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)

(Software)

Density surface models

(Spatial models that account for detectability)

(…and more)

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

Detectability

- Fit to the histogram
- Model \(\mathbb{P} \left[ \text{animal detected } \vert \text{ animal at distance } y\right] = g(y;\boldsymbol{\theta})\)
- Calculate the average probability of detection:

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

- Horvitz-Thompson-type estimators:

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

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

- Extend \(\mathbb{P} \left[ \text{animal detected } \vert \text{ animal at distance } y, \text{ observed covariates}\right] = g(y, \mathbf{z};\boldsymbol{\theta})\)
- Perception bias (\(g(0)<1\))
- Availability bias
- Detection function formulations
- Measurement error

Figure from Marques et al (2007), The Auk

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

- Miller and Thomas (2015)

Spatially explicit models

- raw counts per segment
- model offset is
*effective area*(\(A_j \hat{p}_j\))

- Horvitz-Thompson estimate per segment

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

- model offset is then area (\(A_j\))

If we are modelling counts:

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

- \(\hat{n}_j\) has some count distribution (quasi-Poisson, Tweedie, negative binomial, ziP)
- \(A_j\) is area of segment
- \(f_k\) are
*smooth*functions (splines \(\Rightarrow f_k(x)=\sum_l \beta_l b_l(x)\)) - \(f_k\) can just be fixed effects \(\Rightarrow\) GLM
- Add-in random effects, correlation structures \(\Rightarrow\) GAMM
- Wood (2006) is a good intro book

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).

- “Classically”: quasi-Poisson (I’ve not seen data like this)
- Lately: Tweedie, negative binomial
- Exponential family
*given*power parameter - (
`mgcv`

can now estimate power parameters via`tw()`

and`nb()`

)

- Can use GEE/GAMM structure for autocorrelation along transects
- \(\text{AR}(p)\) process (“obvious” structure)
- In general this is unstable
- Random effects are sparse
- Splines are “dense”
- \(\Rightarrow\) bad for optimisation

Case study:

Black bears in Alaska

Black bears in Alaska

- Area of 26,482 km
^{2}(~28% size of Portugal) - Double observer surveys using Piper Super Cubs
- 1238, 35km transects, 2001-2003

- Surveys in Spring, bears are there, but not too much foliage
- Generally search uphill
- Double observer (Burt et al, 2014)
- Curtain between pilot and observer; light system
- Go off transect and circle to ID

- Truncate at 22m and 450m, leaving 351 groups (out of ~44,000 segments)
- Group size 1-3 (lone bears, sow w. cubs)
- 1402m elevational cutoff

Model selection & checking

- All possible subsets - expensive; stepwise - path dependence
- Term selection by shrinkage to zero effect (Marra & Wood, 2011)
- Approximate \(p\)-values (Marra and Wood, 2012)

- Count data is nasty for goodness of fit
- Dunn & Smyth (1996)
- Back transform for
**exactly**Normal residuals - Fewer problems with artefacts
`dsm::rqgam.check`

- (Thanks to Natalie Kelly at CSIRO for the tip)

`rqgam.check`

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

- Not just correlation!
`mgcv::concurvity()`

computes measures for fitted models

Inference

- H-T estimate: ~1500 black bears
- DSM estimate: ~1200 black bears (968 - 1635, CV ~13%)
- Not a
*huge*difference, so why bother?

- Uncertainty from detection function AND spatial model (and…)
- Refit model with “extra” term

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

*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)

Conclusions

- Detectability
- Flexible spatial models
- GLMs + random effects + smooths + other extras
- autocorrelation can be modelled
- accounting for uncertainty

- Large, heterogeneous areas
- Spatial component is v. helpful for managers
- Two-stage models can be useful!

- Distance for Windows
- Easy to use Windows software
- Len Thomas, Eric Rexstad, Laura Marshall

`Distance`

R package- Simple way to fit detection functions
- Me!

`mrds`

R package- More complex analyses - double observer surveys
- Jeff Laake, me

`dsm`

package- Design “inspired by” (“stolen from”)
`mgcv`

- Easy to build simple models, possible to build complex ones
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

- St Andrews: Eric Rexstad, Len Thomas, Laura Marshall
- CSIRO: Mark Bravington, Natalie Kelly
- Alaska: Earl Becker, Becky Strauch, Mike Litzen, Dave Filkill

Funding from Alaska Department of Fish and Game

Slides available at

http://converged.yt/talks/dsm-talk/talk-Lisbon.html

http://converged.yt/talks/dsm-talk/talk-Lisbon.html

- Burt, ML, DL Borchers, KJ Jenkins, & TA Marques (2014). Using mark-recapture distance sampling methods on line transect surveys. Methods in Ecology and Evolution, 5(11), 1180–1191.
- Dunn, PK, and GK Smyth. Randomized Quantile Residuals. Journal of Computational and Graphical Statistics 5, no. 3 (1996): 236–244.
- Marra, G, & Wood, SN (2011). Practical variable selection for generalized additive models. Computational Statistics and Data Analysis, 55(7), 2372–2387.
- Marra, G and SN Wood (2012). Coverage properties of confidence intervals for generalized additive model components. Scandinavian Journal of Statistics 39(1), 53–74.
- Miller, DL, ML Burt, EA Rexstad and L Thomas. Spatial Models for Distance Sampling Data: Recent Developments and Future Directions. Methods in Ecology and Evolution 4, no. 11 (2013): 1001–1010.
- Miller, DL, & L Thomas (2015). Mixture models for distance sampling detection functions. PLoS ONE.
- Miller, DL, & SN Wood (2014). Finite area smoothing with generalized distance splines. Environmental and Ecological Statistics, 21(4), 715–731.
- Ramsay, T (2002) Spline smoothing over difficult regions. Journal of the Royal Statistical Society, Series B 64, 307-319.
- Williams, R, SL Hedley, TA Branch, MV Bravington, AN Zerbini, & KP Findlay (2011). Chilean Blue Whales as a Case Study to Illustrate Methods to Estimate Abundance and Evaluate Conservation Status of Rare Species. Conservation Biology, 25(3), 526–535.
- Winiarski, KJ, ML Burt, Eric Rexstad, DL Miller, CL Trocki, PWC Paton, and SR McWilliams. Integrating Aerial and Ship Surveys of Marine Birds Into a Combined Density Surface Model: a Case Study of Wintering Common Loons. The Condor 116, no. 2 (2014): 149–161.
- Winiarski, KJ, DL Miller, PWC Paton, and SR McWilliams. A Spatial Conservation Prioritization Approach for Protecting Marine Birds Given Proposed Offshore Wind Energy Development. Biological Conservation 169 (2014): 79–88.
- Wood, SN, MV Bravington, & SL Hedley (2008). Soap film smoothing. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 70(5), 931–955.

Appendices

- “Simple correction factor” for diving animals (Winiarski et al 2014)
- Borchers & co have many solutions using Hidden Markov Models

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

- Calculate within-area distances
- Use multidimensional scaling to project (high usually)
- Use Duchon splines for smoothing
- Use GCV/REML for dimension selection

- “Remove” troublesome parts of the thin plate spline
- Do this
**carefully**(Fourier transform) - Nullspace (plane) terms replaced w. low freq

Miller and Kelly (in prep)