Discussing problems vs. finding solutions:

an operational framework for dealing with

imperfect detection in species distribution modelling

July 2, 2014 | ISEC 2014| Montpellier

Péter Sólymos

Steve Matsuoka

Erin Bayne

Subhash Lele

Estimating abundance is important but tough.

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N=6 Y=2 p=0,33

N=3 Y=2 p=0,66

Estimating abundance is important but tough.

• Why important?

– Indicates magnitude of conservation issues.

– Resonates with public (they understand).

– Triggers legal processes (listing, action).

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N=6 Y=2 p=0,33

N=3 Y=2 p=0,66

Estimating abundance is important but tough.

• Why tough?

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N=6 Y=2 C=0,33

N=3 Y=2 C=0,66

E[Y]=CN

Point count survey

5 5

0–50 50–100 >100 m

0–3 3–5 5–10 perc

time interval

distance band

count

Detection #1

6 6

0–50 50–100 >100 m

0–3 3–5 5–10 perc

time interval

distance band

count

2

1

1

Detection #2

7 7

0–50 50–100 >100 m

0–3 3–5 5–10 perc

time interval

distance band

count

2 3

1 3

1 1

Detection #3

8 8

0–50 50–100 >100 m

0–3 3–5 5–10 perc

time interval

distance band

count

2 3 2

1 3 3

1 1 1

The observation process

9 9

0–50 50–100 >100 m

q

0

1

q(r=50)

q(r=100) q(r=∞)

q(r): probability of detecting an individual that sung within a circle of radius r.

0–3 3–5 5–10 perc

p

0

1

p(t=3) p(t=5)

p(t=10)

p(t): probability of an individual singing within t time interval.

Time (minutes)

Distance (m)

Parameter estimation

10 10

p

0

1

„singing” rate

Time (minutes)

q

0

1

Distance (m)

Effective Detection Radius

E[Y]=NC=(AD)(pq)

Removal sampling

Distance sampling

UdeM Seminar – March 24, 2014 11

Estimating abundance over the whole

range of a species is even tougher!

Why? Because we need a lot more data...

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Boreal Avian Modelling (BAM) Project www.borealbirds.ca

~130 K locations ~200 K sampling events

Assemble and maintain the most complete and current repository of spatially-referenced data for boreal birds and their habitats.

…lot more data = lot more protocol.

• Time and distance intervals: – more information about

the observation process.

• Surveys are not standardized: – time intervals vary,

– distance intervals vary,

– 53 protocols in the data.

• Millions of $$ worth of data from past 20 years Don’t throw it out! Use it!

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# time int.

# distance int.

surveys %

1 1

>1 >1

1 >1

1 >1

75% 1%

12% 12%

Constant singing rate

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Clay-coloured Sparrow (0.63/min)

Blue Jay (0.15/min) Cape May Warbler (39 m)

American Crow (171 m)

Variable importance

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Variable importance

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𝑝 (𝑡𝐽)

species (40%) duration (21%)

JDAY (13%) TSSR (<1%)

Variable importance

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𝑝 (𝑡𝐽) 𝑞 (𝑟𝐾)

species (40%) duration (21%)

JDAY (13%) TSSR (<1%)

species (65%) radius (23%) TREE & LCC (4%)

Variable importance

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𝑝 (𝑡𝐽) 𝑞 (𝑟𝐾)

species (40%) duration (21%)

JDAY (13%) TSSR (<1%)

species (65%) radius (23%) TREE & LCC (4%)

𝑝 (𝑡𝐽)𝑞 (𝑟𝐾)

species (66%) radius (21%)

duration (4%) JDAY (3%)

TREE & LCC (1%) TSSR (<0.01%)

Estimating density

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E[𝑌𝑖𝑗]=𝑁𝑖𝑗 𝑝 (𝑡)𝑖𝑗 𝑞 (𝑟)𝑖𝑗

E[𝑌𝑖𝑗]=𝐷𝑖𝑗 𝐴𝑖 𝑝 (𝑡)𝑖𝑗 𝑞 (𝑟)𝑖𝑗

Sampling area known:

𝐷 𝑖𝑗=𝑌𝑖𝑗/{𝐴𝑖 𝑝 (𝑡)𝑖𝑗 𝑞 (𝑟)𝑖𝑗}

Estimating density

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E[𝑌𝑖𝑗]=𝑁𝑖𝑗 𝑝 (𝑡)𝑖𝑗 𝑞 (𝑟)𝑖𝑗

E[𝑌𝑖𝑗]=𝐷𝑖𝑗 𝐴𝑖 𝑝 (𝑡)𝑖𝑗 𝑞 (𝑟)𝑖𝑗 E[𝑌𝑖𝑗]=𝐷𝑖𝑗 𝐴 𝑖 𝑝 (𝑡)𝑖𝑗 1

Sampling area known: :

Sampling area unknown:

𝐷 𝑖𝑗=𝑌𝑖𝑗/{𝐴𝑖 𝑝 (𝑡)𝑖𝑗 𝑞 (𝑟)𝑖𝑗} 𝐷 𝑖𝑗=𝑌𝑖𝑗/{𝐴 𝑖 𝑝 (𝑡)𝑖𝑗 1}

𝐴 𝑖= 𝜋𝜏 2

QPAD: offset approach

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Conditional ML estimation: 𝜑, 𝜏

y3,y5,y10

y

p,q x

Modell 𝑌𝑖 𝑁𝑖 , 𝑝𝑖 , 𝑞𝑖 ~ Poisson(𝐷𝑖𝐴𝑖𝑝𝑖𝑞𝑖)

log(𝐷𝑖) = α+β 𝑥𝑖

Parameter estimation goodness-of-fit

Prediction: - distribution map, - habitat associations, - population size, - forecasting, - pred. uncertainty.

Observation Offset Predictors

detect R package

GLM, GLMM, BRT, LASSO

Applications

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Stralberg et al. in press. Projecting boreal bird responses to climate change: the signal exceeds the noise. Ecological Applications

Conclusions

• Estimating abundance is important but tough.

• Estimating abundance over the whole range of a species is even tougher.

• But statistical ecologists are the toughest!

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Why?

• Offsets remove protocol and covariate related biases.

• Straightforward variable selection.

• Works with unlimited counts.

• Compatible with a variety of approaches.

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