Recent advances in spatial modelling of distance sampling surveys

David L Miller
CREEM, University of St Andrews

Some institution or conference
The location of said institution or conference
A date when the talk happened  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.

Detectability

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

# Detection functions

• Fit to the histogram
• Model $$\mathbb{P} \left[ \text{animal detected } \vert \text{ object at distance } y\right] = g(y;\boldsymbol{\theta})$$
• 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)

# Detection functions # Distance sampling

• Extend $$\mathbb{P} \left[ \text{animal detected } \vert \text{ object at distance } y, \text{ observed covariates}\right] = g(y, \mathbf{z};\boldsymbol{\theta})$$
• Perception bias ($$g(0)<1$$)
• Availability bias
• Detection function formulations
• Measurement error Figure from Marques et al (2007), The Auk

# Mixture model detection functions Data from Bjarni Mikkelsen, Gísli Vikingsson. Marine Research Institute, Iceland.

# Mixture model detection functions • Miller and Thomas (2015)
Spatially explicit models

# Generalized additive models (in two pages) (I)

If we are modelling counts:

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

• $$n_j$$ 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

# 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

• “Classically”: quasi-Poisson (I’ve not seen data like this)
• Lately: Tweedie, negative binomial
• In mgcv we can now estimate parameters via tw() and nb() # Smoothing in awkward regions Ramsay (2002). Wood, Bravington & Hedley (2008).

# Miller and Wood (2014) • Calculate within-area distances
• Use multidimensional scaling to project (high usually)
• Use Duchon splines for smoothing
• Use GCV/REML for dimension selection

# Smoothing in less awkward regions • “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)

# Autocorrelation

• 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

# Case study - black bears in AK

• Area of 26,482 km2 (~size of VT/MA, ~1/3 size Scotland)
• Double observer surveys using Piper Super Cubs
• 1238, 35km transects, 2001-2003  # 1238 transects # Survey protocol

• 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 # Black bears

• 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       # “Bears don’t like to go too high” # “Bears like to sunbathe” Model checking

# Residual checking # Randomised quantile residuals

• 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 # Concurvity

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

• Not just correlation!
• mgcv::concurvity() computes measures for fitted models Inference

# Abundance estimate for GMU13E

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

# Abundance map # Uncertainty propagation

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

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

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 # Conclusion

• Flexible spatial models
• GLMs + random effects + smooths + other extras
• autocorrelation can be modelled
• Large, heterogeneous areas
• Spatial component is v. helpful for managers
• Two-stage models can be useful!
• Estimating temporal trends

# Distance sampling software

• 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 # The 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

# Acknowledgements

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

Funding from Alaska Department of Fish and Game # Thanks!

Talk available at
http://converged.yt/talks/meta-dsm-all/talk.html
Appendices

# Appendix - REML # Appendix - Availability

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