OPTIMAL FILTERING: RELEVANCE OF THE INTERACTIVE …sting.deis.unibo.it/.../downloads/Slides_lectures_Grivel.pdf · 2012. 5. 28. · Eric Grivel [email protected], [email protected]

OPTIMAL FILTERING: RELEVANCE OF THE

INTERACTIVE MULTIPLE MODEL Eric Grivel

[email protected], [email protected]

SIGNAL & IMAGE Group



About the ENSEIRB-MATMECA

founded in 1920

ranked in the top 20 of

the "Grandes Ecoles" French national educative system

1200 students

4 departments:

Electronics

Computer sciences

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Mechanics

Admission through the selective national exams for the French

“Grandes Ecoles”, after 2 years of preparation in higher education.

For foreign students: please visit http://www.enseirb-matmeca.fr/

Signal and image processing is everywhere

in our everyday life


ENHANCED SPEECH SIGNAL :

FIRST MICROPHONE :

SECOND MICROPHONE :

For speech processing

Processing

Additive noise

Noisy signals

Speech signal

Enhanced speech

When using a mobile phone -after recording the speech signal, sampling the signal and coding it -when transmitting and receiving the speech signal and other data (digital communication issues)

When using a music player -by using various kinds of coding for music, etc.

When using a GPS -for localization and navigation Determining the positions by measuring the propagation delays of signals broadcast by GPS satellite-in-view

-for integrity monitoring 1T

2T

3T 111 ,, zyx

222 ,, zyx

333 ,, zyx

zyx ,,

When watching TV -Digital Video Broadcasting-T (DVB-T)

.

With toys, computers or other electronic devices -Text-to-speech applications -Speech recognition -Texture synthesis for video games, films, etc. -etc.

In other applications -EEG-ECG processing, -image processing to detect defaults (material, etc.), -seismic data processing, -mechanical structures -video processing for traffic surveillance, etc.

In industrial applications -in radar processing -in sonar processing

Radar

Sea clutter

Ground

clutter

jamming

Target Other

target

In financial mathematics -to study, analyse and predict.

Among the issues : • estimation and detection, • extracting information from the observations.

Various ways to address these problems

• In the frequency domain • In the time domain

In each case, selecting a criterion (LS, MV, etc.)

In the frequency domain :spectrum analysis

Off-line estimation:

Subspace methods: ESPRIT, MUSIC, etc.

8

In the frequency domain: time-frequency-analysis


Wigner Ville estimation, Wavelet based estimation, Chirplet based estimation,

EMD, etc.

Recursive method:

For instance, by using a bank of selective filters (Capon,

Borgiotti-Lagunas, etc.)

Recursive method:

For instance, by using a TVAR model

In the time domain, based on a system representation:

Recursive estimation :

Adaptive filter (LMS, RLS), Kalman, H, particule filter


Subspace methods (« playing » with correlation).

Iterative estimation:

Multigrid search, evolutionnary algorithms

Recursive

algorithm

Recursive

algorithm

observation

at time k

Off-line

algorithm

bloc

of observations

Recursive

algorithm

observation

at time k+1

observation

at time k-1 observation

at time k

8

Contents Part 1: about system modeling Part 2: state space representation of the system

• from the continuous to the discrete-time domain: application to motion target

• AR modeling for various applications Part 3: estimating the state vector by Kalman filtering

• in the linear case • In the non Linear case: EKF, SO-EKF, CDKF, QKF

Part 4: relaxing the assumptions on the model by using Interactive Multiple Model

Part 5: applications





Part 4: relaxing the assumption on the model by using Interactive Multiple Model

Part 5: application

System modeling State space rep.. Kalman filter IMM Application

Example 1: modeling a signal by an autoregressive process

with the AR parameter vector,

a white Gaussian zero-mean driving process with variance .

Very popular model used :

• in speech enhancement and coding (IUT norms G.728, G.729), • for Rayleigh fading channel modeling, • in sea clutter rejection in radar processing, • etc.

nuinsansp

ii

1

Tpaa 1

)(nu2u


Example 1: modeling a speech signal by an autoregressive process Analysing the signal properties Selecting a model that matches the spectral and correlation properties of the signal

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

x 104

0.005 0.01 0.015 0.02 0.025-2.5

-2

-1.5

-1

-0.5

0

0.5

1

x 104

0.28 0.29 0.3 0.31 0.32 0.33 0.34 0.35-2.5

-2

-1.5

-1

-0.5

0

0.5

1

x 104

0.005 0.01 0.015 0.02 0.025

-3000

-2500

-2000

-1500

-1000

-500

0


0 0.01 0.02 0.03 0.04 0.05

-4000

-3000

-2000

-1000

0

1000

2000

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.510

20

30

40

50

60

70

80

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.510

20

30

40

50

60

70

80

freq

am

plit

ude

spectre du signal, lpc correspondante

corresponding AR PSD PSD of the

unvoiced

speech frame

0 0.005 0.01 0.015 0.02 0.025 0.03

-150

-100

-50

0

50

100

150

temps

am

plit

ude

representation temporelle du signalTemporal representation of the speech signal

time



0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.530

40

50

60

70

80

90

100

110

0 0.005 0.01 0.015 0.02 0.025 0.03

-6000

-5000

-4000

-3000

-2000

-1000

0

1000

2000

3000

temps

am

plit

ude

representation temporelle du signal

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.530

40

50

60

70

80

90

100

110

freq

am

plit

ude

spectre du signal, lpc correspondante

corresponding AR

PSD

PSD of the

voiced

speech

frame

pitch period of the

voiced frame

fundamental frequency Temporal representation of the speech signal

time



Example 2: sum of sinusoids Example 3: modeling the motion of an aircraft.

2.6 2.605 2.61 2.615 2.62 2.625 2.63 2.635 2.64 2.645

x 104

2.56

2.57

2.58

2.59

2.6

2.61

2.62x 10

4

x

y

position recherchée

2.605 2.61 2.615 2.62 2.625 2.63 2.635 2.64 2.645

x 104

2.57

2.575

2.58

2.585

2.59

2.595

2.6

2.605

2.61

2.615

2.62x 10

4

X

Y

dans le repère aéroporté cartésien Cartesian coordinate system Cartesian coordinate system

position

velocity












Part 4: relaxing the assumption on the model by using Interactive Multiple Model


Definition

Let the state space representation of the system be defined as follows (continuous-time domain):

where

In addition, the observation satisfies:

The observations are often disturbed by an additive measurement noise. Therefore :

)()()()()( twtBtxtAtx

dt

txdtx

)()(

)()( txHty

Tn txtxtx )(,),()( 1

)()()( tbtxHty


Stationary system A, B and H do not vary in time. Hence, one has

If there is no driving process

Searching a solution to this differential equation

)()()( twBtxAtx

)()( txHty

)()( txAtx

)0()( 0 xtx

)0()(exp)( 0 xttAtx

Searching the solution


Searching the general solution

The quantity is often called the transition matrix and is denoted as

)(exp 0ttA

)( 0tt Φ

dwBtAxttAtxt

t

)()(exp)0()(exp)(

0

0

Searching the solution


)()()( twBtxAtx

)()( txHty

)0()( 0 xtx

Example : motion for radar, GPS, traffic control, etc.

Purpose : relating position, velocity and acceleration. Being able to know where a “target” is

Focus on a constant velocity motion

Various examples


Let us assume that the acceleration is a zero-mean white Gaussian sequence with variance q.

Let the state vector be defined as follows:

with

)()()( twtxta

)(.1

0

)(

)(

00

10

)(

)(tw

tx

tx

tx

tx

00

10A

1

0B

Ttxtxtx )(),()(

)()()( tBwtxAtx


Examples in the continuous time domain

Let us now derive the discrete-time state space representation of the system. Given

one has for two successive times, namely

and

or equivalently

dBwtAtxttAtxt

t

)()(exp)()(exp)(

0

00

Tkt )1(

kTt 0

dBwTkAkTxATTkxTk

kT

)())1((exp)(exp))1(()1(

dkTBwTAkTxATTkxT

)()(exp)(exp))1((

0



Given the following Mc Laurin expansion:

One obtains:

or equivalently

or

10

1exp

TATIAT

dkTwT

kTxT

TkxT

)(1

0

10

1)(

10

1))1((

0

dkTwT

kTxT

TkxT

)(1

)(10

1))1((

0

)()()()1( kukxkkx



Then, let us look at the covariance matrix of the driving process (or model noise). 2 cases can be considered depending on the sampling period

1/ is constant during the sampling period

Therefore

T

T

TkTwd

TkTwku

0

2

2).(.1

).()(

)(tw

2

3

34

22

2

24.2

.2.])().([

TT

TT

qTT

T

TqkukuEQ T



Then, let us look at the covariance matrix of the driving process (or model noise). 2 cases can be considered depending on the sampling period

2/ varies during the sampling period

)(tw

TT

TT

qdT

TTq

dTT

qkukuEQ

T

TT

2

23..1

)(.

.1.1

.])()([

2

23

0

2

0



Example 2: A constant acceleration (CA) motion*

In that case, the acceleration derivative (or jerk) satisfies:

-Which state vector ?

-Which continuous–time state space representation?

-Which discrete-time state space representation?

)()()( twtxta

* Or « nearly-constant-acceleration model »



-Which state vector ?

-Which continuous–time state space representation?

-Which discrete-time state space representation?

)()()( twtxta




)()()( twBtxAtx

dBwtAtxttAtxt

t

)()(exp)()(exp)(

0

00

100

102

1

2exp

2

22 T

TT

TAATIAT

Example 2: A constant acceleration (CA) motion*

In that case, the acceleration derivative (or jerk) satisfies:

Therefore, one can consider the following state vector:

It satisfies:

)()()( twtxta

Txxxx ,,

)(.

1

0

0

.

000

100

010

tw

x

x

x

x

x

x

BA




Given the following Mc Laurin expansion:

Thus, one has

100

102

1

2exp

2

22 T

TT

TAATIAT

dkTwT

TT

kTxT

TT

TkxT

)(

1

0

0

100

102

)(1

)(

100

102

1

))1((

0

22

dkTBwTAkTxATTkxT

)()(exp)(exp))1((

0



This leads to:

By defining , one has:

dkTwT

T

kxT

TT

kxT

)(

1

2

)(

)(

100

102

1

)1(

0

22

dkTwT

T

kuT

)(

1

2

)(

)(

0

2

)()()()1( kukxkkx



The covariance matrix of the model noise can be:

when it does not change during the sampling period

TTT

TTT

TTT

qQ

26

238

6820

.

23

234

345

234

345

456

26

2412

61236

.

TTT

TTT

TTT

qQ



Remark: There is an alternative to the white-noise jerk model:

The Wiener-sequence acceleration model assumes that the acceleration increment is an independent (white noise) process

)(

1

2/²

)()()1( kwT

T

kxkkx



Remark 2: There are many other motion models such as the so-called Singer acceleration model.

In that case, the autocorrelation of the acceleration satisfies:

This means that the acceleration results from the filtering of a white noise:

mer mxx

.)( 2

)(*)()()( twthtxta




One can show that:

Therefore:

)()(

)()(

)(1

twdt

txdtx

dt

txdtx

m

)(.

1

0

0

.

00

100

010

tw

x

x

x

x

x

x



Remark 2: There are many other motion models such as the so-called Singer model.

Taking advantage of the Mc Laurin expansion of the « exp » function:

...2

100

...2

10

201

...

00

00

00

2

1

00

00

00

exp

22

2

2

22

2

2

TT

TT

T

T

T

T

T

T

T

IAT

T

T

T

e

e

eTT

00

)1(1

10

)1(1

12




332313

232212

13121122)(

qqq

qqq

qqq

kQ m

)1.(2

1

)1.(2

1

)432.(2

1

)21.(2

1

)2)22(1.(2

1

)143

222.(2

1

233

2

223

2

322

2

313

222

412

23

322

511

T

T

TT

TT

TT

TT

eq

eq

eeTq

eTeq

TTeeTq

eTeT

TTq



Remark 3: MRU, MUA and Singer models

Singer model

Constant velocity motion Constant acceleration

motion

1

2

2 mq

T

T

T

e

e

eTT

00

)1(1

10

)1(1

12

100

102

12

T

TT

22 mq

10

1 T

0



Remark 4: other alternatives concerning the model motion

The Singer model is not very suitable for practical maneuvers in which target acceleration is oscillatory

• Markov Acceleration Model for Coordinated Turns

• Asymmetrically Distributed Normal Acceleration Model

• Coordinated Turn Model



Example 3: AR model identification.

If the purpose is to estimate the parameters from the noise-free observations, the state vector can be defined as follows:

)()(1

kuikyakyp

ii

Tpaakx =)( 1

)()(1)( kxkxΦkx

)()()(

)( )()1()( 1

kukxkH

kuaapkykykyT

p

linear function to update the state vector and linear observation equation


Examples in the discrete-time domain

Example 4: AR process enhancement when AR parametes are known.

If the purpose is the estimation of the process, the state vector can be defined as follows:

with and

)()(1

kuikyakyp

ii

Tpkykykx 1)+-()(=)(

)()(1)( kuGkxΦkx

0100

00

00

0001

11

pp aaa

Φ

T

p

G

1

001



Example 5: AR model with noisy observations: identification and enhancement

If the purpose is to estimate both the parameters and the process from the noise-free observation, the state vector must store the AR parameters and the noise-free process samples

)()(=)z( kbkyk

)()(1

kuikyakyp

ii



Example 5: AR model with noisy observations: identification and enhancement

Tp pkykyaakx 1)+-()(=)( 1

)(

O)1(

0100

00

00

0001=)(

11

kuGkx

aaa

O

OI

kx

pp



Example 5: AR model with noisy observations.

)(

O)1()1(

0

0

1

)1(

0100

00

00

0001

000

)(

O)1(

0100

00

00

0001=)(

T

11

kuGkx

OO

IOkx

O

kxO

OI

kuGkx

aaa

O

OI

kx

pp

Non-linear function updating the state vector



Example 5: AR model with noisy observations.

)(

O)1()1(

0

0

1

)1(

0100

00

00

0001

000

)(

O)1(

0100

00

00

0001=)(

T

11

kuGkx

OO

IOkx

O

kxO

OI

kuGkx

aaa

O

OI

kx

pp

Non-linear function updating the state vector



Example 6: GPS measurements.

Non-linear observation equation

1T

2T

3T 111 ,, zyx

222 ,, zyx

333 ,, zyx

222iiii zzyyxxTc



Some disturbances: -always: the GPS receiver clock bias with

respect to the GPS reference time

-no multipath: just an additive white Gaussian

(with standard deviation = 8m)

-if mutltipath: mean and variance jump to be estimated.

-satellite anomalies which result in biases or drifts corrupting the

GPS measurements

=>integrity monitoring

Conclusions:

linear or non-linear state space representations

Pb: Knowing well the covariance matrices of the model noise and the additive measurement noise in the state space representation of the system.















Kalman

Kalman filter, a recursive alternative to the Wiener filter


)()()()()1( kukGkxkΦkx

)()()()( kvkxkHky

Kalman filtering (linear case)

0)()( lvkuET

0)( kuE

)()()()( lkkQlukuET

0)( kvE

)()()()( lkkRlvkvET

Updating equation

Observation equation

0)()0( kuxET

0)()0( kvxET

0k

0k


Our task: estimating the state vector

By taking into account the information available at time n, which can be before, after or at the instant k.

In so doing, we will consider three different cases:

• k = n: filtering

• k < n: smoothing;

• k > n: prediction.

)(kx



Irrespective of which of the above cases is being treated,

Our aim is to obtain a recursive estimation of the state vector by taking advantage of:

-the state space representation of the system

-the definition of the estimate

We will look closely at the ‘propagation’ and ‘update’ steps

)(,),1()()/(ˆ nyykxEnkx



Propagation step (1/2):

As one has:

One obtains:


)(,),1()()()(,),1()(),1(

)(,),1()1()/1(ˆ

kyykuEkGkyykxEkkΦ

kyykxEkkx

0)(,),1()( kyykuE

)/(ˆ),1()/1(ˆ kkxkkΦkkx


Propagation step (2/2):

Let us look at the relation between a priori error at time k+1 and the a posteriori at time k

The corresponding error covariance matrices hence satisfies:


)()()/(~),1(

)/(ˆ),1()()()(),1(

)/1(ˆ)1()/1(~

kukGkkxkkΦ

kkxkkΦkukGkxkkΦ

kkxkxkkx

)()()(),1()/(),1()/1( kGkQkGkkΦkkPkkΦkkP TT


Update step (1/3):

In that case, the a posteriori error at time k satisfies:


)1/(ˆ)()()()1/(ˆ)/(ˆ kkxkHkykKkkxkkx

)()()1/(~)()(

)()()1/(ˆ)()()(

)1/(ˆ)()()()()()1/(ˆ)(

)1/(ˆ)()()()1/(ˆ)(

)/(ˆ)()/(~

kvkKkkxkHkKI

kvkKkkxkxkHkKI

kkxkHkvkxkHkKkkxkx

kkxkHkykKkkxkx

kkxkxkkx

innovation


Update step (2/3):

The Kalman gain aims at minimizing the following criterion:

where:


)1/(ˆ)()()()1/(ˆ)/(ˆ kkxkHkykKkkxkkx

)()()()()1/()()()/( kRKkKkHkKIkkPkHkKIkkP TT

)/(trace)/(~)/(~trace)( kkPkkxkkxkJT



0)(

)(

kK

kJ

Update step (3/3):

0

)(

)()()()()()1/()()()()()1/(2trace

kK

kKkRkKkKkHkkPkHkKkKkHkkP TTTTT

)()()()1/()()()()1/( kRkKkHkkPkHkKkHkkP TT

1)()()1/()()()1/()(

kRkHkkPkHkHkkPkK TT



Some comments about the Kalman gain:

1

)()()1/()()()1/()(

kRkHkkPkHkHkkPkK TT

Inverse of the covariance matrix

of the innovation

Cross-covariance matrix

between the innovation and

the a priori state prediction


Spectrogramm of the smoothed

signal

Temps (s)

Fré

qu

ence

Noisy and enhanced signals Spectrogram of the noisy signal

Temps (s)

Fré

qu

ence

Spectrogram of the noise-free

signal

Temps (s)

Fré

qu

ence



In the non linear case

))(),(()1( kukxfkx

))(),(()( kvkxgky

Updating equation

Observation equation

Non linear function

Non linear function


How to address the non-linear case?

Various approaches:

Based on Taylor expansion

-first order expansion => Extended Kalman filter (EKF)

-2nd order expansion => Second-Order Extended Kalman filter (SO-EKF)

Based on Sigma Points

-Unscented transform (UKF)

-Central difference Kalman filter (CDKF)

-Quadrature Kalman filter (QKF)


Kalman filtering (non linear case)

Various approaches:

Based on Taylor expansion

-first order expansion => Extended Kalman filter (EKF)

-2nd order expansion => Second-Order Extended Kalman filter (SO-EKF)

Based on Sigma Points

-Unscented transform (UKF)

-Central difference Kalman filter (CDKF)

-Quadrature Kalman filter (QKF)


Using a first order expansion of the non-linear function around the a posteriori estimate at time k-1, one has:

Extended Kalman filter

)())1(()( kukxfkx Updating equation

))1/1(ˆ)1(())1/1(ˆ())1(()1/1(ˆ

kkxkxkkxfkxfkkxf

)())1/1(ˆ)1(())1/1(ˆ(

)())1(()(

)1/1(ˆkuGkkxkxkkxf

kuGkxfkx

kkxf


A first relation can be hence obtained as follows:

The corresponding error satisfies:

)())1(()( kukxfkx Updating equation

))1/1(ˆ(

)1()...1(/)(

))1/1(ˆ)1/(ˆ())1/1(ˆ()1/(ˆ)1/1(ˆ

kkxf

kyykuGE

kkxkkxkkxfkkxkkxf

)()1/1(~

))1/1(ˆ(

)())1/1(ˆ)1(())1/1(ˆ(

)1/(ˆ)()1/(~

)1/1(ˆ

)1/1(ˆ

kuGkkx

kkxf

kuGkkxkxkkxf

kkxkxkkx

kkxf

kkxf



The a priori error satisfies

The a priori error satisfies

The jacobian plays the same role as the updating matrix.

)())1(()( kuGkxfkx Non-Linear case

)()1/1(~)1/(~)1/1(ˆ

kuGkkxkkxkkxf

Linear case )1()1()1()( kuGkxkΦkx

)()()1/1(~)1,()1/(~ kukGkkxkkΦkkx



Let us focus our attention on the a posteriori estimation of the state vector.

A first relation can be hence obtained as follows:

The innovation can hence be expressed by:

)())(()( kvkxgky Observation equation

)())1/(ˆ)(())1/(ˆ(

)())(()(

)1/(ˆkvkkxkxkkxg

kvkxgky

kkxg

)())1/(ˆ)(())1/(ˆ()()1/(ˆ

kvkkxkxkkxgkykkxg



The innovation is defined by:

The innovation is defined by:

The jacobian plays the same role as the observation matrix.

)())1(()( kuGkxfkx Non-Linear case

Linear case )1()1()1()( kuGkxkΦkx

)())1/(ˆ)(()1/(ˆ

kvkkxkxkkxg

)()1/(ˆ)()()1/(ˆ)()()()( kvkkxkxkHkkxkHkvkxkH









Relaxing the assumptions: multiple models (MM)

In the MM estimation, the possible system behavior can be represented by a set of models. A bank of filters runs in parallel at every time, each based on a particular model, to obtain the model-conditional estimates. Overall state estimate is a certain combination of these model-conditional estimates.



Cooperation strategy

)1/1( kkPn

… Filter 2

)/(ˆ2 kkxm

)/(2 kkPm

)1/1(ˆ2 kkx

)1/1(2 kkP

Filter 1

)/(ˆ1 kkxm

)/(1 kkPm

)1/1(ˆ1 kkx

)1/1(1 kkP

Filter n

)/(ˆ kkxmn

)/( kkPmn

)1/1(ˆ kkxn

Output strategy

)1/1(ˆ kkx f

)1/1( kkPf




)1/1( kkPn

… Filter 2

)/(ˆ2 kkxm

)/(2 kkPm

)1/1(ˆ2 kkx

)1/1(2 kkP

Filter 1

)/(ˆ1 kkxm

)/(1 kkPm

)1/1(ˆ1 kkx

)1/1(1 kkP

Filter n

)/(ˆ kkxmn

)/( kkPmn

)1/1(ˆ kkxn

Output strategy

)1/1(ˆ kkx f

)1/1( kkPf

Three main generations


Relaxing the assumptions: multiple models (MM): cooperation strategy

… Filter 2

)/(ˆ2 kkx

)/(2 kkP

)1/1(ˆ2 kkx

)1/1(2 kkP

Filter 1

)/(ˆ1 kkx

)/(1 kkP

)1/1(ˆ1 kkx

)1/1(1 kkP

Filter n

)/(ˆ kkxn

)/( kkPn

)1/1(ˆ kkxn

)1/1( kkPn

Output Strategy

)1/1(ˆ kkx f

)1/1( kkPf

Delay Delay Delay

1st generation: Independant filters



2nd generation: dependant filters

… Filter 2

)/(ˆ2 kkx

)/(2 kkP

)1/1(ˆ2 kkx

)1/1(2 kkP

Filter 1

)/(ˆ1 kkx

)/(1 kkP

)1/1(ˆ1 kkx

)1/1(1 kkP

Filter n

)/(ˆ kkxn

)/( kkPn

)1/1(ˆ kkxn

)1/1( kkPn

Output Strategy

)1/1(ˆ kkx f

)1/1( kkPf

Interactive multiple models (IMM)



3rd generation: The number of filters

may vary in time

)1/1( kkPf

…


Filter 2

)/(ˆ2 kkxm

)/(2 kkPm

)1/1(ˆ2 kkx

)1/1(2 kkP

Filter 1

)/(ˆ1 kkxm

)/(1 kkPm

)1/1(ˆ1 kkx

)1/1(1 kkP

)/(ˆ kkxmn

)/( kkPmn

)1/1(ˆ kkxn

)1/1( kkPn

Output strategy

)1/1(ˆ kkx f

Filter n

)/(ˆ kkx imn

)/( kkP imn

)1/1(ˆ kkx in

)1/1( kkP in

Filter n+i …

« Recursive Adaptive Model Set » (RAMS)




may vary in time

)1/1( kkPf


Model-Group Switching: Using IMM with

an active set of models.

There is a priori the definition of the subsets

M M4

M1

M3

M2

:a model

M M4

M1

M3

M2

Instant k Instant k+1

M2 is active M3 is active




may vary in time

)1/1( kkPf


Likely Model Set: Using IMM with

an active set of models

The subsets change in time

M

:a model

M

Instant k Instant k+1

Mi

Mi

Mimp

Mimp

:Mimp: subset of non-probable models

:Mi :subset of models used for the estimation




may vary in time

)1/1( kkPf


This requires hypothesis test to select which models are active


Relaxing the assumptions: multiple models (MM): output strategy


)1/1( kkPn

… Filter 2

)/(ˆ2 kkxm

)/(2 kkPm

)1/1(ˆ2 kkx

)1/1(2 kkP

Filter 1

)/(ˆ1 kkxm

)/(1 kkPm

)1/1(ˆ1 kkx

)1/1(1 kkP

Filter n

)/(ˆ kkxmn

)/( kkPmn

)1/1(ˆ kkxn

Output strategy

)1/1(ˆ kkx f

)1/1( kkPf

Two main strategies



Two main strategies -Hard decision

… )1/1(ˆ

2 kkx

)1/1(2 kkP

)1/1(ˆ1 kkx

)1/1(1 kkP

)1/1(ˆ kkxn

)1/1( kkPn

)1/1(ˆ2 kkx

)1/1(2 kkP

… … …



Two main strategies -Soft decision

… )1/1(ˆ

2 kkx

)1/1(2 kkP

)1/1(ˆ1 kkx

)1/1(1 kkP

)1/1(ˆ kkxn

)1/1( kkPn

)1/1(ˆ kkx f

)1/1( kkPf

… … …



Two main strategies -Soft decision -Hard decision

How to choose the best filter or how to combine all the filters? Defining weights by taking advantage of the likelihood of each model

that can be defined from the state vector estimate and the error covariance matrix



How to choose the best filter or how to combine all the filters?

Two steps to define the weights

1. a priori step: using a Markov Chain where a state corresponds to a model assumption 2. a posteriori step: updating the weights by using a new observation and more particularly the likelihood corresponding to each estimator



1. a priori step: using a Markov Chain where a state corresponds to a model assumption. Example with 3 models

3/32/31/3

3/22/21/2

3/12/11/1

1

2 3

2112

322331

13

3322

11



1. a priori step: using a Markov Chain where a state corresponds to a model assumption. Example with 3 models

8.01.01.0

2.06.02.0

1.005.085.0

1

8.02.01.0

1.06.005.0

1.02.085.0

321

3213

3212

3211

ppp

pppp

pppp

pppp

39.0

15.0

46.0

3

2

1

p

p

p



1. a priori step: given the Markov Chain with a priori transition probabilities, defining the probability that the ith assumption corresponding to the ith estimator corresponds to the true model at time k: Value of the weight at time k+1 using the Markov Chain:

)(ki 1)(1

n

ii k

If n models

n

iiijj kkk

1

)()/1( 1)/1(1

n

jj kk



1. a priori step: . Remark:

)/1()1/(

)(1

kkkk

kjththeiskatandiththeisktimeatstateP

jji

iij

)/1(

)()1/(

kk

kkk

j

iij

ji

1)/1(

)/1(

)/1(

)()1/(

11

kk

kk

kk

kkk

j

jn

i j

iijn

iji



2. a posteriori step: updating the weights by using a new observation and more particularly the likelihood corresponding to each estimator For a specific filter, let us recall the way the state vector estimation is updated when a new observation is available:

The corresponding likelihood is defined as follows:

)(~)()(~2

1

2/1

1

.)()2(

1)(

kykSky

iNi

iiT

ie

kSk

)(~)()1/(ˆ)(ˆ)()()1/(ˆ

)1/(ˆ)()()()1/(ˆ)/(ˆ

kykKkkxkykykKkkx

kkxkHkykKkkxkkx

Innovation

covariance

matrix



2. a posteriori step: updating the weights If mi(k+1) means that the ith model is the true one at time k+1, one can express the likelihood and the weight as follows

))(),...,3(),2(),1(),1(/)1(~()1( kyyyykmkypk iii

))(),...,3(),2(),1(/)1(()/1( kyyyykmPkk ii



2. a posteriori step: updating the weights Let us recall the Bayes rule in a general case and apply it in a specific case:

n

iii

iiiii

ApAxp

ApAxp

xp

ApAxpxAp

1

)()./(

)()./(

)(

)()./()/(

n

iiii

iii

ii

kyykmPkyykmkyp

kyykmPkyykmkyp

kykyykmP

1

)(),...,1(),1()(),...,1(),1(/)1(~

)(),...,1(/)1()(),...,1(),1(/)1(~

)1(~),(),...,1(/)1(

)1(~ kyxi

)(),...,1(/)1( kyykmA ii



2. a posteriori step: updating the weights

n

iiii

iiiii

kyykmPkyykmkyp

kyykmPkyykmkypkykyykmP

1

)(),...,1(),1()(),...,1(),1(/)1(~

)(),...,1(/)1()(),...,1(),1(/)1(~)1(~),(),...,1(/)1(

)1( ki )/1( kki

)1( kj )/1( kkj



Weight prediction Weight filtering

)1( kj

)(,...,1 kN )1(,...,1 kN

)1/(/ kkji

n

jjj

iii

kkk

kkkk

1

)/1()1(

)/1()1()1(

2. a posteriori step: updating the weights




)1/1( kkPn

… Filter 2

)/(ˆ2 kkxm

)/(2 kkPm

)1/1(ˆ2 kkx

)1/1(2 kkP

Filter 1

)/(ˆ1 kkxm

)/(1 kkPm

)1/1(ˆ1 kkx

)1/1(1 kkP

Filter n

)/(ˆ kkxmn

)/( kkPmn

)1/1(ˆ kkxn

Output strategy

)1/1(ˆ kkx f

)1/1( kkPf

n

iiji

mj

kkxkk

kkx

1

)/(ˆ)1/(

)/(ˆ

n

imjiiji

mj

kPkkPkk

kkP

1

)]()/()[1/(

)1/(

)/(ˆ)/(ˆ)(ˆ

)(ˆ).(ˆ)(

kkxkkxkx

kxkxkP

jmimij

Tmijmijmij




)1/1( kkPn

… Filter 2

)/(ˆ2 kkxm

)/(2 kkPm

)1/1(ˆ2 kkx

)1/1(2 kkP

Filter 1

)/(ˆ1 kkxm

)/(1 kkPm

)1/1(ˆ1 kkx

)1/1(1 kkP

Filter n

)/(ˆ kkxmn

)/( kkPmn

)1/1(ˆ kkxn

Output strategy

)1/1(ˆ kkx f

)1/1( kkPf

n

iii

f

kkxk

kkx

1

)1/1(ˆ)1(

)1/1(ˆ

n

ifiii

f

kPkkPk

kkP

1

)]1()1/1()[1(

)1/1(

)1/1(ˆ)1/1(ˆ)1(~

)1(~).1(~)1(

kkxkkxkx

kxkxkP

iffi

Tfififi














Target tracking


Various models can be considered, but the state vector can change. Some authors suggest using some relations between the state vectors.

2.56 2.58 2.6 2.62 2.64 2.66 2.68

x 104

2.53

2.54

2.55

2.56

2.57

2.58

2.59

2.6

2.61

2.62x 10

4

x

y

position recherchée

2.54 2.56 2.58 2.6 2.62 2.64 2.66

x 104

2.52

2.54

2.56

2.58

2.6

2.62

2.64x 10

4

X

Y

Scénario dans le repère cartésien fixe

valeur réelle

valeur mesurée

valeur estimée

True motion

Observations

Estimated motion

Constant velocity model

2.54 2.56 2.58 2.6 2.62 2.64 2.66

x 104

2.52

2.54

2.56

2.58

2.6

2.62

2.64x 10

4

X

Y


valeur réelle

valeur mesurée

valeur estimée

Target tracking



2.56 2.58 2.6 2.62 2.64 2.66 2.68

x 104

2.53

2.54

2.55

2.56

2.57

2.58

2.59

2.6

2.61

2.62x 10

4

x

y

position recherchée True motion

Observations

Estimated motion

Rotational model assumption

2.54 2.56 2.58 2.6 2.62 2.64 2.66

x 104

2.52

2.54

2.56

2.58

2.6

2.62

2.64x 10

4

X

Y


valeur réelle

valeur mesurée

valeur estimée

2.54 2.56 2.58 2.6 2.62 2.64 2.66

x 104

2.52

2.54

2.56

2.58

2.6

2.62

2.64x 10

4

X

Y


valeur réelle

valeur mesurée

valeur estimée

Target tracking



2.56 2.58 2.6 2.62 2.64 2.66 2.68

x 104

2.53

2.54

2.55

2.56

2.57

2.58

2.59

2.6

2.61

2.62x 10

4

x

y

position recherchée True motion

Observations

Estimated motion

IMM combining Singer and constant velocity models

7.03.0

1.09.0

Questions ?


Documents

OPTIMAL FILTERING: RELEVANCE OF THE INTERACTIVE …sting.deis.unibo.it/.../downloads/Slides_lectures_Grivel.pdf · 2012. 5. 28. · Eric Grivel [email protected], [email protected]