Spatially Explicit Capture–Recapture by Inverse Prediction
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
1 2 3 4 5 6  ip.secr (capthist, predictorfn = pfn, predictortype = "null", detectfn = 0,
mask = NULL, start = NULL, boxsize = 0.2, boxsize2 = boxsize, centre = 3,
min.nsim = 10, max.nsim = 2000, CVmax = 0.002, var.nsim = 1000, maxbox = 5,
maxtries = 2, ncores = 1, seed = NULL, trace = TRUE, ...)
pfn(capthist, N.estimator = c("n", "null","zippin","jackknife") )

Arguments
capthist 
capthist object including capture data and detector (trap) layout 
predictorfn 
a function with two arguments (the first a capthist object) that returns a vector of predictor values 
predictortype 
value (usually character) passed as the second argument of predictorfn 
detectfn 
integer code or character string for shape of detection function 0 halfnormal, 2 exponential, 3 uniform) – see detectfn 
mask 
optional habitat mask to limit simulated population 
start 
vector of np initial parameter values (density, g0 and sigma) 
boxsize 
scalar or vector of length np for size of design as fraction of central parameter value 
boxsize2 
as for 
centre 
number of centre points in simulation design 
min.nsim 
minimum number of simulations per point 
max.nsim 
maximum number of simulations per point 
CVmax 
tolerance for precision of points in predictor space 
var.nsim 
number of additional simulations to estimate variancecovariance matrix 
maxbox 
maximum number of attempts to ‘frame’ solution 
maxtries 
maximum number of attempts at each simulation 
ncores 
integer number of cores available for parallel processing 
seed 
integer random number seed, or NULL 
trace 
logical, set FALSE to suppress progress reports 
... 
further arguments passed to sim.popn 
N.estimator 
character value indicating population estimator to use 
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 2D 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 2D 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 Dhat, but conditioning is not always
justified  seek advice from a statistician if you are unsure.
The simulated 2D 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 variancecovariance 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 Nhat 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
call 
the function call 
IP 
dataframe with estimated density /ha, g0 and sigma (m) 
vcov 
variancecovariance matrix of estimates 
ip.nsim 
total number of simulations 
variance.bootstrap 
dataframe summarising simulations for variance estimation 
proctime 
processor time (seconds) 
For pfn
, a vector of numeric values corresponding to Nhat, phat, 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 
Statistic  Parameter 
Nhat  D 
phat  g0 
RPSV  sigma 
where Nhat and phat are estimates of population size and capture probability from the naive application of a nonspatial population estimator, and RPSV is a traprevealed measure of the scale of movement.
This method provides nearly unbiased estimates of the detection parameter g0 when data are from singlecatch traps (likelihoodbased 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 populationsize in ip.secr
can take noninteger
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 twophase
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 singlecatch 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 livetrapping studies. Oikos 106, 598–610.
Efford, M. G., Borchers D. L. and Byrom, A. E. (2009) Density estimation by spatially explicit capture–recapture: likelihoodbased 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 capturerecapture 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
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26  ## Not run:
## these calculations may take several minutes
## default settings
ip.secr (captdata)
## coarse initial fit, no variance step
ip1 < ip.secr (captdata, boxsize = 0.2, CVmax=0.01, var=0)
## refined fit
ip2 < ip.secr (captdata, start = ip1$IP[,"estimate"],
boxsize = 0.1, CVmax=0.002, var=1000)
ip2
## compare to MLE of same data using multicatch assumption
predict(secrdemo.0)
## improvise another predictor function (dbar instead of RPSV)
pfn2 < function (capthist, v) { ## v is not used
sumni < sum(capthist!=0) ## total detections
n < nrow(capthist) ## number of individuals
nocc < ncol(capthist) ## number of occasions
c(N = n, p = sumni/n/nocc, dbar = dbar(capthist))
}
ip.secr (captdata, predictorfn = pfn2)
## End(Not run)

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