MS

E

The Effect of Resampling on the MSE[1000 particles; 1e3 - Monte-Carlo Simulations]

BSF with Resampling - X-Axis position estimation

BSF No Resampling - X-Axis position estimation

1 2 3 4 5 6 7 8 9 10 110

200

400

600

800

Measurement index

MS

E

BSF with Resampling - Y-Axis position estimation

BSF No Resampling - Y-Axis position estimation

-150 -100 -50 0 50 100 1500

50

100

150Problem Geometry

X-Axis [km]

Y-A

xis

[km

]

Target True Poistion (141,141)

Initial Estimation Of Target Poistion (100,120)

Observer CourseBSF With Resampling

BSF No Resampling

95 100 105 110 115 120 125 130 135 140 145

115

120

125

130

135

140

145

Problem Geometry

X-Axis [km]

Y-A

xis

[km

]

Target True Poistion (141,141)

Initial Estimation Of Target Poistion (100,120)

Observer CourseBSF With Resampling

BSF No Resampling

Simulations

1

3tan

1

k k k

kk k k

k

x Fx Gw

xy x v

x

1 0 0

0 1 0 0

0 0 1

0 0 0 1

T

GT

1 0 0

0 1 0 0

0 0 1

0 0 0 1

T

FT

1

_

_

_

_

k

x position

x velocityx

y position

y velocity

0 50 100 150 200 250 300 350 400 450

-150

-100

-50

0

X-Axis [km]

Y-A

xis

[km

]Initial Conditions: x & y -petrubations among its error variance

Observer Location (0,0)

Target True Location

Initial Estimation Of Target PoistionEKF

BSF

80 100 120 140 160 180

-75

-70

-65

-60

-55

-50

-45

X-Axis [km]

Y-A

xis

[km

]Initial Conditions: x & y -petrubations among its error variance




BSF

0 50 100 150 200 250 300 350 400 450

-140

-120

-100

-80

-60

-40

-20

0

X-Axis [km]

Y-A

xis

[km

]

Initial Conditions: x-petrubation ~0 y-petrubation 5.5




BSF

0 10 20 30 40 50 60-0.55

-0.5

-0.45

-0.4

-0.35

-0.3Measured bearing Vs True Bearing

True Bearing

Measured bearing

0 50 100 150 200 250 300 350

-150

-100

-50

0

50

100

150

200

X-Axis [km]

Y-A

xis

[km

]

Initial Conditions: x & y -petrubations among its error variance




BSF

150 time steps

0 20 40 60 80 100 120 140 160-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6Measured bearing Vs True Bearing

True Bearing

Measured bearing

20 30 40 50 60 70 80 90136

138

140

142

144

146

148

150

152

154

156

X-Axis [km]

Y-A

xis

[km

]Initial Conditions: large x-y petrubations

35 40 45 50 55 60 65 70 75 80 85

142

144

146

148

150

152

154

156

X-Axis [km]

Y-A

xis

[km

]Initial Conditions: large x-y petrubations

0 20 40 60 80 100 120 140

50

100

150

200

250

300

X-Axis [km]

Y-A

xis

[km

]

Initial Conditions: large x-y petrubations

Backup

Markov Chain

Markov property – Given the present state, future states are independent of the past states. The present state fully captures all the information that could influence the

future evolution of the process.

The changes of state are called transitions, and the probabilities associated with various state-changes are called transition probabilities.

210.9

0.1

0.5

0.50.9 0.10.5 0.5

P=

F(X) - calculations

)(

),()|(

BP

BAPBAP

),...,,(),...,(

),...,,(...

)(

),(

1

)()( 21

11

21

1

211n

n

n xxxfxxf

xxxf

xf

xxfxfxf

),...,|()...|()()( 11121 nn xxxfxxfxfxf

)()(

)(...

)(

)(

1

)()( ;1

1:11

:1

11

2:1211nn

nn

nn xfxf

xf

xf

xfxfxf

Where to put

1: 1:

1: 1 1: 1 1: 1: 1

|

| | ,t t

tt t t t t t

fu

f g x

x y

x y y x

1

1: 1 1: 1: 1

| |

| | ,t t t t

tt t t t t t

f y x f x xu

f y g x

y y x

Markov Property Bayes Rule

Normalization Constant

sis sequential importance sampling advanced methods in simulation 096320 winter 2009 presented by:...

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