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The research in this publication was partially or fully funded by Colorado Parks and Wildlife.
Dan Prenzlow, Director, Colorado Parks and Wildlife • Parks and Wildlife Commission: Marvin McDaniel, Chair • Carrie Besnette Hauser, ViceChair
Marie Haskett, Secretary • Taishya Adams • Betsy Blecha • Charles Garcia • Dallas May • Duke Phillips, IV • Luke B. Schafer • James Jay Tutchton • Eden Vardy
�Methods in Ecology and Evolution 2016, 7, 264–273
doi: 10.1111/2041210X.12465
A functional model for characterizing longdistance
movement behaviour
Frances E. Buderman1*, Mevin B. Hooten1,2,3,4, Jacob S. Ivan5 and Tanya M. Shenk6
1
Department of Fish, Wildlife, and Conservation Biology, Colorado State University, Fort Collins, CO, 805231484, USA; 2U.S.
Geological Survey, Colorado Cooperative Fish and Wildlife Research Unit, Colorado State University, Fort Collins, CO, 805231484, USA; 3Department of Statistics, Colorado State University, Fort Collins, CO, 805231484, USA; 4Graduate Degree
Program in Ecology, Colorado State University, Fort Collins, CO, 805231484, USA; 5Colorado Parks and Wildlife, Fort Collins,
CO, 80526, USA; and 6National Park Service, Fort Collins, CO, 80525, USA
Summary
1. Advancements in wildlife telemetry techniques have made it possible to collect large data sets of highly accurate animal locations at a ﬁne temporal resolution. These data sets have prompted the development of a number
of statistical methodologies for modelling animal movement.
2. Telemetry data sets are often collected for purposes other than ﬁnescale movement analysis. These data sets
may diﬀer substantially from those that are collected with technologies suitable for ﬁnescale movement modelling and may consist of locations that are irregular in time, are temporally coarse or have large measurement
error. These data sets are timeconsuming and costly to collect but may still provide valuable information about
movement behaviour.
3. We developed a Bayesian movement model that accounts for error from multiple data sources as well as
movement behaviour at diﬀerent temporal scales. The Bayesian framework allows us to calculate derived quantities that describe temporally varying movement behaviour, such as residence time, speed and persistence in direction. The model is ﬂexible, easy to implement and computationally eﬃcient.
4. We apply this model to data from Colorado Canada lynx (Lynx canadensis) and use derived quantities to
identify changes in movement behaviour.
Keywords: Argos, Bayesian model, Canada lynx, functional data analysis, movement modelling,
splines, telemetry
Introduction
Data sets consisting of animal locations are often collected for
purposes other than movement analysis (e.g., survival analysis,
demographic studies; White & Shenk 2001; Winterstein, Pollock & Bunck 2001) or with technology that prohibits longterm ﬁnescale movement modelling (Yasuda & Arai 2005).
For example, radiotelemetry may be used to estimate survival
(Cowen & Schwarz 2005), but the locations may not be used in
the analysis (e.g., Buderman et al. 2014; Hightower, Jackson
& Pollock 2001). These data sets are costly and timeconsuming to collect, but often contain a wealth of unused spatial
information. The ability to spatially characterize movement
behaviours using data sets that are insuﬃcient for ﬁnescale
movement modelling may help management and conservation
agencies identify critical areas for wildlife movement (Berger
Correction note: Equation 14 appeared incorrectly in the HTML version of this article, (the PDF version has been correct since ﬁrst publication), this has been corrected on 07 October after ﬁrst online
publication.
*Corresponding author. Email: franny.buderman@colostate.edu
2004). In addition, with appropriate temporal data, researchers
can also better understand mechanisms that regulate
movement behaviour (Hays et al. 2014; Scott, Marsh & Hays
2014).
Runge et al. (2014) divide longdistance movements into
four categories: irruption (dispersal), migration, nomadism
and intergenerational relays (which we do not address). Such
movement behaviour can vary among individuals and over an
individual’s lifetime, though some species may be more
inclined to exhibit one kind of longdistance movement behaviour (LDMB) over another (Jonz�en et al. 2011; Mueller et al.
2011; Singh et al. 2012). For most organisms, the causes and
costs of dispersal will vary by individual and in space and time
(Bowler & Benton 2005), resulting in a continuum of movement behaviours (Jonz�en et al. 2011). LDMB may contribute
substantially to population dynamics because it is the main
determinant of population spread and colonization rates
(Greenwood & Harvey 1982; Shigesada & Kawasaki 2002).
Thus, LDMB is an important lifehistory trait for many processes such as species invasions, range shifts and local extinctions, reintroduction programmes, metapopulation dynamics,
connectivity and gene ﬂow (Trakhtenbrot et al. 2005).
© 2015 The Authors. Methods in Ecology and Evolution © 2015 British Ecological Society
This article has been contributed to by U.S. Government employees and their work is in the public domain in the USA.
�Functional models for movement
The spatial location of these behaviours could inform conservation eﬀorts for species capable of longdistance movements, as some behaviours may be more important than
others for population persistence (Runge et al. 2014). In
addition, comparing contemporary movement data with properly analysed historical data may identify changes in movement behaviour resulting from natural and anthropogenic
disturbances. Changes in migratory behaviour could have
wideranging consequences in cases where the species contributes signiﬁcantly to the biological assemblage (Robinson
et al. 2009). Species are usually limited in their range by dispersal ability, foraging ecology or available habitat (Hays & Scott
2013; Wood & Pullin 2002), and as habitat fragmentation and
climate variability increase, the ability of species to traverse
long distance will become critical (Bowler & Benton 2005).
Species that have the capability for longdistance movement
may be able to track habitat as environmental conditions
change. However, individuals usually depend on a network of
suitable habitats for diﬀerent behaviours (e.g., breeding or
migration; Robinson et al. 2009). Longterm survival of the
species can be reduced when the distance between patches
exceeds dispersal ability (Trakhtenbrot et al. 2005), or when
suitable habitat is not available for all of the behaviours that
occur during an annual cycle (Robinson et al. 2009).
Although dispersal, migration and nomadism are all
LDMBs, they may diﬀer in characteristics that can be quantitatively measured, such as residence time, speed and persistence in direction (described using the turning angle). For
example, areas where individuals are foraging or maintaining a
home range may be identiﬁed by longer residence times or
slower speeds (Schoﬁeld et al. 2013) and undirected motion
(Morales et al. 2004). In contrast, movement may be faster
(Dickson, Jenness & Beier 2005) and more directed (Haddad
1999) within corridors. Nomadic individuals may exhibit similar speeds as migrators and dispersers, but they would appear
to be perpetually dispersing, with no consistent activity centre
and a turning angle independent of previous movements
(Lidicker & Stenseth 1992). Dispersal and migration may have
similar speed and directional characteristics, but migration is a
seasonally repeated movement between the same areas (Berger
2004) by individuals within a population (Sawyer, Lindzey &
McWhirter 2005), whereas dispersal is a oneway movement
(Lidicker & Stenseth 1992).
Movement behaviour is typically monitored using very high
frequency (VHF) or satellite telemetry devices. These monitoring devices are more eﬀective at detecting LDMB than plotbased studies, which may underestimate longdistance movement (Koenig et al. 1996). The frequency of VHF data is determined by how often an individual can be located and are
spatially restricted to the actively searched area. Aerial location
accuracy associated with VHF data may be aﬀected by
antenna type, altitude and observer skill, while ground triangulation accuracy may be additionally impacted by terrain, vegetation, power lines, and weather (Mech 1983). In contrast, the
intended ﬁx rate for a satellite telemetry device is preprogrammed and often regularly spaced in time. Fix success rates
and accuracy can be inﬂuenced by animal behaviour, such as
265
diving behaviour, canopy cover, terrain and climatic conditions (e.g., Di Orio, Callas & Schaefer 2003; Dujon, Lindstrom
& Hays 2014; Heard, Ciarniello & Seip 2008; Mattisson et al.
2010). The device’s satellite system (GPS or Argos Satellite
maintained by Service Argos) can also inﬂuence accuracy of
the location observations (Costa et al. 2010; Dujon, Lindstrom
& Hays 2014; Heard, Ciarniello & Seip 2008; Patterson et al.
2010; Vincent et al. 2002). In addition, ﬁx success rate, battery
life and accuracy may all depend on transmitter manufacturer
and model. Both VHF and satellite components can be placed
into the same device or individuals can be outﬁtted with two
separate devices, resulting in data sets consisting of multiple
data types.
Movement modelling often seeks to spatially characterize an
individual’s location as a function of time; however, this function may be highly complex and nonstationary. In addition,
measurement error varies among monitoring methods and can
be large enough to overwhelm smallscale movement patterns
(Breed et al. 2011; Kuhn et al. 2009). Coupled with temporal
irregularity and missing data, these attributes may prohibit the
use of contemporary movement models. We have found that
many available methods do not readily accommodate multiple
sources of data and must impute missing data to obtain locations at regular intervals (e.g., Hanks et al. 2011; Hanks, Hooten & Alldredge 2015; Hooten et al. 2010; Johnson, London &
Kuhn 2011). For example, the continuoustime correlated random walk model presented by Johnson et al. (2008) only
accounts for elliptical error distributions. Breed et al. (2012)
incorporated an augmented particle smoother into a CRW
process model to allow for timevarying parameters; however,
their method does not account for multiple data sources and
its eﬀectiveness was only demonstrated on highly accurate
GPS data at a ﬁne temporal scale (10–30 locations day�1 ).
Winship et al. (2012) incorporated multiple data sources (Argos, GPS and geolocation data) into a statespace model, but
the method performed poorly when there were data gaps,
relied heavily on the estimates of Argos precision presented in
Jonsen, Flemming & Myers (2005) and treated the GPS data
as equivalent to the best Argos location class. Changepoint
models require specifying or estimating the number of change
points, and the change points are discrete in time (Gurarie,
Andrews & Laidre 2009; Hanks et al. 2011; Jonsen, Flemming
& Myers 2005; Jonsen, Myers & James 2007); modelling
smooth transitions in the changepoint framework is more difﬁcult. Given that one individual may exhibit many diﬀerent
LDMBs, we seek a model that is ﬂexible enough to detect different types and degrees of movement behaviour, without specifying or estimating the number of change points. Brownian
bridge movement models, a method commonly used with
highresolution telemetry data, have been shown to work well
only when the measurement error is negligible (Pozdnyakov
et al. 2014), making them unsuitable for data sets obtained
with VHF or Argos technology, which can be subject to substantial error. Recent applications of wavelet analyses also do
not account for location error or uncertainty in the changepoint identiﬁcation and are not feasible with sparse and irregular data sets (Lavielle 1999; Sur et al. 2014).
© 2015 The Authors. Methods in Ecology and Evolution © 2015 British Ecological Society, Methods in Ecology and Evolution, 7, 264–273
�266 F. E. Buderman et al.
Basis functions are a useful set of tools for approximating
continuous functions, such as movement paths, when ordinary
polynomials are inadequate to describe the behaviour of the
function (Rice 1969). Commonly used basis functions include
wavelets, Fourier series and splines. Approximating a function
with splines is computationally easy because the function is just
a weighted sum of simpler functions (Wold 1974), and such
tools have been incorporated into standard statistical software.
Wold (1974) recognized that splines may be most useful in low
information settings where the ultimate goal is to compare
individual estimates of a few characteristic parameters that
describe the curve. Basis functions have been used extensively
in ﬁelds such as physics (e.g., Sapirstein & Johnson 1996), medicine (e.g., Gray 1992) and medical imaging (e.g., Carr, Fright
& Beatson 1997), and climate science (e.g., S�aenzRomero
et al. 2010). However, basis functions and associated statistical
methods are less commonly used in ecology. Most applications
focus on modelling species distributions (e.g., Lawler et al.
2006; Leathwick et al. 2005) and population dynamics (e.g.,
Bjørnstad et al. 1999), though splines have broad applicability
in generalized additive models (Hastie & Tibshirani 1990;
Wood & Augustin 2002). For example, Hanks, Hooten & Alldredge (2015) used Bsplines to model spatial transition rates
as a function of location and directionbased covariates and
timevarying coeﬃcients. In addition, recent eﬀorts have used
Bsplines to estimate density functions associated with movementrelated behavioural states (Langrock et al. 2014). Tremblay et al. (2006) used Bezier, hermite and cubic splines as
strict interpolators of irregular telemetry data from oceanobligate species; however, they assumed the ﬁltered Argos locations were the true locations. There is also precedent in the
statistical literature for the equivalence between stochastic
movement processes, such as the Wiener process, and smoothing polynomial splines (Wahba 1978; Wecker & Ansley 1983).
We describe a functional approach to movement modelling
using basis functions within a Bayesian model that accounts
for multiple data types and their associated error, recognizing
that the observed locations are not the true location. The basis
functions allow us to account for temporal variation in the continuous underlying movement path without specifying movement mechanisms. We then use derived quantities, such as
residence time, speed and persistence in direction, to characterize movement behaviour. In addition, the model is multiscale,
allowing for movement behaviour at multiple biologically
relevant temporal scales. We use this model to describe how
reintroduced Canada lynx (Lynx canadensis) moved throughout Colorado. The two data collection methods, along with
their measurement error and the sampling irregularity, make
this an ideal data set to demonstrate the utility of our model.
Methods
Conventional functional data analysis (FDA) assumes that there is a
continuous underlying process, but the observations are temporally discrete, may be subject to error and are temporally irregular (Ramsay &
Dalzell 1991; Ramsay & Silverman 2002; Ramsay & Silverman 2005).
Unlike traditional time series analysis, FDA does not assume stationar
ity or regularity of time intervals (Levitin et al. 2007). The continuous
function of interest is approximated using basis functions, which are a
set of patterns that capture the main shape of the curve (Ferraty & Vieu
2006; Hastie, Tibshirani & Friedman 2009; Ramsay & Silverman
2005). In our case, diﬀerent sets of basis functions account for complexity in the process at diﬀerent temporal scales, allowing us to detect both
large and smallscale movement. In addition, FDA is useful when the
objectives of an analysis are to estimate the derivatives of a function
(Levitin et al. 2007; Ramsay & Dalzell 1991). In our framework, functions of temporal derivatives, such as residence time, speed and persistence in direction, are derived quantities that can characterize the
movement path. The Bayesian framework allows for inference concerning these derived quantities and their associated uncertainty while
incorporating multiple data sources; for our purposes, we incorporated
VHF and Argos data into a single model.
DATA MODEL
We consider each observed (centred and scaled) location, sj ðtÞ for a
time t 2 T associated with data type j (j = 1,...,6 are Argos error classes
and j = 7 denotes VHF), to arise from a multivariate normal mixture
model with mean, z(t), representing the true location at time t and a
covariance matrix Rj such that
�
NðzðtÞ; Rj Þ; if wj ðtÞ ¼ 1
eqn 1
sj ðtÞ �
~ j Þ; if wj ðtÞ ¼ 0
NðzðtÞ; R
The covariance matrix, Rj � r2j Rj , represents the error variance associated with each data type where the correlation matrix is
�
pﬃﬃﬃ �
1
cq
R � pﬃﬃﬃ
eqn 2
cq
c
for j = 1,...,6 and R � I for j = 7. The prior distribution for the measurement error variance, r2j , was modelled as an inverse gamma, IG(q,r),
where q is the shape parameter and r is the rate parameter. Argos error
for all error classes has been shown to be larger than reported by Argos
and greater in the longitudinal direction (Boyd & Brightsmith 2013;
Costa et al. 2010; Hoenner et al. 2012); therefore, we use the parameter
c, where c � Betaðac ; bc Þ, to scale the error variance to be less in latitude
than longitude. The q parameter scales the degree of covariance
between latitude and longitude and is modelled as Betaðaq ; bq Þ.
The indicator wj ðtÞ determines which mixture component gives rise
to the observed location and is modelled as Bern(0�5). The covariance
~ j , is calculated as Hj Rj H0 where Hj
matrix of the rotated distribution, R
j
is a transformation matrix equal to
�
�
1 0
H�
eqn 3
0 �1
for j = 1,. . .,6, and H � I (the identity matrix) for j = 7. The mixture
model accounts for the fact that Argos error locations do not follow a
symmetric distribution around the true location, but are more likely to
be found in and Xpattern, due to the polar orbit of the satellites (Costa
et al. 2010; Douglas et al. 2012). In preliminary analyses not presented
here, the multivariate normal mixture model ﬁt the data better than a
multivariate normal nonmixture model. Argos locations are commonly modelled with a tdistribution to account for extreme outliers
(following Jonsen, Flemming & Myers 2005), however, the mixture
model allows us to model anisotropic outliers. Though the aforementioned studies have modelled or estimated Argos error, the information
is not directly applicable in the form of priors because the mixture
model is a novel method for modelling Argos error and there is signiﬁcant variability in reported estimates of Argos error (Costa et al. 2010).
Beginning in 2011, the Argos system implemented a new algorithm that
© 2015 The Authors. Methods in Ecology and Evolution © 2015 British Ecological Society, Methods in Ecology and Evolution, 7, 264–273
�Functional models for movement
provides an error ellipse, as opposed to a radius, for each location
(Lopez et al. 2014). Recent work by McClintock et al. 2014b) used the
ellipse parameters provided by the Argos system and a bivariate normal
distribution to model the data.
PROCESS MODEL
In the FDA paradigm, a continuous process for a set of times t ðt 2 T Þ
is written as an expansion of M basis functions of order k:
zðtÞ ¼
M
X
cm /m;k ðtÞ
eqn 4
m¼1
where z(t) is the curve of interest, cm is a coeﬃcient that determines the
weight of each basis function in the construction of the curve, and
/m ðtÞ is a particular basis function (Levitin et al. 2007). The type of
pattern present in the data dictates the best choice of basis function; for
example, splines are often used for nonperiodic data, Fourier series for
periodic data, and wavelet bases for data with sharp localized patterns.
We employed the Bspline basis, which is commonly used in semiparametric regression, because it has local support and stable numerical
properties when the number of knots (the points at which the basis
functions connect) is large (Keele 2008; Ruppert, Wand & Carroll
2003). However, the model we present is general enough to accommodate any type of basis functions. Bspline basis functions are deﬁned
recursively according to the Coxde Boor formula (see De Boor 1978).
Let xm;k denote the mth Bspline basis function of order k (cubic
Bsplines are 4th order and 3rd degree) for the knot sequence s, where
k≤K. Then for m =1,...,N + 2K � k,
xm;k ðtÞ ¼
t � sm
smþk � t
Bm;k�1 ðtÞ þ
Bmþ1; k�1 ðtÞ;
smþk � smþ1
smþk�1 � tm
eqn 5
where N is the number of interior knots (Hastie, Tibshirani & Friedman 2009).
In the spatial statistics and signal processing framework, a continuous stochastic process is often written as a convolution, or a moving
average, of a smoothing kernel function, k(s�t) and a latent process
(e.g., white noise), g(s):
Z
zðtÞ ¼
kðs � tÞgðsÞds
eqn 6
T
for s 2 T (Calder 2007; Higdon 2002; Lee et al. 2002). When discretized, (6) takes on a general formulation (4) (Calder 2007; Higdon
2002; Lee et al. 2002). Nonstationary processes can be modelled by
allowing the kernel to be a function of time (or space) and not just distance (Cressie & Wikle 2011; Higdon 2002; Higdon, Swall & Kern
1999). In the context of animal movement, one can consider the
smoothing kernel as some function that imposes temporal dependence
on the observed locations (the latent process) to create a continuous
and smooth movement path.
In our case, the location of an individual at time t in each direction, z
(t), is a function of an individual’s geographic mean in that direction,
b0 and the summation of M cubic Bsplines evaluated at time t, xm;4 ðtÞ,
and the regularized, directionspeciﬁc coeﬃcient, bm , for that Bspline.
The location in longitude and latitude is
zlon ðtÞ ¼ b0lon þ
M
X
xm;4 ðtÞbmlon
eqn 7
xm;4 ðtÞbmlat
eqn 8
m¼1
zlat ðtÞ ¼ b0lat þ
M
X
m¼1
Using matrix notation, we can write (7) and (8) jointly as
zðtÞ ¼ b0 þ XðtÞb
eqn 9
267
where z(t) is a vector describing the location in space at time t. The
matrix X(t) is a 2by2M matrix where xðtÞ0 is a row vector containing
all of the Bsplines evaluated at time t, such that
�
�
00
xðtÞ0
eqn 10
XðtÞ �
0
0
0
xðtÞ
As such, it can be multiplied by a single 2Mby1 vector of regularized
coeﬃcients
�
�
blon
eqn 11
b �
blon
The regularized coeﬃcients for higherorder splines are not generally
interpreted (Weisberg 2014), but can be thought of as the contribution,
or the directional forcing, of that basis function to the process at that
time. The intercept, b0 , can be interpreted as the geographic centre of
mass for each individual, for which we speciﬁed a relatively uninformative 2dimensional normal prior (Appendix S1). We speciﬁed a normal
prior with mean 0 and covariance matrix Rb for the coeﬃcients such
that
b � Nð0; Rb Þ
eqn 12
We selected three sets of Bsplines and varied the number of knots to
align with temporal scales we believe are biologically important for lynx
movement: year, season (3 months) and month. Including multiple sets
of basis functions allows the continuous function to capture behaviour
at diﬀerent temporal scales without losing predictive capability when
there is an absence of ﬁnescale temporal data. However, the required
number of knots results in a large design matrix of coeﬃcients that is
diﬃcult to visualize; for example, there were 36 and 41 basis functions
for the two Canada lynx presented in the case study. The number of
basis functions will increase as the length of the time series increases.
We used the covariance matrix
!
r2blon I
0
Rb �
eqn 13
0
r2blon I
as a regulator in the ridge regression framework to shrink the b coeﬃcients. The variance terms, r2blon and r2blat , control the smoothing in each
dimension; a very small variance leads to underﬁtting, whereas a large
variance can lead to overﬁtting (Eilers & Marx 1996). We selected the
variance components by calculating the Deviance Information Criterion (DIC; Spiegelhalter et al. 2002) over 10 000 MCMC iterations
and optimizing the DIC over 400 pairs of variance components (Appendix S2). In simulation, we found that DIC and Kfold crossvalidation methods performed similarly. The details of regularization and
ridge regression are beyond the scope of this paper and are explored in
more detail in (Hastie, Tibshirani & Friedman 2009) and Hooten &
Hobbs (2015).
The model described above yields the posterior distribution
½b0 ;b;r2 ;q;c;wjS� /
J Y
Y
½sj ðtÞjb0 ;b;r2j ;q;c;wj ðtÞ�½b0 �½b�½r2 �½q�½c�½w�
j¼1 t2T
eqn 14
w is a vector of the indicators wj ðtÞ, and S is a
where r �
matrix of observed locations. This is the form of a typical ‘integrated’
model where multiple data sources provide information about the same
underlying processes. Similar multidata source models have become
popular in demographic studies (e.g., Barker 1997; Burnham 1993;
Nasution et al. 2001; Schaub & Abadi 2011), but have not been as common in movement studies (but see Winship et al. 2012). If inference for
multiple individuals is desired, the data model can be shared among
individuals while the process model parameters (b0 , b) and regulator
2
½r2j ; :::; r2J �,
© 2015 The Authors. Methods in Ecology and Evolution © 2015 British Ecological Society, Methods in Ecology and Evolution, 7, 264–273
�268 F. E. Buderman et al.
ðRb ) can be allowed to vary by individual. This model can be extended
to account for additional stochasticity using a ﬁrstorder Gaussian process, such that zðtÞ � Nðb0 þ XðtÞb; r2l RÞ, where r2l R accounts for
process error separately. Such Gaussian processes are commonly used
as statistical emulators of complicated nonlinear mechanistic models
(Hooten et al. 2011; O’Hagan & Kingman 1978).
See Appendix S1 for prior speciﬁcations. The model was ﬁt using
Markov chain Monte Carlo (MCMC), and a Gibbs sampler was constructed to sample from the posterior using the fullconditional distributions for all parameters except q and c, because they were not
conjugate. MetropolisHastings was used to sample q and c. See
Appendix S3 for R code (R Core Team 2013).
CHARACTERIZING MOVEMENT
We are interested in quantities derived from z(t) that can be used as
movement descriptors. We describe three relevant derived quantities;
however, our framework can be extended to other systems and conservation questions by modifying these quantities. These derived quantities represent the physical outcome in the movement path from various
movement behaviours. The Bayesian framework allows us to obtain
inference for derived quantities through Monte Carlo integration. We
can visualize these quantities both temporally and spatially. All quantities are calculated in the MCMC algorithm using techniques described
in Appendix S4 (spatial quantities) and Appendix S5 (temporal quantities).
To describe the quantities of interest spatially, we deﬁne a grid of
equally sized regions, Al for l = 1,...,L, that comprise the area for which
we desire inference. This method is similar to that used by Johnson,
London & Kuhn (2011) to describe diving behaviour of northern fur
seals (Callorhinus ursinus). The ﬁrst derived quantity we describe is residence time, rl and is calculated on each MCMC iteration as a per area
frequency of locations in region Al :
r1 ¼ lim
Dt!0
X
DtIfzðtÞ2Al g
eqn 15
t2T
where the indicator I identiﬁes whether location z(t) was in region Al .
The second derived quantity of interest is speed. To calculate the
average speed per unit of area, we ﬁrst need the velocity between the
location at time z(t) and the location at time z(t�Dt). When Dt is suﬃciently small, the ﬁrst derivative of z(t) with respect to t can be approximated by
dzðtÞ
� dðtÞ
dt
eqn 16
where
dðtÞ ¼
zðtÞ � zðt � DtÞ
Dt
eqn 17
In practice, Dt is constant for the entire time series, and velocity is
related to speed m(t) such that
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
mðtÞ ¼ dðtÞ0 dðtÞ
eqn 18
The average speed in Al , given a positive residence time, is
�
ml ¼
limDt!0
P
t2T
DtmðtÞIfzðtÞ2Al g
rl
eqn 19
A large average speed describes areas where the individual was moving
quickly and spending little time. Therefore, large average speeds (19)
identify areas that individuals may use to travel.
Persistence in direction is the third metric of interest and may be useful for describing directed, as opposed to nomadic, movement. We can
describe persistence in direction by deriving the turning angle, h, using
the velocity calculated in (17),
�
!�
�
�
dðt þ 1ÞdðtÞ
�
�
hðtÞ ¼ �arccos pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ �
eqn 20
�
dðtÞdðtÞ dðt þ 1Þdðt þ 1Þ �
Given that residence time is positive, the average turning angle, h(t), in
region Al is
P
�hl ¼ limDt!0 t2T DthðtÞIfzðtÞ2Al g
eqn 21
rl
Alternatively, we can describe these quantities temporally, negating
the need for a spatially deﬁned grid. This decreases computation time
and allows the quantities to be visualized temporally and spatially.
Speed and persistence in direction can be calculated as they were in (18)
and (20) and residence time can be calculated as the inverse of speed:
rðtÞ ¼
1
mðtÞ
eqn 22
CASE STUDY: CANADA LYNX REINTRODUCTION IN
COLORADO
Colorado Division of Wildlife (now Colorado Parks and Wildlife) initiated a reintroduction programme for Canada lynx (Lynx canadensis) in
1997. Between 1999 and 2006, 218 wildcaught lynx from Alaska,
Yukon Territory, British Columbia, Manitoba and Quebec were
released in the San Juan Mountains within 40 km of the Rio Grande
Reservoir (Devineau et al. 2010). Individuals were ﬁtted with either
VHF collars (TelonicsTM , Mesa, AZ, USA) that were active for 12 h
per day or satellite/VHF collars (SirtrackTM , Havelock North, New
Zealand) that were active for 12 h per week with locations obtained
using the Argos system (Devineau et al. 2010). Weekly airplane ﬂights
were conducted over a 20 684 km2 area, which included the reintroduction area and surrounding highelevation sites (>2591 m; Devineau
et al. 2010); attempts were made to locate each VHFcollared individual in the study area once every 2 weeks. Additional ﬂights outside of
the study area were conducted when feasible and during denning season (Devineau et al. 2010). Accuracy of VHF locations was selfreported as 50–500 m (Devineau et al. 2010). Irregular location data
were obtained from 1999 to 2011 due to one or both of the transmitter
components failing, logistical constraints or movement out of the study
area precluding VHF data collection. Therefore, data for each individual vary in the length of the time series, the temporal regularity of locations and the number of locations from each data type and error class.
We have analysed the telemetry data from two Canada lynx (Appendix
S6).
We obtained 10 000 MCMC iterations, with a burnin period of
1000 iterations. All data used in this paper are available in Appendix
S7. Additional results from ﬁtting the model to simulated data are
available in Appendix S8.
Results
To visualize the ﬁt of the model to the data, we calculated standard posterior quantities, such as means and 95% credible
intervals for the marginal location in each direction (Fig. 1a,
b). Increasing uncertainty is evident during long periods of
missing data (Fig. 1a, b). The derived quantities were scaled
relative to the maximum value for that quantity over the indi
© 2015 The Authors. Methods in Ecology and Evolution © 2015 British Ecological Society, Methods in Ecology and Evolution, 7, 264–273
�Functional models for movement
vidual’s lifetime and plotted both spatially, on a map of Colorado (Figs 1c, d, and 2), and temporally (Fig. 2). These relative
values are useful for visualizing the degree of each behaviour at
a given time point, despite the quantities having diﬀerent units;
the degree of shading represents the strength of that behaviour,
with the size corresponding to the spatial uncertainty (Figs 1c,
d, and 2). The optimal variance terms for the regulator matrix
(13) and mean and 95% credible intervals for the covariance
matrix (2) are presented in Appendix S6.
Both individuals had multiple periods of fast speeds, large
turning angles and high residence times (Figs 1c, d and 2). For
these individuals, high residence time often indicated a corresponding large turning angle; however, these behavioural
quantities were not always concurrent (Fig. 2). For example,
individual BC03M04 displayed periods early in the time series
where the turning angle was the strongest quantity, while speed
and residence time were fairly low, suggesting a searching or
nomadic behaviour (Fig. 2a). Both time series culminated with
the individuals residing in two speciﬁc counties (Clear Creek
and Summit), which includes an area that is considered impor(a)
(c)
269
tant lynx habitat (Loveland Pass; Colorado Parks and Wildlife, pers. commun.). These results also indicate that lynx are
capable of consistent longterm movement across large distances without establishing an area of high residence time. For
example, within a period of two months, individual BC03F03
travelled approximately 480 km (posterior mean), from the
southern portion of Colorado (Mineral County) to southern
Wyoming (Medicine Bow National Forest, speciﬁcally the
area located within Carbon and Albany counties; Fig. 2b).
Discussion
The process model we propose falls within the same class of
models as statistical emulators, functional data models and
process convolutions, and we showed that it can be written in
much the same way. The model presented could be written as a
hierarchical model, by allowing the latent process to be
stochastic. However, it is well known that hierarchical models
with two sources of unstructured error and lacking replication
will have identiﬁability issues (Hobbs & Hooten 2015). Given a
(b)
(d)
Fig. 1. Mean and 95% credible intervals of the marginal locations for two Canada lynx [BC03M04 (a) and BC03F03 (b)], with the observed locations. The posterior mean of each movement descriptor, shown with the counties of Colorado, for individuals BC03M04 (c) and BC03F03 (d). The
size of the point corresponds to spatial uncertainty, and the transparency indicates the strength of the behaviour at that location; for visualization
purposes, any value below 25% of the maximum value for that behaviour is not shown. Coordinates correspond to Universal Transverse Mercator
zone 13N.
© 2015 The Authors. Methods in Ecology and Evolution © 2015 British Ecological Society, Methods in Ecology and Evolution, 7, 264–273
�270 F. E. Buderman et al.
(a)
(b)
Fig. 2. Mean relative movement descriptors through time and space for two Canada lynx reintroduced to Colorado [BC03M04’s (a) and BC03F03’s
(b)]. Coordinates correspond to Universal Transverse Mercator zone 13N.
© 2015 The Authors. Methods in Ecology and Evolution © 2015 British Ecological Society, Methods in Ecology and Evolution, 7, 264–273
�Functional models for movement
situation where there are strong constraints on the movement
process, it may be possible to separate data and process error.
For example, Brost et al. (In Press) were able to separately estimate data and process error in a resource selection framework
by constraining the spatial domain of the process. However,
their study focused on a marine mammal, and therefore, the
process can be constrained to the marine environment (Brost
et al. In Press). Constraining movement in a terrestrial environment may be possible but is less intuitive and will impose
strong assumptions.
In addition, one of the beneﬁts of using a functional data
approach is its ﬂexibility, in contrast to more constrained
mechanistic models. The simplicity of the process model results
in greater computational eﬃciency than other available methods for movement modelling. For example, our model can be
ﬁt on the order of minutes for each individual, compared to
other models that require on the order of days (e.g., Hooten
et al. 2010; McClintock et al. 2014a). Although smallscale
movement patterns may be diﬃcult to detect given the coarse
temporal resolution and the large amount of measurement
error associated with Argos locations, largescale movement
patterns are easily discernible and informative. However,
researchers analysing data at a ﬁner temporal scale could discern smallscale movement with properly scaled basis functions (e.g., daily or weekly).
The model can be used to estimate an animal’s movement
path alone, but is especially useful for learning about movement behaviours that describe how individuals are utilizing the
landscape. For example, persistence in direction may be used
to infer when and where an individual is migrating or dispersing, whereas variation in direction may indicate habitat suitable for a home range (Haddad 1999; Morales et al. 2004). In
the data we analysed the movement descriptors corresponded
with anecdotal evidence of lynx movement behaviour. Many
existing methods for analysing location data explicitly model
the quantities that give rise to the movement path (e.g., speed,
turning angle, residence time, velocity), such that the quantities
must be estimated while ﬁtting the model (mechanistic models;
e.g., Breed et al. 2012; Johnson et al. 2008; Jonsen, Flemming
& Myers 2005; McClintock et al. 2012; Morales et al. 2004;
Winship et al. 2012). In contrast, we use the equivariance
property of MCMC to calculate derived quantities as well as
the proper uncertainty associated with each behaviour (Hobbs
& Hooten 2015). Alternative ad hoc methods could be used,
such as calculating derived quantities based on the mean predicted path, but to ensure the validity of those quantities as
estimators with proper uncertainty, a procedure like the one
we describe is necessary. Quantities of interest beyond those
presented can be derived, such as bearing or tortuosity, or summarized with respect to temporal and spatial features. However, our model would need to be adjusted to accommodate
other sources of measurement error (e.g., GPS data).
The model that we developed may be particularly well suited for analysing data sets that have not been collected
explicitly for movement analysis. These data sets may contain multiple data types, have large amounts of error and
have been collected at a coarse temporal resolution. As such,
271
they may not be conducive for ﬁnescale mechanistic movement modelling. We used a data set that embodied these
characteristics, the telemetry data from the Canada lynx
reintroduction to Colorado, to demonstrate that the FDA
approach can be used to estimate movement paths and associated movement descriptors. The biological inference from
the derived movement descriptors can also be extended
beyond what we show here. For example, our framework
could be extended to incorporate spatial and temporal
covariates into the process model, similar to the approach
described by Hanks, Hooten & Alldredge (2015). In addition, the spatial distribution of the movement descriptors
can be used to summarize movement behaviour across linear
landscape features such as roads. Likewise, movement behaviour through nonlinear landscape features, such as National
Parks, can be described with the average posterior mean of a
movement descriptor within a spatial boundary. Our model
can also be generalized for use with multiple individuals. In
this case, the derived quantities can be aggregated to describe
populationlevel movement. This type of population movement model allows the Argos and VHF covariance matrices
to borrow strength across individuals, potentially improving
parameter estimates. Such extensions are the subject of
ongoing research.
Acknowledgements
Any use of trade, ﬁrm or product names is for descriptive purposes only and does
not imply endorsement by the US Government. Funding was provided by Colorado Parks and Wildlife (1304) and the National Park Service (P12AC11099).
Data were provided by Colorado Parks and Wildlife. The authors would like to
thank the anonymous reviewers for their constructive commentary that helped
improve the manuscript.
Data accessibility
Case Study Data: uploaded as online supporting information in Appendix S6:
Case study tables.
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Received 23 January 2015; accepted 10 August 2015
Handling Editor: Jason Matthiopoulos
Supporting Information
Additional Supporting Information may be found in the online version
of this article.
Appendix S1. Prior speciﬁcations.
Appendix S2. Deviance information criterion calculation.
Appendix S3. MCMC algorithm for ﬁtting the splinebased movement
model and calculating derived behavioral.
Appendix S4. Spatial quantities.
Appendix S5. Temporal quantities.
Appendix S6. Case study tables.
Appendix S7. Centered and scaled location data for two Canada lynx,
BC03M04 and BC03F03, released in Colorado. Data type and Argos
error classiﬁcation is listed for each location.
Appendix S8. Simulations.
© 2015 The Authors. Methods in Ecology and Evolution © 2015 British Ecological Society, Methods in Ecology and Evolution, 7, 264–273
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A functional model for characterizing longdistance movement behaviour
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<p class="articlesection__header section__title main">Summary</p>
<div class="articlesection__content en main">
<ol>
<li>Advancements in wildlife telemetry techniques have made it possible to collect large data sets of highly accurate animal locations at a fine temporal resolution. These data sets have prompted the development of a number of statistical methodologies for modelling animal movement.</li>
<li>Telemetry data sets are often collected for purposes other than finescale movement analysis. These data sets may differ substantially from those that are collected with technologies suitable for finescale movement modelling and may consist of locations that are irregular in time, are temporally coarse or have large measurement error. These data sets are timeconsuming and costly to collect but may still provide valuable information about movement behaviour.</li>
<li>We developed a Bayesian movement model that accounts for error from multiple data sources as well as movement behaviour at different temporal scales. The Bayesian framework allows us to calculate derived quantities that describe temporally varying movement behaviour, such as residence time, speed and persistence in direction. The model is flexible, easy to implement and computationally efficient.</li>
<li>We apply this model to data from Colorado Canada lynx (<i>Lynx canadensis</i>) and use derived quantities to identify changes in movement behaviour.</li>
</ol>
</div>
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Buderman, F. E., M. B. Hooten, J. S. Ivan, and T. M. Shenk. 2016. A functional model for characterizing longdistance movement behavior. Methods in Ecology and Evolution 7:264273. <a href="https://doi.org/10.1111/2041210X.12465" target="_blank" rel="noreferrer noopener">https://doi.org/10.1111/2041210X.12465</a>
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Buderman, Frances E.
Hooten, Mevin B.
Ivan, Jacob S.
Shenk, Tanya M.
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Argos
Bayesian model
Canada lynx
Functional data analysis
Movement modelling
Splines
Telemetry
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10 pages
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20150831
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<a href="http://rightsstatements.org/vocab/InCNC/1.0/" target="_blank" rel="noreferrer noopener">In Copyright  NonCommercial Use Permitted</a>
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application/pdf
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English
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Methods in Ecology and Evolution
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Article