Kalman filtering techniques for parameter estimation Jared Barber Department of Mathematics, University of Pittsburgh Work with Ivan Yotov and Mark Tronzo.

Kalman filtering techniques for parameter estimation

Jared BarberDepartment of Mathematics, University of Pittsburgh

Work with Ivan Yotov and Mark TronzoMarch 17, 2011

Outline

• Motivation for Kalman filter• Details for Kalman filter• Practical example with linear Kalman filter• Discussion of other filters

– Extended Kalman filter– Stochastic Collocation Kalman filter– Karhunen-Loeve SC Kalman filter

• Results for simplified NEC model

Necrotizing Enterocolitis (NEC) Model

• Ten nonlinear PDEs• Four layers• Time consuming

simulations for normal computational

refinements• Approximately fifty free

parameters– Diffusion rates– Growth and death rates– Interaction rates

0

0.5

1

21

0

21

0

00.20.4

Computational Domain-

21

0

21

0

00.20.4

Distribution of macrophages

Maximum Likelihood Estimate• Recall formula (for normal distributions):

• Disadvantages:– Consider all times simultaneously—larger optimization problem and

generally slower– Don’t take into account measurement error and model error separately– To be more efficient: Want derivative information– Get only one answer…not a distribution (which tells you how good

your answer is)– May be hard to parallelize

n

i i

ii

xMLE

n

i

xMy

i

xMyxexL i

ii

12

2

1

))(())((

minarg;2

1)(

2

2

Kalman Filter

• Various versions: Linear KF, Extended KF, Ensemble KF, Stochastic Collocation/Unscented KF, Karhunen-Loeve Kalman Filter

• Advantages of some of these methods (to a lesser or greater extent)– Consider each time separately– Keep track of best estimates for your parameters (means) and your uncertainties (covariances)

– Consider data and measurement error separately– Don’t need derivative information– Easy to parallelize

Kalman Filter Picture: Initial State

Measurement

Model Estimate

Kalman Filter Picture: Analysis/Adjustment Step

Model Estimate

Measurement

True SolutionEstimate

Kalman Filter Picture: Prediction/Forecast Step

Model Estimate

Measurement

True SolutionEstimate

Kalman Filter Picture: Measurement Step

Model Estimate

Measurement

True SolutionEstimate

Kalman Filter: Analysis/Adjustment Step

Model Estimate

Measurement

True SolutionEstimate

Advancing two things: Mean and covariance

Adjusted state vector =Model State Vector + Kk*(Measured Data-Model Data)

Kk = f(Model Covariance, Data Covariance)

Kalman Filter: General Algorithm, Quantities of interest

• Measured data = true data plus measurement noise

• Measurement function

• Optimal “blending factor” Kalman Gain:

• Model/forecast and adjusted state vectors

• Forecast/model function:

• Best/Analyzed model estimate

ktk

mk vdd

ak

fk xx ,

tk

tk dxh )(

))(( fk

mkk

fk

ak xhdKxx

)( 1ak

fk xfx

kK

Kalman Filter: General Algorithm, Uncertainties/Covariances of interest

• Prescribed: Measured data and application of model covariance

• Forecast state covariances

• Adjusted state covariances

))((,Tkk

mkdd vvEP model

kxxP ,

])][)(][[(

])][)(][[(

])][)(][[(

,

,

,

Tfk

fk

fk

fk

fkdd

Tfk

fk

fk

fk

fkxd

Tfk

fk

fk

fk

fkxx

ddEddEEP

ddExxEEP

xxExxEEP

])][)(][[(,Ta

kak

ak

ak

akxx xxExxEEP

Kalman Filter: General Algorithm, Kalman Gain

• Recall to adjust the model’s state vector:

• Minimize the sum of the uncertainties associated with the adjusted state to find the right blending factor

))(( fk

mkk

fk

ak xhdKxx

1

,,

,

)(

minarg

fkdd

fkxdk

akxx

Kk

PPK

PtraceKk

Linear Kalman Filter

• Special Case:

• Covariances given by:

fk

fk

fk

kakkk

ak

fk

Hxxhd

wxAwxfx

)(

)( 1111

Tfkxdk

fkxx

akxx

measkdd

Tfkxx

fkdd

Tfkxx

fkxd

modelkxx

Tk

akxxk

fkxx

PKPP

PHHPP

HPP

PAPAP,,,,

,,,

,,

,,,

Example: Exact system

• Physical system modeled exactly by:

• Exact solution for this physical system is then:2)0(

1)0(

;24'

;122'

2

1

212

211

y

y

yyy

yyy

.1)2cos(3

;2/)1)2cos(3)2sin(3(

2

1

ty

tty

Example: Model

• Pretend we have a model that we think might work for this system:

• We have three unknown variables, y1, y2, and a.

• We wish to find a that makes the model fit the data.

2)0(

1)0(

;24'

;22'

2

1

212

211

y

y

yyy

ayyy

Example: State vector

• Define the state vector as:

• Note, must make reasonable starting guess for unknown parameter. Here we guessed 1.5 which is close to the actual value of 1.

5.1

2

1

?

2

1

02

1 guessfx

a

y

y

x

Example: Data vector

• Assume we can actually measure y1 and y2 in our system

• Note our measurement function becomes:

995.1

01.1

005.0

01.0

2

1; 000

2

1 vddy

yd tm

)(010

0012

1

2

1 xhxH

a

y

y

y

yd

Example: Model/forecast function

• Assuming that the parameter a is a quantity that should not change with time, we can rewrite the system of equations as:

• Forward euler for our system gives us:

xM

a

y

y

a

y

y

x

2

1

2

1

000

024

122

'

'

'

11111 )( kkkkkk xAxdtMIxdtMxx

Example: Algorithm step through: Initialization

• Start with state vector data:

• Start with data vector data:

1.000

001.00

0001.0

5.1

2

1

,0model

kxxf Px

01.00

001.0

995.1

01.1,0

measkdd

m Pd

Example: Algorithm step through: Initialization

• Initial forecast model uncertainty as initial uncertainty in model state vector:

1.000

001.00

0001.0

0,f

xxP

Example: Algorithm step through: Obtaining other covariances

• Use formulas to find other state vector covariances:

00

01.00

001.0

00

10

01

1.000

001.00

0001.0

0,0,Tf

xxf

xd HPP

Example: Algorithm step through:Obtaining other covariances

• Use formulas to find other state vector covariances:

02.00

002.0

01.00

001.0

00

10

01

1.000

001.00

0001.0

010

001

0,0,0,meas

ddTf

xxf

dd PHHPP

Example: Algorithm step through: Obtaining Kalman Gain

• Find Kalman Gain:

00

5.00

05.0

02.00

002.0

00

01.00

001.0

)(

1

10,0,0

fdd

fxd PPK

Example: Algorithm step through: Obtaining adjusted state vector

• Best estimate, note: halfway in between data and model…this is because the data uncertainties and model uncertainties are the same size

9975.1

005.1

5.1

2

1

010

001

995.1

01.1

00

5.00

05.0

5.1

2

1

)( 00000fmfa HxdKxx

Example: Algorithm step through: Find adjusted state’s covariance

• Note: Uncertainties are smaller than other covariances…used both data and model to get an estimate with less uncertainty than before

1.000

0005.00

00005.0

001.00

0001.0

00

5.00

05.0

1.000

001.00

0001.0

,0,00,0,Tf

xdf

xxa

xx PKPP

Example: Algorithm step through

• Predict step, assume dt = 0.1:

5.1

2

6565.0

5.1

9975.1

005.1

100

02.01)4.0(

1.02.0)2.0(1

)( 01af xdtMIx

Example: Algorithm step through

• New state covariance matrix:

2.0001.0

0014.00016.0

01.00016.00184.0

1.000

001.00

0001.0

001.0

02.12.0

04.02.1

1.000

0005.00

00005.0

000

08.04.0

1.02.02.1

1,11,11,model

xxTa

xxf

xx PAPAP

Example: Algorithm step through

• “Measure” data

004.2

009.1or

995.1

01.1

002.0

028.0

1))1.0(2cos(3

2/)1))1.0(2cos(3))1.0(2sin(3(011

measured

simulated

tm vdd

Extended Kalman Filter:Nonlinear equations

• Special Case:

• Covariances given by:)(;)( 11

fk

fkk

ak

fk xhdwxfx

fkxxk

akxx

measkdd

Tfkxx

fkdd

xx

Tfkxx

fkxd

modelkxx

Tk

akxxk

fkxx

xxk

PHKIP

PHHPP

x

hH

HPP

PAPAP

x

fA

ak

ak

,,

,,,

,,

,1,,

)(

11

Problems with KF/Ext KF?

• KF only works for linear problems.• Ext KF

– Works for mildly nonlinear problems– Must find and store Jacobians

• State vector size >> data vector size– Reminder: Need to find– Don’t need any Pxx’s.

– Pxx’s can be big and hard to calculate/keep track of

1,, )( fkdd

fkxdk PPK

Ensemble Kalman Filter: The Ensemble

• Create an ensemble:

.4.2

7.0

3.2

5.4

3.1

4.2

5.4

3.1

2.4

6.3

4.3

2.2

1.4

6.4

9.1

4.4

9.3

9.0

,4.2

7.0

3.2

,5.4

3.1

4.2

,5.4

3.1

2.4

,6.3

4.3

2.2

,1.4

6.4

9.1

,4.4

9.3

9.0

,

etc

etcx fenk

Ensemble Kalman Filter: Ensemble Properties

• The ensemble carries mean and covariance information

system of statemean

4.4

5.3

3.2

)(

.4.2

1.4

3.2

5.4

3.1

4.2

5.4

3.1

2.4

6.3

4.3

2.2

1.4

6.4

9.1

4.4

9.3

9.0

,

fk

fenk

xE

etcx

Ensemble Kalman Filter: New algorithm

Linear KF Ensemble KF

fkxxk

akxx

fk

mkk

fk

ak

fkdd

fkxdk

measkdd

Tfkxx

fkdd

Tfkxx

fkxd

modelkxx

Tk

akxxk

fkxx

fk

fk

ak

fk

PHKIP

xhdKxx

PPK

PHHPP

HPP

PAPAP

xhdxfx

,,

1,,

,,,

,,

,1,,

1

)(

))((

)(

);();(

))((

)(

)1/(1

)1/(1

)),(,(

)),(,(

)(

)(

,,,,

1,,

,,,,

,,,,

,,,,,

,,,,,

,,,

,,1,

fenk

menkk

fenk

aenk

fkdd

fkxdk

Tfkd

fkd

fkdd

Tfkd

fkx

fkxd

fienk

fienk

fkd

fienk

fienk

fkx

enkf

enkf

enk

enka

enkf

enk

xhdKxx

PPK

EEqP

EEqP

dEdE

xExE

vxhd

wxfx

Ensemble KF Advantages

• Handles nonlinearity (better than Ext KF)• Don’t need Jacobian• Don’t need Pxx’s

Ensemble KF Disadvantages

• Need many ensemble members to be accurate– Mean and variances used in algorithm are within

O(1/sqrt(q)) of actual (MC integration)– Accurate representation can affect convergence rate

and end error on parameter estimate• Using many ensemble members requires many

model/forecast function evaluations (slow)Can we use fewer points and obtain more accuracy?Note: Don’t need a lot of accuracy, just enough to

get the job done.

Stochastic Collocation Kalman Filter:Stochastic Collocation

• Consider the expected value of the function g(z) on stochastic space

• ci and zi are collocation weights and locations, respectively

• Collocation exact for linear functions of the components of z and normal pdfs

12

1

2/)()(2/

)(

)det(2

1)()]([

1

n

iii

zz

zzPzzn

zgc

dzP

ezgzgE

n

Tzz

Ensemble and Stochastic Collocation Comparison

• En mean (q≈1000 pts) • SC mean (N≈200 pts)

q

i

fik

fk x

qxE

1,1

1][

-10 -8 -6 -4 -2 0 2 4 6 8 10-10

-8

-6

-4

-2

0

2

4

6

8

10

N

i

fiki

fk xcxE

1,][

Stochastic Collocation:Kalman Filter Algorithm

iena

kxxak

aienk

Tfkxdk

fkxx

akxx

fk

mkk

fk

ak

fkdd

fkxdk

N

i

measxx

Tfk

fienk

fk

fienki

fkdd

N

i

Tfk

fienk

fk

fienki

fkxd

N

i

modelxx

Tfk

fienk

fk

fienki

fkxx

N

i

fienki

fk

N

i

fienki

fk

fenk

fenk

aenk

fenk

zPxx

PKPP

xhdKxx

PPK

PddddcP

ddxxcP

PxxxxcP

dcd

xcx

xhd

xfx

,,,,

,,,,

1,,

1,,,,,

1,,,,,

1,,,,,

1,,

1,,

,,

,1,

))((

)(

))((

))((

))((

)(

)(

Stochastic Collocation Advantages and Disadvantages

• Faster than En KF for small numbers• Slower than En KF for large numbers• Usually more accurate than En KF• Can handle nonlinearities• Curse of dimensionality for pdes

– 20x20x20 grid needs 16001 collocation points• Is there any way to get around this?

Stochastic Collocation Kalman Filter with Karhunen-Loeve Expansion

• On 3-d grids, above methods assume error is independent of location in computational grid

• Instead, assume error is spatially correlated:

• Hope: Most of the error is captured by the most dominant eigenfunctions

• Idea: Keep only the first twenty-five

n

jjkejkw

n

jjkejkwk fcyxfcw

1,,,,

1,,,, ),(

Karhunen-Loeve SC:Kalman Filter Algorithm

ikeikea

kxxT

ikeiwenak

aienk

Tfkxdk

fkxx

akxx

fk

mkk

fk

ak

fkdd

fkxdk

N

i

measxx

Tfk

fik

fk

fiki

fkdd

N

i

Tfk

fik

fk

fiki

fkxd

N

i

modelxx

Tfk

fik

fk

fiki

fkxx

N

i

fiki

fk

N

i

fiki

fk

fenk

fenk

aenk

fenk

ffPfzxx

PKPP

xhdKxx

PPK

PddddcP

ddxxcP

PxxxxcP

dcd

xcx

xhd

xfx

,,,,,,,,,,,

,,,,

1,,

1,,,

1,,,

1,,,

1,

1,

,,

,1,

)(

))((

)(

))((

))((

))((

)(

)(

Simplified Necrotizing Enterocolitis Model: The experiment

Epithelial Layer

Create woundwith pipette

Epithelial Layer

Wound

≈150 µm

Simplified Necrotizing Enterocolitis Model: The Model and Equation

-5 0 0 0 5 0 0

-3 0 0

0

3 0 0

)1()1( 22

2

ccpccc

cc eekeee

eD

t

e

Simplified Necrotizing Enterocolitis: Perfect Simulated Data

En KF; SC KF; KL SC KF.

0 1 2 31e-10

1e-6

D E

stim

ate

Err

or

Time (hrs)

0.0001

0.1

kp E

stim

ate

Err

or

0 1 2 3

2e-6

3e-6

D E

stim

ate

Time (hrs)

0.6

0.8

1

kp E

stim

ate

Simplified Necrotizing Enterocolitis: Imperfect Simulated Data

En KF; SC KF; KL SC KF.

0 1 2 3

1e-8

1e-6

D E

stim

ate

Err

or

Time (hrs)

0.001

0.1

kp E

stim

ate

Err

or

0 1 2 3

2e-6

3e-6

D E

stim

ate

Time (hrs)

0.6

0.8

1

kp E

stim

ate

Simplified Necrotizing Enterocolitis: Real Data

En KF; SC KF; KL SC KF.

0 1 2 3

0

1e-5

D E

stim

ate

Time (hrs)

1

2

3

kp E

stim

ate

Are parameter estimates good?

• Produce qualitatively correct results t = 0 h t = 0.5 h t = 1 h t = 1.5 h

t = 2 h t = 2.5 h t = 3 h t = 3.5 h

200 m

Comparisons

• With perfect measurements and a pretty good model, SC does best, then KL, then En

• With imperfect measurements, all are comparable• With real data, KL fails. Why? Guess: Too much error

associated with D• Additional real data info

– Gives temporal information about the parameters– Gives uncertainty estimates

• All run significantly faster than the direct optimization method used