Recent advances in spatial modelling of distance sampling surveys

David L Miller (@millerdl)
Integrated Statistcs, Woods Hole
CREEM, University of St Andrews

Point process workshop
Seattle, Washington
2-3 July 2016

Density surface models

(Spatial models that account for detectability)

(…and more)

\(\geq 2\)-stage models

Hedley and Buckland (2004). Miller et al (2013).


Distance sampling - line transects

Code for animation at

Detection functions

Detection functions

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

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

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

Distance sampling (extensions)

Figure from Marques et al (2007)

Spatially explicit data

Data setup

Ursus from PhyloPic.

Case study - black bears in AK

1238 transects

Spatially explicit models

Spatial model

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

Back to those bears…

“Bears don’t like to go too high”

“Bears like to sunbathe”

Abundance map

What could go wrong?

“Of course our response distribution is correct…”

Response distributions

“We selected the right covariates!”

Model selection

“We removed correlated covariates!”


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

“Variance was estimated correctly”

Uncertainty propagation

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


The dsm package

Distance sampling software




Funding from Alaska Department of Fish and Game


Slides (with extra content) available at



“Our spatial smoother fit well”

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)

“Our detection functions look great!”

Mixture model detection functions

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

Mixture model detection functions

“Our parameter estimates are fine!”

Smoothing parameter estimation by REML

Taken from Wood (2011).

“Our residuals are fine!”

Residual checking (gam.check)

Residual checking

Randomised quantile residuals


“Nope, no problems with availability”


“What spatial autocorrelation?”


More references