SEEC Toolbox seminars - Spatial Capture-Recapture (SCR) models · 2017-06-14 · Spatial...

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Transcript of SEEC Toolbox seminars - Spatial Capture-Recapture (SCR) models · 2017-06-14 · Spatial...

Page 1: 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
Page 2: 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

Brief outline

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

SCR Model ComponentsObservation processState process

Demonstration with Kruger Park leopards

SCR Extensions

Page 3: 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

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

Wikimedia Commons (Andreas Trepte)

Page 4: 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

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

pbsg.npolar.no (Andrew Derocher)

Page 5: 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

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

CC Derek Ramsey

Page 6: 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

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

www.realscience.org.uk

Page 7: 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

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.

Page 8: 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

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

Page 9: 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

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

Page 10: 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

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

Page 11: 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

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

Page 12: 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

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

Page 13: 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

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)

Page 14: 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

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

Page 15: 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

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

Page 16: 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

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 )

Page 17: 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

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 ))

Page 18: 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

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 )))

Page 19: 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

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 )))

Page 20: 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

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.

Page 21: 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

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.

Page 22: 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

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 )

Page 23: 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

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)).

Page 24: 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

Thinned PP

0 10 20 30 40 50 60

s

D(s)

+ + +p(s)

D(s) p(s)

Page 25: 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

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.

Page 26: 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

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 27: 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

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

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

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

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

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

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.

Page 33: 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

Habitat suitability

Easting

Nor

thin

g

1.0

1.5

2.0

2.5

3.0

Page 34: 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

(Some) SCR extensions

I Ecological distance

I Continuous-time SCR

I Acoustic SCR

I Open Population SCR

Page 35: 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

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

Page 36: 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

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.