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
Hedley and Buckland (2004). Miller et al (2014).
\[ \hat{p}_i = \frac{1}{w} \int_0^w g(y; \mathbf{z}_i, \boldsymbol{\hat{\theta}}) \text{d}y \]
\[ \hat{N} = \sum_{i=1}^n \frac{s_i}{\hat{p}_i} \]
(where \(s_i\) are group/cluster sizes)
Figure from Marques et al (2007), The Auk
Data from Daniel Pike, Bjarni Mikkelsen and Gísli Vikingsson. Marine Research Institute, Iceland.
\[ \hat{n}_j = \sum_{i \text{ in segment } j} \frac{s_i}{\hat{p}_i} \]
If we are modelling counts:
\[ \mathbb{E}(\hat{n}_j) = A_j\exp \left\{ \beta_0 + \sum_k f_k(z_{jk}) \right\} \]
Minimise distance between data and model while minimizing:
\[ \lambda_k \int_\Omega \frac{\partial^2 f_k(z_k)}{\partial z_k^2} \text{ d}z_k \]
Fitting via REML, see Wood (2011).
mgcv
can now estimate power parameters via tw()
and nb()
)dsm::rqgam.check
rqgam.check
\[ \text{Altitude} = f(x,y) + \epsilon \quad \text{or} \quad \text{Chlorophyll A} = f(\text{SST}) + \epsilon \]
mgcv::concurvity()
computes measures for fitted models\[ \log\left[ \mathbb{E}(n_j) \right] = \log\left[A_j p_j(\boldsymbol{\hat{\theta}})\right] + \color{red}{\left[\frac{ \partial \log p(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}} \Big\vert_{\boldsymbol{\theta} = \hat{\boldsymbol{\theta}}}\right] \boldsymbol{\gamma}} + \beta_0 + \sum_k f_k(z_{jk}) \]
Williams et al (2011). Bravington, Hedley and Miller (in prep)
Distance
R package
mrds
R package
dsm
packagemgcv
Syntax example:
model <- dsm(count ~ s(x,k=10) + s(depth,k=6),
detection_function,
segment_data,
observation_data,
family=tw())
Utility functions: variance estimation, plotting, prediction etc
Funding from Alaska Department of Fish and Game
Ramsay (2002). Wood, Bravington & Hedley (2008).
Miller and Kelly (in prep)