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-
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
Roland Langrock HMMs for animal telemetry data
-
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
Roland Langrock HMMs for animal telemetry data
-
Simulated trajectory (two-state HMM)
State 1:exploratory1State 2:encamped
●
−1500 −1000 −500 0 500 1000 1500
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Roland Langrock HMMs for animal telemetry data
-
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...
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
Roland Langrock HMMs for animal telemetry data
-
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
Roland Langrock HMMs for animal telemetry data
-
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
Roland Langrock HMMs for animal telemetry data
-
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
Roland Langrock HMMs for animal telemetry data
-
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
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
Roland Langrock HMMs for animal telemetry data
-
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
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
Roland Langrock HMMs for animal telemetry data
-
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
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
Roland Langrock HMMs for animal telemetry data
-
Feedback models
basic HMM: state depends only on previous state
feedback HMM: additional dependence on previous actualobservation (or derived process)
consider likelihood conditional on initial location
St−1 0St 0 St+1
Zt−1 0Zt 0 Zt+1
. . . behavioural states
step lengths
& directions
St+1
St+1
St+1
Roland Langrock HMMs for animal telemetry data
-
Feedback models
basic HMM: state depends only on previous state
feedback HMM: additional dependence on previous actualobservation (or derived process)
consider likelihood conditional on initial location
St−1 0St 0 St+1
Zt−1 0Zt 0 Zt+1
. . . behavioural states
step lengths
& directions
St+1
St+1
St+1
St+1
Roland Langrock HMMs for animal telemetry data
-
Feedback models
basic HMM: state depends only on previous state
feedback HMM: additional dependence on previous actualobservation (or derived process, e.g. body condition)
consider likelihood conditional on initial location
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
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
Roland Langrock HMMs for animal telemetry data
-
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)
Roland Langrock HMMs for animal telemetry data
-
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)
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
Roland Langrock HMMs for animal telemetry data
-
Bison application
GPS telemetry data on 9 bison in Prince Albert NationalPark, Canada
showcase of a hierarchical hidden semi-Markov model
! HSMM outperforms HMM in terms of the AIC
! see our Ecology paper (Langrock, King, Matthiopoulos,Thomas, Fortin & Morales, 2012)
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
Roland Langrock HMMs for animal telemetry data
-
Bison application
GPS telemetry data on 9 bison in Prince Albert NationalPark, Canada
showcase of a hierarchical hidden semi-Markov model
! inclusion of random e�ects further improves the AIC
! see our Ecology paper (Langrock, King, Matthiopoulos,Thomas, Fortin & Morales, 2012)
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
Roland Langrock HMMs for animal telemetry data
-
Bison application
GPS telemetry data on 9 bison in Prince Albert NationalPark, Canada
showcase of a hierarchical hidden semi-Markov model
! inclusion of random e�ects further improves the AIC
! see our Ecology paper (Langrock, King, Matthiopoulos,Thomas, Fortin & Morales, 2012)
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
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
Roland Langrock HMMs for animal telemetry data
-
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
Roland Langrock HMMs for animal telemetry data
-
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
0 500 1000 1500
0.0
0.1
0.2
0.3
0.4
Switching probability as a function of depth
depth in metres
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
Roland Langrock HMMs for animal telemetry data
-
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
0 100 200 300 4000.0
000.
002
0.00
40.
006
0.00
8
duration of stay in state 3
time units
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
Roland Langrock HMMs for animal telemetry data
-
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
0 2000 4000 6000 8000 10000
−15
00−
1000
−50
00
dive profile simulated from fitted model
time
dept
h
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
Roland Langrock HMMs for animal telemetry data
-
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!
Roland Langrock HMMs for animal telemetry data
-
Group movement (simulation)
Roland Langrock HMMs for animal telemetry data
-
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)
Roland Langrock HMMs for animal telemetry data
Some HMM basicsSome possible extensionsSemi-Markovian state processesBiased random walksIncorporation of feedback
Outline of three applicationsBison movementDive paths of whalesCorrelated movement paths/group movement
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