extensions of hidden markov models for animal telemetry data · 2012-07-09 · showcase of a...

Extensions of hidden Markov models for animal

telemetry data

HA Roland Langrock

HA CREEMHAHAblablabla

Roland Langrock HMMs for animal telemetry data

1 Some HMM basics

2 Some possible extensionsSemi-Markovian state processesBiased random walksIncorporation of feedback

3 Outline of three applicationsBison movementDive paths of whalesCorrelated movement paths/group movement


Basic HMM structure

St−1 0St 0 St+1

Zt−1 0Zt 0 Zt+1

. . . . . .

hidden(behavioural state:

e.g. foraging)

observed(movement behaviour:

directions & step lengths)

discrete time, continuous space

includes multi-state random walks �a la Morales et al. (2004):\Extracting more out of relocation data: [...]."

special case of a state-space model


Simulated trajectory (two-state HMM)

State 1:exploratory1State 2:encamped

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HMM features

estimation via numerical maximization of the likelihood:

L = �P(z1)�P(z2) � : : : � P(zT�1)�P(zT )1t

or, alternatively, using MCMC

model checking via residuals feasible

con�dence intervals can be obtained (Hessian or bootstrap)

underlying hidden states can be estimated (Viterbi algorithm)

incorporating covariates/seasonality straightforward { well, intheory...


1 Some HMM basics




Semi-Markovian state processes

in a basic HMM the statedwell-times are necessarilygeometrically distributed

! mode is 1 (often unrealistic)0 10 20 30 40 50

0.00

0.02

0.04

0.06

0.08

0.10

geometric

duration of stay

prob

abili

ty

hidden semi-Markov modelsrelax this restrictive condition:any distribution on the positiveintegers can be modelled

! e.g. negative binomial! estimation a bit more challenging,but any HSMM can be framed as an HMM


Semi-Markovian state processes

in a basic HMM the statedwell-times are necessarilygeometrically distributed

! mode is 1 (often unrealistic)0 10 20 30 40 50

0.00

0.02

0.04

0.06

0.08

0.10

geometric

duration of stay

prob

abili

ty

hidden semi-Markov modelsrelax this restrictive condition:any distribution on the positiveintegers can be modelled

! e.g. negative binomial! estimation a bit more challenging,but any HSMM can be framed as an HMM

0 10 20 30 40 50

0.00

0.01

0.02

0.03

0.04

0.05

negative binomial

duration of stay

prob

abili

ty


Biased random walk components

Biased random walk:! general preference for some direction (e.g., East)! or bias towards some location/centre of attraction

consider likelihood conditional on initial location

0





St−1 0St 0 St+1

Zt−1 0Zt 0 Zt+1

. . . behavioural states

step lengths

& directions

St+1

St+1

St+1

Figure: Basic HMM





St−1 0St 0 St+1

Zt−1 0Zt 0 Zt+1

Lt−1 0Lt 0 Lt+1

. . .

. . .

behavioural states

step lengths

& directions

locations

St+1

St+1

St+1

St+1

St+1

Figure: HMM including BRW components


Feedback models

basic HMM: state depends only on previous state

feedback HMM: additional dependence on previous actualobservation (or derived process)


St−1 0St 0 St+1

Zt−1 0Zt 0 Zt+1


step lengths

& directions

St+1

St+1

St+1


Feedback models


feedback HMM: additional dependence on previous actualobservation (or derived process)


St−1 0St 0 St+1

Zt−1 0Zt 0 Zt+1


step lengths

& directions

St+1

St+1

St+1

St+1


Feedback models


feedback HMM: additional dependence on previous actualobservation (or derived process, e.g. body condition)


St−1 0St 0 St+1

Zt−1 0Zt 0 Zt+1

Dt−1 0Dt 0 Dt+1

. . .

. . .

behavioural states

step lengths

& directions

derived process

St+1

St+1

St+1

St+1

St+1


1 Some HMM basics




Bison application

GPS telemetry data on 9 bison in Prince Albert NationalPark, Canada

showcase of a hierarchical hidden semi-Markov model

! detects switching between \encamped" and \exploratory" state

! see our Ecology paper (Langrock, King, Matthiopoulos,Thomas, Fortin & Morales, 2012)


Bison application



! detects switching between \encamped" and \exploratory" state


0.0 0.1 0.2 0.3 0.4 0.5 0.6

01

23

45

6

step length

dens

ity

−3 −2 −1 0 1 2 3

0.0

0.1

0.2

0.3

0.4

0.5

turning angle

dens

ity


Bison application



! HSMM outperforms HMM in terms of the AIC


Table: Number of parameters (p), log-likelihood and �AIC values.

p logL �AIC

HMM with common parameter 10 -19874.70 57.3

set for all bison

HSMM with common parameter 12 -19846.55 5.0

set for all bison

HSMM with random e�ects for 14 -19842.06 0

Weibull scale parameters


Bison application



! inclusion of random e�ects further improves the AIC


Table: Number of parameters (p), log-likelihood and �AIC values.

p logL �AIC

HMM with common parameter 10 -19874.70 57.3

set for all bison

HSMM with common parameter 12 -19846.55 5.0

set for all bison

HSMM with random e�ects for 14 -19842.06 0

Weibull scale parameters


1 Some HMM basics




Beaked whale application (current joint work with Marques & Thomas)

0 2000 4000 6000 8000 10000

−15

00−

1000

−50

00

beaked whale dive profile (data collected by Baird et al./Cascadia Research Collective)

time

dept

h

state description transition type

1 ! (at surface) Markov

2 & (deep dive) feedback from depth

3 ! (deep dive) semi-Markov

4 % (deep dive) feedback from depth

5 & (shallow dive) Markov

6 ! (shallow dive) Markov

7 % (shallow dive) feedback from depth



0 2000 4000 6000 8000 10000

−15

00−

1000

−50

00


time

dept

h

0 500 1000 1500

0.0

0.1

0.2

0.3

0.4

Switching probability as a function of depth

depth in metres











0 2000 4000 6000 8000 10000

−15

00−

1000

−50

00


time

dept

h

0 100 200 300 4000.0

000.

002

0.00

40.

006

0.00

8

duration of stay in state 3

time units











0 2000 4000 6000 8000 10000

−15

00−

1000

−50

00


time

dept

h

0 2000 4000 6000 8000 10000

−15

00−

1000

−50

00

dive profile simulated from fitted model

time

dept

h


1 Some HMM basics




Group movement model (current joint work with a lot of individuals, includingBlackwell, King, Patterson & Pedersen)

movement of individuals is driven by some centroid/centre ofgravity (not necessarily a single \leader") of entire group

each individual can switch between behavioural states such as

! \staying in the group" (following the centroid) or! \moving independently of the group" (solitarily exploring)

for the centroid any model can be used (e.g. HMM)

using an approximation for the centroid, such a model can be�tted using relatively standard HMM techniques!


Group movement (simulation)


Langrock, R., Zucchini, W., 2011. Hidden Markov models with arbitrary

dwell-time distributions. Computational Statistics and Data Analysis, 55,

pp. 715-724.

Langrock, R., King, R., Matthiopoulos, J., Thomas, L., Fortin, D.,

Morales, J. M., 2012. Flexible and practical modeling of animal telemetry

data: hidden Markov models and extensions. Ecology, preprint online.

Morales, J. M., Haydon, D. T., Frair, J. L., Holsinger, K. E., Fryxell, J.

M., 2004. Extracting more out of relocation data: building movement

models as mixtures of random walks. Ecology, 85, pp. 2436{2445.

Patterson, T. A., Basson, M., Bravington, M. V., Gunn, J. S., 2009.

Classifying movement behaviour in relation to environmental conditions

using hidden Markov models. Journal of Animal Ecology, 78, pp.

1113{1123.

(The slides of this talk will soon be available on my web page at StAndrews' University)


Some HMM basicsSome possible extensionsSemi-Markovian state processesBiased random walksIncorporation of feedback

Outline of three applicationsBison movementDive paths of whalesCorrelated movement paths/group movement

extensions of hidden markov models for animal telemetry data · 2012-07-09 · showcase of a...

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