Recent advances in spatial modelling of distance sampling surveys

David L Miller
CREEM, University of St Andrews

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Ecological questions

How many animals are there?

Where are all the animals?

Why are they there?

Density surface models

(Spatial models that account for detectability)

(…and more)

\(\geq 2\)-stage models

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

Counts and transects

Line transects

Right: Rhode Island/Block Island coast. Left: GMU 13E Alaska.

Data setup

Ursus from PhyloPic.


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

Figure from Marques et al (2007), The Auk

Mixture model detection functions

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

Mixture model detection functions

Spatially explicit models

Generalized additive models (in two pages) (I)

If we are modelling counts:

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

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).

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

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

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

Response distributions

Smoothing in awkward regions

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

Miller and Wood (2014)

Smoothing in less awkward regions

Miller and Kelly (in prep)


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 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

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

Extra term:

\[ \left[\frac{ \partial \log p(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}} \Big\vert_{\boldsymbol{\theta} = \hat{\boldsymbol{\theta}}}\right] \boldsymbol{\gamma} \]

random effect – fix the corresponding variance matrix \(\boldsymbol{\gamma} \sim N(0,-\mathbf{H}^{-1}_\theta)\)

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


Talk available at



Appendix - REML

Appendix - Availability