Recent advances in spatial modelling of distance sampling surveys

David L Miller (`@millerdl`

)

Integrated Statistcs, Woods Hole

CREEM, University of St Andrews

converged.yt

Point process workshop

Seattle, Washington

2-3 July 2016

Density surface models

(Spatial models that account for detectability)

(…and more)

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

Detectability

Code for animation at https://gist.github.com/dill/2b0c120d5484d338d8ef

- “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:

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

- Horvitz-Thompson-type estimators:

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

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

- Covariates that affect detectability (Marques et al, 2007)
- Perception bias (\(g(0)<1\)) (Burt et al, 2014)
- Availability bias (Winiarski et al, 2013; Borchers et al, 2013)
- Detection function formulations (Miller and Thomas, 2015)
- Measurement error (Marques, 2004)

Figure from Marques et al (2007)

Spatially explicit data

- Area of 26,482 km
^{2}(~area of VT or MA) - Double observer surveys using Piper Super Cubs
- 1238, 35km transects, 2001-2003

Spatially explicit models

- Generalized Additive Models (GAMs)

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

- \(n_j\) has some count distribution (Horvtiz-Thompson estimate)
- \(A_j\hat{p}_j\) is area of segment \(\times\) “detectability”
- \(f_k\) 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
- GAMs: Wood (2006)

Back to those bears…

What could go wrong?

- “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()`

)

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

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

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

computes measures for fitted models

- Major criticism of \(\geq2\)-stage models
- Uncertainty from detection function AND spatial model (and…)
- Refit model with “extra” term – zero mean effect, variance contribution

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

Software

`dsm`

packageEasy to build simple models, possible to build complex ones

`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

`Distance`

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

`mrds`

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

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

Conclusions

- Existing statistical framework (GAM)
- Flexible spatial models
- Detectability
- GLMs + random effects + smooths + other extras
- accounting for uncertainty

- Two-stage models can be useful!
- Distribute tasks
- Modular model checking

- 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 (with extra content) available at

`converged.yt`

- Borchers, DL, Zucchini, W, Heide-Jørgensen, MP, Cañadas, A, Langrock, R, Buckland, ST, & Marques, TA (2013). Using hidden Markov models to deal with availability bias on line transect surveys. Biometrics, 69(3), 703–713.
- 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.
- Marques, TA (2004). Predicting and correcting bias caused by measurement error in line transect sampling using multiplicative error models. Biometrics, 60(3), 757–763.
- 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.
- 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.
- Wood, SN (2011). Fast stable restricted maximum likelihood and marginal likelihood estimation of semiparametric generalized linear models. Journal of the Royal Statistical Society: Series B, 73(1), 3–36.

Appendices

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)

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

- Miller and Thomas (2015)

- GCV tends to undersmooth (Reiss & Ogden, 2009)
- REML much better, esp. with correlated covariates

Taken from Wood (2011).

`gam.check`

)- 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`

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

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

- Dunn, PK, and GK Smyth (1996). Randomized Quantile Residuals. Journal of Computational and Graphical Statistics 5(3) 236–244.
- 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.
- Winiarski, KJ, DL Miller, PWC Paton, and SR McWilliams (2014). A Spatial Conservation Prioritization Approach for Protecting Marine Birds Given Proposed Offshore Wind Energy Development. Biological Conservation 169 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.