Recent developments in Kalman Filter analysis of track data · PFRP Principal Investigators Workshop 9. 266 268 270 272 274 8 10 12 14 16 18 Longitude Latitude Most probable track

Recent developments in Kalman Filter analysisof track data

Anders Nielsen

December 2002

Acknowledgments: John R. Sibert, Michael K. Musyl, Yonat Swimmer, and Molly Lutcavage.

c© Anders Nielsen, December 2002

PFRP Principal Investigators Workshop

1

Background: Fish stock assessment

• Deterministic models are still used

• Stochastic models are under development

• Hundreds of model parameters

• Huge uncertainties

• Migration and heterogeneity is most often ignored.



2

Models for migration

• Stochastic differential equations at the individual level(

dxt = αdt +√

2DdBt

)

• Partial differential equations at population level(

∂∂t Nx,t = −α ∂

∂xNx,t +D ∂2

∂x2 Nx,t

)

One fish Numerous fish Population



3

266 268 270 272 274

810

1214

1618

Longitude

Latit

ude

Track of Olive Ridley 13112



4

State equation (describing the movements)

αi = αi−1 + ci +ηi

Here:

• αi is current position (x,y) in Nautical miles from (0,0)

• αi−1 is previous position

• ci = ∆ti(u

v

)

is drift (or preferred movement)

• ηii.d.∼ N(0,2D∆ti · I2×2) is random movement



5

Space equation (translation from position to observation)

yi = z(αi)+di + εi

• z(αi) = 160

( αi,1cos(αi,2π/180/60)

αi,2

)

converts state coordinates to longitude and latitude

• di =(

bxby

)

is observation bias

• εii.d.∼ N

(

0,

(

σ2x 0

0 σ2yi

))

is observation error

z

0



6

The Kalman Filter

• First observation is know without error

• Second observation

– predict from model, parameters and previous observation

– calculate the difference w2 = (obs2−pred2) and its variance F2

– find best prediction as weighted average of prediction and observation

• Third, then fourth, ...

– like second.

The negative log likelihood becomes:

− logL(X|(u,v,D,bx,by,σ2x,σ

2y, . . .)) = T log(2π)+0.5∑ log(|Fi|)+0.5∑w′

iFiwi



7

266 268 270 272 274

810

1214

1618

Longitude

Latit

ude

Predicted track of Olive Ridley 13112



8

Most probable track

• Best predicted track points are calculated from previous points only.

ai = E{αi|y1 . . .yi}

• Most probable track points use entire track.

ai|T = E{αi|y1 . . .yT}



9

266 268 270 272 274

810

1214

1618

Longitude

Latit

ude

Most probable track of Olive Ridley 13112



10

Searching for premature pop–off point

• Important in itself

• First step in identifying behavior switching



11

The pop–off model

• Different movement pattern only changes the state equation

• New parameters are: u1, u2, v1, v2, D1, D2 and the pop–off time τ

• Consider for instance D in the interval from ti−1 to ti

– if τ ≤ ti−1 then D = D2

– if τ ≥ ti then D = D1

– if ti−1 < τ < ti then D =τ−ti−1ti−ti−1

D1 + ti−τti−ti−1

D2

t i−2 t i−1 τ t i t i+1

• Same for u and v.



12

Reminder: Likelihood ratio test

• Assume model M2 is a sub–model of model M1

– for instance like the basic model is a sub–model of the pop–off model

• Test ratio:

QM1→M2 =LM2(X |θ2)

LM1(X |θ1)

• Test probability:

P-valueM1→M2

app.= P(χ2

nM1−nM2≥−2log(QM1→M2))

• Reject the hypothesis that M2 describes the data as well as M1 if the P-value is below some threshold

(often 5%).



13

0 50 100 150

290

310

Days at liberty

Long

itude

Latit

ude

0 50 100 150

3040

Days at liberty

−lo

g lik

elih

ood

415

425

8Oct

99

27N

ov99

16Ja

n200

0

6Mar

2000

Date

290 300 310 320

3035

4045

Logitude

Latit

ude

Estimated track of Bluefin.4934



14

Future plans for the KF-track

• Pack it as a documented R-package

• Several segments

• Several tracks

• Include temperature information

• Robust estimation

• . . .



15

Preview of the R-package

A typical session could be:

• Load the package into R

R> library(kftrack)

• Read in data, for instance from a simple text file

R> OR.13112 <- read.table("13112.dat", header = TRUE)

R> OR.13112[1:3, ]

day month year lon lat

1 29 11 2001 273.4148 10.4557

2 3 12 2001 273.6600 13.1300

3 4 12 2001 274.0100 13.6800

• Run and save the desired model fit

R> fit.13112 <- kftrack(OR.13112, fix.last = TRUE, var.struct = "uniform")

possibly with more options.



16

R> fit.13112

#R-KFtrack fit

#Tue Dec 3 12:57:23 2002

#Number of observations: 57

#Negative log likelihood: 211.815

#The convergence criteria was met

Parameters:

u v D bx by sx sy

Estimates: -6.32409 -1.37745 449.382 0.0683247 1.33927 0.894642 2.01884

Std. dev.: 3.90260 3.90300 195.680 0.6452700 0.85168 0.120750 0.21439

This object contains the following sub-items:

[1] "npar" "nlogL" "max.grad.comp" "estimates"

[5] "std.dev" "nominal.track" "pred.track" "most.prob.track"

[9] "days.at.liberty" "date" "spd" "hdg"

[13] "nobs" "header" "fix.first" "fix.last"

[17] "theta.active" "theta.init" "var.struct" "dev.pen"

[21] "call" "data.name"



17

R> plot(fit.13112)

R> addmap(fit.13112)

0 10 20 30 40 50 60

266

270

274

Days at liberty

Long

itude

Latit

ude

812

16

29N

ov20

01

9Dec

2001

19D

ec20

01

29D

ec20

01

8Jan

2002

18Ja

n200

2

28Ja

n200

2

Date

266 268 270 272 2748

1012

1416

18

Logitude

Latit

ude

Estimated track of OR.13112



18

Documents

Recent developments in Kalman Filter analysis of track data · PFRP Principal Investigators Workshop 9. 266 268 270 272 274 8 10 12 14 16 18 Longitude Latitude Most probable track