Jottings: Open population models with SECR

HOME Updated 20 May - My thoughts section added; 15 May - plots for fishers; 13 May - more info in several places; 8 May - black bears; 7 May 2018 - Field vole plots added.

Material from this page is now part of this blog post.

We are thinking about models for multi-year SECR analysis for sea cucumbers and other species. For that the usual closure assumption is implausible. Recall the closure assumption for SECR:

  • no births, deaths, immigration or emigration, and
  • no change in activity centre (AC) location.

Various implementations of multi-year SECR have relaxed the first part of this, allowing for recruitment and deaths/emigration, with a "robust" design (Pollock 1982). The second part is more difficult. Options include:

  • model AC as the same every year anyway - "all models are wrong" - (eg, pampas cats, grizzlies, tigers); simulations by Royle et al 2016 (see black bears below) suggest that the bias will be small. This is likely a good solution for 2 or 3 years, but won't do for the long term.
     
  • complete independence in AC for each year (ovenbirds, fishers), which is also implausible. With this option, we have the problem of estimating the probability of detection when the animal is not caught - is it gone or alive but not detected - as detection probability depends on the AC location and thus, indirectly, on the size of the state space used.
     
  • ACs in successive  seasons move as a random walk, with movement modelled as bivariate normal (shrikes, black bears) or direction and distance (field voles).

See some conclusions below.

Examples from the literature

Pampas cats

Gardner, B., Reppucci, J., Lucherini, M., & Royle, J.A. (2010) Spatially explicit inference for open populations: estimating demographic parameters from camera-trap studies. Ecology, 91, 3376-3383.

The real data is only for 2 years, but they provide code for simulations for 3 years. The data are sparse, so assume AC doesn't change, but suggest a random walk for changes in AC with bivariate-normal steps.

The pampas cats also get a mention in Royle, J.A. & Gardner, B. (2011). Hierarchical spatial capture–recapture models for estimating density from trapping arrays. In Camera traps in animal ecology: methods and analyses (eds A.F. O'Connell, J.D. Nichols & K.U. Karanth), pp. 163–190. Springer, New York.

Ovenbirds

Royle, J.A., Chandler, R.B., Sollmann, R., & Gardner, B. (2014) Spatial capture-recapture. Elsevier (SCR book), Chapter 16 give a number of analyses of the ovenbird data collected by Dawson and Efford from 2005 to 2009 and included in the secr package as an example data set. In his example analysis in secr, Murray Efford uses a buffer of 350m around the traps, but Royle et al use just 150m: I tried both.

They present code for a spatial multi-season model (p.375, this models changes in population density but does not track individual birds) and both non-spatial robust JS (p.408) and spatial JS (p.413) models.

For their spatial models they use independent uniform priors for the ACs in each year. The posterior distributions for these for a bird caught in years 1, 3, 4 and 5 but not in year 2 are shown below with a 150m buffer.

Clearly the birds do move around and attempting to fit a single AC location for all years would be suboptimal. The parameter pStar given is the probability of detecting the bird at least once in the year - any occasion, any trap - if it is alive and in the study area. If a bird is not caught in a specific year, the AC could be anywhere in the state space except near to a trap; this posterior is the same for all birds not captured and pStar is also the same for all birds not captured. The results are affected by the width of the buffer; the plots below show the ACs for a bird with a 350m buffer:

For birds captured, the pStar values are almost the same, but for those not captured, pStar = 0.35 for 150m buffer and 0.07 for 350m buffer, a huge difference. This clearly affects the estimated status of birds not caught: apparent survival, phi, is 0.53 with a 150m buffer but 0.74 with a 350m buffer. If the state space is big enough, no birds need die, they just move to the outer edges! Looked at another way, changing the state space also changes the definition of "emigration".

Having completely independent locations in each year, with no constraint on the location when the bird is not captured, won't work.

Royle et al (p.426) discuss modelling movement, suggesting either a movement kernel (eg, bivariate normal) or an autoregressive model. They comment: "At this point, very little work has been done to model movement using SCR models; however, we expect that in the near future this will be one of the most exciting areas of research."

Field voles

Ergon, T. & Gardner, B. (2014) Separating mortality and emigration: modelling space use, dispersal and survival with robust-design spatial-capture-recapture data. Methods in Ecology and Evolution, 5, 1327-1336.

They model changes in AC as a random walk, with movement from one year to the next occurring in a random direction, then a distance drawn from a distribution, which may be exponential, gamma or log-normal. They also tried zero-inflated movement models, with a certain probability of no movement. They give lots of code for simulation and estimation in JAGS, including goodness of fit procedures.

A few unusual features of this analysis (compared with other SCR models):

  • Like the classic (non-robust) CJS models, they only consider animals captured, and only survival/dispersal after the first capture, they don't try to model density or recruitment, so...
  • No data augmentation and no all-zero capture histories, and thus...
  • No bounded state space; there are uniform priors for the AC during the first period the animal was captured, but no limits on its subsequent movements.

As a result of that, I'm not sure whether we will be able to combine these ideas with our usual density-estimation methods. Still, I ran the code and got some plots of ACs. Below are plots for a male and a female, both captured in primary occasions 1 and 2 and not captured in 3 and 4:

In session 3, the male is probably dead (mean z is the posterior probability of being alive), but if alive it has moved out of the trapping array, and remains there in session 4. The female is almost certainly dead, but if alive stays at the same place. This is consistent with the conclusion that males on average disperse further than females between primary sessions (6m vs 3m; trap spacing is 7m), have higher survival probability (0.91 vs 0.75), and higher capture probability if inside the array (as indicated by pStar, the probability of capture if alive).

These models seem to all have poor mixing and high cross-correlations. For the voles, I'm getting n.eff's of <600 from 400k iterations (12hrs x 20 cores); such a small proportion usually indicates a misspecified model, but I don't see how it can be reparametrised. It doesn't help that for the field voles they use a generalised power detection function instead of half-normal, which means another parameter to estimate; they have a total of 11 main parameters to estimate with 158 animals caught.

Red-backed shrikes

Schaub, M. & Royle, J.A. (2014) Estimating true instead of apparent survival using spatial Cormack-Jolly-Seber models. Methods in Ecology and Evolution, 5, 1316-1326.

As with field voles, this is a modified CJS model, conditioning on first capture. They used area search instead of traps, and if the bird is detected the nest is found and this is the AC. The detection function is categorical: 1 if the AC is inside the search area, 0 if outside. Their random walk is bivariate normal, as with black bears below, instead of direction plus distance (though they comment that the latter may be more biologically meaningful). Code for simulations given, not real data. I didn't explore this in detail as very different to the data we collect.

Grizzlies

Whittington, J. & Sawaya, M.A. (2015) A comparison of grizzly bear demographic parameters estimated from non-spatial and spatial open population capture-recapture models. PLoS ONE, 10, e0134446. Their data set has > 250,000 hair-snare samples (budget?) for 80 individual bears from 630 traps over 3 years, so likely a long run, which I haven't attempted. Animal activity centres were assumed not to change over the 3 years (= pampas cats).

Black bears

Royle, J.A., Fuller, A.K., & Sutherland, C. (2016) Spatial capture–recapture models allowing Markovian transience or dispersal. Population Ecology, 58, 53-62.

Single season model, but allowing for changes in occasion-specific AC. For some or all of the population (transients) the ACs move as a random walk controlled by a bivariate normal. A similar model for dispersal involves a single large move to a new location. They did lots of simulations, and the good news is that the bias in the density estimates due to ignoring AC movement is remarkably small (see Tables 1-3 in the paper). The somewhat bad news is that the estimate of σ increases with time, so be careful if you are using this as a measure of home range size.

They analyse  the Gardner et al (2010, J Wildlife Mant, see also the SCR book, chapter 4) Fort Drum bear hair snare data as their example, where the occasions are 1 week in length. Mixing is poor, even with an overnight run with 400k iterations: n.eff's are down to 145 for the transience parameter. Nevertheless, the plots suggest that something sensible is happening.

The first plot is for a bear detected on the first occasion at a location on the edge of the trap array and not thereafter (yellow dot is the location of capture, the black dot the centroid of the posterior estimates of the AC):

It's AC is modelled as outside the array, and it moves slightly further out in subsequent occasions, with less and less certainty about the location. Animals caught only on the last occasion show the reverse pattern, lots of uncertainty at the beginning and gradually homing in on the location of capture. The next animal was caught twice in the northern part of the array and a third time in the south, and is the one with the most movement (the blue dots indicate the previous home range centroids):

Most animals tended to stay in much the same place and the next plots are more typical:

One of my reservations about the usual camera trap analysis that we do is that captures are not actually independent: if an animal was captured in one trap last night, it's not going to appear tonight at the opposite end of its territory. Over a short period (say, 1 week) captures are likely to be clustered, and this may be one way to model that.

Fishers

Green, D.S., Matthews, S.M., Swiers, R.C., Callas, R.L., Yaeger, J.S., Farber, S.L., Schwartz, M.K., Powell, R.A., & Griffen, B. (2018) Dynamic occupancy modelling reveals a hierarchy of competition among fishers, grey foxes and ringtails. Journal of Animal Ecology, 87, 813-824.

In addition to the occupancy modelling for the 3 species, they also ran SCR analysis for fishers, individually identified from hair samples, annually from 2006 to 2013. They modelled ACs as independent for each year (= ovenbirds). The model with all the covariates runs very slowly (25 secs per iteration!) but with 20 cores and 3 days I managed to get n.eff's > 2000; unfortunately augmentation was insufficient, but I'm not about to run it again with more, which will be even slower. The general pattern is similar to the ovenbirds, but with a trap array much bigger than the home range we get a better idea of movement. The plot below is for an animal captured in 5 years out of 8:

Looking at the AC locations in successive years, it seems that a random walk would be a good model. In that case, the location in year 6 would be not far from the array and detection probability would be higher and probability of survival lower. ACs in years 7 and 8 would be less constrained but not scattered over the entire state space.

Tigers

Wang, T., Royle, J.A., Smith, J.L.D., Zou, L., Lü, X., Li, T., Yang, H., Li, Z., Feng, R., Bian, Y., Feng, L., & Ge, J. (2018) Living on the edge: Opportunities for Amur tiger recovery in China. Biological Conservation, 217, 269-279. This was a 2-year study, appears they used the Gardner et al 2010 method (= pampas cats).

Jaguars

Glennie, R., Borchers, D.L., Murchie, M., Harmsen, B.J., & Foster, R.J. Open population maximum likelihood spatial capture-recapture. (submitted). They have a 12-year data set for male jaguars in Belize and (like it says) want to do an ML estimation. They do not allow for change in AC, which seems bold given the length of the period, but they do use the CMSA parameterisation for recruitment (see below).

My thoughts and experiments

CJS vs JS

Cormack-Jolly-Seber (CJS) methods only consider animals captured and only their survival after first capture, as with field voles and shrikes above. That literature has interesting ideas about modeling survival and movement, but is generally not what we want.

Jolly-Seber (JS) methods, in particular the robust versions, fit best with our over approach, where we are interested in abundance and recruitment as well as survival.

Modelling AC movement over time

Fixed ACs are in general implausible, as animals are almost certain to 'drift' as the years go by. So we need to be able to model changes in AC over time; that does not mean that movement models will be the best in every case, and I have worked with some sparse data where AC drift and the usual SCR scale parameter were not separable.

Independent ACs each year are easy to implement but hard to estimate. In particular, survival estimates depend on the area of the state space, as seen with the ovenbirds.

A random walk, where this year's AC depends on last year's AC, seems the way to go; no one has chosen the "hyper-AC' idea. Movement is modelled independently as drawn from a bivariate normal (or similar) between primary sessions.

Modeling the walk as a random direction plus a distance drawn from a half-normal (or similar) distribution, seems biologically sensible, as used for field voles and suggested for shrikes. Unfortunately, JAGS has no sampler to deal with circular distributions. (The Egon & Gardner (2014) prior for direction is coded as theta ~ dunif(-3.141593, 3.141593), which is fine for simulating data but not for MCMC. For MCMC, the sampler needs to be able to move easily between a direction of 359˚ and 1˚.)

Recruitment parameterisation

Recruitment gets tangled with data augmentation. With our SECR models, we use data augmentation to be able to estimate abundance, and an inclusion parameter (usually \(\psi\) or \(\Omega\)) controls the probability that individuals in the augmented population are in the population available for detection. The parameter has no biological significance and depends on the degree of data augmentation.

An early approach, used for all the above except jaguars, is to add recruits from the augmented population, with a parameter \(\gamma_t\) controlling the probability that an individual in the augmented population which has not yet been added to the real population will be added in year \(t\). Again, the parameter has no biological meaning, and it would be difficult to put any hierarchical structure on it or even to enforce constant recruitment.

More efficient is to borrow the idea of a superpopulation from the Crosbie-Manly-Schwarz-Arneson (CMSA) literature (see Link & Barker 2010 p.255). The superpopulation consists of all the individuals in the population - and available for detection - at any primary session. Membership of the superpopulation is determined by \(\psi\) or \(\Omega\) as usual. For these individuals, recruitment is a question of when, not if. The timing of recruitment is a categorical variable determined by a vector of probabilities, where \(\alpha_t\) is the probability that an individual will be recruited just before primary session \(t\), with \(\alpha_1\) the probability of recruitment at some time before the first survey, and all the \(\alpha_t\) summing to 1. Now it's clear how to model recruitment as a function of covariates or how to reduce the number of parameters, for example with:

alpha[1] ~ dbeta(1,1)
for(t in 2:T) {
  alpha[t] <- (1 - alpha[1]) / (T - 1)
}

Survival parameterisation

The usual apparent survival parameter (\(\phi_t\)) works fine here, giving the probability that an individual will leave the population through death or permanent emigration before primary session \(t\).

Slices for sessions

The standard single-season SCR study assumes the population is closed (no births/deaths/immigration/emigration). For many rare and elusive species, passive detectors need to be deployed for long periods to get enough data We often discard data outside a restricted period to try to meet closure assumptions and still hope that infringements will not cause too big a bias.

Some projects have moved to continuous monitoring of populations. Slicing these extended periods short sessions and using a robust analysis method may be the best way to utilise the data.