continuous-time spatially explicit capture-recapture models, with an application to a jaguar...

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Date of submission: 11th April 2014

Running title: Continuous-time SECR

Number of words: approx 7000

Number of tables: 5

Number of gures: 3

Number of referen es: 43

Continuous-time spatially expli it

apture-re apture models, with an appli ation

to a jaguar amera-trap survey

David Bor hers

1

Greg Distiller

2 *

Rebe a Foster

3,4

Bart Harmsen

3,4

Lorenzo Milazzo

5

1. S hool of Mathemati s and Statisti s, Centre for Resear h into E ologi al

and Environmental Modelling, The Observatory, Bu hanan Gardens, Uni-

versity of St Andrews, Fife KY16 9LZ, UK

2. Statisti s in E ology, Environment and Conservation (SEEC), Department

of Statisti al S ien es, University of Cape Town, Private Bag X3, Rondebos h

7701, South Afri a

3. Panthera, 8 West 40th Street, 18th Floor, New York, NY 10018, USA

4. Environmental Resear h Institute (ERI), University of Belize, PO Box 340,

Belmopan, Belize, Central Ameri a

5. University of Cambridge, Cambridge Infe tious Diseases (CID), Madingley

Road, Cambridge CB3 0ES UK

*Corresponding author. Greg Distiller

This article has been accepted for publication and undergone full peer review but has not been

through the copyediting, typesetting, pagination and proofreading process, which may lead to

differences between this version and the Version of Record. Please cite this article as doi:

10.1111/2041-210X.12196

This article is protected by copyright. All rights reserved.

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le Summary

1. Many apture-re apture surveys of wildlife populations operate

in ontinuous time but dete tions are typi ally aggregated into

o asions for analysis, even when exa t dete tion times are avail-

able. This dis ards information and introdu es subje tivity, in

the form of de isions about o asion denition.

2. We develop a spatio-temporal Poisson pro ess model for spatially

expli it apture-re apture (SECR) surveys that operate ontinu-

ously and re ord exa t dete tion times. We show that, ex ept in

some spe ial ases (in luding the ase in whi h dete tion prob-

ability does not hange within o asion), temporally aggregated

data do not provide su ient statisti s for density and related

parameters, and that when dete tion probability is onstant over

time our ontinuous-time (CT) model is equivalent to an exist-

ing model based on dete tion frequen ies. We use the model

to estimate jaguar density from a amera-trap survey and on-

du t a simulation study to investigate the properties of a CT

estimator and dis rete-o asion estimators with various levels of

temporal aggregation. This in ludes investigation of the ee t on

the estimators of spatio-temporal orrelation indu ed by animal

movement.

3. The CT estimator is found to be unbiased and more pre ise than

dis rete-o asion estimators based on binary apture data (rather

than dete tion frequen ies) when there is no spatio-temporal or-

relation. It is also found to be only slightly biased when there


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le is orrelation indu ed by animal movement, and to be more ro-

bust to inadequate dete tor spa ing, while dis rete-o asion es-

timators with binary data an be sensitive to o asion length,

parti ularly in the presen e of inadequate dete tor spa ing.

4. Our model in ludes as a spe ial ase a dis rete-o asion estim-

ator based on dete tion frequen ies, and at the same time lays a

foundation for the development of more sophisti ated CT mod-

els and estimators. It allows modelling within-o asion hanges

in dete tability, readily a ommodates variation in dete tor ef-

fort, removes subje tivity asso iated with user-dened o asions,

and fully utilises CT data. We identify a need for developing

CT methods that in orporate spatio-temporal dependen e in de-

te tions and see potential for CT models being ombined with

telemetry-based animal movement models to provide a ri her in-

feren e framework.

Key-words: density estimation, data aggregation, su ien y, animal

movement, statisti al methods


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le Introdu tion

Monitoring elusive spe ies like forest-dwelling arnivores that o ur at low

densities and range over wide areas is di ult, as is monitoring spe ies that

are di ult to trap or dete t visually, but many su h spe ies are of high

onservation on ern. The estimation of density and dete tability are om-

mon in population monitoring and are ru ial parameters for elusive spe ies

(Thompson, 2004; Stuart et al., 2004).

Te hnologi al advan es in non-invasive sampling methods su h as s at

or hair olle tion for geneti analysis, amera-trapping, and a ousti dete -

tion have led to new ways of apturing individuals, that are parti ularly

useful for spe ies that are di ult to survey using more onventional meth-

ods. Camera-trap arrays in parti ular have be ome an important tool for

assessment and management of spe ies that are hard to dete t by other

means. They have been used with apture-re apture (CR) methods for vari-

ous spe ies that are individually identiable from photographs, in luding

felids, anids, ursids, hyaenids, pro ynoids, tapirids and dasypodids (see ref-

eren es in Foster & Harmsen, 2012), and are in reasingly used with spatially

expli it apture-re apture (SECR) methods. SECR in orporates spatial in-

formation on apture lo ations and allows inferen es to be drawn about the

underlying pro ess governing the lo ations of individuals, as well as providing

estimates of density without the need for separate estimates of the ee tive

sampling area (Eord, 2004; Bor hers & Eord, 2008; Eord, Bor hers &

Byrom, 2009a; Royle et al., 2009).

When a CR study ( onventional or SECR) involves physi ally apturing

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le individuals, the survey pro ess generates well-dened sampling o asions.

But when there is no physi al apture and release, as in the ase of amera-

trap studies or a ousti CR surveys that re ord times of dete tion, there is no

well-dened sampling o asion. In su h ases, aggregating data into apture

o asions introdu es subje tivity (the o asion length hosen by the analyst)

and dis ards some information (the exa t times of apture). Continuous-time

(CT) models avoid these problems. The earliest CT model is probably that

of Craig (1953). It assumed equal at hability for all individuals at all times,

but has sin e been extended to allow individual heterogeneity (Boy e et al.,

2001). Most CT estimators use martingale estimating fun tions (Be ker,

1984; Yip, 1989; Be ker & Heyde, 1990; Yip, Huggins & Lin, 1995; Yip

et al., 1999; Lin & Yip, 1999; Yip & Wang, 2002; Hwang & Chao, 2002),

although Nayak (1988), Be ker & Heyde (1990) and Hwang, Chao & Yip

(2002) developed maximum likelihood estimators (MLEs) and Chao & Lee

(1993) developed a CT estimator based on sample overage. Of the CT mod-

els in the literature, only the models of Nayak (1988), Yip (1989) and Hwang,

Chao & Yip (2002) estimate dete tability and abundan e simultaneously, in

ommon with the estimator we develop below.

There is no onsensus on the utility of CT estimators. Some authors

on luded that the use of a dis rete-o asion model to analyze a CT data-

set will bias estimates (Yip & Wang, 2002; Barbour, Pon iano & Lorenzen,

2013); while others found no benet over dis rete-o asion estimators (e.g.

Chao & Lee, 1993; Wilson & Anderson, 1995). We show in Appendix C,

that Chao & Lee's (1993) on lusion that apture times are uninformative

about abundan e does not hold in general. We develop CT SECR likelihoods

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le and MLEs whi h, unlike any existing CT models, are parameterised in terms

of population density rather than abundan e, and ru ially, in orporate a

model for (unobserved) individual a tivity entres. An a tivity entre is the

entre of gravity of an individual's lo ations over the duration of the sur-

vey. It may have biologi al interpretation, e.g. as a home range entre, but

need not. Our model is obtained by formulating the SECR survey pro ess

as a nonhomogeneous spatio-temporal Poisson pro ess, in whi h events are

dete tions at points in spa e and time (see Cook & Lawless, 2007, for a om-

prehensive overview of re urrent event pro esses, of whi h Poisson pro esses

are a spe ial ase). We note that the intera tion of re urrent event pro esses

and CR methods was identied by Chao & Huggins (2005, p85) as a fruitful

area for future resear h, whi h we explore in this paper.

We fo us on the development of SECR methods for data from amera-trap

surveys, but our models are also appropriate for any CR survey with on-

tinuous sampling in whi h apture times are re orded. This in ludes a ousti

SECR surveys su h as those des ribed in Eord, Dawson & Bor hers (2009b),

Dawson & Eord (2009) or Bor hers et al. (2013), although re apture iden-

ti ation may be di ult in su h surveys. CT SECR models also provide

the basis for varying-eort SECR methods su h as those of Eord, Bor hers

& Mowat (2013) and we show how dis rete-o asion models arise from a CT

formulation.

We assume independen e between aptures, onditional on a tivity entre

lo ation. This is almost ertainly violated to some degree in amera-trap

studies be ause individuals are only dete ted when they move within range

of a amera, and an individual's lo ations are temporally orrelated. We

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le investigate sensitivity to violation of this assumption.

We develop a general CT SECR likelihood and then evaluate models un-

der the assumptions of onstant dete tion probability over time and onstant

intensity of the spatial point pro ess governing the number and lo ations of

a tivity entres. In this ase the CT likelihood is equivalent to a single-

o asion Poisson ount likelihood spanning the whole survey, in whi h the

data are the apture frequen ies of ea h individual at ea h dete tor (see Ap-

pendix C). Simulations are used to ompare the CT estimator for this ase

with dis rete-o asion estimators based on binary data within o asions and

dete tors (i.e. in whi h ea h individual's apture frequen y at ea h dete tor

within an o asion is redu ed to a binary indi ator of dete tion or not). We

ompare various levels of temporal aggregation both when the assumption of

independen e holds and when it is violated by simulating orrelated animal

movement.

Methods

Study site and amera trapping of jaguars

Jaguars (Panthera on a) are near threatened and their global population is

de lining (Caso et al., 2008), and population monitoring is di ult be ause

they o ur at low densities, range widely and are elusive, often inhabiting

thi k habitat. Over the past de ade, the hallenge of dete ting jaguars for

population estimation has been fa ilitated by amera traps, following the

work of Silver et al. (2004); however, reliable and robust density estimates of

jaguars are rarely obtained (see Foster & Harmsen, 2012; Tobler & Powell,

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le 2013).

We estimate male jaguar density in the Co ks omb Basin Wildlife San -

tuary in Belize from amera-trap data. The san tuary en ompasses 490 km

2

of se ondary tropi al moist broadleaf forest at various stages of regenera-

tion following anthropogeni and natural disturban e (for more details see

Harmsen et al., 2010b). To the west, the san tuary forms a ontiguous forest

blo k with the prote ted forests of the Maya Mountain Massif (≈ 5,000 km

2

of forest). To the east the san tuary is partially buered by unprote ted

forest beyond whi h is a mosai of pine savannah, shrub land and broadleaf

forest, inter-dispersed with villages and farms. Jaguars are found through-

out this lands ape (Foster, Harmsen & Don aster, 2010). There are 65 km

of trails, all within the eastern part of the san tuary (see Figure 1). Male

jaguars routinely walk the trail system; trail use overlaps extensively. Al-

though they frequently leave the trails to move through the forest, they are

rarely dete ted o-trail by amera traps (Harmsen et al., 2010a). Nineteen

paired amera stations (Panthera am v3) were deployed along the trail net-

work within the eastern basin and maintained for 90 days (April to July

2011). Neighbouring stations had an average spa ing of 2.0 km (1.1 to 3.1

km). The ameras used have an enfor ed 8 se ond delay between onse utive

photos and digital photographi data were downloaded every two weeks.

Continuous-time models for proximity dete tors

We refer to both apture and dete tion as dete tion, and to all dete tors,

in luding traps, as dete tors.

We model the pro ess generating dete tions as a nonhomogeneous Poisson

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le pro ess (NHPP) in spa e and time, in whi h the events are dete tions. This

model is governed by a hazard of dete tion, whi h an vary with lo ation and

time and is equal to the expe ted number of dete tions per unit time at the

lo ation and time. The model allows this hazard to hange with lo ation (to

a ommodate lower apture probability for individuals with a tivity entres

farther from dete tors, for example), time (to a ommodate dierent animal

a tivity levels at dierent times of day, for example) and other explanatory

variables, but we assume initially that the hazard for any individual at any

time is independent of the individual's apture history up to that time. We

develop a model and estimator under this assumption and then investigate

its performan e when the hazard of dete tion does depend on where and

when individuals were previously dete ted (as a onsequen e of orrelated

animal movement). Parameters and standard errors were estimated using a

Newton-type algorithm with fun tion nlm in R (R Core Team, 2013).

Continuous-time proximity dete tor likelihood

We onsider a survey in whi h we have an array of K ontinuously-sampling

proximity dete tors. The key dieren e between the likelihood presented here

and a dis rete-o asion SECR likelihood is that there are no longer o asions,

but instead a ontinuous survey running for a period (0, T ). Instead of a

apture history of length S (if there were S o asions) for the ith individual at

dete tor k, we have ωik aptures of the individual at dete tor k at times tik =

(tik1, . . . , tikωik). For brevity, we let t = tik (i = 1, . . . , n; k = 1, . . . , K)

denote the set of all dete tion times.

To simplify presentation of the likelihood, we start by onsidering a single

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le individual (the ith individual), with a tivity entre at xi, and a single de-

te tor (the kth dete tor). By onsidering tik to be a realisation of an NHPP

in time, we obtain an expression for the probability density fun tion (pdf)

of tik. This is analogous to the probability of obtaining a apture history at

dete tor k for individual i in a dis rete-o asion formulation, but it is worth

noting that both the length of tik (i.e. ωik) and the times that it ontains

are random variables, whereas for a dis rete-o asion survey the length of a

apture history is xed by the number of o asions used.

Using the pdf of tik, we obtain an expression for the probability of dete t-

ing individual i at all. Then, following Bor hers & Eord (2008), we assume

that the (unobserved) a tivity entres of dete ted individuals are realisations

of a ltered spatial NHPP, with lter at a lo ation x being the dete tion

probability at that lo ation. This gives rise to a joint probability model for

the number of individuals dete ted (n) and their dete tion times at dete tors

(t). When onsidered as a fun tion of the unknown model parameters, this

is the CT SECR likelihood fun tion.

The pdf of tik

In a CT formulation, the dis rete-o asion expression for the probability

of dete ting individual i at dete tor k on an o asion is repla ed by a de-

te tion hazard fun tion that spe ies the expe ted dete tion rate per unit

time at dete tor k of an individual, at time t, given the lo ation of its a tiv-

ity entre (xi for individual i). This hazard fun tion hk(t,xi; θ) depends on

the distan e from dete tor k to the individual's a tivity entre (xi), and we

make this expli it when we spe ify models for hk(t,xi; θ), but for notational

brevity we do not make the dependen e on distan e expli it otherwise. The

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le hazard fun tion has an unknown ve tor of dete tion fun tion parameters θ.

The survivor fun tion for individual i at dete tor k over the whole survey

(the probability of individual i not being dete ted on the dete tor by time

T ) is Sk(T,xi; θ) = exp(

−∫ T0 hk(u,xi; θ) du

)

, and hen e 1 − Sk(T,xi; θ) is

the probability of dete tion in (0, T ).

In a similar vein, the link between a CT dete tion hazard and a dis rete-

o asion dete tion probability, pks(xi; θ) for dete tor k on o asion s, of

length Ts = ts−ts−1 (running from ts−1 to ts) is: pks(xi; θ) = 1−exp(

−∫ tsts−1

hks(t,xi; θ))

.

Under the assumption of a onstant hazard, solving this for hks(t,xi; θ) leads

to a hazard with the following form:

hks(xi; θ) = − log 1− pks(xi; θ) /Ts (1)

In our analyses, we use hazard fun tions in whi h pks(xi; θ) for a spe ied

o asion length has a half-normal form: pks(xi; θ) = g0,Tsexp−dk(xi)2/(2σ2)

,

where dk(xi) is the distan e from xi to dete tor k and g0,Tsand σ are paramet-

ers to be estimated. The parameter g0,Tsis the probability that an individual

with a tivity entre lo ated at a dete tor is dete ted in a time period of length

Ts, while σ is the s ale parameter of the half-normal dete tion fun tion, and

determines the range over whi h an individual is dete table.

Assuming that, onditional on the a tivity entre lo ation, the times of

dete tions are independent, we an model the number of dete tions, ωik, and

the dete tion times tik as a nonhomogeneous Poisson pro ess with pdf as

follows (see Cook & Lawless, 2007, Theorem 2.1 on page 30, and Equation

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le (2.13) on page 32):

fk(tik|xi; θ) = Sk(T,xi; θ)ωik∏

r=1

hk(tikr,xi; θ) (2)

(We omit ωik from the LHS for brevity, be ause given tik, ωik is known.)

For those familiar with the denition of hazard fun tions in terms of the pdf

and survivor fun tion, we note that hk(tikr,xi; θ) = fk(tikr|xi; θ)/Sk((tikr −

tik(r−1)),xi; θ).

Equation (2) is a generalisation of the expression obtained by Hwang,

Chao & Yip (2002, p43) for their model Mt (given by their expression for

Li), and of the expression of Chao & Huggins (2005, p80) for the ase in

whi h hk(tikr,xi; θ) is onstant. Both sets of authors use λ for the hazard,

whereas we use h( ). In addition to lo ating these models in the Poisson

pro ess literature, our generalisation involves the in lusion of x as a latent

variable (see below), and extension to multiple dete tors.

Multiple dete tors and individuals

An individual's overall dete tion frequen y a ross the K dete tors is de-

noted ωi· =∑K

k=1 ωik and the event ωi· > 0 indi ates dete tion by some

dete tor. The ombined dete tion hazard over all dete tors at time t is

h·(t,xi; θ) =∑K

k=1 hk(t,xi; θ), and the overall probability of dete tion in

(0, T ) over all dete tors is p·(xi; θ) = p(ωi·>0|xi; θ) = 1−S·(T,xi; θ), where

S·(T,xi; θ) = exp(

−∫ T0 h·(t,xi; θ)

)

is the overall survivor fun tion.

Following Bor hers & Eord (2008), we assume that a tivity entres o ur

independently in the plane a ording to an NHPP with intensity D(x;φ) at

x, where x is any point in the survey region and φ is a ve tor of unknown

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le parameters that govern the intensity and hen e the distribution of a tivity

entres. D(x;φ) is the expe ted a tivity entre density at x. A tivity entres

are assumed not to move during the survey.

The likelihood for φ and θ is the joint distribution of the number of

animals aptured n, and the density of the out ome ωik events, at times

tik1 < . . . < tikωikr, for all i and k. This an be written in terms of the

marginal distribution of n and the onditional distribution of t given n.

L(θ,φ | n, t) = P (n | φ, θ)ft(t|n, θ,φ) (3)

If individuals are dete ted independently, n is a Poisson random variable

with parameter λ(θ,φ) =∫

A D(x;φ)p·(x; θ) dx, where p·(x; θ) is the prob-

ability of an individual with a tivity entre at x being dete ted at all and

the integral is over all points x in the area A in whi h a tivity entres o ur.

We obtain an expression for ft(t|n, θ,φ) in Appendix A and show that,

given the dete tion times of all dete ted individuals, t, the likelihood for φ

and θ is

L(φ, θ|n, t) =exp−λ(φ,θ)

n!

n∏

i=1

∫

AD(xi;φ)S·(T,xi; θ)

K∏

k=1

ωik∏

r=1

hk(tikr,xi; θ) dx

(4)

The integral is over all possible a tivity entre lo ations that ould have led

to a dete tion on the survey. (See se tion 4.2 of Bor hers & Eord, 2008, for

a brief dis ussion of this issue.)

It is sometimes useful to write this likelihood in terms of the marginal

distribution of ωik and the onditional distribution of tik given ωik. The

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le apture frequen y over the survey for individual i at dete tor k has a Pois-

son distribution with parameter Hk(xi; θ) =∫ T0 hk(t,xi; θ) dt (see Cook &

Lawless, 2007, Equation (2.15) on page 32):

Pk(ωik|xi; θ) =Hk(xi; θ)

ωik exp−Hk(xi;θ)

ωik!=

Hk(xi; θ)ωikSk(T,xi; θ)

ωik!(5)

It follows from this and Equation (2) that the onditional pdf of dete tion

times tik, given ωik, for the ith animal is

ft|ω,k(tik|ωik,xi; θ) = ωik!ωik∏

r=1

hk(tikr|xi; θ)

Hk(xi; θ)(6)

and hen e that

L(φ, θ|n, t) =e−λ(θ,φ)

n!

n∏

i=1

∫

AD(xi;φ)

K∏

k=1

Pk(ωik|xi; θ)ωik∏

r=1

ft|ω,k(tik|ωik,xi; θ) dx

(7)

Constant density, onstant hazard likelihood

One simple spe ial ase arises when density is uniform i.e. the spatial Poisson

pro ess governing a tivity entre lo ations is homogeneous, and the dete -

tion hazards do not hange with time. In this ase D(x;φ) is a onstant

(D), λ(θ,φ) = Da(θ) (where a(θ) =∫

A p·(x; θ) dx is the ee tive survey

area), hk(t,xi; θ) = hk(xi; θ) and so Hk(xi; θ) = Thk(xi; θ). As a result,

Equation (6) be omes ft|ω,k(tik|ωik,xi; θ) = ωik!∏ωik

r=1 T−1

and the likelihood

Equation (7) simplies to


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le L(D, θ|n, t) =

(

Dne−Da(θ)

n!

n∏

i=1

∫

A

K∏

k=1

Pk(ωik|xi; θ) dx

)(

n∏

i=1

K∏

k=1

ωik!ωik∏

r=1

1

T

)

(8)

The se ond term in large bra kets is the pdf of the dete tion times t, given

the dete tion frequen ies ωik (i = 1, . . . , n; k = 1, . . . , K). Sin e this term

does not ontain any model parameters, and the term in the rst large bra k-

ets ontains the parameters and the dete tion frequen ies ωik but not the

dete tion times, it follows (from the Fisher-Neyman Fa torization Theorem

for su ien y) that in this ase the dete tion frequen ies are su ient stat-

isti s for the parameters θ and D, i.e., that no information about D is lost

by dis arding the dete tion times. Chao & Lee (1993) obtained this result

for the ase with a single dete tor, no x, and a model parameterised in terms

of abundan e N rather than density D but the result does not hold in

general, as we dis uss below.

Relationship to Poisson ount models

One an see from Equation (8) that under the above assumptions, the CT

model gives rise to a Poisson ount model, sin e Pk(ωik) is a Poisson prob-

ability mass fun tion (pmf) with rate parameter Tshks(xi; θ) and the term

in the se ond large bra ket an be negle ted be ause it is onstant with re-

spe t to the parameters. This model was proposed in Appendix A of Eord,

Dawson & Bor hers (2009b), and a similar model was proposed by Royle

et al. (2009). The CT model provides an interpretation of the Poisson rate

parameter of these models as the umulative dete tion hazard, Tshks(xi; θ).

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le Both Eord, Dawson & Bor hers (2009b) and Royle et al. (2009) proposed

modelling Tshks(xi; θ) using the same fun tional forms as are used for the

probability of dete ting an individual within an o asion, but allowing an

inter ept greater than 1. This may be a reasonable strategy but model-

lers should be aware that pks(xi; θ) and hks(xi; θ) ne essarily have dierent

shapes be ause pks(xi; θ) = 1 − exp −Tshks(xi; θ). So for example, as-

suming a half-normal shape for the Poisson rate parameter (as do Royle

et al., 2009): Tshks(xi; θ) = λ0 exp −dk(xi)2/σ2 generates a shape for the

dete tion fun tion pks(xi; θ) that is not half-normal.

La k of su ien y of dete tion frequen ies

In Appendix C we show that the dis rete-o asion Bernoulli, binomial and

Poisson models are redu ed-data versions of the CT model and that the

dete tion histories or dete tion frequen ies Ω = ωiks (i = 1, . . . , n; k =

1, . . . , K; s = 1, . . . , S, where S is the number of o asions in the survey) are

not in general su ient statisti s for θ and φ, i.e. that in general dete tion

time is informative. A notable ex eption is when the dete tion hazard is

onstant within o asions. In this ase, dis arding dete tion times does not

dis ard information about population density. Redu ing ounts to a binary

event does however dis ard information and the onsequen es of this redu -

tion are explored in the rst simulation below.

Models with variable effort

CT models readily a ommodate dete tors that are operational for dierent

periods of time, by setting the dete tion hazard at a dete tor to zero while

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le the dete tor is out of operation (i.e. hk(t,x; θ) = 0 if dete tor k was not

operating at time t). This is the basis of the varying eort model for binary

dete tors and ounts in Eord, Bor hers & Mowat (2013). It is more di ult

to a ommodate dete tors when their time of failing is unknown, and we do

not address that problem here.

Simulation testing amera-trap estimators

If animals are dete ted as a onsequen e of their moving lose to a dete tor,

the expe ted dete tion rate of the animal at any given dete tor depends on

when and where the animal was last dete ted, be ause animals' lo ations

are temporally orrelated. But if times between dete tions are su iently

long that the pdf of an animal's lo ation at time t is independent of where it

was last dete ted, then we an model the pro ess generating dete tion times

using the model derived above.

In addition, to investigate the properties of the CT estimator for amera-

trap surveys when dete tions are temporally orrelated, we ondu ted sim-

ulations using a generating pro ess in whi h probability of dete tion at any

dete tor depends on when and where an animal was last dete ted. Be ause

our fo us is on the appli ation of CT estimators to jaguar amera-trap sur-

vey data, we model dependen e by onstru ting individual movement models

that are plausible for jaguar movement.

For all simulation s enarios, we also investigated the properties of a binary

dis rete-o asion proximity dete tor SECR model (in whi h ωiks is binary)

using a half-normal dete tion fun tion and a range of aggregation levels: (i)

360 o asions of length Ts = 6 hours (ii) 90 o asions of length Ts = 24 hours

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le (iii) 30 o asions of length Ts = 72 hours, and (iv) 18 o asions of length

Ts = 120 hours. In all ases we assumed onstant intensity, D, in the survey

region.

The density estimates obtained from tting a variety of models to the jag-

uar survey data (see below) were used to inform the simulation s enarios and

led to the use of D = 4 individuals per 100 km

2. The movement model (be-

low) is not parameterised in terms of the parameters g0 and σ that are used

for the independen e simulation model. In order to fa ilitate omparison of

results a ross the two kinds of simulations, the values used for these para-

meters in the independen e simulations were based on estimates obtained by

tting the CT model to data from the movement simulations. All simula-

tions involved twenty dete tors arranged on a 4x5 grid. Three dierent grid

spa ings were used: 1,500 m (62.5% of σ), 3,000 m (125% of σ), and 4,500

m (187.5% of σ).

Independen e simulations

Fun tion sim.popn from the R pa kage se r (Eord, 2013) was used to sim-

ulate populations of N individuals with a tivity entres x1, . . . ,xN , where N

is sto hasti and varied from simulation to simulation (see Fewster & Bu k-

land, 2004, for details of the simulation method). A time- onstant hazard

hk(xi; θ) (k = 1, . . . , K) was assumed for the duration of the survey (0, T )

and the frequen y with whi h individual i was dete ted by dete tor k (ωik)

was obtained as a draw from a Poisson distribution with rate parameter

T × hk(xi; θ). Dete tion times were generated by independent draws from a

uniform distribution on (0, T ).

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le Separate dete tion hazards were used to simulate data for the dierent

aggregation levels, so as to have dis rete-o asion dete tion fun tions with

half-normal forms at ea h level (to fa ilitate dis rete-o asion estimation with

the se r pa kage). The orresponding CT model estimator was also applied

to ea h simulated dataset. For an o asion of length Ts, hks(xi; θTs) is equal

to Equation (1), where pks(xi; θTs) = g0,Ts

exp−dk(xi)2/(2σ2), dk(xi) is the

distan e from xi to dete tor k, θ6 = (g0,6 = 0.01, σ = 2, 400), θ24 = (g0,24 =

0.05, σ = 2, 400), θ72 = (g0,72 = 0.15, σ = 2, 400) and θ120 = (g0,120 =

0.25, σ = 2, 400).

Movement model simulations

A tivity entres were simulated as above. Animals were moved in time steps

of one hour a ording to the movement model des ribed below. Whenever

a path rossed a 250 × 250 m grid ell ontaining a dete tor, dete tion o -

urred with probability PGC (where subs ript GC stands for Grid Cell as

these probabilities operate at the level of grid ells). The lower PGC , the

less temporally lustered are dete tions and the loser the model is to the

independen e model. This is be ause a orrelated time series be omes less

orrelated when randomly thinned.

The movement model: a mixture of random walks There is onsid-

erable literature on the modeling of animal movement using various types

of random walks (Morales et al., 2004; Codling et al., 2008; Smouse et al.,

2010; Langro k et al., 2012). Movement an onveniently be simulated by

generating a step length and a turning angle from appropriate distributions

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le at ea h time step (Morales et al., 2004). Dierent turning angle distributions

produ e dierent types of random walks. For example, for a biased random

walk (BRW) the turning angle for the next step is drawn from a distribu-

tion with mean oriented towards some fo al point whereas for a orrelated

random walk (CRW) the mean is the urrent dire tion of movement.

In order to reprodu e both the tenden y of individuals to persist in the

dire tion in whi h they are urrently moving and the tenden y not to stray

too far from their a tivity entre, we onstru ted a movement model in whi h

individuals transition sto hasti ally between two states. In the rst state

(state s = 0) they follow a CRW, and in the se ond state (state s = 1) a

BRW with bias towards an a tivity entre. The probability of being in a

parti ular state depends on the distan e the individual is from its a tivity

entre. Spe i ally, the state s of an individual at distan e d from its a tivity

entre has the following Bernoulli distribution s|d ∼ γ(d)s(1−γ(d))1−s, where

γ(d) is the probability of being in state 1, whi h is modelled as a logisti

fun tion of distan e: γ(d) = (1+expα+βd)−1. For example, the parameter

values used in the simulation (Table 1) result in a probability of being in state

0 (CRW) of 0.95 at a distan e of zero, 0.5 at a distan e of 2,500 m, and 0.05

at a distan e of 5,000 m from the a tivity entre.

The size of ea h movement step (l) is a draw from a gamma distribution

with mean step length a and standard deviation b: l ∼ Gamma(a2

b2, b

2

a). The

turning angle (φ), given that the individual is in state s, is drawn from a

Von Mises distribution φ|s ∼ von Mises(µ(s), κ(s)) where µ(s) is the mean

and κ(s) the on entration parameter. Values for κ(s) lose to zero gener-

ate something lose to a uniform distribution on the ir le, whereas larger

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le values will pla e in reasing mass around the mean value. Table 1 shows the

movement model parameter values used in the movement simulations and

Figure 2 shows a realisation of the model over T = 2, 160 hours (90 days).

Be ause the obje tive is to generate orrelated dete tion times rather than to

make inferen es about the underlying behaviour, a large standard deviation

for step length was used to ree t movement of a range of behavioural states

(resting, hunting, travelling et ).

Results

Belize Jaguar survey

Females have very dierent home range sizes to males, so that modeling

both sexes would require separate parameters for ea h sex. Be ause very few

female jaguars were dete ted, this analysis only uses male dete tions. A total

of 207 dete tions of 17 identiable males were made, of whi h 6 males (35%)

were dete ted at least twi e in a day at the same dete tor. Not all dete tors

operated for the 90 days of the survey therefore we set the dete tion hazard

for the dete tor to zero for all of the failed days in the ase of the CT model,

and removed the dete tor from o asions on whi h it was out of a tion in the

ase of the dis rete-o asion model.

In addition to models with no ovariates, models with a behavioural re-

sponse in the form of separate g0 and/or σ parameters before and after rst

dete tion were onsidered. This led to models with a behavioural response

for g0 but not σ being sele ted on the basis of AICc (see Table 2). Estimates

from a dis rete-o asion model with Ts = 24 hours are shown in Table 3,

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le together with estimates from the omparable CT model. Varian es were

estimated using the inverse of the observed Fisher information matrix and

normality was assumed to estimate onden e intervals.

Simulation Results

Independen e simulation results The estimated biases of estimators as

a per entage of true parameter values (%Bias) are shown together with their

standard errors for the three dierent dete tor spa ings in Table 4. The per-

entage bias in density for the dis rete-o asion models was parti ularly high

for losely spa ed traps, more so when many short o asions were used. The

number of individuals dete ted per survey in reased with dete tor spa ing

be ause greater spa ing leads to a larger area being sampled.

With 1,500 m dete tor spa ing 12% of individuals were dete ted at least

twi e a day at the same dete tor, on average. This redu ed to 9% and 6% with

spa ing of 3,000 m and 4,500 m, respe tively. These values are substantially

lower than the 35% observed in the survey data.

Movement model simulation results Table 5 summarises the results

from the movement simulations. The pattern of bias is similar to that from

the independent simulations though the bias is worse when PGC is high ( om-

pare results from Table 4 for 125%σ with the orresponding spa ing in Table 5

when PGC = 0.9).

Redu ing PGC in the movement simulation ee tively thins the dete tion

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le time pro ess and onsequently redu es temporal orrelation. For spa ing of

3,000 m, when PGC = 0.9, on average 85% of individuals were dete ted at

least twi e in a day at the same dete tor and this proportion drops to 52%

when PGC = 0.3. Results were similar for the larger dete tor spa ing.

Dis ussion

Belize jaguar survey results

The per entage of individuals that are dete ted at least twi e in a day at the

same amera gives a rough indi ation of spatio-temporal lustering. The dif-

feren es between the real data (35%) and the simulated movement data (52%

and 85%) suggest that the Belize jaguar data are loser to satisfying the inde-

penden e assumptions of the ontinuous-time (CT) model than are either of

the movement simulation s enarios. Although the assumption of independ-

ent a tivity entres is not realisti for territorial spe ies, no SECR model

yet exists that a ommodates the repulsive ee t that a tivity entres of

territorial spe ies exert on one another's a tivity entres, and modelling this

is hallenging. Eord, Bor hers & Byrom (2009a) found SECR estimators

of density to be robust to violation of the independent a tivity entre distri-

bution assumption in the form of lustering (overdispersion), and they may

also be robust to violation in the form of underdispersion.

Sin e no bait or lure was used, it is unlikely that the behavioural re-

sponse in g0 is a true trap-happiness ee t. The amera stations are lo ated

on trails resulting in a higher probability of dete ting the individuals that

habitually use those trails, and hen e the behavioural response parameter

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le may be a ting as a proxy for individual heterogeneity (Royle et al., 2009;

Soisalo & Caval anti, 2006).

Estimator properties

Like Eord, Dawson & Bor hers (2009b), we found that a SECR ount model

estimator with a single sampling interval (equivalent to our CT estimator

with onstant hazards) an give pre ise and unbiased density estimates. In

addition, we show that when there is inadequate spa ing between dete tors,

the estimator from a ount model with a single sampling interval performs

substantially better than binary model estimators in whi h the interval is

divided into multiple o asions.

Information loss with binary data

In general, information is lost if exa t dete tion times are dis arded, although

this is not the ase if the hazard of dete tion is onstant and ount data are

retained. There is a loss if ount data are redu ed to binary data although

Eord, Dawson & Bor hers (2009b) found the loss to be small in the s enarios

they onsidered. Similarly we found little dieren e between the perform-

an e of binary dis rete-o asion estimators and CT estimators with onstant

dete tion hazard when using dete tor spa ings of 1.25σ and 1.88σ. However,

when dete tor spa ing was redu ed to 0.625σ, the CT estimator performed

substantially better than the binary dis rete-o asion estimators.

To understand this behaviour, we investigated simulated datasets in whi h

no ount was greater than 1 in any interval. The binary and CT estimat-

ors produ ed dierent estimates in these ases, although they operate on

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le identi al data. This is be ause the binary likelihood models the probability

of at least one dete tion in an interval whereas the CT likelihood models the

probability of exa tly one dete tion, and the latter is more informative about

individual a tivity entre lo ation. For example, the fa t that an individual

was dete ted exa tly on e by the dete tor suggests that the entre was not

very lose to the dete tor (if it was, more than one dete tion would be likely),

whereas the fa t that an individual was dete ted at least on e in ludes the

possibility that it was dete ted many times, in whi h ase the entre ould

be very lose to the dete tor. Consequently, the binary data model tends to

attra t entres to dete tors, whi h an lead to negatively biased ee tive

sampling area estimates and positively biased density estimates (Figure 3).

This ee t is more pronoun ed for a tivity entres that are on the edge of

the dete tor array, whi h explains why the bias is worse for smaller dete tor

arrays. When an a tivity entre is surrounded by dete tors, the presen e of

other dete tors limits the extent to whi h the binary model an attra t the

entre to any dete tor (Figure 3).

Data aggregation

Ex ept in some spe ial ases, data aggregation involves information loss.

Aggregation an be useful when modelling unaggregated data is di ult,

but aggregation without good reason is not good pra ti e. When SECR

data are re orded in ontinuous time, we are not aware of any reason not

to use a CT model if dete tions are independent events. If the dete tion

hazard is onstant the CT model is identi al to a ount model spanning a

single o asion but the CT formulation an be extended to in orporate time-

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le varying hazards within o asion, and therefore enables extensions that are

not available without a CT formulation. When dete tion hazards depend on

observable time-varying ovariates (time of day, temperature or some other

time-varying environmental variables for example) there may be advantage

in modelling this using exa t dete tion times even if there is no inherent

interest in the nature of the varying hazard, and there is ertainly benet

in modelling the ee t of time-varying ovariates on dete tion hazard when

there is interest in how the hazard fun tion depends on su h ovariates.

When there is spatio-temporal orrelation in dete tions (e.g. due to move-

ment) and a design with adequate dete tor spa ing is used, there may be

merit in aggregating data a ross time and using binary dis rete-o asion

models with appropriate o asion lengths. However, our simulations suggest

that (a) the bias of binary dis rete-o asion density estimators an be quite

sensitive to the length of o asions, parti ularly when dete tors are too lose

together, and (b) that the CT point estimator of density is moderately robust

to inadequate dete tor spa ing and levels of spatio-temporal orrelation that

are substantially higher than is apparent in the jaguar eld data.

We have not in this paper onsidered the adequa y of interval estimators

based on the observed Fisher information matrix. If dete tions of the same

animal are not independent a ross dete tor lo ations and time (due to orrel-

ated animal movement, for example), varian e estimates may be biased and

onden e interval estimates may have in orre t overage probability. One

way of dealing with this would be to onsider the likelihood Equation (4)

to be an approximation to the orre t likelihood and use a version the ABC

bootstrap method of Efron (1987) that in orporates an animal movement

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le model (as per our movement simulation model, for example), to obtain or-

re ted point and interval estimates.

Con lusion

We have shown that CT SECR methods, unlike dis rete-o asion methods,

fully use the available data when dete tions are made ontinuously, and they

have a number of other advantages over methods that aggregate data into

dis rete o asions. We therefore re ommend that CT methods be onsidered

when dete tion times are available, and we re ommend olle ting dete tion

times whenever possible.

Although CT SECR methods are not yet as well developed as dis rete-

o asion methods (e.g. models in orporating individual ovariates or un-

observed individual-level heterogeneity remain to be developed), this is a

temporary obsta le that will soon be over ome. A more substantial obsta le

in the ase of amera-trap studies may be the development of CT models

that in orporate spatio-temporal dependen e in dete tions due to individual

movement. This seems hallenging but we see potential for CT models being

ombined with telemetry-based animal movement models to provide a ri her

inferen e framework than is urrently available.

A knowledgements

This work was part-funded by EPSRC grant EP/I000917/1. We would like to

thank the Editor and Asso iate Editor of MEE and three referees, in luding

Torbjørn Ergon, for useful suggestions that substantially improved the larity

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le of the paper.

Data A essibility

Data deposited in the Dryad repository: http://datadryad.org/resour e/doi:10.5061/dryad.mg5kv

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with random removals in ontinuous time. Biometri s, 58, 192199.

Yip, P., Zhou, Y., Lin, D. & Fang, X. (1999) Estimation of population size based on

additive hazards models for ontinuous-time re apture experimets. Biometri s,

55, 904908.


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le A Continuous-time proximity likelihood deriv-

ation

Consider a SECR survey of duration T with ontinuously-sampling prox-

imity dete tors and suppose that the ith dete ted individual is dete ted by

dete tor k at times tik = (tik1, . . . , tikωik), where ωik is the number of times

individual i is dete ted by dete tor k. Assuming that, onditional on its

a tivity entre lo ation (xi), the times of dete tions are independent, we an

model the dete tion times as a nonhomogeneous Poisson pro ess with pdf

fk(tik|xi; θ) = Sk(T,xi; θ)∏ωik

r=1 hk(tikr,xi; θ), where hk(t,xi; θ) is the non-

homogeneous event intensity or hazard at time t, θ is an unknown parameter,

and Sk(T,xi; θ) = e−∫

T

0hk(u,xi;θ) du

is the survivor fun tion for dete tor k

(the probability of individual i not being dete ted on the dete tor by time

T ).

We need to dene a few more things for onvenient expression of the likeli-

hood fun tion. Noting that ωi· =∑K

k=1 ωik is the individual's overall dete tion

frequen y and the event ωi·>0 indi ates dete tion by some dete tor, we let

ω· = ωi· (i = 1, . . . , n) and let ω·>0 indi ate dete tion by some dete tor

of individuals i = 1, . . . , n. t = tik (i = 1, . . . , n; k = 1, . . . , K) denotes the

set of all dete tion times. We also note that the ombined dete tion hazard

over all dete tors at time t is h·(t,xi; θ) =∑K

k=1 hk(t,xi; θ) and the prob-

ability of dete tion in (0, T ) is p·(xi; θ) = p(ωi·>0|xi; θ) = 1 − S·(T,xi; θ),

where S·(T,xi; θ) =∏

k Sk(T,xi; θ) = exp−∫

T

0h·(u,xi;θ)du

.

This likelihood derivation is very similar to that in Bor hers & Eord

(2008) and Equations (10) and (11) below are also derived in that paper.

31


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le The reader may nd it useful to refer to Bor hers & Eord (2008) when

reading this derivation. We assume that, onditional on a tivity entre lo a-

tions of dete ted individuals (X = (x1, . . . ,xn)), the times and lo ations of

dete tions of individuals are independent, and that individuals are dete ted

independently. Suppose also that a tivity entres are realisations of a non-

homogeneous Poisson pro ess (NHPP) in the plane, with intensity D(x;φ)

at x, and that φ is an unknown parameter ve tor. Then the likelihood for

the parameters φ of the NHPP governing a tivity entre lo ations and the

parameters θ of the dete tion pro ess, is obtained from the pdf of the number

of dete ted individuals n (P (n|φ, θ)), the pdf of the a tivity entre lo ations

given dete tion (fX(X|ω·>0;φ, θ)), and the pdf of dete tion times t, given

X and dete tion (ft(t|X,ω· > 0; θ)), after integrating over the unknown

lo ations X, as follows:

L(φ, θ|n, t) = P (n|φ, θ)ft(t|n, θ,φ)

= P (n|φ, θ)∫

AfX(X|ω·>0;φ, θ)ft(t|X,ω·>0; θ) dX (9)

where

P (n|φ, θ) =λ(θ,φ)ne−λ(θ,φ)

n!(10)

with λ(θ,φ) =∫

A D(x;φ)p·(x; θ) dx (where the integral is over all points x

in the area A in whi h a tivity entres o ur), and

fX(X|ω·>0;φ, θ) =n∏

i=1

D(xi;φ)p·(xi; θ)

λ(θ,φ)(11)


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le ft(t|X,ω·>0; θ) =n∏

i=1

∏Kk=1 fk(tik|xi; θ)

p·(xi; θ)(12)

=n∏

i=1

1

p·(xi; θ)

K∏

k=1

Sk(T,xi; θ)ωik∏

r=1

hk(tikr,xi; θ).

If we substitute Equations (11) and (12) into the integral in Equation (9),

note that

∏

k Sk(T,xi; θ) =∏

k e−∫

T

0hk(u,xi;θ) du = e−

∫

T

0h·(u,xi;θ) du = S·(T,xi; θ)

and an el p·(xi; θ) we get

ft(t|n, θ,φ) =∫

A

n∏

i=1

D(xi;φ)

λ(θ,φ)S·(T,xi; θ)

K∏

k=1

ωik∏

r=1

hk(tikr,xi; θ) dX

=n∏

i=1

∫

A

D(xi;φ)

λ(θ,φ)S·(T,xi; θ)

K∏

k=1

ωik∏

r=1

hk(tikr,xi; θ) dX.

(We an take the produ t over n outside the integral be ause the n xis are

independent.) Substituting this in Equation (9) and an elling λ(θ,φ) gives

Equation (4).

B Dis rete-o asion proximity likelihood de-

rivation

The likelihood for φ and θ for the dis rete-o asion ase in whi h there are S

o asions an be obtained in a similar way to that in whi h the ontinuous-

time (CT) likelihood was obtained, but repla ing the pdf of dete tion times,

given dete tion: ft(t|X,ω·>0; θ) with the pdf of dete tion histories, given

dete tion, PΩ(Ω|X,ω·>0; θ), where Ω = ωik (i = 1, . . . , n; k = 1, . . . , K)

and ωik = (ωik1, . . . , ωikS) is individual i's dete tion history on dete tor k

33


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le over the S o asions:

L(φ, θ|n,ωik) = P (n|φ, θ)∫

AfX(X|ω·>0;φ, θ)PΩ(Ω|X,ω·>0; θ) dX. (13)

This is the likelihood obtained by Bor hers & Eord (2008).

In 13 above PΩ(Ω|X; θ) is onditioned on being aught (ω·>0) and leads

to another term entering the likelihood p.(x, θ)−1. Two forms of PΩ(Ω|X; θ)

have been proposed for proximity dete tors: one in whi h ωiks is binary,

indi ating dete tion or not of individual i on dete tor k on o asion s, and

one in whi h ωiks is a ount of the number of times individual i was dete ted

by dete tor k on o asion s. Both have the form

PΩ(Ω|X; θ) =n∏

i=1

S∏

s=1

K∏

k=1

P (ωiks|xi; θ). (14)

The two forms of dis rete-o asion SECR model dier in the form they use

for P (ωiks|xi; θ). We show that when a CT dete tion pro ess is dis retised

into dis rete o asions this gives rise to a Bernoulli model in the ase of

binary data, and to binomial and Poisson SECR models in the ase of ount

data. (Binomial and Poisson models have been proposed for the ase in

whi h there is only one o asion (S = 1) but are easily extended to multi-

o asion s enarios, as we show below.) To do this, we divide the time interval

(0, T ) into S subintervals with interval s running from ts−1 to ts, of length

Ts = ts − ts−1, and with t0 = 0.

With a CT model, the probability of dete ting individual i at least

on e in dete tor k in this interval is pks(xi; θ) = 1 − e−Hks(xi;θ), where

34


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le Hks(xi; θ) =∫ tsts−1

hk(u,xi; θ)du. Spe ifying a model for pks(xi; θ) implies

a model for hk(t,xi; θ), although not a unique one. The mean value of the

dete tion hazard in interval s must be hks(xi; θ) = − log 1− pks(xi; θ) /Ts

and any hk(t,xi; θ) with this mean is onsistent with pks(xi; θ). When

the hazard is onstant in the interval there is a one-to-one relationship

between the dete tion hazard and the dete tion probability: hks(xi; θ) =

− log 1− pks(xi; θ) /Ts.

Bernoulli model This is obtained from a ontinuous-time model as PBern(ωiks|xi; θ) =

pks(xi; θ)ωiks 1− pks(xi; θ)

1−ωiks, with pks(xi; θ) = 1− e−Hks(xi;θ)

and bin-

ary ωiks.

Binomial ount model Eord, Dawson & Bor hers (2009b) proposed this

for the ase in whi h the S original intervals are ollapsed into S∗(< S) in-

tervals and ωiks∗ is the ount in the new interval s∗, whi h omprisesNs∗ adja-

ent original intervals. In this ase PBinom(ωiks∗|xi; θ) =(

Ns∗

ωiks∗

)

pks∗(xi; θ)ωiks∗ 1− pks∗(xi; θ)

1−ωik,

where pks∗(xi; θ) = 1− e−Hks∗(xi;θ), and Hks∗(xi; θ) =

∑Ns∗

s=1 Hks(xi; θ).

Poisson ount model A NHPP with intensity hk(t,xi; θ) at time t gives

rise to an event ount (ωiks) in a time interval (ts−1, ts) that has the Poisson

pmf PP (ωiks|xi; θ) = Hks(xi; θ)ωiks exp−Hks(xi;θ) /(ωiks!), whereHks(xi; θ) =

∫ tsts−1

hk(u,xi; θ)du and event ounts in non-overlapping intervals are inde-

pendent.


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le C La k of su ien y of Ω

To investigate the su ien y of Ω for the unknown parameters θ and φ, we

onsider the term fk(tik|xi; θ).

The onditional distribution of dete tion times, t(s)ik = t

(s)ik1, . . . , t

(s)ikωiks

, in

interval s, given that ωiks dete tions o urred in the interval is as follows:

fts(t(s)ik |ωiks; θ) = ωiks!

ωiks∏

r=1

hk(tikr,xi; θ)

Hks(xi; θ). (15)

We an now fa torise fk(tik|xi; θ) into the pmf for the ount ωiks in in-

terval s and the pdf for the dete tion times, given the ount, as follows:

fk(tik|xi; θ) =S∏

s=1

Hks(xi; θ)ωiks exp−Hks(xi;θ)

(ωiks!)

[

ωiks!ωiks∏

r=1

hk(tikr,xi; θ)

Hks(xi; θ)

]

=S∏

s=1

PP (ωiks|xi; θ)fts(t(s)ik |ωiks; θ) (16)

The likelihood Equation (4) an then be written as

L(φ, θ|n, t) =e−λ(φ,θ)

n!

n∏

i=1

∫

AD(xi;φ)

K∏

k=1

S∏

s=1

PP (ωiks|xi; θ)fts(t(s)ik |ωiks; θ) dx(17)

The likelihood for the Poisson ount model proposed by Eord, Dawson &

Bor hers (2009b) is this likelihood with a single o asion (S = 1) and without

fts(t(s)ik |ωiks; θ)). It ignores the times of dete tion and be ause fts(t

(s)ik |ωiks; θ)

involves θ, the dete tion frequen ies alone (without the times of dete tion)

are not in general su ient for θ. And be ause φ o urs in a produ t inside

the integral, we an't fa torise the likelihood into a omponent with φ and

without t(s)ik , so that (by the Fisher-Neyman Fa torization Theorem) Ω is

36


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le neither su ient for θ nor φ. Finally, be ause neither the binomial nor the

Bernoulli models involve the dete tion times, Ω is not su ient for θ and φ

with any of these models either.

There are two notable ex eptions. The rst is when hk(t,xi; θ) is on-

stant within intervals. In this ase fts(t(s)ik |ωiks; θ) = ωiks!/T

ωiks

s , whi h

does not involve θ (or the dete tion times) and so ounts with the Poisson

model are su ient for θ and φ. Note that in this ase, the multi-o asion

(S > 1) likelihood is identi al to a single-o asion likelihood with o asion

duration T =∑S

s=1 Ts (the only dieren e being the multipli ative onstant

ωiks!/Tωiks

s ). A onsequen e of this is that when dete tion hazards do not

depend on time, the notion of o asion is redundant when using a Poisson

ount model sin e the likelihood is identi al whether or not it involves

o asions.

The se ond is when density is onstant, i.e. D(x;φ) = D. In this ase D

an be fa torised out of the integral and n is onditionally su ient for φ,

given θ.


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Table 1 Movement model parameter values used in the movement simu-

lations. CRW refers to a Correlated Random Walk and BRW to a Biased

Random Walk. Only the Von Mises parameters are state dependent; 0o in-di ates movement in the same dire tion while φt represents a turning angle

from the urrent lo ation ba k towards the a tivity entre.

Parameter CRW (state s = 0) BRW (state s = 1)α 3 3

β -0.0012 -0.0012

a 750 750

b 450 450

κ(s) 6 0.5

µ(s) 0o φt


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Table 2 Model sele tion summary for the Belize jaguar data, (b) indi -

ates that a behavioural response was in luded. ∆AICc: dieren e in AICc

between the urrent and the best model (AICc is Akaike Information Cri-

terion orre ted for small sample size); wt: Akaike weight, ; npar: number

of estimated parameters.

Model ∆AICc wt npar

Dis rete-o asion models

D(.) g0(.) σ(.) 1.29 0.28 3

D(.) g0(b) σ(.) 0.00 0.53 4

D(.) g0(.) σ(b) 3.83 0.08 4

D(.) g0(b) σ(b) 3.04 0.12 5

Continuous-time models

D(.) g0(.) σ(.) 8.97 0.01 3

D(.) g0(b) σ(.) 0.00 0.81 4

D(.) g0(.) σ(b) 8.18 0.01 4

D(.) g0(b) σ(b) 3.16 0.17 5


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Table 3 Estimates from the sele ted SECR proximity model with Ts = 24hours and a ontinuous-time (CT) model tted to the male jaguar data.

Estimated density (D) is individuals per 100 km

2; g00 and g01 are estimated

dete tion fun tion inter epts before and after initial dete tion, respe tively;

σ (in km) is estimated dete tion fun tion s ale parameter. The CT model

used a hazard that is onsistent with a half normal dete tion fun tion shape

over an interval of 24 hours.

Ts D (95% CI) g00 (95% CI) g01 (95% CI) σ (95% CI)

24 3.6 (2.14 ; 6.05) 0.034 (0.016 ; 0.068) 0.067 (0.055 ; 0.094) 2.737 (2.419 ; 3.095)

CT 3.4 (2.02 ; 5.68) 0.028 (0.014 ; 0.055) 0.077 (0.061 ; 0.096) 2.942 (2.624 ; 3.299)


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le Table 4 %Bias and asso iated standard error of density and other estimat-

ors, from simulations with independent data, with dis rete-o asion models

with interval length Ts ranging from 6 to 120 hours, and the orrespond-

ing ontinuous-time (CT) model. `CT (t)' indi ates that a CT model with

dete tion hazard form that generates a half-normal dete tion shape over t

hours was used. True density was 4 individuals per 100 km

2, σ = 2, 400

m, g0 = 0.01 for Ts = 6, g0 = 0.05 for Ts = 24, g0 = 0.15 for Ts = 72,g0 = 0.25 for Ts = 120. Twenty dete tors spa ed at 1,500 m, 3,000 m and

4,500 m apart were used and the survey duration was T = 2, 160 hours. De-te tions is mean number of dete tions per survey, Unique is mean number

of individuals dete ted per survey and Reps is number of surveys that did

not result in estimation error or warning messages (su h estimates were ex-

luded).

Dete tor spa ing Ts %Bias D (se) %Bias g0 (se) %Bias σ (se) Dete tions Unique Reps

62.5% σ (1,500)

6 85.43 (2.51) -41.13 (0.69) -25.43 (0.44) 103.7 10.0 979

CT (6) 8.35 (1.31) -3.01 (0.75) -2.28 (0.46) 103.8 9.9 1000

24 69.35 (2.21) -36.94 (0.73) -21.62 (0.41) 130.6 10.6 988

CT (24) 8.69 (1.16) -4.10 (0.65) -2.49 (0.38) 132.3 10.5 998

72 36.15 (1.66) -21.17 (0.77) -13.49 (0.43) 129.6 10.4 995

CT (72) 6.64 (1.19) -3.40 (0.64) -1.92 (0.41) 134.7 10.4 999

120 21.75 (1.39) -11.99 (0.75) -8.01 (0.41) 130.0 10.5 998

CT (120) 6.42 (1.15) -3.55 (0.62) -1.27 (0.40) 139.1 10.5 1000

125% σ (3,000)

6 1.81 (0.81) -2.97 (0.53) -1.35 (0.25) 101.8 16.0 995

CT (6) -0.51 (0.79) -0.01 (0.54) 0.10 (0.25) 102.0 16.0 1000

24 0.01 (0.82) -0.31 (0.44) -0.68 (0.22) 128.1 16.7 991

CT (24) -0.95 (0.81) 0.61 (0.44) -0.14 (0.22) 129.5 16.6 1000

72 -0.84 (0.78) 0.42 (0.46) -0.05 (0.23) 127.6 16.7 981

CT (72) -1.14 (0.77) 0.17 (0.44) -0.06 (0.23) 132.4 16.6 1000

120 -0.24 (0.81) 0.23 (0.43) 0.15 (0.22) 128.0 16.8 950

CT (120) -0.86 (0.80) 0.24 (0.40) 0.06 (0.21) 136.6 16.7 1000

187.5% σ (4,500)

6 1.25 (0.67) 1.22 (0.52) -0.02 (0.23) 105.2 24.6 994

CT (6) 1.18 (0.67) 1.18 (0.51) -0.02 (0.23) 105.4 24.6 1000

24 0.89 (0.65) 0.74 (0.45) -0.09 (0.20) 130.9 25.6 982

CT (24) 0.84 (0.64) 0.84 (0.44) -0.11 (0.20) 132.8 25.6 1000

72 1.07 (0.68) -0.02 (0.45) 0.29 (0.19) 130.9 25.8 924

CT (72) 0.89 (0.65) 0.22 (0.41) 0.12 (0.18) 136.3 25.7 1000

120 1.51 (0.69) 1.29 (0.46) -0.20 (0.21) 131.6 25.9 865

CT (120) 1.15 (0.64) 1.16 (0.41) -0.25 (0.19) 141.0 25.8 1000


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le Table 5 % Bias and asso iated standard error of density estimators is

shown for two levels of PGC and for two dierent dete tor spa ings (in

meters), using half normal dete tion fun tions, from simulations with an an-

imal movement model, with o asion lengths ranging from Ts = 6 to Ts = 120hours spanning a survey duration of T = 2, 160 hours. Simulated survey

design is as per Table 4. PGC is the probability that an individual is dete ted

when it is in a 250m×250m ell ontaining a amera. Dete tions is mean

number of dete tions per survey, Unique is mean number of individuals

dete ted per survey and Reps is number of surveys that did not result in

estimation error or warning messages (su h estimates were ex luded). The

ontinuous-time (CT) model uses a hazard that is onsistent with a half

normal dete tion fun tion over an interval size of 24 hours.

Dete tor spa ing Ts %Bias D (se) Dete tions Unique Reps

3000 m

PGC = 0.9

6 35.33 (1.21) 367.17 19.81 990

24 13.76 (0.90) 332.07 19.77 997

72 3.82 (0.76) 303.64 19.78 995

120 0.82 (0.81) 283.49 19.77 865

CT (24) -4.33 (0.71) 468.17 19.77 1000

PGC = 0.3

6 8.00 (0.86) 143.20 17.24 992

24 2.48 (0.81) 138.09 17.25 998

72 1.49 (0.81) 133.16 17.25 995

120 1.50 (0.81) 129.35 17.26 990

CT (24) 0.38 (0.85) 153.63 17.28 898

4500 m

PGC = 0.9

6 5.39 (0.64) 361.73 29.15 997

24 -0.03 (0.60) 327.11 29.12 976

72 -0.12 (0.60) 299.25 29.16 987

120 -0.98 (0.62) 278.07 28.93 853

CT (24) -4.45 (0.57) 461.89 29.16 1000

PGC = 0.3

6 -0.91 (0.64) 142.02 25.56 1000

24 -0.26 (0.65) 137.00 25.65 960

72 -0.47 (0.64) 131.85 25.56 988

120 -0.47 (0.65) 128.05 25.56 969

CT (24) -1.83 (0.64) 153.64 25.56 986

42


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Fig. 1 Camera trap lo ations within Co ks omb Basin Wildlife San tuary,

Belize.


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X axis (km)

−10

−5

0

5

10

Y axis

(km

)

−10

−5

0

5

10

Density

0.000

0.005

0.010

0.015

0.020

Fig. 2 Example of simulated movement for a 90 day period in hourly steps,

showing the number of times ea h ell is visited in the period and smoothed

with a nonparametri smoother.

44


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x axis (km)

y a

xis

(km

)

−5 0 5 10

−4

−2

02

46

810

0 0 0 0 1

0 0 0 0 0

0 0 0 1 0

0 0 0 0 1

True parameters(Binary=black; CT with constant hazard=blue)

Animal #4

x axis (km)

y a

xis

(km

)

−5 0 5 10

−4

−2

02

46

810

0 0 0 0 1

0 0 0 0 0

0 0 0 1 0

0 0 0 0 1

Estimated parameters(Binary=black; CT with constant hazard=blue)

Animal #4

x axis (km)

y a

xis

(km

)

−5 0 5 10

−4

−2

02

46

810

0 0 0 0 0

0 0 0 1 2

0 0 0 1 2

0 0 0 0 1


Animal #5

x axis (km)

y a

xis

(km

)

−5 0 5 10

−4

−2

02

46

810

0 0 0 0 0

0 0 0 1 2

0 0 0 1 2

0 0 0 0 1


Animal #5

x axis (km)

y a

xis

(km

)

−5 0 5 10

−4

−2

02

46

810

0 4 2 3 2

0 1 4 7 3

1 2 5 4 3

0 0 2 1 5


Animal #8

x axis (km)

y a

xis

(km

)

−5 0 5 10

−4

−2

02

46

810

0 4 2 3 2

0 1 4 7 3

1 2 5 4 3

0 0 2 1 5


Animal #8

Fig. 3 Probability ontours for the lo ation of a tivity entres, given the

spatial apture histories for sele ted individuals from a simulated dataset,

generated from both a binary likelihood and a ontinuous-time (CT) likeli-

hood with a onstant hazard. The left panels show the ontours when the

true dete tion fun tion parameters are used for both models, whereas the

right panels do the same using the estimated dete tion fun tion paramet-

ers. Numbers show lo ations and dete tion frequen ies for ea h individual

on ea h dete tor.

45


continuous-time spatially explicit capture-recapture models, with an application to a jaguar...

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