Wildlife Survey Models: Thinned spatial point processes ... · Spatial Capture-Recapture Distance sampling and Capture-Recapture methods the most widely used wildlife survey methods

Wildlife Survey Models:Thinned spatial point processes with unknown thinning

probabilities

David Borchers 1

Janine Illian 1 Steve Buckland 1

Finn Lindgren 2 Fabian Bachl 2 Joyce Yuan 1

1University of St Andrews

2University of Edinburgh

Borchers et al. (UStA & UE) Wildlife Survey Models 1 / 30

Abundance Estimation vs Spatial Modelling

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N = 2275

Abundance/Density estimation Spatial Modelling



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Abundance/Density estimation

Spatial Modelling



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N = 2275

Abundance/Density estimation Spatial Modelling


Abundance EstimationSampling Theory paradigm (Design-based inference)

Sampling Frame

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text


Abundance EstimationSampling Theory paradigm (Design-based inference)

Random or systematic sample

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Each cell treated as representative of all cells (no modelling).


Abundance EstimationSampling Theory paradigm inadequte if miss some in sampled cells


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More efficient to search from grid centre than comb whole cell.


Abundance EstimationSampling Theory paradigm inadequte if miss some in sampled cells


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The cost of efficientcy is that you miss some in the cell.


Abundance EstimationSampling Theory paradigm inadequte if miss some in sample

An unknown detection probability function is operating

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Distance

Det

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If detection probability is 1 at distance zero, we can use the drop innumber detected as distance increases to estimate the detection function.

But only if we know how animal density changes with distance.(Individuals’ locations are usually assumed to be iid uniform within

detectable range: this is a very simple local spatial model.)




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Distance

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But only if we know how animal density changes with distance.

(Individuals’ locations are usually assumed to be iid uniform withindetectable range: this is a very simple local spatial model.)




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Abundance EstimationDistance Sampling: Mixture of model- and design-based inference

Spatial model within cells, design-based inference between cells.

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Abundance EstimationSpatial Capture-Recapture

Distance sampling and Capture-Recapture methods the most widelyused wildlife survey methods.

Distance sampling spatial models often neglect spatial correlation andaggregate points into gird cells.

Until very recently (2004 – 2008), capture-recapture had no spatialcomponent.

Advent of Spatial Capture-Recapture (SCR) introduced (simple)spatial models to Capture-Recapture.

Can view SCR methods as Distance Sampling with (sometimes very)noisy observation of locations, and the addition of recapture data forestimation of detection probability.






























Distance Sampling with Spatial Modelling

Two kinds of approaches to modelling distribution and abundance usingdistance sampling data have been developed to date:

1 Discretize samplers:I Treat samplers as points.I Apply GAMM or Generalised Estimating Equation approaches to deal

with correlation in response (count usually) between points.

2 Thinned Poisson processI Do not discretize samplers; treat observed locations as realizations of a

thinned Poisson process.I No method for dealing with spatial correlation.



















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Search along lines (Line TransectDistance Sampling method).

Discretizing involves breaking lines intosegments and treating each segment asa sample point.


Thinned Spatial Poisson Process

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We assume that the points (locatedat x1, . . . , xN) in a survey area A aregenerated by a Poisson process withintensity (density) λ(s) at s ∈ A.

f (x1, . . . , xN) = e−ΛN∏i=1

λ(xi ).


Thinned Poisson Process: Consider one spatial dimension to illustrate thethinning process.

0 10 20 30 40 50

s

λ(s)


Thinned Poisson Process

0 10 20 30 40 50

s

λ(s)

p(s)



0 10 20 30 40 50

s

λ(s)

p(s)

λ(s)p(s)



0 10 20 30 40 50

050

100

200

s

λ(s)

λ(s)p(s)

Observations are from a thinned Poisson process with intensity

λ(s)p(s) = exp [−β0 + βzz(s) + log{p(s)

}] .



0 10 20 30 40 50

050

100

200

s

λ(s)

λ(s)p(s)

Observations are from a thinned Poisson process with intensity

λ(s)p(s) = exp [−β0 + βzz(s) + log{p(s)

}] .


Modelling the Thinned Poisson Process

First done without discretising samplers by Johnson et al. (2010)1.An example:

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6030

6040

6050

6060

660 680 700x

y

−5.0

−2.5

0.0

2.5

0.00

0.25

0.50

0.75

1.00

0.00 0.05 0.10 0.15 0.20 0.25distance

dete

ctio

n pr

obab

ility

Fully model-based inference

... but this does not allow for spatialcorrelation.

1Johnson, D. S., Laake, J. L., and Ver Hoef, J. M. (2010). A model-based approach for making ecological inference from

distance sampling data. Biometrics, 66:310aAS318.


Modelling the Thinned Poisson Process

First done without discretising samplers by Johnson et al. (2010)1.An example:

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6030

6040

6050

6060

660 680 700x

y

−5.0

−2.5

0.0

2.5

0.00

0.25

0.50

0.75

1.00

0.00 0.05 0.10 0.15 0.20 0.25distance

dete

ctio

n pr

obab

ility

Fully model-based inference ... but this does not allow for spatialcorrelation.

1Johnson, D. S., Laake, J. L., and Ver Hoef, J. M. (2010). A model-based approach for making ecological inference from

distance sampling data. Biometrics, 66:310aAS318.


A fully spatial modelEastern Tropical Pacific shipboard line transect survey

K vertices, with random variable ζk on vertex k, generating a Gauss MarkovRandom Field:

(ζ1, . . . , ζK ) ∼ N (0,Σ) .

Continuous model obtained using SPDE approach of Lindgren et al. (2011)2.

2Lindgren, F., Rue, H., and Lindstom, J. (2011). An explicit link between Gaussian fields and Gaussian Markov random

fields: the SPDE approach (with discussion). JRSS B, 73:423–498.



K vertices, with random variable ζk on vertex k, generating a Gauss MarkovRandom Field:

(ζ1, . . . , ζK ) ∼ N (0,Σ) .

Continuous model obtained using SPDE approach of Lindgren et al. (2011)2.

2Lindgren, F., Rue, H., and Lindstom, J. (2011). An explicit link between Gaussian fields and Gaussian Markov random

fields: the SPDE approach (with discussion). JRSS B, 73:423–498.



Observations are from a log-Gaussian Cox (LGCP) process with intensity

λ(s)p(s) = exp [−β0 + βzz(s) + ζ(s) + log{p(s)

}] .

and ζ(s) is a Gaussian random field.

The model is fitted using inlabru (a customised version of R-INLA):https://sites.google.com/r-inla.org/inlabru/home.



Observations are from a log-Gaussian Cox (LGCP) process with intensity

λ(s)p(s) = exp [−β0 + βzz(s) + ζ(s) + log{p(s)

}] .

and ζ(s) is a Gaussian random field.The model is fitted using inlabru (a customised version of R-INLA):https://sites.google.com/r-inla.org/inlabru/home.



Posterior median density

... and credible interval width.



Posterior median density ... and credible interval width.



The estimated Gaussian random field

... and its estimated Standard Error.



The estimated Gaussian random field ... and its estimated Standard Error.


A fully spatial model, with individual-level covariate modelETP shipboard line transect survey, with groupsize

0.00

0.25

0.50

0.75

1.00

0 2 4 6distance

haza

rd(d

ista

nce,

log(

10),

df.g

ssca

le, d

f.lsi

gma,

df.l

b)

Detection probability of small groups

0.4

0.6

0.8

1.0

0 2 4 6distance

haza

rd(d

ista

nce,

log(

1000

), d

f.gss

cale

, df.l

sigm

a, d

f.lb)

Detection probability of large groups



0.0

0.1

0.2

0.3

0.4

0 2 4 6lgrpsize

dnor

m(lg

rpsi

ze, m

ean

= g

s.m

ean,

sd

= e

xp(g

s.ls

d))

Group size

distribution estimate

−20

0

20

40

−150 −125 −100 −75lon

lat

3.0

3.5

4.0

col

lgsFALSETRUE

Group size - space mean estimates (col=log(E[size]); TRUE:

size<25; FALSE: size≥25)



0.0

0.1

0.2

0.3

0.4

0 2 4 6lgrpsize

dnor

m(lg

rpsi

ze, m

ean

= g

s.m

ean,

sd

= e

xp(g

s.ls

d))

Group size

distribution estimate

−20

0

20

40

−150 −125 −100 −75lon

lat

3.0

3.5

4.0

col

lgsFALSETRUE

Group size - space mean estimates (col=log(E[size]); TRUE:

size<25; FALSE: size≥25)


Spatial Capture-RecaptureCamera trapping leopards in Kruger Park

Thinned Poisson process (no spatial correlation).









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Explanatory variables: habitattype, distance to water (smooth,df = 3)

Point process with territorialityis challenging.



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Explanatory variables: habitattype, distance to water (smooth,df = 3)

Point process with territorialityis challenging.


Movement and connectivitySpatial Capture-Recapture example: Camera trapping snow leopardsData form Snow Leopard Trust and Snow Leopard Conservation Foundation-Mongolia, Cameratrap study by Koustubh Sharma, Purevjav Lkhagvajav and Lkhagvasumberel Tumursukh.

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Terrain ruggedness. Numbers are camera traps.


Movement and connectivitySpatial Capture-Recapture example: Camera trapping snow leopards

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Least-cost paths: Discretized space as a network.



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Some estimated least-cost paths for the snow leopards (from green to red dots).



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Habitat connectivity: Probabilities of getting to dot from anywghere in survey region.


Summary and Conclusion

Wildlife survey methods have tended not to involve fully spatialmodels,

but this is changing.

Wildlife surveys are difficult because they involve unknown, andspatially varying, thinning.

Development of methods to deal with unknown, spatially varyingthinning opens up possibilities for

I more realistic spatial modelling of wildlife distribution,I understanding what drives the distribution,I understanding what drives change in distribution,I understanding relationship between individual’s characteristics and

spatial variables (e.g. group size),I understanding what drives individuals’ movements,I understanding habitat connectivity,I and productive collaborations between statistical ecologists and spatial

statisticians.

THE END



Wildlife survey methods have tended not to involve fully spatialmodels, but this is changing.





statisticians.

THE END








statisticians.

THE END








statisticians.

THE END






I more realistic spatial modelling of wildlife distribution,

I understanding what drives the distribution,I understanding what drives change in distribution,I understanding relationship between individual’s characteristics and


statisticians.

THE END






I more realistic spatial modelling of wildlife distribution,I understanding what drives the distribution,

I understanding what drives change in distribution,I understanding relationship between individual’s characteristics and


statisticians.

THE END






I more realistic spatial modelling of wildlife distribution,I understanding what drives the distribution,I understanding what drives change in distribution,

I understanding relationship between individual’s characteristics andspatial variables (e.g. group size),

I understanding what drives individuals’ movements,I understanding habitat connectivity,I and productive collaborations between statistical ecologists and spatial

statisticians.

THE END







spatial variables (e.g. group size),

I understanding what drives individuals’ movements,I understanding habitat connectivity,I and productive collaborations between statistical ecologists and spatial

statisticians.

THE END







spatial variables (e.g. group size),I understanding what drives individuals’ movements,

I understanding habitat connectivity,I and productive collaborations between statistical ecologists and spatial

statisticians.

THE END







spatial variables (e.g. group size),I understanding what drives individuals’ movements,I understanding habitat connectivity,

I and productive collaborations between statistical ecologists and spatialstatisticians.

THE END








statisticians.

THE END








statisticians.

THE END


Documents

Wildlife Survey Models: Thinned spatial point processes ... · Spatial Capture-Recapture Distance sampling and Capture-Recapture methods the most widely used wildlife survey methods