A multivariate mixed hidden Markov model for blue whale
behaviour and responses to sound exposure
Stacy L. DeRuiter, Roland Langrock∗ , Tomas Skirbutas, Jeremy A. Goldbogen, John
Calambokidis, Ari S. Friedlaender, Brandon L. Southall
∗ Bielefeld
Roland Langrock | Bielefeld University
University, Germany
1 / 14
Aim of the study
Investigate behaviour changes by blue whales in response to military sonar.
The approach we took:
find a baseline model that accurately describes undisturbed behaviour
quantify any change of the baseline model under sonar influence
Roland Langrock | Bielefeld University
2 / 14
Data
From tag data and visual observations, the following variables were
calculated on a dive-by-dive basis:
maximum depth
dive time
post-dive surface duration
number of feeding lunges
variability of whale heading
(horizontal) step length
(horizontal) turning angle
→ 37 individual whales
→ 6-93 dives per animal, 1054 dives in total
→ sonar exposure during 168 of the dives
Roland Langrock | Bielefeld University
3 / 14
Roland Langrock | Bielefeld University
4 / 14
Baseline (hidden Markov) model
xd−1
xd 2
xd+1
sd−1
sd 2
sd+1
multivariate observations
(dive time, max. depth, ...)
...
hidden (behavioural) states
during each dive d, the whale is in one of N states
(which will often be good proxies for behavioural states)
observed variables generated by one of N component distributions, as
selected by the state process
main assumptions:
• Markov property for the state process
• given the states, observations are independent across dives
Roland Langrock | Bielefeld University
5 / 14
Specification of the observation process
Within a dive, the data streams are assumed cond. independent, given the state.
(they are still correlated under the model, since the states induce dependence)
Distributions assumed for the different data streams:
gamma for maximum depth
gamma for dive time
gamma for post-dive surface duration
Poisson for number of feeding lunges
beta for variability of whale heading
gamma for (horizontal) step length
von Mises for (horizontal) turning angle
In each case, there is one set of parameters for each of the N states.
Roland Langrock | Bielefeld University
6 / 14
Fitted state-dependent distributions (3-state model)
dive time
surface time
0
50
100
150
200
250
300
0.015
0.010
0.005
0.000
0.000
0.000
0.002
0.010
0.004
0.020
0.006
maximum depth
0
200
depth in mtrs.
400
600
800
0
50
100
time in sec.
150
200
250
time in sec.
lunges
heading variance
●
●
●
●
●
0
●
●
●
●
●
2
●
●
4
●
●
●
●
6
3
2
●
0
●
1
●
0.2
0.0
shallow foraging
travelling
deep foraging
●
0.4
0.6
0.8
4
●
8
0.0
0.2
0.4
0.6
0.8
1.0
no. lunges
turning angle
0.0
0.000
0.2
0.002
0.4
0.004
0.6
step length
0
200
Roland Langrock | Bielefeld University
400
600
800
1000
−3
−2
−1
0
1
2
3
7 / 14
Fitted state-dependent distributions (4-state model)
dive time
surface time
0
50
100
150
200
250
300
0.015
0.010
0.005
0.000
0.000
0.000
0.002
0.010
0.004
0.020
0.006
maximum depth
0
200
depth in mtrs.
400
600
800
0
50
100
time in sec.
150
200
250
time in sec.
lunges
heading variance
●
●
●
●
●
0
●
●
●
●
2
●
●
4
●
●
●
●
6
3
2
●
0
●
1
●
0.2
0.0
shallow foraging
travelling
deep foraging I
deep foraging II
●
0.4
0.6
0.8
4
●
8
0.0
0.2
0.4
0.6
0.8
1.0
no. lunges
turning angle
0.0
0.000
0.2
0.002
0.4
0.004
0.6
step length
0
200
Roland Langrock | Bielefeld University
400
600
800
1000
−3
−2
−1
0
1
2
3
8 / 14
Remarks on the choice of the number of states
both AIC and BIC favour models with > 7 (!!) states
(typical behaviour when fitting HMMs to complex real data...)
this is due to minor violations of some of the modelling assumptions1
the additional states are included to compensate for these violations, but
have no clear biological interpretation anymore
we made the pragmatic choice to continue with the easy-to-interpret,
biologically sensible and computationally feasible 3-state model
crucially, the lack of fit is not pertinent to the study aim
1
including the conditional independence assumptions, the shape of the state-dep. distributions, etc.
Roland Langrock | Bielefeld University
9 / 14
Extension I: accounting for heterogeneity using discrete random effects
There is substantial heterogeneity across the different whales’ series.
Explanation: variation in prey abundance drives whales to
adopt a corresponding behavioural “context”.
We assume that, for whale w = 1, . . . , 37, the state transition prob. matrix is
Γ (w ) =

Γ1 with probability π1




Γ2 with probability π2
..


.



ΓK with probability πK
K possible Γs, each individual whale’s time series driven by exactly one of them.
AIC selects K = 4.
Roland Langrock | Bielefeld University
10 / 14
Extension II: including the sonar exposure covariate
Transition probabilities as function of the covariate via row-wise multinomial logits:
exp αijk + βij zwd
(w )
γijkd =
where
(
zwd =
Roland Langrock | Bielefeld University
1+
P
l 6=i
exp αilk + βil zwd
,
1
if whale w was exposed to sonar during dive d;
0
otherwise.
11 / 14
Difference
Exp. − Base.
Exposure
Baseline
Results
1
2
3
1
2
3
1
2
3
1
2
3
1
0
0
1
1
0.99
0
0.01
1
0.57
0.08
0.36
1
0.31
0.17
0.52
2
0
1
0
2
0
0.82
0.18
2
0
0.89
0.11
2
0
0.45
0.55
3
0.03
0
0.97
3
0.05
0.16
0.79
3
0.86
0.06
0.08
3
0.24
0.15
0.62
1
2
3
1
2
3
1
2
3
1
2
3
1
1
0
0
1
1
0
0
1
0.97
0.03
0
1
0.94
0.06
0
2
0.22
0.78
0
2
0
0.79
0.21
2
0
0.87
0.13
2
0
0.36
0.64
3
0.01
0
0.99
3
0.01
0.23
0.75
3
0.25
0.1
0.66
3
0.07
0.21
0.72
1
2
3
1
2
3
1
2
3
1
2
3
1
1
0
−1
1
0.01
0
−0.01
1
0.4
−0.05
−0.36
1
0.63
−0.11
−0.52
2
0.22
−0.22
0
2
0
−0.03
0.03
2
0
−0.02
0.02
2
0
−0.09
0.09
3
−0.02
0
0.02
3
−0.04
0.07
−0.04
3
−0.61
0.04
0.58
3
−0.17
0.06
0.1
Roland Langrock | Bielefeld University
12 / 14
Discussion
Model without sonar covariate vs. model with sonar covariate:
∆AIC = 5.1 (in favour of the model with sonar)
p-value (LR test) = 0.009 (fairly strong indication of an effect)
According to the final model, when exposed to sonar, whales ...
... do not switch from “shallow foraging” to “deep foraging” anymore
This finding is consistent with previous analyses of a subset of the data.
Can’t draw firm conclusions without additional data:
Is this significant, baby, or is it confusion?
Roland Langrock | Bielefeld University
(Jimi Hendrix)
13 / 14
References
The (almost up-to-date version of the) paper:
DeRuiter et al. (2015), A multivariate mixed hidden Markov model to analyze
blue whale diving behaviour during controlled sound exposures, arXiv
HMM methodology, including discrete random effects:
Zucchini, MacDonald & Langrock (2016),
Hidden Markov Models for Time Series:
An Introduction Using R, 2nd Edition
cat approved! 0000000000
Roland Langrock | Bielefeld University
14 / 14