Density surface models

David L Miller (CREEM, University of St Andrews)


Cornell University
5 September 2014






Spatial modelling

Ecological questions

Ecology \(\Rightarrow\) statistics

Density surface models

Density surface models

Compatible survey data sources

Line transects

Data setup

Ursus from PhyloPic.

Detectability

Distance sampling

Detection functions

\[ P_a = \frac{1}{w} \int_0^w g(y;\boldsymbol{\theta}) \text{d}y \]

Detection functions

Distance sampling

 

Figure from Marques et al (2007), The Auk

Spatially explicit models

Two pages generalized additive models (I)

If we are modelling counts:

\[ \mathbb{E}(n_j) = A_j\exp \left\{ \beta_0 + \sum_k f_k(z_{jk}) \right\} \]

Two pages generalized additive models (II)

Minimise distance between data and model while minimizing:

\[ \lambda_k \int_\Omega \frac{\partial^2 f_k(z_k)}{\partial z_k^2} \text{ d}z_k \]

“just wiggly enough”

Two options for response

\(n_j\) - raw counts per segment

\[ \mathbb{E}(n_j) = A_j \hat{p}_j \exp \left\{ \beta_0 + \sum_k f_k(z_{jk}) \right\} \]

 

\(\hat{n}_j\) - H-T estimate per segment

\[ \hat{n}_j = \sum_{i \text{ in segment } j} \frac{s_i}{\hat{p}_i} \]

\[ \mathbb{E}(\hat{n}_j) = A_j \exp \left\{ \beta_0 + \sum_k f_k(z_{jk}) \right\} \]

Case study I - Seabirds in RI waters

Case study I - Seabirds in RI waters

RI seabirds - Aims

Photo by jackanapes on flickr (CC BY-NC-ND)

RI seabirds - Detection function modelling

RI seabirds - Spatial covariates (I)

RI seabirds - Spatial covariates (II)

RI seabirds - spatial model

RI seabirds - raw data

RI seabirds - Covariate effects

RI seabirds - Results

RI seabirds - Uncertainty

Case study II - black bears in Alaska

Case study II - black bears in AK

1238 transects

Survey protocol

Black bears

“Bears don’t like to go too high”

“Bears like to sunbathe”

Abundance estimate for GMU13E

Abundance map

CV map

Conclusions

Distance sampling software

The dsm package

Acknowledgements

Thanks!

Talk available at
http://converged.yt/talks/cornell-dsm/talk.html

References

Randomised quantile residuals

gam.check

rqgam.check