Brief outline

From CR to SCRCapture-RecaptureSpatial Capture-Recapture (SCR) Models

SCR Model ComponentsObservation processState process

Demonstration with Kruger Park leopards

SCR Extensions

Traditional Capture-Recapture (CR) Models⇒ Multiple sampling occasions⇒ Animals captured, marked and released⇒ Models for open vs closed populations.

Wikimedia Commons (Andreas Trepte)

Traditional Capture-Recapture (CR) Models⇒ Multiple sampling occasions⇒ Animals captured, marked and released⇒ Models for open vs closed populations.

pbsg.npolar.no (Andrew Derocher)

Traditional Capture-Recapture (CR) Models⇒ Multiple sampling occasions⇒ Animals captured, marked and released⇒ Models for open vs closed populations.

CC Derek Ramsey

Traditional Capture-Recapture (CR) Models⇒ Multiple sampling occasions⇒ Animals captured, marked and released⇒ Models for open vs closed populations.

www.realscience.org.uk

Spatial Capture-Recapture (SCR) Models

I Often an estimate of density is required.

I An estimate of the effective trapping area (ETA) is required toestimate density with CR.

I Several ad hoc methods used to estimate ETA but widelyrecognised as problematic.

I SCR models extend capture-recapture (CR) models to includespatial location information.

I SCR models solve the problem by directly estimating density.

I Can be implemented in a maximum likelihood or Bayesianframework.

Spatial Capture-Recapture (SCR) Models

I The standard approach for estimating and modelling animaldensity.

I Essentially a hierarchical CR model.I Spatial point process model that describes abundance and

distribution of animals in spaceI Observation model that deals with the detection process

Observation processI The expected frequency of encountering an individual depends

on the individual’s location in space.

0

2

4

6

8

1 1 2 2 3

4

5

6

7

8

1

2

3

4

5

Activity centre

Observation process

I The expected frequency of encountering an individual is adecreasing function of distance.

0 200 400 600 800

0.00

0.02

0.04

0.06

0.08

0.10

SCR detection functions

Distance (m)

Exp

ecte

d en

coun

ter

rate

Half normalExponentialHazard rate

Observation process

I The observation component of the model can be written in ageneral form:

P(Ω|S) =n∏

i=1

P(ωi|si)

I Different types of detectors gather different types of data

Observation processI The observation component of the model can be written in a

general form:

P(Ω|S) =n∏

i=1

P(ωi|si)

I Different types of detectors gather different types of data

David Borchers

Observation processI The observation component of the model can be written in a

general form:

P(Ω|S) =n∏

i=1

P(ωi|si)

I Different types of detectors gather different types of data

CC (Albert Herring)

Observation processI The observation component of the model can be written in a

general form:

P(Ω|S) =n∏

i=1

P(ωi|si)

I Different types of detectors gather different types of data

Eric Rexstad

Observation process

I The observation component of the model can be written in ageneral form:

P(Ω|S) =n∏

i=1

P(ωi|si)

I Different types of detectors gather different types of data

John Measey

Observation process - encounter rate model

I For a camera trap survey of duration T :

cij ∼ Poisson(λ(dij)T )

I A suitable model (assuming n caught individuals, J traps,independence of captures):

P(Ω|S) =n∏

i=1

P(ωi|si) =n∏

i=1

J∏j=1

Poisson(cij ;λ(dij)T )

Observation process - binary model

I For a hair snare survey of duration T :

δij ∼ Bernoulli(p(dij ,T ))

I Can still use the EER model:

P(cij > 0) = 1− P(cij = 0) = 1− e−λ(dijT )

I A suitable model (assuming n caught individuals, J traps,independence of captures):

P(Ω|S) =n∏

i=1

P(ωi|si) =n∏

i=1

J∏j=1

Bernoulli(δij ; p(dij ,T ))

Observation process - multiple occasionsI So far no mention of occasions, SCR uses spatial capture

histories.

I EER model (assuming n caught individuals, K occasions, Jtraps, independence of captures):

P(Ω|S) =n∏

i=1

K∏k=1

J∏j=1

Poisson(cijk ;λk(dij)T )

I Binary model (assuming n caught individuals, K occasions, Jtraps, independence of captures):

P(Ω|S) =n∏

i=1

K∏k=1

J∏j=1

Bernoulli(δijk ; pk(dij ,T )))

Observation process - multiple occasionsI So far no mention of occasions, SCR uses spatial capture

histories.I EER model (assuming n caught individuals, K occasions, J

traps, independence of captures):

P(Ω|S) =n∏

i=1

K∏k=1

J∏j=1

Poisson(cijk ;λk(dij)T )

I Binary model (assuming n caught individuals, K occasions, Jtraps, independence of captures):

P(Ω|S) =n∏

i=1

K∏k=1

J∏j=1

Bernoulli(δijk ; pk(dij ,T )))

Spatial Models

I The goal is to draw inferences about the density, abundanceand spatial distribution of the activity centres of apopulation BUT:

I don’t observe the locations of any of themI and don’t even know how many there are

I A spatial point process (SPP) model can be used. It is astatistical model that describes how the number and locationsof points in space arise.

Spatial Models

I The goal is to draw inferences about the density, abundanceand spatial distribution of the activity centres of apopulation BUT:

I don’t observe the locations of any of themI and don’t even know how many there are

I A spatial point process (SPP) model can be used. It is astatistical model that describes how the number and locationsof points in space arise.

Spatial ModelsI We assume that the points in a survey area A are generated

by a Poisson process with intensity (density) D(s) at s ∈ A.I The number of points in a region:

N ∼ P(λ) where λ =

∫AD(s) ds

I The density for locations given N:

f (s1, . . . , sN|N) =N∏i=1

f (si ) =

∏Ni=1D(si )

λ

I Combining these:

f (s1, . . . , sN) = e−λN∏i=1

D(si )

Thinned PP

I When points from a point process are detectedprobabilistically→ the detected points comprise a “thinned” point process.

I For a Poisson point process: the thinned point process is alsoa Poisson point process.

I If X ∼ P(λ(s)) then XThinned ∼ P(λ(s)p(s)).

Thinned PP

0 10 20 30 40 50 60

s

D(s)

+ + +p(s)

D(s) p(s)

Covariates

I Covariates on different levels:I TrapI IndividualI Points in space (mask)

I GLM type transformations to ensure constraints:I σ, λ→ log link.I g0 → logit link.

Summary of components needed for SCR

I Spatial locations of traps.

I Spatial capture histories (for one or more occasions).

I Region to be specified from where individuals couldconceivably be detected.

I Model for encounter rate / detection probability (as afunction of distance)

I Model for density (as a function of location s in space).

I Software for estimation.

Summary of components needed for SCR

I Spatial locations of traps.

I Spatial capture histories (for one or more occasions).

I Region to be specified from where individuals couldconceivably be detected.

I Model for encounter rate / detection probability (as afunction of distance)

I Model for density (as a function of location s in space).

I Software for estimation.

Summary of components needed for SCR

I Spatial locations of traps.

I Spatial capture histories (for one or more occasions).

I Region to be specified from where individuals couldconceivably be detected.

I Model for encounter rate / detection probability (as afunction of distance)

I Model for density (as a function of location s in space).

I Software for estimation.

Summary of components needed for SCR

I Spatial locations of traps.

I Spatial capture histories (for one or more occasions).

I Region to be specified from where individuals couldconceivably be detected.

I Model for encounter rate / detection probability (as afunction of distance)

I Model for density (as a function of location s in space).

I Software for estimation.

Summary of components needed for SCR

I Spatial locations of traps.

I Spatial capture histories (for one or more occasions).

I Region to be specified from where individuals couldconceivably be detected.

I Model for encounter rate / detection probability (as afunction of distance)

I Model for density (as a function of location s in space).

I Software for estimation.

Summary of components needed for SCR

I Spatial locations of traps.

I Spatial capture histories (for one or more occasions).

I Region to be specified from where individuals couldconceivably be detected.

I Model for encounter rate / detection probability (as afunction of distance)

I Model for density (as a function of location s in space).

I Software for estimation.

Example data

Kruger Park leopards

We are going to use (part of) theleopard data from the SANParks andAFW Kruger National Park camera-trapphotographic survey 2010-2012 a todevelop ideas.

aSouth African National Parks Board (SANParks) and the

African Wildlife Foundation (AWF);Maputla, N.W. 2014. Drivers of leopard population dynamics in the KrugerNational Park, South Africa. PhD Thesis, University of Pretoria, Pretoria, RSA.

Habitat suitability

Easting

Nor

thin

g

1.0

1.5

2.0

2.5

3.0

(Some) SCR extensions

I Ecological distance

I Continuous-time SCR

I Acoustic SCR

I Open Population SCR

ReferencesI Royle, J.A. and Young, K.V. (2008). A hierarchical model for spatial

capture-recapture data. Ecology 89 2281-2289.

I Borchers, D.L. and Efford, M.G. (2008). Spatially explicit maximumlikelihood methods for capture-recapture studies. Biometrics 64 377-385.

I Efford, M.G, Borchers, D.L. and Byrom, A.E. (2009). Density estimation byspatially explicit capture-recapture: Likelihood-based methods. In Thomson,D., Cooch, E., and Conroy, M., editors, Modeling Demographic Processes inMarked Populations, pages 255–269. Springer, New York, New York, USA.

I Efford, M.G., Dawson, D.K., and Borchers, D.L. (2009) Population densityestimated from locations of individuals on a passive detector array. Ecology,90(10):2676–2682.

I Borchers, D. (2012). A non-technical overview of spatially explicitcapture-recapture models. Journal of Ornithology, 152(2):435–444.

I Borchers, D. and Fewster, R. (2016). Spatial capture-recapture models.Statistical Science, 31(2) 219-232.

I Efford, M. (2016). secr: Spatially explicit capture-recapture models. Rpackage version 2.10.3

ReferencesI Gardner, B., Repucci, J., Lucherini, M. and Royle, J.A. (2010). Spatially

explicit inference for open populations: estimating demographic parametersfrom camera-trap studies. Ecology 91 3376-3383.

I Stevenson, B.C., Borchers, D.L., Altwegg, R., Swift, R.J., Gillespie, D.M. andMeasey, G.J. (2014) A general framework for animal density estimation fromacoustic detections across a fixed microphone array. Methods in Ecology andEvolution, 6 (1) 38-48.

I Borchers, D., Distiller, G., Foster, R., Harmsen, B. and Milazzo, L. (2014).Continuous-time spatially explicit capture-recapture models, with anapplication to a jaguar camera-trap survey. Methods in Ecology andEvolution, 5(7) 656-665.

I Sutherland, C., Fuller, A.K. and Royle, J.A. (2014) Modelling non-Euclideanmovement and landscape connectivity in highly structured ecologicalnetworks. Methods in Ecology and Evolution, 6(2) 167-177.