Recent advances in spatial modelling of distance sampling surveys

David L Miller
CREEM, University of St Andrews

Universidade de Lisboa
Lisbon, Portugal
11 June 2015

Ecological questions

How many animals are there?

Where are all the animals?

Why are they there?

Practical question

How can we do this?

(Statistical methods)


Density surface models

(Spatial models that account for detectability)

(…and more)

\(\geq 2\)-stage models

Hedley and Buckland (2004). Miller et al (2014).


Distance sampling

Code for animation at

Detection functions

\[ \hat{p}_i = \frac{1}{w} \int_0^w g(y; \mathbf{z}_i, \boldsymbol{\hat{\theta}}) \text{d}y \]

\[ \hat{N} = \sum_{i=1}^n \frac{s_i}{\hat{p}_i} \]

(where \(s_i\) are group/cluster sizes)

Detection functions

Distance sampling (extensions)

Figure from Marques et al (2007), The Auk

Mixture model detection functions

Data from Daniel Pike, Bjarni Mikkelsen and Gísli Vikingsson. Marine Research Institute, Iceland.

Mixture model detection functions

Spatially explicit models

Data setup

Ursus from PhyloPic.

Two options for response




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

Generalized additive models (in two pages) (I)

If we are modelling counts:

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

Generalized additive models (in two pages) (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”

Fitting via REML, see Wood (2011).

Response distributions


Case study:
Black bears in Alaska

Case study - black bears in AK

1238 transects

Survey protocol

Black bears

“Bears don’t like to go too high”

“Bears like to sunbathe”

Model selection & checking

Model selection

Residual checking

Residual checking

Randomised quantile residuals



\[ \text{Altitude} = f(x,y) + \epsilon \quad \text{or} \quad \text{Chlorophyll A} = f(\text{SST}) + \epsilon \]


Abundance estimate for GMU13E

Abundance map

Uncertainty propagation

\[ \log\left[ \mathbb{E}(n_j) \right] = \log\left[A_j p_j(\boldsymbol{\hat{\theta}})\right] + \color{red}{\left[\frac{ \partial \log p(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}} \Big\vert_{\boldsymbol{\theta} = \hat{\boldsymbol{\theta}}}\right] \boldsymbol{\gamma}} + \beta_0 + \sum_k f_k(z_{jk}) \]

Williams et al (2011). Bravington, Hedley and Miller (in prep)

CV map



Distance sampling software

The dsm package


Funding from Alaska Department of Fish and Game


Slides available at



Appendix - REML

Appendix - Availability

Appendix - Smoothing in awkward regions

Ramsay (2002). Wood, Bravington & Hedley (2008).

Appendix - Miller and Wood (2014)

Appendix - Smoothing in less awkward regions

Miller and Kelly (in prep)