seec toolbox seminars - spatial capture-recapture (scr) models · 2017-06-14 · spatial...
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Brief outline
From CR to SCRCapture-RecaptureSpatial Capture-Recapture (SCR) Models
SCR Model ComponentsObservation processState process
Demonstration with Kruger Park leopards
SCR Extensions
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Traditional Capture-Recapture (CR) Models⇒ Multiple sampling occasions⇒ Animals captured, marked and released⇒ Models for open vs closed populations.
Wikimedia Commons (Andreas Trepte)
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Traditional Capture-Recapture (CR) Models⇒ Multiple sampling occasions⇒ Animals captured, marked and released⇒ Models for open vs closed populations.
pbsg.npolar.no (Andrew Derocher)
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Traditional Capture-Recapture (CR) Models⇒ Multiple sampling occasions⇒ Animals captured, marked and released⇒ Models for open vs closed populations.
CC Derek Ramsey
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Traditional Capture-Recapture (CR) Models⇒ Multiple sampling occasions⇒ Animals captured, marked and released⇒ Models for open vs closed populations.
www.realscience.org.uk
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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.
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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
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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
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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
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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
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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
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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)
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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
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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
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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 )
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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 ))
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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 )))
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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 )))
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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.
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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.
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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 )
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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)).
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Thinned PP
0 10 20 30 40 50 60
s
D(s)
+ + +p(s)
D(s) p(s)
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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.
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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.
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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.
![Page 28: SEEC Toolbox seminars - Spatial Capture-Recapture (SCR) models · 2017-06-14 · Spatial Capture-Recapture (SCR) Models I Often an estimate of density is required. I An estimate of](https://reader030.vdocuments.us/reader030/viewer/2022040309/5f1f217f2b217060b273c932/html5/thumbnails/28.jpg)
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.
![Page 29: SEEC Toolbox seminars - Spatial Capture-Recapture (SCR) models · 2017-06-14 · Spatial Capture-Recapture (SCR) Models I Often an estimate of density is required. I An estimate of](https://reader030.vdocuments.us/reader030/viewer/2022040309/5f1f217f2b217060b273c932/html5/thumbnails/29.jpg)
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.
![Page 30: SEEC Toolbox seminars - Spatial Capture-Recapture (SCR) models · 2017-06-14 · Spatial Capture-Recapture (SCR) Models I Often an estimate of density is required. I An estimate of](https://reader030.vdocuments.us/reader030/viewer/2022040309/5f1f217f2b217060b273c932/html5/thumbnails/30.jpg)
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.
![Page 31: SEEC Toolbox seminars - Spatial Capture-Recapture (SCR) models · 2017-06-14 · Spatial Capture-Recapture (SCR) Models I Often an estimate of density is required. I An estimate of](https://reader030.vdocuments.us/reader030/viewer/2022040309/5f1f217f2b217060b273c932/html5/thumbnails/31.jpg)
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.
![Page 32: SEEC Toolbox seminars - Spatial Capture-Recapture (SCR) models · 2017-06-14 · Spatial Capture-Recapture (SCR) Models I Often an estimate of density is required. I An estimate of](https://reader030.vdocuments.us/reader030/viewer/2022040309/5f1f217f2b217060b273c932/html5/thumbnails/32.jpg)
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.
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Habitat suitability
Easting
Nor
thin
g
1.0
1.5
2.0
2.5
3.0
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(Some) SCR extensions
I Ecological distance
I Continuous-time SCR
I Acoustic SCR
I Open Population SCR
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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
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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.