Description of state-space model structure, fitting, performance and
parameter estimates
Model structure and fitting
The first component of the state-space model is the process equation, which in our case describes
how the movement process evolves over time. The process equation used here is the firstdifference correlated random walk model with switch between behavioral states (DCRWS)
developed by [1]. The equation describing the dynamics of the DCRWS movement process was
specified as:
(1)
where d t is the difference between positions x t and x t-1 (i.e., two-dimensional vectors of
coordinates); d t-1 is the difference between positions x t-1 and x t-2 ; bt is the behavioral state at
( )
time t ; T qbt is a transition matrix describing the rotational component of the movement,
where q bt is the mean turn angle under behavioral state bt ; g bt is the degree of correlation (0 <
g bt < 1) in both direction and speed under behavioral state bt ; and N 2 is a bivariate normal
distribution with mean zero vector and covariance matrix represented by
that describes the
variability in the movement process.
Two behavioral states termed here as transiting (b = trs) and exploratory (b = exp) were
estimated. The transiting behavior is characterized by relatively fast moves and persistency in
direction; the exploratory behavior is characterized by relatively slow moves with frequent
changes in direction. A total of four transitions between behavioral states are possible but only
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two of these need to be estimated. The probabilities of these two transitions are termed a trs|trs ,
which is the probability of a fish transiting at time t , given that it was also transiting at time t -1
; and a trs|exp , which is the probability of a fish transiting at time t , given that it was at the
exploratory state at time t -1.
The second component of the state-space model is the observation equation, which
describes how the unobserved states predicted by the process equation relate to the observed
data. The observation equation enabled us to account for the irregularity, variable quality, and
error in the position data. The irregularly observed data were modeled directly within the statespace framework with the use of an index i for observed positions between time t and t +1,
where i = (0, 1, 2, … , nt ). An estimate for each of the irregularly observed positions, y t,i , was
interpolated as:
(2)
where ji (0 < ji <1) is the proportion of the regular time interval between x t-1 and x t at which
the ith observation is made and
is a t-distributed random variable (see details below)
representing the error in the positions estimated by the telemetry system. The ji values were
calculated from the data based on date and time stamps associated with each observed position,
and ji for regular time intervals where observations were missing were set to 0.5 and i = 1.
Independent t-distributions, which have been shown to be robust to extreme values [1],
were used to model the easting and northing components of error in Eq. 2. That is, for each
component et,q ~ t (mt,q , t t,q , nt,q ) , where q ( q = 1, … , 7) is the quality class (see below) of the
position estimate,
is the location parameter,
is the scale parameter and
are the
degrees of freedom. Rather than estimate 42 parameters for the t-distributions within the state-
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space model, the parameters of the t-distributions were fixed to values estimated from errors in
the easting and northing components of the position estimates of three beacon tags with fixed
known location (Additional file 1). Because of the potential for error in position estimates to vary
over time (e.g, due to environmental conditions, movement of log boom receivers), a scaling
parameter (
) was used to inflate or deflate the parameter
of the t-distributions as needed for
each track.
The bull trout elevation (computed by subtracting fish depth from reservoir elevation)
and body temperature data were also modeled within the state-space framework to account for
missing data and errors in sensor readings. The process equation used in this case was an autoregressive model of order 1 (AR1) and was specified as:
wt,s = rwt-1,s + ut,s
(3)
where wt,s and wt-1,s are the values of the variable s (s = fel [fish elevation] or s = btp [body
temperature]) at time t and t -1, respectively; r is the degree of correlation between the values
of
s ; and ut,s is a random variable representing the process variability in s , with
ut,s ~ N ( 0, ws ) .
The observation equation used for interpolating the irregularly observed fish elevation
and body temperature data, zt,i,s , is similar to Eq. 2 and was specified as:
zt,i,s = (1- ji ) wt-1,s + ji wt,s + zt,s
(4)
where z t,s is a random variable representing the error in the sensor readings for
s , with
z t,s ~ N ( 0, ht,s ) .
A Bayesian approach was used to fit the state-space models to the data with the software
JAGS [2] and R [3], utilizing codes modified from the R package “bsam” [4]. The model was
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fitted using a hierarchical framework, with the parameters of all models being estimated for the
entire dataset, except the scaling parameter y , which was estimated individually for each track.
Uninformative priors were specified for all parameters, except for the movement parameters g b
and q b , and for the variance of the sensor error distributions hs . The priors for the movement
parameters were: qtrs ~ 2p Beta ( 20, 20) - p ; qexp ~ 2p Beta (10, 10) ; gtrs ~ Beta ( 2, 1.5) ; and
gexp ~ gtrs Beta (1, 1) . These priors are based on the fact that turn angles typically are close to zero
and correlation is high while transiting, whereas turn angles are typically greater than zero and
correlation is low while in the exploratory state [1, 5]. The priors for the variance of the sensor
error distributions were hrel ~ U(0, 2.26) and hbtp ~ U(0, 0.12) . These priors were based on the
accuracy of the sensors reported by the tag manufacturer, with the upper limit equaling
æ sensor accuracy ö
ç
÷ .
è
ø
2.33
2
The model was fitted to the data using a total of 250 000 Markov Chain Monte Carlo
(MCMC) samples per chain (two chains were used), with the first 200 000 discarded as a burn-in
and the remaining 50 000 samples thinned out to 500 by retaining every 100th sample [6].
During the MCMC sampling, a sample from the posterior distribution of positions was retained
only if it occurred at a location where the elevation of the forebay bottom (from bathymetry data)
was lower than the reservoir water elevation for the date the track was observed (indicating that
the sample was in water); and the associated sample from the posterior distribution of fish
elevation was greater than the elevation of the forebay bottom at the same location (indicating
that the sample was above the bottom). The posterior distribution for fish elevation was truncated
at the reservoir water elevation value for the date a track was observed to force the sample to be
underwater. Sampling was repeated if a sample did not meet these conditions. The approach
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effectively created a dynamic 3-dimensional land mask (grid size resolution used: 6 × 6 m) that
informed the model of locations to where a fish could not move.
The median and 2.5% and 97.5% percentiles (i.e., to form 95% credible intervals) of the
distribution of model parameters, fish positions, elevations, body temperatures, and behavioral
states were calculated from the resulting 1 000 MCMC samples. The proportion of behavioral
states estimated as exploratory at each position was computed and interpreted as as the
probability of bull trout being in the exploratory state (Pexp). Convergence of the MCMC chains
was assessed graphicaly using trace, density and autocorrelation plots for the model parameters,
and QQ-plots of the standard Normal Z-scores of the Geweke’s simple test for convergence
applied to all fish positions, behavioral state, elevations and body temperatures [4].
Model performance
The performance of the DCRWS state-space model was assessed by comparing the estimates of
positions and behavioural states for three test tags with those obtained from a differential GPS
(DGPS) device (GeoXH, Trimble, Sunnyvale, CA, USA). The DGPS device was mounted on a
styrofoam platform that was floated across the forebay along with the three test tags hanging
from a line attached to the platform (Figure 1a).
The assessment revealed mean absolute error of 12.8 m (± 5.9 m) in the DCRWS
estimates. This represents a modest improvement (~1.5–5 m) compared with the mean absolute
error of 17.8 m (± 18.6 m) using all observations or of 14.2 m (± 11.8 m) using those
observations with reliability number above the threshold of 2.5 [7]. However there was
substantial improvement in the variability of the error – the % coefficient of variation for the
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DCRWS estimates was substantially smaller (46.1%) than those computed from all observations
(104.5%) and from observations with reliability number > 2.5 (83.1%).
The DCRWS model estimated the “behavioral state” of the test tags adequately, yielding
a high Pexp value for locations where the tags were allowed to drift and low Pexp value for
locations where the tags were moved in a persistent direction (Figures 1b–d). Indeed state-space
models have been previously shown to effectively estimate behavioral states from movement
data [8].
Parameter estimates
Parameter estimates of the DCRWS state-space model indicated good separation between bull
trout transiting and exploratory behavioral states, with no overlap between the 95% credible
intervals for turning angles (i.e., q trs and qexp ) and between correlations (i.e., g trs and gexp ) in
direction and speed for each behavioral state (Table 1). Arbitrarily defining locations with Pexp <
0.25 as transiting and Pexp > 0.75 as exploratory revealed more variable turning angles and much
lower speeds (in body lengths per second) for the exploratory state (Figures 2a−b). Estimates of
speed in the transiting and exploratory states were consistent with speeds measured for bull trout
in laboratory and field studies [9, 10]. The probability of bull trout remaining in the same
behavioral state was high (see a trs|trs and 1- atrs|exp in Table 1). The estimates of scaling
parameters y ranged from 0.04 to 603.89 (median of 2.22) across tracks.
Parameter estimates of the AR1 models revealed high correlation between sequential
values for fish elevation and body temperature, indicating strong persistence in these variables
(Table 1). Converting fish elevation to depth revealed that bull trout typically remained within 15
m of the surface, were usually deeper during the summer and fall, and closer to the surface in the
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spring and winter (Figure 2c). Body temperatures were typically between 0 °C and 12 °C, being
highest in the fall and summer, and lowest in the winter (Figure 2d). Variability in body
temperatures was greatest in the spring and lowest during the winter (Figure 2d).
References
1. Jonsen ID, Mills-Flemming J, Myers RA: Robust state-space modeling of animal
movement data. Ecology 2005, 86:2874–2880.
2. Plummer M: JAGS: a program for analysis of Bayesian graphical models using Gibbs
sampling. In Procedings of the 3rd International Workshop on Distributed Statistical
Computing (DSC 2003): 20–22 March 2003; Vienna. Edited by Hornik K, Leisch F, Zeileis A.
2003.
3. R Development Core Team: R: A language and environment for statistical computing.
[http://www.r-project.org]
4. Jonsen ID, Basson M, Bestley S, Bravington MV, Patterson TA, Pedersen MW, Thomson R,
Thygesen UH, Wotherspoon SJ: State-space models for bio-loggers: A methodological
road map. Deep Sea Res Part II 2013, 88-89:34–46.
5. Pedersen MW, Patterson TA, Thygesen UH, Madsen H: Estimating animal behavior and
residency from movement data. Oikos 2011, 120:1281–1290.
6. Lunn D, Jackson C, Best N, Thomas A, Spiegelhalter D: The BUGS Book: A Practical
Introduction to Bayesian Analysis. Boca Raton: CRC Press; 2013.
7. Niezgoda G, Benfield M, Sisak M, Anson P: Tracking acoustic transmitters by code
division multiple access (CDMA)-based telemetry. Hydrobiologia 2002, 483:275–286.
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8. Beyer HL, Morales JM, Murray D, Fortin M-J: The effectiveness of Bayesian state-space
models for estimating behavioural states from movement paths. Methods Ecol Evol 2013,
4:433–441.
9. Mesa MG, Welland LK, Zydlewski GB: Critical swimming speeds of wild bull trout.
Northwest Sci 2004, 78:59–65.
10. Taylor MK, Hasler CT, Findlay CS, Lewis B, Schimidt DC, Hinch SG, Cooke SJ:
Hydrologic correlates of bull trout (Salvelinus confluentus) swimming activity in a
hydropeaking river. River Res Appl, in press.
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Table 1 Posterior medians and 95% credible interval for the parameters of the DCRWS
model and the AR1 models for fish elevation and body temperature
Percentile
Parameter
2.5%
50%
97.5%
q trs
-0.03
-0.01
0.01
qexp
3.17
3.24
3.31
gtrs
0.89
0.91
0.93
gexp
0.43
0.47
0.52
a trs|trs
0.76
0.78
0.81
a trs|exp
0.13
0.16
0.18
s easting
30.52
32.62
34.89
s northing
28.11
30.26
32.50
rfel
0.99
0.99
0.99
wfel
0.71
0.73
0.75
hfel
1.13
1.15
1.17
rbtp
0.99
0.99
0.99
wbtp
4.2 × 10−2
4.3 × 10−2
4.4× 10−2
hbtp
4.9 × 10−4
5.0 × 10−4
5.1 × 10−4
DCRWS
AR1 (fish elevation)
AR1 (body temperature)
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q is the mean turning angle; g is the degree of correlation in direction and speed; a is the
conditional probability of switching between behavioral states;
s is the process variance in
movement; r is the degree of correlation in fish elevation and body temperature;
w is the
process variance in fish elevation and body temperature; h is the variance in sensor reading
errors. trs: transiting; exp: exploratory; fel: fish elevation; btp: body temperature.
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Figure 1 Assessment of the state-space model ability to estimate true positions and
behavioral states. (a) Track recorded by the DGPS, with dashed circles denoting
locations where the tags were allowed to drift to simulate the exploratory behavior. (b−d)
State-space estimates of true tag positions and associated Pexp (filled circles). The grey
line denotes the track estimated by the acoustic telemetry system and the black solid line
denotes the track recorded by the DGPS. In all panels, the dashed line denotes the water
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line at the time the tracking data were recorded; the black rectangle denotes the top of the
powerhouse; and the black polygon denotes part of the dam.
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Figure 2 State-space model estimates. (a) Speed (body lengths per second) and (b)
turning angle by behavioral state, (c) depth and (d) body temperature by season.
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