spatially explicit models of black bears in Alaska

David L Miller (CREEM)

CREEM Seminar, 18 December 2013

- 5
^{th}largest dam in the world - On the Susitna river in South-Central Alaska
- AK Fish & Game contracted to provide black and brown bear numbers
- Earl Becker & DLM to investigate data from 2001-2003

- Need reliable and
*spatially explicit*estimates - Use existing data (no $ for new surveys)
- Biological interpretability
- HUGE area
- Understandable to managers

- Game Management Unit 13E (plus a bit extra)
- Area of 26,482 km
^{2} - ~1/3 size of Scotland
- ~ size of Vermont/Massachusetts

- 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
- Curtain between pilot and observer; light system
- Go off transect and circle to ID

- Segment transects (GIS, 43,838)
- Allocate counts to segments
- Correct for detectibility (and \(g(0)\neq1\))
- Fit a GAM

- Segment transects (GIS, 43,838)
- Allocate counts to segments
- Correct for detectibility (and \(g(0)\neq1\))
- Fit a GAM

- Segment transects (GIS, 43,838)
- Allocate counts to segments
- Correct for detectibility (and \(g(0)\neq1\))
- Fit a GAM

- Segment transects (GIS, 43,838)
- Allocate counts to segments
- Correct for detectibility (and \(g(0)\neq1\))
- Fit a GAM

- Double observer model
*a la*Borchers et al (2006) - Independent observers, with point independence
- Partial likelihood:
- Mark-recapture: estimate \(\mathbb{P}[\text{detection at apex}]\)
- Detection function: estimate detection probabilities

- Use ML for both.
- Estimate per-segment abundance using Horvitz-Thompson \[ \hat{N}_j = \sum_{i \in \text{ transect } j}\frac{s_i}{p_i} \]

- 2-part normal detection function (Becker & Christ, in prep)
- Avoid heavy left truncation (discard ~30% data)
- Modelling by EB

- Saw 373 groups 8.4-711.8m
- Truncate at 22m and 450m, leaving 351 groups
- Group size 1-3 (lone bears, sow w. cubs)
- 1402m elevational cutoff

- covariates:
- distance
- indicator for distance greater than the mode (required to make the distribution gamma-like)
- log of search distance
- pilot search type

- Mode at about 129 metres

Search distance: distance from line to the furthest location.

Pilot search type: 2 groups, with one group searching further out.

- Using a density surface model (of course!)
- Just a GAM
- Model adjusted counts per segment

\[ \mathbb{E}(\hat{N}_j) = A_j \exp \left( \sum_k f_k(\zeta_{jk}) \right) \]

where \(f_k\) are smooths of covariates \(\zeta_k\), segment area \(A_j\)

and

\[ \hat{N}_j = \sum_{i \in \text{ transect } j}\frac{s_i}{p_i} \]

- Giant zero-inflation (~350 out of ~44,000 segments)
- Tried both Tweedie and Negative Binomial response
- Negative binomial best fit
- “Primative parallelisation” to find parameter

- Goodness of fit testing
- Dunn and Smyth (1996)
- Back transform for
**exactly**Normal residuals - Less problems with artefacts
- (Thanks to Natalie Kelly at CSIRO for the tip)

`gam.check`

`rqgam.check`

- REML smoothness estimation
- AIC model selection (usual REML constraints)
- (approximate) \(p\)-values
- extra penalty
- biological plausability

- bivariate smooth of location
- smooth of elevation
- bivariate smooth of slope and aspect

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

- Two-stage models can be useful!
- split modelling
- split work

- Large areas, makes sense
- Variance estimation is still a bit ropey
- Spatial component is v. helpful for managers

- Paper write-up
- Integrate all this stuff into Distance proper
- Already a lot of spatially explicit data available

- Becker, EF, and PX Quang. A Gamma-Shaped Detection Function for Line-Transect Surveys with Mark-Recapture and Covariate Data. Journal of Agricultural, Biological, and Environmental Statistics 14, no. 2 (2009): 207–223.
- Borchers, DL, JL Laake, C Southwell, and CGM Paxton. Accommodating Unmodeled Heterogeneity in Double‐Observer Distance Sampling Surveys. Biometrics 62, no. 2 (2006): 372–378.
- Dunn, PK, and GK Smyth. Randomized Quantile Residuals. Journal of Computational and Graphical Statistics 5, no. 3 (1996): 236–244.
- 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.

Talk available at http://dill.github.com/talks/akbears/talk.html

- Earl Becker
- Becky Strauch
- Mike Litzen
- Dave Filkill