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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
�Statistical Methodology 17 (2014) 82–98
Contents lists available at ScienceDirect
Statistical Methodology
journal homepage: www.elsevier.com/locate/stamet
Temporal variation and scale in movementbased resource
selection functions
M.B. Hooten a,b,c,∗ , E.M. Hanks c , D.S. Johnson d , M.W. Alldredge e
a
U.S. Geological Survey, Colorado Cooperative Fish and Wildlife Research Unit, Fort Collins, CO, USA
b
Department of Fish, Wildlife, and Conservation Biology, Colorado State University, Fort Collins, CO, USA
c
Department of Statistics, Colorado State University, Fort Collins, CO, USA
d
National Marine Mammal Laboratory, National Oceanic and Atmospheric Administration, Seattle, WA, USA
e
Colorado Parks and Wildlife, Fort Collins, CO, USA
article
info
Article history:
Received 9 June 2012
Received in revised form
8 December 2012
Accepted 9 December 2012
Keywords:
Animal movement
Kullback–Leibler
Telemetry data
abstract
A common population characteristic of interest in animal ecology
studies pertains to the selection of resources. That is, given
the resources available to animals, what do they ultimately
choose to use? A variety of statistical approaches have been
employed to examine this question and each has advantages and
disadvantages with respect to the form of available data and
the properties of estimators given model assumptions. A wealth
of high resolution telemetry data are now being collected to
study animal population movement and space use and these data
present both challenges and opportunities for statistical inference.
We summarize traditional methods for resource selection and
then describe several extensions to deal with measurement
uncertainty and an explicit movement process that exists in studies
involving highresolution telemetry data. Our approach uses a
correlated random walk movement model to obtain temporally
varying use and availability distributions that are employed in a
weighted distribution context to estimate selection coefficients.
The temporally varying coefficients are then weighted by their
contribution to selection and combined to provide inference at
the population level. The result is an intuitive and accessible
statistical procedure that uses readily available software and is
computationally feasible for large datasets. These methods are
∗ Corresponding author at: Department of Fish, Wildlife, and Conservation Biology, Colorado State University, Fort Collins,
CO 805231484, USA.
Email address: Mevin.Hooten@colostate.edu (M.B. Hooten).
15723127/$ – see front matter. Published by Elsevier B.V.
doi:10.1016/j.stamet.2012.12.001
�M.B. Hooten et al. / Statistical Methodology 17 (2014) 82–98
83
demonstrated using data collected as part of a largescale mountain
lion monitoring study in Colorado, USA.
Published by Elsevier B.V.
1. Introduction
An explosion of recent papers on the statistical analysis of animal movement indicates rapid
growth in this emerging area of animal ecology. The formal mathematical description of animal
movement is quite old, dating back hundreds of years, and even though the seminal text on the
topic written by Turchin [47] is quite relevant, it is currently out of print and lacks a contemporary
statistical perspective. Despite the existence of highly technical literature describing ways to model
animal movement (e.g., [7,15]), most applied studies that have actual management or conservation
objectives have sought to focus more on what is termed ‘‘space use’’, in which they seek to characterize
the geographical and/or environmental space used by either individual animals or populations or both.
Types of space use analyses vary widely and include: (1) describing an individual’s home range or
core area (e.g., [52]) (2) describing the spatial distribution (probability density function or ‘‘utilization
distribution’’, e.g. [34]) from which individual’s locations (st , for time t ∈ T ) might arise and (3) the
estimation of resource selection functions (or RSFs, e.g., [32]). In the latter, statistical inference is
focused on identifying the probability of resource use given resource availability (i.e., selection).
1.1. Traditional resource selection
The basic statistical approach put forth by Manly et al. [32], and since extended in several different
directions (e.g., [23,30,24,29]), specifies that the distribution of use [x]u is equal to a weighted
distribution of availability [x]a :
[x]u =
g (x, β)[x]a
g (ν, β)[ν]a dν
,
= c g (x, β) [x]a ,
= [xβ]u ,
(1)
where, the square bracket notation ‘[. . .]’ denotes a probability density function, x corresponds to
a vector of resource covariates, β is a set of selection parameters (often regression coefficients), c
is a normalizing constant, and g (x, β) is the resource selection function (often the inverse logit or
exponential function, depending on the desired inference). The last line in (1) is not commonly used
elsewhere, but is a notation that we will make use of in what follows.
Importantly, as others point out (e.g., [40,30,49,12]), the equation in (1) is referred to as a ‘‘weighted
distribution’’ and can be arrived at by an application of Bayes rule. Making a slight modification to the
typical weighted distribution specification, we index the resource observations by the location st at
which they were observed at time t. Thus, without loss of generality we may write:
[x(st )]u =
g (x(st ), β)[x(st )]a
g (x(s), β)[x(s)]a ds
= [x(st )β]u .
,
(2)
Now, assuming independent observations x(st ) for t = 1, . . . , T and a known availability distribution
[x(st )]a , the likelihood can be written as the product of the righthandside of (2):
T
[x(st )β]u ,
(3)
t =1
and can be maximized with respect to the selection coefficients β, given the data, as long as the integral
in the denominator of (2) can be computed. Conveniently, it has been shown that the likelihood in
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M.B. Hooten et al. / Statistical Methodology 17 (2014) 82–98
(3) can be maximized, under certain conditions, using logistic regression and a ‘‘casecontrol’’ design
(e.g., [23,49]). Thus, computational methods for fitting resource selection models to telemetry data
are accessible to animal ecologists. All that is needed is a binary regression data set consisting of
ones and zeros, where ones correspond to the observed telemetry locations (and associated covariates
x(s)) and the zeros correspond to a background sample that is typically drawn from a uniform spatial
distribution over the study area or homerange of an individual animal. Then this most commonly
employed method for estimating resource selection proceeds by fitting a logistic regression model to
the augmented binary data set and associated set of covariates while omitting the intercept term (β0 )
in the model. It has been shown that the intercept is not identifiable as a model parameter; this arises
because of the standardization of g (x, β) in the likelihood.
These ideas are certainly not new in the statistical literature, but their application to animal space
use studies has only relatively recently become common [32]. Further, it should be noted that this
basic approach to estimating resource selection is actually equivalent to many of the approaches
proposed for species distribution modeling (e.g., [50,12]) where socalled presenceonly data are
collected and a background sample is taken to numerically compute the necessary integral in the
denominator of (2). In fact, [50] show that the integral in the denominator of (2) can be more efficiently
computed using numerical quadrature rather than Monte Carlo integration as is implicit in the random
background data sampling used in the logistic regression method.
1.2. Emerging issues with contemporary telemetry data
Animals move in a nonindependent fashion with respect to resources and our monitoring
techniques are becoming so sophisticated that the temporal resolution of fixes (i.e., location
observations in time, st ) is quite fine; hours or minutes typically, using global positioning system
(GPS) telemetry devices. Thus, wildlife biologists have a movement process evident in most of their
recent telemetry data and conventional methods for estimating resource selection are not equipped
to explicitly address this. Furthermore, in programming telemetry devices (which is the only control
over the design once the animal is tagged) most studies opt for regular fix rates, though not all fixes
may be successful given terrain, atmospheric conditions, or other landscape features [35]. Therefore,
irregularly spaced temporal data at a very fine resolution are common and result in hundreds or
thousands of observations depending on the extent of the study. Fig. 1 shows a subset of telemetry
locations from two of the animals used in our study. Notice how, even with a fine temporal fix rate
(3 h between locations) there exists much uncertainty pertaining to the use of the resource shaded in
gray (let alone selection).
The wellcited work of Swihart and Slade [45,46] provides a means to assess autocorrelation
under certain types of monitoring designs (e.g., known home range). However, if these methods are
employed at all, they are often only used to ‘‘thin’’ data sets prior to further modeling. The downside to
thinning is that the full potential of the monitoring technology cannot be utilized. Also, the selection
of resources can depend on the behavioral state of the animal and other exogenous factors that are
dynamic in time. Examples of these could include diurnal activity patterns, satiation, and weather
events. Although becoming less of an issue as the monitoring technology improves, another problem
common with satellite telemetry observations (e.g., Argos and to a lesser extent GPS) is the potential
presence of measurement error or location uncertainty.
To some extent, many (but not all) of these problems have been addressed in animal movement
models that explicitly consider a movement process with builtin environmental effects on movement
as a form of selection function (e.g., [42,19,17,33]). This is a rapidly growing area of statistical modeling
research, but so far these methods have not been widely adopted by the wildlife biologists who need
them most. This lack of adoption may be due to the sophistication and specialization of the models
for certain species and the lack of available software. Thus, most applied ecological studies of animal
space use still rely on logistic regression routines to estimate resource selection functions [27].
Johnson et al. [24] and Forester et al. [14] describe methods for estimating resource selection while
considering the autocorrelation due to movement, and while others (e.g., [38,13]) have argued that
�M.B. Hooten et al. / Statistical Methodology 17 (2014) 82–98
85
Fig. 1. A portion of the path and telemetry data for two different mountain lions. The gray shaded cells in the background
represent one landtype (specifically, shrub type) used as a covariate in the analyses. Lines between telemetry locations are only
shown to indicate the individual and sequence of fixes.
population level studies need not be concerned with such forms of autocorrelation, conventional
weighted distribution approaches are simply not appropriate for use with modern forms of telemetry
data.
1.3. Roadmap
In what follows, we describe an approach that: (1) allows for resource selection inference while
utilizing all of the available data, (2) explicitly accommodates location uncertainty, (3) allows for the
scale of selection to be user defined before and after the data are collected, and (4) provides a very
flexible framework for incorporating more sophisticated modeling if desired. The methodology that
we present relies on a constraint (i.e., model for movement) that is placed on the movement process
which allows for both interpolation and prediction of locations at a regular set of time intervals (the
scale of which is a user decision). Specifically, movementrelated use and availability distributions
are estimated from telemetry data and then put in a timespecific weighted distribution context to
obtain selection coefficients βt . These βt coefficients represent timevarying effects of the resource
covariates on selection over the period of (t − 1t , t ] for all t of interest. The simplest form of desired
inference often pertains to the mean and variance of βt , but much more complicated models on βt
may allow for inference on populationlevel selection, changes in individual and population behavior
over time, intra and interspecific interaction, and synchrony of selection with other timevarying
environmental variables. These methods are intuitive and relatively simple to implement using readily
available software, falling somewhere between the oversimplified logistic regression approaches to
estimating resource selection and the more sophisticated and specialized animal movement models
that have appeared in the recent literature.
Beginning in the Methods section, we describe how to frame the selection problem in a movement
context directly, how to estimate timevarying selection, and how to summarize selection over
the temporal extent of interest. In the Application section, we demonstrate the utility of this new
approach using telemetry data from a population of mountain lions (Puma concolor) in Colorado, USA.
Finally, in the Conclusion section, we summarize our approach and provide several promising novel
methodological directions as well as potential future applications.
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M.B. Hooten et al. / Statistical Methodology 17 (2014) 82–98
2. Methods
2.1. Modeling movement
One of the simplest models for animal movement is based on a random walk. Typically this takes
the form of a statespace model where the random walk is correlated and framed in either discrete
time (e.g., [25]) or in continuous time (e.g., [22]). In principle, either approach could be employed here,
however our focus is on determining the two critical distributions for estimating resource selection:
use and availability. As we show next, both distributions arise naturally in a local time setting as a
byproduct of the movement model.
Consider the continuoustime correlated random walk (CTCRW) model of Johnson et al. [22] where
the true underlying continuous movement of an individual is assumed to be dynamic according to
an Ornstein–Uhlenbeck process (i.e., stochastic differential equation involving Brownian motion).
When the dynamical system is discretized to accommodate a finite set of data as well as to facilitate
computation, the geographical location process (µt ) is not Markovian over the interval (t − 1t , t ].
However, a combined dynamic process containing the locations and velocities can be constructed
such that it is Markov. That is, a Brownian motion process can be specified in terms of a set of coupled
velocity (vt ) and location (µt ) equations (using a parenthetical indexing notation to handle nonregular time intervals and ignoring the multivariate notation for simplicity):
θ2 e−θ3 t
ω(e2θ3 t ),
v(t ) = θ1 + √
2θ3
t
µ(t ) = µ(0) +
v(u)du,
(4)
0
where the θ parameters control the motion and ω(t ) is a standard Brownian motion process with
min(t ,τ )
E (ω(t )) = 0, Var(ω(t )) = t, and Corr(ω(t ), ω(τ )) = max(t ,τ ) (e.g., [22]). When discretized, this serves
as a model for the true underlying movement process where, for zt = (µt , vt )′ we have:
zt = Mt zt −∆t + ηt ,
(5)
such that ηt ∼ N(0, 6z ) and Mt is a propagator matrix consisting of elements that perform the
necessary combination of previous zt −∆t using nonlinear functions of θ and ∆t as weights in the sums
resulting from the matrix multiply: Mt zt −∆t .
To construct a statistical model, we describe a likelihood that gives rise to the data. In this case,
we assume the observed telemetry locations st depend stochastically on the true locations µt and
some measurement error (e.g., perhaps due to inaccuracy of the sensor and/or weather and terrain
conditions). Thus, Johnson et al. [22] show that the data model can be written:
st = Ht zt + ϵt ,
(6)
where, ϵt ∼ N(0, 6s ) and the transformation matrix Ht simply maps the statespace to the dataspace.
In the 2D movement situation, each Ht will be a 2 × 4 matrix with rows (1, 0, 0, 0) and (0, 1, 0, 0),
respectively. Note that (5) and (6) combine to form a statespace model that fits within the general
spatiotemporal modeling framework proposed by Wikle and Hooten [51].
2.2. Use and availability distributions
Given its convenient statespace form, the hierarchical movement model in (5) and (6) can be
implemented using likelihood and Kalman methods for state estimation [22]. Using either Kalman
methods or finding the associated fullconditional distributions for zt given different portions of the
entire latent state process, we can arrive at the filter, smoother, and predictor distributions for zt .
These terms are commonly used in the time series literature (e.g., [43]) and specifically, the smoother
distribution corresponds to our best understanding of zt given all of the data, while the predictor
distribution describes the uncertainty concerning zt given only the preceding data. Though the filter
�M.B. Hooten et al. / Statistical Methodology 17 (2014) 82–98
87
distribution has great utility in time series modeling, we are only concerned with the smoother and
predictor distributions herein.
In translating the smoother and predictor concepts to the animal movement setting, the smoother
distribution pertaining to the marginal location process µt can be thought of as our understanding
of an animal’s true location at time t given all of the data, and the predictor distribution of µt
represents a prediction of where the animal might be at time t given the best knowledge of its
location at times t − 1t and earlier. To be consistent with the animal spaceuse literature, we
refer to these two distributions, smoother and predictor, as the use, [µt ]u , and availability, [µt ]a ,
distributions, respectively. Using the Kalman filtersmoother algorithm one can obtain the parameters
φt = E [µt {st , ∀t }] and t = Var[µt {st , ∀t }] for, [µt ]u = N (φt , t ) (details presented in [22]).
Thus from (5), we have [µt ]a = N (M̃t φt −1t , M̃t t −1t M̃′t + 6̃z ), where M̃t and 6̃z are the upper 2 × 2
submatrices of Mt and 6z , respectively.
Note that these two distributions give rise to a geographical vector rather than a resource vector
as in (2). Fig. 2 illustrates the use and availability distributions for a single individual at a given time
t. Due to the differences between the Kalman smoother and predictor, both the mode and diffuseness
of the distributions can vary, though more temporal information is used by the smoother and this
induces a more precise use distribution than availability distribution.
As a point of clarification, we note that the term ‘‘smoother’’ is somewhat misleading in this context
because in fact, the use distribution (i.e., smoother distribution) will be less diffuse than the availability
distribution (i.e., predictor distribution). The term ‘‘smoother’’ arises in the time series literature to
denote that the process is being updated from both sides (from behind as well as ahead in time), thus
it should be less diffuse in the space of the latent state than the predictor distribution, which is based
on forecasting only (hence inducing extra variation in the distribution). In our case, the latent state
of interest is the animal’s true location, thus the smoother provides the most precise information we
have about the use of space by the animal, while the predictor indicates where the animal is forecast
to be at some time in the future. This forecast identifies the likely spatial locations that are available
to the animal at a future time step.
2.3. Estimating the selection coefficients
Employing the use and availability (i.e., smoother and predictor) distributions with respect to
location, we can rewrite the weighted distribution from (2) as:
[µt ]u ≈
g (x(µt ), βt )[µt ]a
g (x(νt ), βt )[νt ]a dνt
,
≈ [µt βt ]u .
(7)
Aside from the use and availability distributions depending on covariates rather than locations
(e.g., [1]), note that the other inherent difference between (7) and (2), is the timespecific nature of
selection as denoted by the time index on βt . Thus, the left hand side of (7) should be approximated
with the right hand side depending on the selection coefficients βt . In conventional resource selection
approaches, the product over the weighted distribution equations (similar to the right hand side of
(7)) results in a likelihood that can be maximized with respect to β because the left hand side of (7)
is not known. Here, we actually know (or can estimate) both [µt ]u and [µt ]a based on the available
data and assumed movement model in (5) and (6). Thus, we need only find the timespecific selection
coefficients βt that minimize the discrepancy between [µt ]u and [µt βt ]u from (7). In what follows,
we approach the necessary optimization problem from an information theoretic perspective.
The Kullback–Leibler divergence from [µt βt ]u to [µt ]u can be written as:
[µt ]u
[µt ]u dµt ,
DKL (βt ) =
log
[µ β ]u
t t
[µt ]u
= Eu log
,
[µt βt ]u
= Eu (log[µt ]u ) − Eu (log[µt βt ]u ),
(8)
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M.B. Hooten et al. / Statistical Methodology 17 (2014) 82–98
Fig. 2. An example of the use (smoother) and availability (predictor) distributions for temporal prediction grain of 1 h for a
single individual. The more diffuse surface (dark gray) represents the availability distribution while the less diffuse surface
(light gray) represents the use distribution for one time point.
and, since the Kullback–Leibler divergence is nonnegative by Gibbs’ inequality [31] and finite since
µt is Gaussian in our case, a numerical algorithm can be constructed to obtain an estimator of βt based
on Kullback–Leibler loss:
β̂t = argmin{DKL (βt )},
βt
(9)
such that [µt β̂t ]u is as close as possible to [µt ]u as desired in (7). In fact, since Eu (log[µt ]u ) does not
contain βt , to minimize DKL (βt ), we only need to maximize Eu (log[µt βt ]u ). That is, we maximize the
criterion:
q(βt ) =
log[µt βt ]u [µt ]u dµt ,
(10)
with respect to βt .
The implications of this movementconstrained weighted distribution approach are that we can
characterize timespecific selection coefficients βt using the inherent relationship between the use
and availability distributions at each time of interest. Further, since we can find the use and availability
distributions at any time throughout the extent of the study, we can obtain β̂t over a quasicontinuous
set of times controlled computationally by 1t.
2.4. Infinite point process equivalence
As noted by Warton and Shepherd [50] and Aarts et al. [1], many of the species distribution models
and resource selection approaches can be equivalently thought of in a spatial point process framework.
As we show below, the same result arises here, though because of the animal movement constraint,
we have continuous knowledge of the use distribution [µt ]u in space. This continuity implies that,
at each time t, instead of a finite point process of animal locations in space, we ‘‘observe’’ an entire
spatial distribution due to the uncertainty associated with the true individual location µt . One way
to illustrate the equivalence is to show that maximizing the optimization criterion q(βt ) in (10) is
�M.B. Hooten et al. / Statistical Methodology 17 (2014) 82–98
89
equivalent to maximizing the likelihood from an infinite inhomogeneous point process. Recall, to
minimize the Kullback–Leibler divergence DKL (βt ), and thus maximize the efficiency of representing
[µt ]u with [µt βt ]u , we can maximize the criterion q(βt ) with respect to βt . Expanding q(βt ), we have,
q(βt ) =
log[µt βt ]u [µt ]u dµt ,
log
=
g (x(µt ), βt )[µt ]a
=
g (x(νt ), βt )[νt ]a dνt
[µt ]u dµt ,
log(g (x(µt ), βt )[µt ]a )[µt ]u dµt −
log
g (x(νt ), βt )[νt ]a dνt
= q1 (βt ) − q2 (βt ).
[µt ]u dµt ,
(11)
We note that q2 (βt ) can be written as
log
g (x(νt ), βt )[νt ]a dνt
[µt ]u dµt ,
which is the same as log
g (x(νt ), βt )[νt ]a dνt only, since [µt ]u dµt = 1. Thus, rather than
maximize q(βt ) directly with respect to βt , it is equivalent to maximize
Eu (log(g (x(µt ), βt ))) − log Ea (g (x(µt ), βt )),
(12)
which has a similar form as the point process likelihood described by Cressie [10, p. 651] and referred
to as the ‘‘conditional inhomogeneous Poisson point process’’ (CIPP) by Aarts et al. [1]. In fact, since
Warton and Shepherd [50] show that maximizing the likelihood of the Poisson generalized linear
model results in the same inference as the logistic regression approaches that approximate a weighted
distribution likelihood, and, Aarts et al. [1] show that inference under the CIPP model is equivalent,
then we only have to show that the fit using the weighted distribution likelihood yields the same
objective function as (12).
To show this, consider a likelihood based on (7), where we are interested in obtaining inference
for βt :
L(βt ) =
M
[µi,t β]u ,
i =1
=
M
g (x(µi,t ), βt )[µi,t ]a
,
g (x(νt ), βt )[νt ]a dνt
i =1
where the corresponding log likelihood is
l(βt ) =
M
log[µi,t β]u ,
i=1
∝
M
i=1
log(g (x(µi,t ), βt )) −
M
log
g (x(νt ), βt )[νt ]a dνt
.
i=1
Since we know the use distribution [µt ]u in this context, we can sample a large point process from
it such that µi,t ∼ [µt ]u for i = 1, . . . , M. Now, if M → ∞, then the sums in the log likelihood
approximate the required integrals in (12) and the log likelihood is linearly related to (12) and will
maximize at the same value of βt .
The utility of this equivalence between our Kullback–Leibler minimization and the infinite point
process is that we can use the approaches of Warton and Shepherd [50] to find β̂t . That is, using a
Poisson generalized linear model approach on a fine grid with a very large set of grid cell counts, we
obtain the same values for β̂t as if we numerically performed the integration in (8).
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2.5. Selection inference
The selection of resources by animals varies with temporal scale in both grain and extent [21,6].
From the grain perspective, when 1t is small and an animal moves from location µt −1t to µt it
has less opportunity to actually select resources because of the inherently correlated nature and
physics of movement. However, as 1t increases, the animal has more of an opportunity to make
decisions based on environmental cues. Thus, temporal selection grain is an important, but relatively
unaddressed aspect in most spaceuse studies. Therefore, we have the opportunity to study resource
selection across different temporal scales by varying 1t in our model. Further, the inferential scale
often pertains to the extent of the study. For example, it is common to want inference about selection
summarized over the temporal extent of the study. That is, if an animal is tracked during a breeding
season, say for a set of time T , and breeding season level inference is desired, then we may be
interested in E (β̂t ) for all t ∈ T .
As an animal moves through its environment, it will perform varying amounts of resource selection
based on its momentum (or memory) and available resources. The extent to which the animal is
influenced by our chosen environmental covariates at any given time is expressed in our estimate
for selection β̂t and the contribution of those coefficients in transforming the availability distribution
into the use distribution. After reconciling use and availability by optimizing βt , we are left with the
(t )
(t )
(t )
(t )
minimized Kullback–Leibler divergence (Dmin ) such that 0 ≤ Dmin ≤ Dau , where Dau represents the
(t )
Kullback–Leibler divergence from [µt ]a to [µt ]u . Note that when Dau is small (i.e., close to zero) the use
and availability distributions are very similar, implying that the movement model is able to reconcile
selection without any influence from covariates. Therefore, we are only able to learn about selection
(t )
(t )
due to our chosen covariates when there is a large difference between Dau and Dmin . Thus consider
(t )
(t )
the quantity ∆(t ) = Dau − Dmin ; this ∆(t ) provides us with a measure of the potential amount of
selection due to our chosen covariates after accounting for the environmentally homogeneous physics
of movement. For example, when ∆(t ) is large, it implies that the covariates were able to help reconcile
use and availability, but when ∆(t ) is close to zero, it implies that either the movement model was able
to predict selection by itself, or that our covariates were not very helpful in reconciling the use and
availability distributions.
In this case, we would want an estimator for E (β̂t ) to be more heavily influenced by β̂t when ∆(t )
is large. Thus, consider the weighted estimator β̄:
β̄ =
wt β̂t ,
(13)
t ∈T
where, wt are weights indicating which β̂t coefficients contribute the most to our knowledge
(t )
of selection during period T . To construct the weights themselves, consider the quantity e−∆ ;
(t )
(t )
after some trivial algebra, this yields the quantity e−Dau /e−Dmin which can be recognized as the
‘‘ratio of evidence’’ discussed by Burnham and Anderson [9, p. 78]. Note that this quantity can also
approximately be considered a (1) likelihood ratio [2,8] where the Kullback–Leibler divergences are
approximated using an information criterion multiplied by 1/2 and (2) a Bayes factor [26,18] where
(t )
(t )
the prior model weights are equal. In this case, since Dau ≥ Dmin , the ratio will be positive and bounded
above by 1. Furthermore, it pertains to the ratio of evidence in favor of the homogeneous CTCRW
(t )
model over the resource selection model. Thus, when e−∆ = 1 we have no evidence of selection. This
(t )
implies that the quantity 1 − e−∆ provides a Kullback–Leibler measure of the evidence for selection
at time point t and can be normalized over all times t ∈ T to create weights for temporal averaging:
1 − e−∆
wt =
(t )
1 − e−∆
(τ )
.
(14)
τ ∈T
We note that our approach directly relates to information theory and maximum entropy concepts.
When we minimize the Kullback–Leibler divergence from the availability distribution to the use
�M.B. Hooten et al. / Statistical Methodology 17 (2014) 82–98
91
distribution, we are, in effect, obtaining a model that loses a minimum amount of information in
the data as compared to the null model of environmentally homogeneous movement. Further, given
the wellknown negative relationship between Kullback–Leibler divergence and entropy, we are also
implicitly maximizing the mean entropy, or maximizing the ‘‘noise’’ associated with movement,
leaving only the information about selection [20,3]. The Kullback–Leibler differences ∆(t ) measure
the amount of information we gain about movement when covariates are used in a resource selection
framework.
Returning now to the estimator of mean selection (13), it is easily shown that β̄ yields an unbiased
estimator of E (β̂t ) under certain conditions (e.g.,
t wt = 1 for wt ≥ 0). Similarly, the unbiased
weighted covariance estimator of 6̂β̂ ≡ var(β̂t ) is
6̂β̂ =
t ∈T
wt (β̂t − β̄)(β̂t − β̄)′
2
,
1−
wt
(15)
t ∈T
where the covariance matrix of the weighted mean, β̄, can then be estimated with
6̂β̄ =
wt2 6̂β̂ .
(16)
t ∈T
For populationlevel inference, each selection estimator β̄j will be indexed by individual j for
j = 1, . . . , J (and hence 6̂β̄,j will also be indexed by j). Conventional frequentist inference can be
obtained for the population by computing summary statistics using the β̄j over all individuals. That is,
for populationlevel inference we desire the estimates β̄pop and v
ar(β̄pop ) given the individuallevel
information β̄j and 6̂β̄,j for j = 1, . . . , J. This is essentially a metaanalysis problem and there are
numerous approaches for combining the information across individuals, many of which are analytical
approximations or require further modelbased estimation methods (e.g., [16]). For the analysis that
follows, we obtain β̄pop and v
ar(β̄pop ) using a parametric bootstrap procedure (Appendix) primarily
for its relative simplicity as compared with other methods.
Many further extensions can be made to compare subpopulation selection (e.g., differences among
varying demographics) as well as temporal subsets such as varying day periods or biologically
relevant cycles (e.g., breeding vs. nonbreeding season). Further, given the propensity for some
species to exhibit periodic behavior, functional data analysis methods can be applied within this same
framework.
3. Application: mountain lion resource selection
3.1. Background
The mountain lion (Puma concolor) is a carnivore native to North America whose range has
diminished substantially since European settlement. Despite the dramatic reduction in the spatial
extent of mountain lions during the past two centuries, thriving populations currently exist in
various portions of the country. There are numerous mountain lion studies across the Western United
States, however our understanding of mountain lion ecology is nascent in this area largely due
to the difficulty and expense of studying such an elusive, wideranging, and solitary species [39].
Technological advances, such as GPS telemetry, have increased the ability of researchers to gather
valuable information on mountain lions, but such research has just begun, such as the Uncompahgre
Plateau research project. Even less information is known about mountain lion spatial ecology within
urban and exurban environments.
Other studies have documented the impacts of urban environments on mountain lion temporal and
spatial use patterns. Ordenana et al. [36] documented an overall decrease in mountain lion occurrence
associated with proximity and density of urban landscapes. Other studies have shown that dense
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M.B. Hooten et al. / Statistical Methodology 17 (2014) 82–98
housing developments can act as movement barriers to mountain lions [37] or that mountain lions will
become more nocturnal in urban areas [28]. Similarly, studies have shown selection of home ranges,
use within home ranges, general movements, and dispersal can be affected by roads, road densities
and/or traffic volumes [48,5,4,44,11,37]. Preliminary investigations suggest that mountain lions in the
frontrange of Colorado are similarly affected by urbanization as nocturnal behaviors and changes in
use relative to human density have been observed with GPS collared mountain lions during the study.
3.2. Telemetry and covariate data
As part of a larger study on Colorado FrontRange mountain lions, GPS telemetry data have been
obtained for 66 individuals at a fairly regular fix interval of approximately 3 h in an ongoing Colorado
Parks and Wildlife (CPW) monitoring effort. The observation windows for these 66 mountain lions
vary; some individuals have location histories for the past 5 years, while others have only recently
been captured and fitted with a telemetry device. We focused on the month of June, 2011 for
our analysis of populationlevel resource selection and examined telemetry data collected from 25
mountain lions in the Colorado FrontRange study during this month.
One of the main objectives of the Colorado FrontRange mountain lion study is to examine how the
population of mountain lions are selecting resources in a mosaic of urban and exurban environments.
The availability of fine scale, highly accurate, GPS data provides the potential to look at selection
patterns in great detail. For spatial covariates, we used a set of land cover types (i.e., classified
as urban, agricultural, shrub, and bare ground) as well as continuous elevation to assess resource
selection (Fig. 3); note that a sixth covariate (i.e., forest land cover type) is implicitly considered in
the analysis as a baseline category. For each individual, the telemetry devices were programmed
to provide approximately 7 fixes per day for the month of June, 2011. In our analysis, we desired
inference on a relatively small temporal scale, and thus chose a one hour temporal grain for selection.
This implies that we seek to learn about how mountain lions select resources during a one hour time
period.
3.3. Data analysis
The full estimation procedure for mountain lion selection involves a sequence of steps that are
easily summarized. First, we fit the continuoustime correlated random walk movement model
described in (5) and (6) using the telemetry data from June, 2011 (T ) for each of the 25 individuals in
our study. Second, in fitting the movement model, we obtain the use and availability distributions for
all t ∈ T and individuals; these arise naturally as the Kalman smoother and predictor distributions
pertaining to the fitted movement path. This can all be accomplished using the ‘crawl’ package [22]
in the R Statistical Computing Environment [41].
Next, we reconcile the use and availability distributions by minimizing the Kullback–Leibler
divergence (8) between them. Fortunately, this step can be accomplished using a standard generalized
linear model fitting algorithm [50]. Specifically, we sample a large spatial point process from each of
the use and availability distributions then sum the points over a fine grid and fit a Poisson regression
model with the use counts (u) as the response variable and the availability counts (a) as an offset in
the model (i.e., u ∼ Pois(aλ) where log(λ) = β0 + x′ β) for each animal at t ∈ T . The result is a
set of selection coefficients β̂j,t for each individual j at every time t. The resource selection for each
individual was then estimated with β̄j (13) and combined for populationlevel inference using the
bootstrapping approach described in the previous section.
In Fig. 4 we provide the estimated selection coefficient β̂t for the urban land type covariate and the
weight wt over the complete set of hourly selection periods in June, 2011 for a single individual (i.e.,
mountain lion # 2, an adult female). These results indicate that an individual’s resource selection likely
varies over time (as we expected) as does the amount of selection (as evidenced by wt ). The ‘‘spikes’’ in
these time series of β̂t and wt coincide with times where telemetry location data exist. This behavior
can be explained by the fact that the use and availability distributions are quite similar at points distant
in time from telemetry data (at least in this example) and the Kullback–Leibler minimization cannot
�M.B. Hooten et al. / Statistical Methodology 17 (2014) 82–98
93
Fig. 3. Maps showing the spatial covariates used in the mountain lion analysis as well as the full set of telemetry locations for
all individuals in the study.
make use of the covariate information, hence estimating the coefficients to be near zero. At points
near the telemetry data, the opposite occurs: at these times, we have a large discrepancy between use
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M.B. Hooten et al. / Statistical Methodology 17 (2014) 82–98
Fig. 4. Time series plot showing the estimated β̂t for mountain lion # 2 (adult female) and the associated weight. Large values
for wt indicate important times for inferring selection based on our chosen covariates.
and availability distributions and the RSF model has a chance to learn about the selection parameters.
An advantage of the weighted estimator (13) is that the coefficients at times when little information
is available about selection will be downweighted appropriately. We note that as the number of
observed telemetry locations in the original dataset increases, we will see a corresponding increase
in the wt spikes and this will, in turn, provide more precise estimates of β̄j with lower variance due
to the increased effective sample size.
In our analysis, the results for each individual showed similar variation in selection and divergence,
although each displayed somewhat different patterns. In this study, our analysis was focused on the
estimation of resource selection at the populationlevel, but it is important to point out that various
data mining methods could be employed to better illuminate patterns of resource selection resulting
from output similar to that shown in Fig. 4.
Table 1 summarizes the estimates for populationlevel resource selection based on all 25
individuals in our study. Note that the estimates for all land types (i.e., urban, agricultural, shrub, and
bare ground), as well as the elevation coefficient, are significantly negative. The negative populationlevel coefficients for land types imply that this population of mountain lions select most strongly for
the forest land type (not shown because it is implicit in the baseline category). In fact, in order of
selection for the categorical covariates (i.e., selection for that resource), Table 1 indicates that forest
is first, with agricultural land next, then urban land, then bare ground, with shrub last. These results
suggest that, during June, 2011, the selection for urban land type by this mountain lion population is
significantly less than the selection for forest. Similar implications hold for the other land types and
elevation as well.
4. Conclusion
We have proposed a statistical method for estimating resource selection using highresolution
animal telemetry data. Though alternative methods for assessing movementbased RSFs exist
(e.g., [24,22,14]), our approach differs in several critical features. First and foremost, by conditioning
on movement, we obtain naturally arising use and availability distributions that allow us to estimate
individualbased selection at each time of interest for a prespecified selection period. Further, our
estimation of individualbased selection using Kullback–Leibler minimization provides insight into
�M.B. Hooten et al. / Statistical Methodology 17 (2014) 82–98
95
Table 1
Populationlevel inference for selection coefficients based on monthly analysis and a 1 h temporal selection grain.
Parameter
Est.
Std. error
95% CI
β1 (urban)
β2 (agricultural)
β3 (shrub)
β4 (bare)
β5 (elevation)
−1.19
−0.22
−1.95
−1.76
−0.002
0.042
0.021
0.059
0.057
0.0004
(−1.28, −1.01)
(−0.26, −0.17)
(−2.07, −1.83)
(−1.87, −1.64)
(−0.003, −0.001)
when a given animal appears to be selecting resources (based on our chosen covariates) and can be
used to weight the selection coefficients based on how much information each is contributing to our
understanding of global selection. To our knowledge, this is the first resource selection approach
that allows the researcher to investigate the degree of selection learning possible throughout an
individual’s path.
Finally, the methods we present are intuitive, easy to implement, and highly generalizable. Many
recent models for animal movement (e.g., [19,17,33]) are perhaps useful, but, due to their complexity,
may be inaccessible to wildlife biologists and managers who need the inference for scientific inquiry
and for wildlife and resource management. Our method couples wellunderstood animal movement
models [22] with commonly used RSF approaches to provide inference about resource selection at fine
temporal scales using highresolution telemetry data. Despite the complex optimization procedure
that is required (10), we show that readily available and familiar statistical software can be used to
perform the estimation.
In terms of computational efficiency, the bulk of required calculations are ‘‘embarrassingly parallel’’
in that the individual animals can be processed separately to find the sequences of use and availability
distributions. The Kullback–Leibler optimization can also be parallelized and performed using sparse
matrix calculations reducing computing burden even further by distributing the load among several
processors. Still, in our case, without parallelizing the code, computing for each animal occurred on the
order of a few minutes. Given the massive size of the covariate grids (approximately 1.5 million pixels),
the compute times we experienced indicate that this methodology can be applied to real datasets
spanning large spatial landscapes.
It is important to point out that this method is heavily based on an underlying mechanistic
movement model. In fact, these methods are similar in spirit to a metaanalysis and exploit the fact
that the current (i.e., use) and future (i.e., availability) location distributions of the animals under
study can be estimated using the available telemetry data. If the observed telemetry locations are too
temporally distant, then a fine scale movement model like the continuoustime correlated random
walk model used herein may not provide the information needed to identify timevarying use and
availability distributions. As a general rule of thumb we would not advise using any timevarying
resource selection method unless the data include a distinct ‘‘movement’’ signal. That is, if one can
envision a realistic animal path connecting the telemetry location data, then typically these sorts
of models work well. In terms of guidance on scales for inference, if the temporal change ∆t is too
small then there will be very little difference between use and availability distributions and thus
only minimal information will be available about selection. Finally, if seasonal (i.e., periodic) animal
behavior is expected, it should be accounted for so that one does not obtain misleading inference.
For example, animals may exhibit varying amounts and types of resource selection during different
photoperiods (e.g., day or night). A simple way to deal with this might be to summarize results from
night and day periods separately. Finally, like with all statistical models, if the telemetry fix rate (i.e.,
sampling) is itself correlated with environmental variables then there is a potential to obtain biased
inference for resource selection. A potential area of future research with this type of analysis would
be to simultaneously model fix rate and resource selection.
Acknowledgment
Funding for this project was provided by Colorado Parks and Wildlife (#1201). The use of trade
names or products does not constitute endorsement by the US Government.
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M.B. Hooten et al. / Statistical Methodology 17 (2014) 82–98
Appendix. Implementation
The approach to fitting resource selection models for animal movement data described in the
METHODS section requires a sequence of steps that are outlined below. Due to the complexities
of specific datasets, we provide a mixture of software syntax in the R Statistical Computing
Environment [41] and pseudocode that demonstrate how one would actually implement the fitting
procedure on their own dataset. First we assume that the user has already obtained telemetry data
(containing a ‘‘movement signal’’ as discussed in CONCLUSION) for J individuals, gridded covariate
data over the spatial domain of interest, and has a temporal scale for selection in mind (∆t ). Then, the
following steps can be used to implement the described inference procedure:
1. Fit the CTCRW model (i.e., ‘crawl’ R package; [22]) to each set of telemetry observations from
each individual using the ‘crwMLE’ R function. Then use the ‘crwPredict’ R function to obtain
the posterior predictive distribution for the complete path of each individual at selection scale
of interest. By setting the ‘getUseAvail=TRUE’ flag in the ‘crwPredict’ function one can output the
use and availability (i.e., smoother and predictor, respectively) distributions at each time point
t in terms of the first two moments of the distributions (since they are both Gaussian). See the
help documentation for the ‘crawl’ R package for further details on syntax. Once these use and
availability moments (mean vector and covariance matrix) have been stored the remainder of
output can be discarded. Furthermore, it is important to point out that each animal is processed
individually, thus this step can be parallelized.
2. Once the use and availability distributions have been obtained using ‘crawl,’ the Kullback–Leibler
(t )
for
divergence (Dau ) between the use and availability distributions can
be approximated
each time point t by approximating the necessary integrals (i.e.,
log [µt ]u [µt ]u dµ and
log [µt ]a [µt ]u dµt ) using either numerical quadrature or Monte Carlo integration. The Monte
Carlo integration approach would proceed by sampling several location realizations µt from the
use distribution, computing the quantities log([µt ]u ) and log([µt ]a ), arithmetically average each
of these, and then subtract them.
3. Minimize the Kullback–Leibler divergence using the Poisson point process model described in
Section 2.4. One method for implementation is, for each time t, sample two very large spatial
point processes (on the order of thousands of samples from each to approximate the distribution);
one from the use distribution and the other from the availability distribution. Compute the cell
frequencies for each point process over a very fine grid spanning the convex hull of all points.
Let the ‘use’ cell frequencies be the response variable in a Poisson regression (e.g., using the ‘glm’
R function) where the ‘availability’ cell frequencies serve as the offset and the covariates are
associated with each grid cell. The estimates β̂t resulting from this fit serve as our temporally
indexed selection coefficients.
(t )
4. Given the coefficients β̂t , the minimized Kullback–Leibler divergence Dmin can be approximated
numerically as described in step 2 above.
5. Obtain the individuallevel averaged selection coefficients by first subtracting the Kullback–Leibler
(t )
(t )
divergences to get ∆(t ) = Dau − Dmin , computing the weights (14), then using the estimators (13),
(15) and (16) to get β̄j and 6̂β̄,j for all individuals j = 1, . . . , J.
6. Finally, populationlevel inference can be obtained via a metaanalysis of the individuallevel estimates. One approach for estimating the populationlevel mean and standard error
is a parametric bootstrap. It is trivial to implement such a bootstrap and should not be too
computationally intensive if the number of individuals is relatively small. Within a bootstrap loop
with K iterations, the first step is to sample a realization from each individuallevel multivariate
Gaussian distribution with mean β̄j and covariance 6̂β̄,j such that we have J temporary samples
in each iteration of the bootstrap loop. We then simply compute the sample mean and covariance
of these samples and save for each of the K bootstrap iterations. This approach uses a Gaussian
assumption for the β̄j , but given that these coefficient vectors are means, Gaussianity is reasonable.
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<span>A common population characteristic of interest in animal ecology studies pertains to the selection of resources. That is, given the resources available to animals, what do they ultimately choose to use? A variety of statistical approaches have been employed to examine this question and each has advantages and disadvantages with respect to the form of available data and the properties of estimators given model assumptions. A wealth of high resolution telemetry data are now being collected to study animal population movement and space use and these data present both challenges and opportunities for statistical inference. We summarize traditional methods for resource selection and then describe several extensions to deal with measurement uncertainty and an explicit movement process that exists in studies involving highresolution telemetry data. Our approach uses a correlated random walk movement model to obtain temporally varying use and availability distributions that are employed in a weighted distribution context to estimate selection coefficients. The temporally varying coefficients are then weighted by their contribution to selection and combined to provide inference at the population level. The result is an intuitive and accessible statistical procedure that uses readily available software and is computationally feasible for large datasets. These methods are demonstrated using data collected as part of a largescale mountain lion monitoring study in Colorado, USA.</span>
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Hooten, Mevin B.
Hanks, Ephraim M.
Johnson, Devin S.
Alldredge, Mathew W.
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English
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Statistical Methodology
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application/pdf
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17 pages
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Hooten, M. B., E. M. Hanks, D. S. Johnson, and M. W. Alldredge. 2014. Temporal variation and scale in movementbased resource selection functions. Statistical Methodology 17:82–98. <a href="https://doi.org/10.1016/j.stamet.2012.12.001" target="_blank" rel="noreferrer noopener">https://doi.org/10.1016/j.stamet.2012.12.001</a>
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Article