Description

Estimate population density by simulation and inverse prediction (Efford 2004; Efford, Dawson & Robbins 2004). A restricted range of SECR models may be fitted (detection functions with more than 2 parameters are not supported, nor are covariates).

Usage

Arguments

Details

‘Inverse prediction’ uses methods from multivariate calibration (Brown 1982). The goal is to estimate population density (D) and the parameters of a detection function (usually g0 and sigma) by ‘matching’ statistics from predictorfn(capthist) (the target vector) and statistics from simulations of a 2-D population using the postulated detection model. Statistics (see Note) are defined by the predictor function, which should return a vector equal in length to the number of parameters (np = 3). Simulations of the 2-D population use sim.popn. The simulated population is sampled with sim.capthist according to the detector type (e.g., ‘single’ or ‘multi’) and detector layout specified in traps(capthist), including allowance for varying effort if the layout has a usage attribute.

... may be used to control aspects of the simulation by passing named arguments (other than D) to sim.popn. The most important arguments of sim.popn to keep an eye on are ‘buffer’ and ‘Ndist’. ‘buffer’ defines the region over which animals are simulated (unless mask is specified) - the region should be large enough to encompass all animals that might be caught. ‘Ndist’ controls the number of individuals simulated within the buffered or masked area. The default is ‘poisson’. Use ‘Ndist = fixed’ to fix the number in the buffered or masked area A at N = DA. This conditioning reduces the estimated standard error of D-hat, but conditioning is not always justified - seek advice from a statistician if you are unsure.

The simulated 2-D distribution of animals is Poisson by default. There is no ‘even’ option as in Density.

Simulations are conducted on a factorial experimental design in parameter space - i.e. at the vertices of a cuboid ‘box’ centred on the working values of the parameters, plus an optional number of centre points. The size of the ‘box’ is specified as a fraction of the working values, so for example the limits on the density axis are D*(1–boxsize) and D*(1+boxsize) where D* is the working value of D. For g0, this computation uses the odds transformation (g0/(1–g0)). boxsize may be a vector defining different scaling on each parameter dimension.

A multivariate linear model is fitted to predict each set of simulated statistics from the known parameter values. The number of simulations at each design point is increased (doubled) until the residual standard error divided by the central value is less than CVmax for all parameters. An error occurs if max.nsim is exceeded.

Once a model with sufficient precision has been obtained, a new working vector of parameter estimates is ‘predicted’ by inverting the linear model and applying it to the target vector. A working vector is accepted as the final estimate when it lies within the box; this reduces the bias from using a linear approximation to extrapolate a nonlinear function. If the working vector lies outside the box then a new design is centred on value for each parameter in the working vector.

Once a final estimate is accepted, further simulations are conducted to estimate the variance-covariance matrix. These also provide a parametric bootstrap sample to evaluate possible bias. Set var.nsim = 0 to suppress the variance step.

See Efford et al. (2004) for another description of the method, and Efford et al. (2005) for an application.

The value of predictortype is passed as the second argument of the chosen predictorfn. By default this is pfn, for which the second argument (N.estimator) is a character value from c("n", "null","zippin","jackknife"), corresponding respectively to the number of individuals caught (Mt+1), and N-hat from models M0, Mh and Mb of Otis et al. (1978).

If not provided, the starting values are determined automatically with autoini.

Linear measurements are assumed to be in metres and density in animals per hectare (10 000 m^2).

If ncores > 1 the parallel package is used to create processes on multiple cores (see Parallel for more).

Value

For ip.secr, a list comprising

For pfn, a vector of numeric values corresponding to N-hat, p-hat, and RPSV, a measure of the spatial scale of individual detections.

Warning

Simulation becomes unreliable with very sparse populations, or sparse sampling, because some simulated datasets will have no recaptures or even no captures. Adjustments were made in secr 2.3.1 to make the function more stable in these conditions (e.g., allowing a failed simulation to be repeated, by setting the ‘maxtries’ argument > 1), but results probably should not be relied upon when there are warning messages regarding failed simulations.

Note

Each statistic is expected to have a monotonic relationship with one parameter when the other parameters are held constant. Typical statistics are -

where N-hat and p-hat are estimates of population size and capture probability from the naive application of a nonspatial population estimator, and RPSV is a trap-revealed measure of the scale of movement.

This method provides nearly unbiased estimates of the detection parameter g0 when data are from single-catch traps (likelihood-based estimates of g0 are biased in this case - Efford, Borchers & Byrom 2009).

The implementation largely follows that in Density, and it may help to consult the Density online help. There are some differences: the M0 and Mb estimates of population-size in ip.secr can take non-integer values; the simulation design used by ip.secr uses odds(g0) rather than g0; the default boxsize and CVmax differ from those in Density 4.4. There is no provision in ip.secr for two-phase estimation, using a different experimental design at the second phase. If you wish you can achieve the same effect by using the estimates as starting values for a second call of ip.secr (see examples).

Maximum likelihood estimates from secr.fit are preferable in several respects to estimates from inverse prediction (speed*; more complex models; tools for model selection). ip.secr is provided for checking estimates of g0 from single-catch traps, and for historical continuity.

* autoini with thin = 1 provides fast estimates from a simple halfnormal model if variances are not required.

References

Brown, P. J. (1982) Multivariate calibration. Journal of the Royal Statistical Society, Series B 44, 287–321.

Efford, M. G. (2004) Density estimation in live-trapping studies. Oikos 106, 598–610.

Efford, M. G., Borchers D. L. and Byrom, A. E. (2009) Density estimation by spatially explicit capture–recapture: likelihood-based methods. In: D. L. Thompson, E. G. Cooch and M. J. Conroy (eds) Modeling Demographic Processes in Marked Populations. Springer. Pp. 255–269.

Efford, M. G., Dawson, D. K. and Robbins C. S. (2004) DENSITY: software for analysing capture-recapture data from passive detector arrays. Animal Biodiversity and Conservation 27, 217–228.

Efford, M. G., Warburton, B., Coleman, M. C. and Barker, R. J. (2005) A field test of two methods for density estimation. Wildlife Society Bulletin 33, 731–738.

Otis, D. L., Burnham, K. P., White, G. C. and Anderson, D. R. (1978) Statistical inference from capture data on closed animal populations. Wildlife Monographs 62.

See Also

capthist, secr.fit, RPSV, autoini, sim.popn, Detection functions

Examples