This vignette explains some of the functions in the package overdispsim and provides code for simulations in the paper of Efford and Fletcher (2024 and later versions). overdispsim uses the R packages secr (Efford 2025a) and secrdesign (Efford 2025b) available from the CRAN repository. overdispsim itself is available on GitHub and Zenodo.
The research question concerns the effect of the distribution of animal activity centres (AC) on estimates from spatially explicit capture–recapture models, specifically the maximum-likelihood estimates of population density provided by R package secr.
The scenarios considered are a superset of those reported by Efford and Fletcher (2024). To reproduce the specific results in Efford and Fletcher (2024) see the file ‘Figures.R’ on Zenodo.
Three base processes are simulated for the distribution of AC:
Each process is simulated both unconditionally and conditional on \(N(A)\) (i.e. fixed number in a buffered area \(A\)). The unconditional simulations result in Poisson-distributed N(A). Conditional simulations are identified by the suffix ‘f’. Thus there are 6 distinct processes (1, 2, 3, 1f, 2f, 3f). Each process is simulated over a range of parameter values.
This vignette provides both the code to run the simulations, given the R package overdispsim, and tables summarising the resulting simulations. Simulations are re-run if the variable ‘runsimulations’ is set to TRUE; otherwise, summaries are based on previous simulations downloaded from Zenodo (Efford 2025c) or possibly a local directory.
Packages secr and secrdesign are used throughout for simulation and model fitting.
Functions rLGCP and rThomas from the
spatstat package (Baddeley et al. 2015) are used where
possible. The algorithm for rLGCP in
spatstat.random changed in version 3.2.1 (Oct 2023) to
avoid dependence on RandomFields (Schlather et
al. 2015). Some simulations here were run with the earlier code and may
differ (see warning in ?spatstat.random::rLGCP).
Conditional simulations with rLGCP and rThomas
require spatstat.random version >= 3.3.3.12 and secr version >=
5.2.2. These versions may be installed from R-universe if they are not
yet on CRAN.
Package zen4R (Blondel 2024) is required to download simulations from Zenodo.
library(overdispsim, quietly = TRUE)
options(digits = 5)
runsimulations <- FALSE # set TRUE to run afresh
runexamples <- FALSE
nrepl_n <- 0 # 10000 zero causes load from localrepo or zenodoDOI
nrepl_M <- 0 # 1000
csvdir <- NULL # folder to save summaries as csv
zenodoDOI <- "10.5281/zenodo.15455288" # optionally retrieve all simulations
localrepo <- ""For testing set runsimulations <- TRUE and the number
of replicates to a small number e.g. 5.
We next set some options by assigning variables to an environment (‘.local’) that is available to other functions in the package, and incidentally keep a copy in a list named ‘localpar’.
The function is run with these arguments automatically when the package is attached (i.e. in .onAttach), so these are default settings. The actual number of cores is capped at the number available if that is less than the given maxncores.
localpar <- setparameters(
lambda0 = 0.5,
sigma = 1.0,
detectfn = 'HHN',
noccasions = 5,
traps = make.grid(12,12, detector = 'proximity', spacing = 2.0),
maskspacing = 0.5,
maskbuffer = 4,
N = 256,
maxncores = 24
)The list and environment include some derived quantities.
Start by specifying some vectors with varying levels of parameters:
Vlevels <- c(0, 0.125, 0.25, 0.5, 0.75, 1.0) # LGCP variances
mulevels <- c(1,2,4,8,16,32) # expected number per cluster
Alevels <- c(0.0625, 0.125, 0.25, 0.5, 0.75, 1.0) # randomHabitat ANext, for each process generate a list of arguments, one component for each combination of parameter values. Each component is ultimately passed internally to the secr function ‘sim.popn’ to simulate realisations of the distribution of AC.
eps <- spacing(localpar$mask)
# 1. LGCP
basepopargs1 <- list(
D = localpar$D,
core = localpar$mask,
buffer = eps/2,
model2D = "rLGCP",
details = list(var = 0, scale = 5 * localpar$sigma,
eps = eps, saveLambda = TRUE))
popargs1 <- extend(basepopargs1,
values = list(var = Vlevels, scale = c(2,5,10) * localpar$sigma))
t(sapply(popargs1, '[[', 'details'))## var scale eps saveLambda
## [1,] 0 2 0.5 TRUE
## [2,] 0.125 2 0.5 TRUE
## [3,] 0.25 2 0.5 TRUE
## [4,] 0.5 2 0.5 TRUE
## [5,] 0.75 2 0.5 TRUE
## [6,] 1 2 0.5 TRUE
## [7,] 0 5 0.5 TRUE
## [8,] 0.125 5 0.5 TRUE
## [9,] 0.25 5 0.5 TRUE
## [10,] 0.5 5 0.5 TRUE
## [11,] 0.75 5 0.5 TRUE
## [12,] 1 5 0.5 TRUE
## [13,] 0 10 0.5 TRUE
## [14,] 0.125 10 0.5 TRUE
## [15,] 0.25 10 0.5 TRUE
## [16,] 0.5 10 0.5 TRUE
## [17,] 0.75 10 0.5 TRUE
## [18,] 1 10 0.5 TRUE# 2. Thomas clustering
basepopargs2 <- list(
D = localpar$D,
core = localpar$mask,
buffer = eps/2,
model2D = "rThomas",
details = list(mu = 1, scale = localpar$sigma, eps = eps, saveLambda = TRUE))
# rThomas requires scale>0, so use tiny value instead 1e-4
popargs2 <- extend(basepopargs2,
values = list(mu = mulevels, scale = c(1e-4, 1, 2, 4)*localpar$sigma))
t(sapply(popargs2, '[[', 'details'))## mu scale eps saveLambda
## [1,] 1 1e-04 0.5 TRUE
## [2,] 2 1e-04 0.5 TRUE
## [3,] 4 1e-04 0.5 TRUE
## [4,] 8 1e-04 0.5 TRUE
## [5,] 16 1e-04 0.5 TRUE
## [6,] 32 1e-04 0.5 TRUE
## [7,] 1 1 0.5 TRUE
## [8,] 2 1 0.5 TRUE
## [9,] 4 1 0.5 TRUE
## [10,] 8 1 0.5 TRUE
## [11,] 16 1 0.5 TRUE
## [12,] 32 1 0.5 TRUE
## [13,] 1 2 0.5 TRUE
## [14,] 2 2 0.5 TRUE
## [15,] 4 2 0.5 TRUE
## [16,] 8 2 0.5 TRUE
## [17,] 16 2 0.5 TRUE
## [18,] 32 2 0.5 TRUE
## [19,] 1 4 0.5 TRUE
## [20,] 2 4 0.5 TRUE
## [21,] 4 4 0.5 TRUE
## [22,] 8 4 0.5 TRUE
## [23,] 16 4 0.5 TRUE
## [24,] 32 4 0.5 TRUE# 3. random habitat
basepopargs3 <- list(
D = randomDensity,
core = localpar$mask,
buffer = 0,
model2D = "IHP",
details = list(D = localpar$D, p = 0.5, A = 0.25, rescale = TRUE))
popargs3 <- extend(basepopargs3,
values = list(A = Alevels, p = c(0.25, 0.5)) )
t(sapply(popargs3, '[[', 'details'))## D p A rescale
## [1,] 2844.4 0.25 0.0625 TRUE
## [2,] 2844.4 0.25 0.125 TRUE
## [3,] 2844.4 0.25 0.25 TRUE
## [4,] 2844.4 0.25 0.5 TRUE
## [5,] 2844.4 0.25 0.75 TRUE
## [6,] 2844.4 0.25 1 TRUE
## [7,] 2844.4 0.5 0.0625 TRUE
## [8,] 2844.4 0.5 0.125 TRUE
## [9,] 2844.4 0.5 0.25 TRUE
## [10,] 2844.4 0.5 0.5 TRUE
## [11,] 2844.4 0.5 0.75 TRUE
## [12,] 2844.4 0.5 1 TRUEModify each of the preceding scenarios by telling
sim.popn to use fixed N.
fix <- function(x) {
x$Ndist <- "fixed"
x$Nbuffer <- localpar$N
x
}
# 1f. Conditional LGCP requires spatstat.random >= v3.3.3.2 and secr >= 5.2.2
popargs1f <- lapply(popargs1, fix)
# 2f. Conditional Thomas process requires spatstat.random >= v3.3.3.12 and secr >= 5.2.2
popargs2f <- lapply(popargs2, fix)
# cf Bischof et al. fixed clusters (clone = "constant", scale = 0)
# 3f. random habitat, fixed-N
popargs3f <- lapply(popargs3, fix)Rather than interrupt the flow here, refer to Appendix 1.
Here we list the commands used to generate the main results. Sampling with each process (1, 2, 3, 1f, 2f, 3f) and parameter set is initially run many times without fitting a model in order to approximate the expected number of detected individuals etc. Then each sampling scenario is simulated and a model is fitted by maximising the full SECR likelihood (1M, 2M, 3M, 1Mf, 2Mf, 3Mf). A subset of scenarios is further simulated fitting only the conditional SECR likelihood (1MCL, 2MCL, 3MCL).
The function run_all from overdispsim
is a wrapper for secr and secrdesign
simulation functions that uses settings in the local environment. The
argument ‘extractfn’ of secrdesign::run.scenarios is a
different function for each of the three groups of simulations; these
functions are defined in overdispsim.
Each of these takes substantial time to run (hours). They should be run separately and saved for later processing (i.e. summary).
Table 1. Simulations.
| Code | AC distribution | Fit | extractfn |
|---|---|---|---|
| 1 | LGCP | none | extract_n |
| 2 | Thomas process | none | extract_n |
| 3 | random habitat | none | extract_n |
| 1f | fixed-N(A) LGCP | none | extract_n |
| 2f | fixed-N(A) Thomas process | none | extract_n |
| 3f | fixed-N(A) random habitat | none | extract_n |
| 1M | LGCP | full likelihood | extract_M |
| 2M | Thomas process | full likelihood | extract_M |
| 3M | random habitat | full likelihood | extract_M |
| 1fM | fixed-N(A) LGCP | full likelihood | extract_M |
| 2fM | fixed-N(A) Thomas process | full likelihood | extract_M |
| 3fM | fixed-N(A) random habitat | full likelihood | extract_M |
| 1MCL | LGCP | conditional likelihood | extract_MCL |
| 2MCL | Thomas process | conditional likelihood | extract_MCL |
| 3MCL | random habitat | conditional likelihood | extract_MCL |
Here we generate SECR samples, count the number of detected individuals \(n\) and compute the \(\hat c\) measure of overdispersion using the known detection parameters. Models are not fitted.
# Poisson N(A)
sims1 <- run_all(nrepl_n, popargs1, fit = FALSE)
sims2 <- run_all(nrepl_n, popargs2, fit = FALSE)
sims3 <- run_all(nrepl_n, popargs3, fit = FALSE)
# Fixed N(A)
sims1f <- run_all(nrepl_n, popargs1f, fit = FALSE)
sims2f <- run_all(nrepl_n, popargs2f, fit = FALSE)
sims3f <- run_all(nrepl_n, popargs3f, fit = FALSE)Here we select a subset of the defined population scenarios.
# Poisson N(A)
sims1M <- run_all(nrepl_M, popargs1[13:18], fit = TRUE)
sims2M <- run_all(nrepl_M, popargs2[13:18], fit = TRUE,
start = list(D = 3000, lambda0 = 0.4, sigma = 2.2))
sims3M <- run_all(nrepl_M, popargs3[7:12], fit = TRUE)
# Fixed N(A)
sims1fM <- run_all(nrepl_M, popargs1f[13:18], fit = TRUE, distribution = "binomial",
start = list(D = 5000))
sims2fM <- run_all(nrepl_M, popargs2f[13:18], fit = TRUE, distribution = "binomial")
sims3fM <- run_all(nrepl_M, popargs3f[7:12], fit = TRUE, distribution = "binomial")The SECR model was fitted by maximising the conditional likelihood as a convenient way in secr to simulate the coverage of confidence intervals for the effective sampling area \(a(\hat \theta)\) and the proportion of variance due to \(n\) and \(a(\hat \theta)\).
Simulation results have been archived on Zenodo (Efford 2025c). If necessary, we retrieve them with the R package zen4R (Blondel 2024).
tmpfolder <- tempdir()
if (file.exists(localrepo)) {
files_to_copy <- list.files(localrepo, full.names = TRUE)
file.copy(files_to_copy, tmpfolder)
reloadedfrom <- localrepo
} else {
if (!requireNamespace('zen4R')) stop ("Package zen4R is required to download from Zenodo")
zen4R::download_zenodo(zenodoDOI, path = tmpfolder, quiet = TRUE)
reloadedfrom <- zenodoDOI
}
files <- list.files(path = tmpfolder, full.names = TRUE)
files <- files[grepl('.rdata', tolower(files))]
reloaded <- unlist(lapply(files, load, envir = .GlobalEnv))
required <- c('sims1', 'sims2', 'sims3',
'sims1f', 'sims2f', 'sims3f',
'sims1M', 'sims2M', 'sims3M',
'sims1fM', 'sims2fM', 'sims3fM',
'sims1MCL', 'sims2MCL', 'sims3MCL',
'sims2C', 'sims2CCL', 'sims2Cf', 'sims2CfCL'
)
if (!all(required %in% reloaded)) {
warning ("Could not reload all required simulations")
} else {
cat ("successfully reloaded all required files from ", reloadedfrom, "\n")
}## successfully reloaded all required files from 10.5281/zenodo.15455288Now summarise the simulations. The summary tables are printed in Appendix 2.
# no fit
sum1 <- summary_n(sims1)
sum2 <- summary_n(sims2)
sum3 <- summary_n(sims3)
sum1f <- summary_n(sims1f)
sum2f <- summary_n(sims2f)
sum3f <- summary_n(sims3f)
# vs detection-weighted local density
sum1M <- summary_M(sims1M)
sum2M <- summary_M(sims2M)
sum3M <- summary_M(sims3M)
sum1fM <- summary_M(sims1fM)
sum2fM <- summary_M(sims2fM)
sum3fM <- summary_M(sims3fM)
# vs global density
sum1MT <- summary_M(sims1M, true = localpar$D)
sum2MT <- summary_M(sims2M, true = localpar$D)
sum3MT <- summary_M(sims3M, true = localpar$D)
sum1fMT <- summary_M(sims1fM, true = localpar$D)
sum2fMT <- summary_M(sims2fM, true = localpar$D)
sum3fMT <- summary_M(sims3fM, true = localpar$D)
# conditional likelihood (for COV(a) etc.)
sum1MCL <- summary_MCL(sims1MCL, true = localpar$D)
sum2MCL <- summary_MCL(sims2MCL, true = localpar$D)
sum3MCL <- summary_MCL(sims3MCL, true = localpar$D)Summaries are optionally output to .csv:
if (!is.null(csvdir)) {
# no fit
write.csv(sum1, file= paste0(csvdir, '/sum1.csv'))
write.csv(sum2, file= paste0(csvdir, '/sum2.csv'))
write.csv(sum3, file= paste0(csvdir, '/sum3.csv'))
# no fit, fixed N
write.csv(sum1f, file= paste0(csvdir, '/sum1f.csv'))
write.csv(sum2f, file= paste0(csvdir, '/sum2f.csv'))
write.csv(sum3f, file= paste0(csvdir, '/sum3f.csv'))
# fitted model vs detection-weighted local density
write.csv(sum1M, file= paste0(csvdir, '/sum1M.csv'))
write.csv(sum2M, file= paste0(csvdir, '/sum2M.csv'))
write.csv(sum3M, file= paste0(csvdir, '/sum3M.csv'))
# fitted model vs detection-weighted local density, fixed N
write.csv(sum1fM, file= paste0(csvdir, '/sum1fM.csv'))
write.csv(sum2fM, file= paste0(csvdir, '/sum2fM.csv'))
write.csv(sum3fM, file= paste0(csvdir, '/sum3fM.csv'))
# fitted model vs global density
write.csv(sum1MT, file= paste0(csvdir, '/sum1MT.csv'))
write.csv(sum2MT, file= paste0(csvdir, '/sum2MT.csv'))
write.csv(sum3MT, file= paste0(csvdir, '/sum3MT.csv'))
# fitted model vs global density, fixed N
write.csv(sum1fMT, file= paste0(csvdir, '/sum1fMT.csv'))
write.csv(sum2fMT, file= paste0(csvdir, '/sum2fMT.csv'))
write.csv(sum3fMT, file= paste0(csvdir, '/sum3fMT.csv'))
# conditional likelihood (for COV(a) etc.)
write.csv(sum1MCL, file= paste0(csvdir, '/sum1MCL.csv'))
write.csv(sum2MCL, file= paste0(csvdir, '/sum2MCL.csv'))
write.csv(sum3MCL, file= paste0(csvdir, '/sum3MCL.csv'))
}Additional simulations were performed with overdispersion in the
detection process. A novel function (sim.cohesion)
is defined in overdispsim for simulating detection with
variable within-cluster cohesion, ranging from none (gamma = 0) to
complete (gamma = 1) (Bischof et al. 2020). This applies only
to clustered AC.
Table 2. Simulations with complete within-cluster cohesion.
| Code | AC distribution | Fit | extractfn |
|---|---|---|---|
| 2C | Thomas process | none | extract_n |
| 2Cf | fixed-N(A) Thomas process | none | extract_n |
| 2CCL | Thomas process | conditional likelihood | extract_M |
| 2CfCL | fixed-N(A) Thomas process | conditional likelihood | extract_M |
detargs <- list (savepopn = TRUE, gamma = 1) # complete cohesion
# No fit
sims2C <- run_all(nrepl_n, popargs2, CH.function = "sim.cohesion", detargs = detargs,
fit = FALSE)
sims2Cf <- run_all(nrepl_n, popargs2f, CH.function = "sim.cohesion", detargs = detargs,
fit = FALSE)
# Conditional likelihood fit (effect of cohesion on coverage of a-hat)
sims2CCL <- run_all(nrepl_M, popargs2[1:6], CH.function = "sim.cohesion", detargs = detargs,
fit = TRUE, CL = TRUE)
sims2CfCL <- run_all(nrepl_M, popargs2f[1:6], CH.function = "sim.cohesion", detargs = detargs,
fit = TRUE, CL = TRUE)Baddeley, A., Rubak, E., and Turner, R. (2015) Spatial Point Patterns: Methodology and Applications with R. Chapman and Hall/CRC Press, London.
Bischof, R., Dupont, P., Milleret, C., Chipperfield, J., and Royle, J. A. (2020) Consequences of ignoring group association in spatial capture–recapture analysis. Wildlife Biology wlb.00649. DOI 10.2981/wlb.00649
Blondel, E. (2024) zen4R: Interface to ‘Zenodo’ REST API. R package version 0.10. https://CRAN.R-project.org/package=zen4R/
Borchers, D. L. and Efford, M. G. (2008) Spatially explicit maximum likelihood methods for capture–recapture studies. Biometrics 64, 377–385.
Efford, M. G. (2025a) secr: Spatially explicit capture–recapture models. R package version 5.2.2. https://CRAN.R-project.org/package=secr/
Efford, M. G. (2025b) openCR: Spatially explicit capture–recapture models. R package version 2.9.3. https://CRAN.R-project.org/package=openCR/
Efford, M. G. (2025c) Simulations of overdispersed activity centres in spatially explicit capture-recapture [Data set]. Zenodo. https://doi.org/10.5281/zenodo.14873669
Efford, M. G. and Fletcher, D. (2024) The effect of spatial overdispersion on confidence intervals for population density estimated by spatially explicit capture–recapture. bioRxiv https://doi.org/10.1101/2024.03.12.584742.
Saura, S., and Martínez-Millán, J. (2000) Landscape patterns simulation with a modified random clusters method. Landscape Ecology 15, 661–678.
Schlather, M., Malinowski, A., Menck, P. J., Oesting, M., and Strokorb, K. (2015) Analysis, simulation and prediction of multivariate random fields with package RandomFields. Journal of Statistical Software 63, 1–25. https://www.jstatsoft.org/v63/i08/
This is an aside, intended to show the code in action.
First select a subset of cluster (Thomas process) population scenarios and extract the parameter levels for later use.
sapply(popargs2, '[[', 'details')[1:2,] # display parameter values
pops <- popargs2[13:18] # select mu = 1..32, scale = 2And plot an example with mu = 32 and scale = 2:
set.seed(12345)
pop <- do.call(sim.popn, popargs2[[18]])
plot(pop) # AC
plot(localpar$traps, add = TRUE) # detectorsFurther execution of the example is suppressed by default, but can be cut-and-pasted into an R session.
Messages are suppressed.
| Column | Description |
|---|---|
| [var, scale etc.] | Parameters specific to the scenarios |
| varration | Ratio var(n) / mean(n) |
| chatF | Mean ‘Fletcher-chat’ for count \(n_k\) (individuals per detector) |
| chatW | Mean ‘Wedderburn-chat’ for count \(n_k\) (individuals per detector) |
| Nsim | number of datasets simulated |
| N | mean number of individuals in simulated population |
| varN | variance of number of individuals in simulated population |
| n | mean number of individuals detected |
| varn | variance of number of individuals detected |
| localD | mean of detection-weighted density |
| varlocalD | variance of detection-weighted density |
| VRlocalD | variance ratio from localD : (varlocalD/localD^2 + 1/En) / (1/En) |
## var scale varration chatF chatW Nsim N varN n varn localD
## [1,] 0.000 2 0.99245 1.0236 1.0235 10000 255.88 258.74 180.64 179.27 2844.4
## [2,] 0.125 2 1.68665 1.1656 1.1679 10000 256.17 443.26 180.86 305.04 2845.7
## [3,] 0.250 2 2.46564 1.3213 1.3299 10000 256.56 652.35 181.19 446.76 2849.1
## [4,] 0.500 2 4.03615 1.6491 1.6700 10000 255.91 1081.94 180.82 729.82 2842.6
## [5,] 0.750 2 5.62483 2.0036 2.0478 10000 256.11 1514.58 180.65 1016.13 2841.0
## [6,] 1.000 2 7.61201 2.4179 2.5222 10000 256.71 2087.58 181.19 1379.22 2851.0
## [7,] 0.000 5 0.99076 1.0216 1.0210 10000 256.11 254.15 180.71 179.04 2844.4
## [8,] 0.125 5 4.20222 1.2630 1.2684 10000 255.79 1166.92 180.65 759.13 2844.8
## [9,] 0.250 5 7.49596 1.5160 1.5349 10000 256.26 2108.89 180.72 1354.64 2847.6
## [10,] 0.500 5 14.08327 2.0564 2.1461 10000 256.65 4009.15 181.08 2550.26 2848.5
## [11,] 0.750 5 22.49622 2.6528 2.9097 10000 257.53 6392.31 182.11 4096.83 2867.5
## [12,] 1.000 5 31.36740 3.3012 3.7434 10000 255.81 8878.25 180.17 5651.36 2840.1
## [13,] 0.000 10 1.02480 1.0207 1.0197 10000 255.86 262.32 180.52 185.00 2844.4
## [14,] 0.125 10 8.49390 1.3160 1.3254 10000 256.39 2555.69 180.97 1537.10 2846.2
## [15,] 0.250 10 16.14817 1.6140 1.6444 10000 256.09 4912.57 180.65 2917.14 2842.4
## [16,] 0.500 10 33.38755 2.2638 2.4123 10000 255.63 10113.74 180.28 6019.12 2838.9
## [17,] 0.750 10 50.46441 2.9385 3.2699 10000 253.18 15161.13 178.56 9010.90 2815.8
## [18,] 1.000 10 75.04071 3.7679 4.5916 10000 256.99 22816.81 181.41 13613.44 2856.2
## varlocalD VRlocalD
## [1,] 0 1.0000
## [2,] 31769 1.7089
## [3,] 66398 2.4780
## [4,] 136368 4.0495
## [5,] 206878 5.6313
## [6,] 298914 7.6450
## [7,] 0 1.0000
## [8,] 144762 4.2323
## [9,] 292922 7.5272
## [10,] 594044 14.2292
## [11,] 982467 22.5895
## [12,] 1355672 31.3683
## [13,] 0 1.0000
## [14,] 334931 8.4706
## [15,] 673894 16.0722
## [16,] 1455533 33.6346
## [17,] 2195648 51.0380
## [18,] 3321925 74.5787## mu scale varration chatF chatW Nsim N varN n varn localD
## [1,] 1 1e-04 1.9515 1.5942 1.5979 10000 255.97 525.71 180.77 352.77 2846.1
## [2,] 2 1e-04 2.8197 2.1514 2.1543 10000 256.26 759.87 180.64 509.34 2844.4
## [3,] 4 1e-04 4.6758 3.2979 3.3087 10000 255.89 1264.75 180.56 844.26 2840.0
## [4,] 8 1e-04 8.4963 5.5645 5.5728 10000 255.52 2375.23 180.39 1532.60 2841.3
## [5,] 16 1e-04 15.7522 10.1470 10.1935 10000 256.85 4370.05 181.45 2858.29 2855.6
## [6,] 32 1e-04 30.8829 19.7672 19.4955 9988 257.89 8499.70 182.20 5626.97 2873.5
## [7,] 1 1e+00 1.8845 1.3501 1.3517 10000 256.10 491.33 180.71 340.55 2845.8
## [8,] 2 1e+00 2.8045 1.6905 1.6928 10000 255.97 741.55 180.69 506.74 2843.7
## [9,] 4 1e+00 4.4429 2.3584 2.3670 10000 256.11 1187.99 180.92 803.84 2845.6
## [10,] 8 1e+00 8.0496 3.6882 3.7002 10000 256.15 2172.81 181.08 1457.60 2851.1
## [11,] 16 1e+00 14.9667 6.3834 6.3463 10000 255.51 4036.53 180.19 2696.87 2836.0
## [12,] 32 1e+00 29.0646 12.6467 11.7497 9993 255.41 7834.09 180.06 5233.44 2835.9
## [13,] 1 2e+00 1.7829 1.1637 1.1603 10000 255.69 468.04 180.29 321.43 2840.8
## [14,] 2 2e+00 2.5815 1.3139 1.3178 10000 256.45 686.65 181.02 467.31 2848.2
## [15,] 4 2e+00 4.3688 1.6088 1.6143 10000 256.31 1166.32 180.93 790.46 2845.5
## [16,] 8 2e+00 7.4237 2.1859 2.1763 10000 255.58 2019.52 180.27 1338.29 2841.2
## [17,] 16 2e+00 13.7286 3.4020 3.3624 10000 255.64 3739.45 180.12 2472.85 2839.0
## [18,] 32 2e+00 27.1992 5.9977 5.7396 9999 256.96 7381.74 181.25 4929.97 2851.9
## [19,] 1 4e+00 1.6799 1.0680 1.0687 10000 256.30 447.67 180.88 303.86 2848.7
## [20,] 2 4e+00 2.3468 1.1123 1.1122 10000 256.21 630.24 180.80 424.30 2845.2
## [21,] 4 4e+00 3.7444 1.2023 1.2031 10000 256.20 1016.82 181.00 677.71 2848.0
## [22,] 8 4e+00 6.3141 1.3877 1.3814 10000 256.30 1724.33 180.84 1141.87 2846.2
## [23,] 16 4e+00 11.4893 1.7612 1.7380 10000 256.07 3194.12 180.85 2077.86 2848.8
## [24,] 32 4e+00 22.5649 2.6180 2.4849 10000 257.24 6322.40 181.65 4098.83 2861.0
## varlocalD VRlocalD
## [1,] 41457 1.9248
## [2,] 80883 2.8065
## [3,] 164846 4.6931
## [4,] 335048 8.4992
## [5,] 662692 15.6847
## [6,] 1338423 30.2892
## [7,] 40093 1.8946
## [8,] 80718 2.8036
## [9,] 153265 4.4200
## [10,] 313790 7.9752
## [11,] 618597 14.8978
## [12,] 1244243 28.9563
## [13,] 35576 1.7966
## [14,] 72455 2.6139
## [15,] 149770 4.3425
## [16,] 287085 7.4263
## [17,] 564954 13.6654
## [18,] 1175481 27.1160
## [19,] 30660 1.6827
## [20,] 59222 2.3219
## [21,] 118003 3.6288
## [22,] 235601 6.2552
## [23,] 464264 11.3367
## [24,] 971294 22.4419## p A varration chatF chatW Nsim N varN n varn localD
## [1,] 0.25 0.0625 10.1904 5.7224 5.5442 10000 244.55 2669.49 172.23 1755.11 2711.8
## [2,] 0.25 0.1250 5.3219 3.2687 3.1591 10000 246.45 1394.87 173.62 923.98 2733.3
## [3,] 0.25 0.2500 2.9727 2.0106 1.9656 10000 250.52 772.83 176.53 524.77 2783.0
## [4,] 0.25 0.5000 1.6728 1.3531 1.3548 10000 256.20 439.67 180.81 302.45 2846.2
## [5,] 0.25 0.7500 1.2025 1.1264 1.1334 10000 257.60 311.23 181.86 218.69 2864.5
## [6,] 0.25 1.0000 1.0210 1.0206 1.0205 10000 256.10 255.65 180.78 184.57 2844.4
## [7,] 0.50 0.0625 82.0432 14.6344 15.4264 10000 243.85 21086.70 171.72 14088.11 2700.5
## [8,] 0.50 0.1250 38.9567 7.9088 7.7991 10000 245.06 9960.17 172.45 6718.08 2714.9
## [9,] 0.50 0.2500 17.7696 4.1351 4.0193 10000 250.56 4648.48 176.77 3141.19 2785.4
## [10,] 0.50 0.5000 6.3848 2.0811 2.0437 10000 256.04 1707.66 180.69 1153.69 2847.1
## [11,] 0.50 0.7500 2.8202 1.3666 1.3645 10000 257.99 751.02 182.32 514.19 2865.6
## [12,] 0.50 1.0000 1.0090 1.0221 1.0214 10000 255.84 257.99 180.57 182.19 2844.4
## varlocalD VRlocalD
## [1,] 394297.8 10.6884
## [2,] 186531.9 5.5116
## [3,] 87158.9 3.0334
## [4,] 29269.0 1.6529
## [5,] 9469.7 1.2085
## [6,] 0.0 1.0000
## [7,] 3435483.1 86.1249
## [8,] 1624819.1 40.8331
## [9,] 739692.8 18.2270
## [10,] 245677.3 6.4765
## [11,] 81988.9 2.8041
## [12,] 0.0 1.0000## var scale varration chatF chatW Nsim N varN n varn localD varlocalD
## [1,] 0.000 2 0.29472 1.0090 1.0087 10000 256 0 180.66 53.244 2844.4 0
## [2,] 0.125 2 0.36986 1.1449 1.1459 10000 256 0 180.70 66.835 2841.4 19261
## [3,] 0.250 2 0.43128 1.2885 1.2891 10000 256 0 180.60 77.887 2840.1 29111
## [4,] 0.500 2 0.56122 1.5800 1.5809 10000 256 0 180.43 101.262 2836.1 41958
## [5,] 0.750 2 0.74428 1.9168 1.9223 10000 256 0 180.61 134.424 2837.6 51114
## [6,] 1.000 2 0.90422 2.2630 2.2705 10000 256 0 180.21 162.954 2831.9 60226
## [7,] 0.000 5 0.29611 1.0119 1.0120 10000 256 0 180.71 53.509 2844.4 0
## [8,] 0.125 5 0.42046 1.1997 1.1969 10000 256 0 180.44 75.866 2837.7 35588
## [9,] 0.250 5 0.55801 1.3968 1.3957 10000 256 0 180.43 100.683 2837.5 44841
## [10,] 0.500 5 0.85296 1.8135 1.8089 10000 256 0 179.97 153.507 2826.8 59986
## [11,] 0.750 5 1.16479 2.2458 2.2374 10000 256 0 179.50 209.082 2817.2 74005
## [12,] 1.000 5 1.47639 2.7350 2.7313 10000 256 0 179.51 265.032 2818.1 90358
## [13,] 0.000 10 0.29366 1.0068 1.0065 10000 256 0 180.65 53.049 2844.4 0
## [14,] 0.125 10 0.42407 1.1820 1.1795 10000 256 0 180.47 76.534 2836.6 39128
## [15,] 0.250 10 0.58069 1.3537 1.3485 10000 256 0 180.13 104.600 2832.0 46074
## [16,] 0.500 10 0.90157 1.7221 1.7074 10000 256 0 179.53 161.863 2823.2 61645
## [17,] 0.750 10 1.18220 2.1031 2.0815 10000 256 0 179.23 211.891 2816.2 73503
## [18,] 1.000 10 1.56242 2.4980 2.4598 10000 256 0 178.50 278.896 2806.9 90877
## VRlocalD
## [1,] 1.0000
## [2,] 1.4311
## [3,] 1.6521
## [4,] 1.9426
## [5,] 2.1471
## [6,] 2.3570
## [7,] 1.0000
## [8,] 1.7986
## [9,] 2.0064
## [10,] 2.3564
## [11,] 2.6849
## [12,] 3.0558
## [13,] 1.0000
## [14,] 1.8787
## [15,] 2.0381
## [16,] 2.3975
## [17,] 2.6747
## [18,] 3.0843## mu scale varration chatF chatW Nsim N varN n varn localD varlocalD
## [1,] 1 2 0.36914 1.1459 1.1461 10000 256 0 180.65 66.683 2845.2 21293
## [2,] 2 2 0.43387 1.2782 1.2783 10000 256 0 180.72 78.407 2844.8 30640
## [3,] 4 2 0.58498 1.5456 1.5468 10000 256 0 180.66 105.685 2847.5 42370
## [4,] 8 2 0.85208 2.0814 2.0825 10000 256 0 180.64 153.918 2846.5 58968
## [5,] 16 2 1.43674 3.1561 3.1518 10000 256 0 180.36 259.131 2846.2 86831
## [6,] 32 2 2.60334 5.2964 5.2967 10000 256 0 180.48 469.863 2845.5 138297
## VRlocalD
## [1,] 1.4753
## [2,] 1.6841
## [3,] 1.9442
## [4,] 2.3150
## [5,] 2.9369
## [6,] 4.0863## p A varration chatF chatW Nsim N varN n varn localD
## [1,] 0.25 0.0625 2.26640 5.9074 5.9198 10000 256 0 179.94 407.823 2702.5
## [2,] 0.25 0.1250 1.23438 3.3139 3.3161 10000 256 0 180.28 222.540 2734.8
## [3,] 0.25 0.2500 0.68621 1.9983 1.9966 10000 256 0 180.37 123.769 2778.7
## [4,] 0.25 0.5000 0.41695 1.3399 1.3390 10000 256 0 180.64 75.317 2845.1
## [5,] 0.25 0.7500 0.33455 1.1153 1.1159 10000 256 0 180.82 60.493 2865.6
## [6,] 0.25 1.0000 0.28844 1.0098 1.0099 10000 256 0 180.67 52.112 2844.4
## [7,] 0.50 0.0625 7.91690 15.0237 14.6784 10000 256 0 174.22 1379.295 2705.4
## [8,] 0.50 0.1250 4.00903 7.9921 7.8031 10000 256 0 176.81 708.833 2734.8
## [9,] 0.50 0.2500 1.87218 4.0732 4.0103 10000 256 0 178.93 334.986 2780.9
## [10,] 0.50 0.5000 0.81551 2.0284 2.0108 10000 256 0 180.17 146.928 2844.3
## [11,] 0.50 0.7500 0.46354 1.3426 1.3374 10000 256 0 180.42 83.633 2862.5
## [12,] 0.50 1.0000 0.29686 1.0097 1.0083 10000 256 0 180.46 53.572 2844.4
## varlocalD VRlocalD
## [1,] 405161.3 11.0243
## [2,] 189515.2 5.5786
## [3,] 85666.7 3.0048
## [4,] 29171.9 1.6512
## [5,] 9406.1 1.2070
## [6,] 0.0 1.0000
## [7,] 3338345.8 83.4196
## [8,] 1682055.3 41.6374
## [9,] 723218.1 17.8984
## [10,] 245287.9 6.4785
## [11,] 82804.1 2.8261
## [12,] 0.0 1.0000| Column | Description |
|---|---|
| [var, scale etc.] | Parameters specific to the scenarios |
| n | mean number of individuals detected |
| N | mean number of individuals in simulated population |
| nvalid | number of successful simulations |
| estimate | mean estimated density |
| SE.estimate | mean SE of estimate |
| RSE | ratio of preceding |
| trueD | true density; either detection-weighted (default) or global as specified by the ‘true’ argument |
| RB | estimated relative bias relative to trueD |
| seRB | SE of RB |
| COV | unadjusted coverage of 95% interval relative to trueD |
| COVF | adjusted coverage |
| chatF | mean ‘Fletcher-chat’ for count \(n_k\) (individuals per detector) |
| varration | Ratio var(n) / mean(n) |
Summarise relative to local density –
## var scale n N nvalid estimate SE.estimate RSE trueD RB
## [1,] 0.000 10 180.41 255.85 1000 2839.7 212.93 0.074857 2844.4 -0.00165807
## [2,] 0.125 10 180.28 255.56 1000 2839.1 211.83 0.075970 2845.1 -0.00254059
## [3,] 0.250 10 178.61 254.10 1000 2813.4 209.68 0.077459 2828.8 -0.00556826
## [4,] 0.500 10 180.48 256.00 1000 2842.6 208.67 0.079521 2856.5 -0.00482499
## [5,] 0.750 10 178.91 253.83 1000 2818.0 205.94 0.082149 2822.0 -0.00335182
## [6,] 1.000 10 182.52 258.94 1000 2874.7 206.05 0.084302 2877.3 0.00079733
## seRB COV COVF chatF varration
## [1,] 0.0023270 0.949 0.955 1.0028 0.95833
## [2,] 0.0024735 0.946 0.962 1.1682 8.94715
## [3,] 0.0024919 0.955 0.980 1.3598 18.45895
## [4,] 0.0025272 0.950 0.984 1.7308 35.68167
## [5,] 0.0025814 0.958 0.988 2.0670 50.03398
## [6,] 0.0026938 0.954 0.992 2.5461 68.79657## mu scale n N nvalid estimate SE.estimate RSE trueD RB
## [1,] 1 2 180.78 256.33 1000 2848.5 213.25 0.075030 2848.5 0.00003581
## [2,] 2 2 180.14 254.89 1000 2835.2 212.51 0.075336 2832.7 0.00069200
## [3,] 4 2 181.92 257.38 1000 2865.1 213.47 0.075198 2864.8 0.00016706
## [4,] 8 2 180.42 255.41 1000 2842.4 212.14 0.075760 2853.2 -0.00401549
## [5,] 16 2 180.29 256.52 1000 2840.7 211.01 0.077462 2840.3 0.00029217
## [6,] 32 2 180.59 254.65 1000 2843.8 208.91 0.080513 2848.6 -0.00240334
## seRB COV COVF chatF varration
## [1,] 0.0024017 0.952 0.966 1.1440 1.7964
## [2,] 0.0023799 0.938 0.964 1.2668 2.6201
## [3,] 0.0023800 0.954 0.986 1.5516 4.2845
## [4,] 0.0023878 0.951 0.995 2.0623 7.6649
## [5,] 0.0024258 0.954 0.998 3.1192 14.8511
## [6,] 0.0025481 0.960 0.999 5.1320 27.4153## p A n N nvalid estimate SE.estimate RSE trueD RB
## [1,] 0.5 0.0625 173.59 246.59 1000 2733.8 198.29 0.092845 2732.6 0.00344502
## [2,] 0.5 0.1250 176.92 249.66 1000 2785.4 205.34 0.082412 2775.6 0.00093416
## [3,] 0.5 0.2500 177.19 251.34 1000 2789.3 208.64 0.078612 2784.5 0.00197242
## [4,] 0.5 0.5000 179.97 254.85 1000 2834.0 211.89 0.076073 2831.0 0.00055428
## [5,] 0.5 0.7500 181.95 257.93 1000 2865.7 213.66 0.075016 2863.9 0.00043249
## [6,] 0.5 1.0000 180.39 255.56 1000 2840.7 213.00 0.074884 2844.4 -0.00131566
## seRB COV COVF chatF varration
## [1,] 0.0031768 0.950 1.000 12.5557 79.9847
## [2,] 0.0027466 0.943 1.000 7.1100 41.2310
## [3,] 0.0025757 0.944 0.999 3.7833 17.3613
## [4,] 0.0024379 0.943 0.988 1.9548 6.7732
## [5,] 0.0024436 0.945 0.967 1.3330 3.0225
## [6,] 0.0024268 0.945 0.950 1.0084 1.0495## var scale n N nvalid estimate SE.estimate RSE trueD RB
## [1,] 0.000 10 180.53 256 1000 2836.2 117.44 0.041289 2844.4 -0.00288660
## [2,] 0.125 10 180.55 256 1000 2837.1 117.49 0.041736 2826.4 0.00737808
## [3,] 0.250 10 180.94 256 1000 2842.6 117.56 0.041569 2841.7 0.00427704
## [4,] 0.500 10 179.57 256 1000 2821.5 117.15 0.041746 2823.0 0.00374049
## [5,] 0.750 10 178.63 256 1000 2806.0 116.76 0.041696 2819.2 -0.00067265
## [6,] 1.000 10 177.98 256 1000 2796.6 116.61 0.041932 2801.9 0.00196996
## seRB COV COVF chatF varration
## [1,] 0.0012856 0.955 0.960 1.0058 0.28361
## [2,] 0.0023048 0.727 0.765 1.1722 0.43590
## [3,] 0.0023200 0.731 0.792 1.3535 0.53914
## [4,] 0.0024148 0.721 0.821 1.6926 0.82003
## [5,] 0.0023429 0.738 0.877 2.0995 1.16323
## [6,] 0.0023323 0.742 0.913 2.4541 1.47605## mu scale n N nvalid estimate SE.estimate RSE trueD RB
## [1,] 1 2 180.87 256 1000 2843.4 117.73 0.041468 2846.3 1.1849e-03
## [2,] 2 2 181.10 256 1000 2846.2 117.69 0.041404 2851.7 7.4964e-04
## [3,] 4 2 179.95 256 1000 2829.4 117.45 0.041522 2841.7 -7.7040e-04
## [4,] 8 2 180.21 256 1000 2832.1 117.36 0.041490 2843.2 -5.4217e-04
## [5,] 16 2 180.75 256 1000 2842.0 117.66 0.041625 2845.2 2.1757e-03
## [6,] 32 2 181.17 256 1000 2845.5 117.42 0.041555 2857.5 -6.0318e-05
## seRB COV COVF chatF varration
## [1,] 0.0018688 0.843 0.864 1.1391 0.36090
## [2,] 0.0020438 0.785 0.831 1.2770 0.47138
## [3,] 0.0022310 0.764 0.844 1.5261 0.60383
## [4,] 0.0022181 0.776 0.899 2.0714 0.86592
## [5,] 0.0022865 0.746 0.938 3.0704 1.43629
## [6,] 0.0024376 0.727 0.977 5.2926 2.57225## p A n N nvalid estimate SE.estimate RSE trueD RB
## [1,] 0.5 0.0625 175.42 256 636 2706.5 114.51 0.042322 2840.7 -0.031199
## [2,] 0.5 0.1250 176.53 256 749 2764.8 115.62 0.040939 2860.6 -0.032500
## [3,] 0.5 0.2500 179.87 256 872 2812.9 116.93 0.040584 2909.3 -0.029165
## [4,] 0.5 0.5000 179.25 256 985 2813.8 116.91 0.040369 2903.4 -0.030488
## [5,] 0.5 0.7500 180.57 256 999 2838.4 117.61 0.040273 2923.7 -0.029139
## [6,] 0.5 1.0000 181.25 256 1000 2847.7 117.68 0.040212 2928.1 -0.027488
## seRB COV COVF chatF varration
## [1,] 0.00122320 0.95440 1.000 12.3827 8.13516
## [2,] 0.00092689 0.97196 1.000 7.1206 3.96755
## [3,] 0.00082460 0.99083 1.000 3.8178 1.90863
## [4,] 0.00071098 0.98376 1.000 2.0137 0.80661
## [5,] 0.00068789 0.99399 0.998 1.3329 0.46176
## [6,] 0.00069386 0.98800 0.992 1.0008 0.30812Summarise relative to global density –
## var scale n N nvalid estimate SE.estimate RSE trueD RB
## [1,] 0.000 10 180.41 255.85 1000 2839.7 212.93 0.074857 2844.4 -0.00165807
## [2,] 0.125 10 180.28 255.56 1000 2839.1 211.83 0.074470 2844.4 -0.00188017
## [3,] 0.250 10 178.61 254.10 1000 2813.4 209.68 0.073716 2844.4 -0.01090800
## [4,] 0.500 10 180.48 256.00 1000 2842.6 208.67 0.073361 2844.4 -0.00063536
## [5,] 0.750 10 178.91 253.83 1000 2818.0 205.94 0.072402 2844.4 -0.00929904
## [6,] 1.000 10 182.52 258.94 1000 2874.7 206.05 0.072439 2844.4 0.01064051
## seRB COV COVF chatF varration
## [1,] 0.0023270 0.949 0.955 1.0028 0.95833
## [2,] 0.0070575 0.515 0.544 1.1682 8.94715
## [3,] 0.0100575 0.357 0.418 1.3598 18.45895
## [4,] 0.0140470 0.272 0.362 1.7308 35.68167
## [5,] 0.0165683 0.238 0.325 2.0670 50.03398
## [6,] 0.0196247 0.217 0.326 2.5461 68.79657## mu scale n N nvalid estimate SE.estimate RSE trueD RB
## [1,] 1 2 180.78 256.33 1000 2848.5 213.25 0.074972 2844.4 0.00141947
## [2,] 2 2 180.14 254.89 1000 2835.2 212.51 0.074709 2844.4 -0.00324955
## [3,] 4 2 181.92 257.38 1000 2865.1 213.47 0.075048 2844.4 0.00726087
## [4,] 8 2 180.42 255.41 1000 2842.4 212.14 0.074582 2844.4 -0.00072018
## [5,] 16 2 180.29 256.52 1000 2840.7 211.01 0.074184 2844.4 -0.00131664
## [6,] 32 2 180.59 254.65 1000 2843.8 208.91 0.073446 2844.4 -0.00023791
## seRB COV COVF chatF varration
## [1,] 0.0031668 0.860 0.891 1.1440 1.7964
## [2,] 0.0038008 0.784 0.838 1.2668 2.6201
## [3,] 0.0049038 0.672 0.778 1.5516 4.2845
## [4,] 0.0065224 0.517 0.684 2.0623 7.6649
## [5,] 0.0090643 0.383 0.624 3.1192 14.8511
## [6,] 0.0123188 0.282 0.619 5.1320 27.4153## p A n N nvalid estimate SE.estimate RSE trueD RB
## [1,] 0.5 0.0625 173.59 246.59 1000 2733.8 198.29 0.069711 2844.4 -0.0388946
## [2,] 0.5 0.1250 176.92 249.66 1000 2785.4 205.34 0.072191 2844.4 -0.0207409
## [3,] 0.5 0.2500 177.19 251.34 1000 2789.3 208.64 0.073349 2844.4 -0.0193934
## [4,] 0.5 0.5000 179.97 254.85 1000 2834.0 211.89 0.074492 2844.4 -0.0036849
## [5,] 0.5 0.7500 181.95 257.93 1000 2865.7 213.66 0.075116 2844.4 0.0074584
## [6,] 0.5 1.0000 180.39 255.56 1000 2840.7 213.00 0.074884 2844.4 -0.0013157
## seRB COV COVF chatF varration
## [1,] 0.0206043 0.160 0.561 12.5557 79.9847
## [2,] 0.0149577 0.230 0.601 7.1100 41.2310
## [3,] 0.0097142 0.354 0.628 3.7833 17.3613
## [4,] 0.0061204 0.545 0.710 1.9548 6.7732
## [5,] 0.0041230 0.730 0.795 1.3330 3.0225
## [6,] 0.0024268 0.945 0.950 1.0084 1.0495## var scale n N nvalid estimate SE.estimate RSE trueD RB
## [1,] 0.000 10 180.53 256 1000 2836.2 117.44 0.041289 2844.4 -0.00288660
## [2,] 0.125 10 180.55 256 1000 2837.1 117.49 0.041304 2844.4 -0.00259471
## [3,] 0.250 10 180.94 256 1000 2842.6 117.56 0.041329 2844.4 -0.00065155
## [4,] 0.500 10 179.57 256 1000 2821.5 117.15 0.041184 2844.4 -0.00806882
## [5,] 0.750 10 178.63 256 1000 2806.0 116.76 0.041048 2844.4 -0.01350721
## [6,] 1.000 10 177.98 256 1000 2796.6 116.61 0.040997 2844.4 -0.01680326
## seRB COV COVF chatF varration
## [1,] 0.0012856 0.955 0.960 1.0058 0.28361
## [2,] 0.0015732 0.893 0.916 1.1722 0.43590
## [3,] 0.0017458 0.864 0.902 1.3535 0.53914
## [4,] 0.0021492 0.766 0.875 1.6926 0.82003
## [5,] 0.0025528 0.691 0.857 2.0995 1.16323
## [6,] 0.0028613 0.623 0.810 2.4541 1.47605## mu scale n N nvalid estimate SE.estimate RSE trueD RB
## [1,] 1 2 180.87 256 1000 2843.4 117.73 0.041388 2844.4 -0.00035378
## [2,] 2 2 181.10 256 1000 2846.2 117.69 0.041374 2844.4 0.00062985
## [3,] 4 2 179.95 256 1000 2829.4 117.45 0.041291 2844.4 -0.00528330
## [4,] 8 2 180.21 256 1000 2832.1 117.36 0.041260 2844.4 -0.00434261
## [5,] 16 2 180.75 256 1000 2842.0 117.66 0.041364 2844.4 -0.00086732
## [6,] 32 2 181.17 256 1000 2845.5 117.42 0.041281 2844.4 0.00035765
## seRB COV COVF chatF varration
## [1,] 0.0014377 0.925 0.946 1.1391 0.36090
## [2,] 0.0016430 0.887 0.918 1.2770 0.47138
## [3,] 0.0018490 0.829 0.902 1.5261 0.60383
## [4,] 0.0022333 0.746 0.902 2.0714 0.86592
## [5,] 0.0028425 0.636 0.884 3.0704 1.43629
## [6,] 0.0038039 0.505 0.880 5.2926 2.57225## p A n N nvalid estimate SE.estimate RSE trueD RB
## [1,] 0.5 0.0625 175.42 256 636 2706.5 114.51 0.040257 2844.4 -0.048506
## [2,] 0.5 0.1250 176.53 256 749 2764.8 115.62 0.040649 2844.4 -0.027992
## [3,] 0.5 0.2500 179.87 256 872 2812.9 116.93 0.041107 2844.4 -0.011100
## [4,] 0.5 0.5000 179.25 256 985 2813.8 116.91 0.041100 2844.4 -0.010771
## [5,] 0.5 0.7500 180.57 256 999 2838.4 117.61 0.041349 2844.4 -0.002136
## [6,] 0.5 1.0000 181.25 256 1000 2847.7 117.68 0.041370 2844.4 0.001151
## seRB COV COVF chatF varration
## [1,] 0.0083433 0.28302 0.81289 12.3827 8.13516
## [2,] 0.0053235 0.45527 0.84112 7.1206 3.96755
## [3,] 0.0034914 0.57913 0.86812 3.8178 1.90863
## [4,] 0.0021316 0.76954 0.91675 2.0137 0.80661
## [5,] 0.0016157 0.89790 0.93994 1.3329 0.46176
## [6,] 0.0013360 0.94800 0.95200 1.0008 0.30812| Column | Description |
|---|---|
| [var, scale etc.] | Parameters specific to the scenarios |
| n | mean number of individuals detected |
| nvalid | number of successful simulations |
| estimate | mean estimated density |
| SE.estimate | mean SE of estimate |
| RSE | ratio of preceding |
| trueD | true density; either detection-weighted (default) or global as specified by the ‘true’ argument |
| RB | estimated relative bias relative to trueD |
| COV | unadjusted coverage of 95% interval relative to trueD |
| chatF | mean ‘Fletcher-chat’ for count \(n_k\) (individuals per detector) |
| varration | Ratio var(n) / mean(n) |
| a | mean a-hat effective sampling area |
| SEa | SE of a-hat |
| RBa | RB(a-hat) |
| RSEa | RSE(a-hat) |
| COVa | coverage of 95% CI for a-hat |
| pCVn | fraction of var(D-hat) attributable to var(n) |
## var scale n nvalid estimate SE.estimate RSE trueD RB COV
## [1,] 0.000 10 180.41 1000 2839.7 212.71 0.075106 2844.4 0.00165462 0.949
## [2,] 0.125 10 180.28 1000 2839.1 211.61 0.076330 2844.4 0.00187665 0.515
## [3,] 0.250 10 178.61 1000 2813.4 209.45 0.078064 2844.4 0.01090464 0.357
## [4,] 0.500 10 180.48 1000 2842.6 208.44 0.080050 2844.4 0.00063189 0.272
## [5,] 0.750 10 178.91 1000 2818.0 205.69 0.082677 2844.4 0.00929566 0.238
## [6,] 1.000 10 182.52 1000 2874.7 205.79 0.084337 2844.4 -0.01064404 0.216
## chatF varration a SEa RBa RSEa COVa pCVn
## [1,] 0.99571 0.95833 0.063535 0.00055086 -0.00015474 0.0086737 0.956 0.98659
## [2,] 1.15995 8.94715 0.063513 0.00056111 0.00020169 0.0088384 0.946 0.98653
## [3,] 1.35020 18.45895 0.063492 0.00058084 0.00053530 0.0091530 0.944 0.98621
## [4,] 1.71866 35.68167 0.063493 0.00059641 0.00050757 0.0093983 0.953 0.98620
## [5,] 2.05242 50.03398 0.063489 0.00061907 0.00057851 0.0097581 0.950 0.98610
## [6,] 2.52815 68.79657 0.063491 0.00063768 0.00054452 0.0100531 0.961 0.98586## mu scale n nvalid estimate SE.estimate RSE trueD RB COV chatF
## [1,] 1 2 180.78 1000 2848.5 213.04 0.075165 2844.4 -0.00142354 0.860 1.1359
## [2,] 2 2 180.14 1000 2835.2 212.29 0.075425 2844.4 0.00324558 0.784 1.2579
## [3,] 4 2 181.92 1000 2865.1 213.26 0.075313 2844.4 -0.00726474 0.671 1.5407
## [4,] 8 2 180.42 1000 2842.4 211.93 0.076208 2844.4 0.00071646 0.516 2.0478
## [5,] 16 2 180.29 1000 2840.7 210.79 0.077573 2844.4 0.00131319 0.382 3.0973
## [6,] 32 2 180.59 1000 2843.8 208.67 0.080924 2844.4 0.00023491 0.282 5.0959
## varration a SEa RBa RSEa COVa pCVn
## [1,] 1.7964 0.063470 0.00055321 0.00087121 0.0087201 0.950 0.98648
## [2,] 2.6201 0.063538 0.00055272 -0.00018870 0.0087032 0.951 0.98662
## [3,] 4.2845 0.063501 0.00055623 0.00038337 0.0087633 0.947 0.98640
## [4,] 7.6649 0.063482 0.00056422 0.00068177 0.0088915 0.943 0.98634
## [5,] 14.8511 0.063472 0.00057905 0.00085095 0.0091275 0.958 0.98613
## [6,] 27.4153 0.063513 0.00062685 0.00020532 0.0098765 0.948 0.98541## p A n nvalid estimate SE.estimate RSE trueD RB COV
## [1,] 0.5 0.0625 170.47 1000 2684.1 196.21 0.092636 2844.4 0.05638411 0.165
## [2,] 0.5 0.1250 175.64 1000 2765.6 204.51 0.083118 2844.4 0.02770617 0.226
## [3,] 0.5 0.2500 179.40 1000 2823.9 209.58 0.078271 2844.4 0.00723175 0.345
## [4,] 0.5 0.5000 180.79 1000 2847.2 212.26 0.075915 2844.4 -0.00097145 0.565
## [5,] 0.5 0.7500 181.55 1000 2858.2 213.16 0.075161 2844.4 -0.00484925 0.765
## [6,] 0.5 1.0000 180.51 1000 2842.3 212.85 0.075092 2844.4 0.00073771 0.956
## chatF varration a SEa RBa RSEa COVa pCVn
## [1,] 12.40657 78.65684 0.063502 0.00073237 3.7137e-04 0.0115561 0.919 0.98453
## [2,] 7.02766 38.29193 0.063509 0.00063511 2.5739e-04 0.0100162 0.922 0.98559
## [3,] 3.77776 18.04944 0.063538 0.00058288 -1.9502e-04 0.0091783 0.938 0.98623
## [4,] 1.93655 6.27424 0.063506 0.00056026 3.0983e-04 0.0088258 0.954 0.98642
## [5,] 1.30621 2.70141 0.063526 0.00055286 1.5263e-06 0.0087070 0.944 0.98651
## [6,] 0.99349 0.98934 0.063515 0.00055202 1.7452e-04 0.0086953 0.948 0.98651Thomas cluster process
## mu scale varration chatF chatW Nsim N varN n varn localD
## [1,] 1 1e-04 1.9714 2.0406 2.0442 10000 255.92 501.92 180.69 356.21 2843.4
## [2,] 2 1e-04 2.9922 3.0495 3.0670 10000 256.46 782.01 180.79 540.97 2848.5
## [3,] 4 1e-04 4.9533 5.0723 5.0756 10000 255.19 1280.16 180.34 893.30 2838.3
## [4,] 8 1e-04 9.0612 9.1509 9.1809 10000 256.34 2319.68 180.72 1637.56 2848.3
## [5,] 16 1e-04 17.1158 17.3421 17.2080 10000 253.74 4326.44 179.45 3071.43 2820.8
## [6,] 32 1e-04 32.4194 33.9636 33.9090 9967 257.06 8260.15 180.92 5865.30 2859.7
## [7,] 1 1e+00 1.9803 2.0301 2.0290 10000 255.63 478.52 180.08 356.61 2841.1
## [8,] 2 1e+00 2.9434 3.0515 3.0500 10000 255.29 728.99 179.61 528.66 2837.1
## [9,] 4 1e+00 4.9435 5.0711 5.0751 10000 255.82 1187.97 180.07 890.20 2839.7
## [10,] 8 1e+00 8.9155 9.1458 9.1541 10000 256.03 2145.97 180.33 1607.78 2844.1
## [11,] 16 1e+00 16.6991 17.2916 17.2731 10000 256.49 4052.56 180.25 3010.02 2846.3
## [12,] 32 1e+00 32.7225 33.7846 33.2691 9964 256.08 7661.98 179.19 5863.46 2842.4
## [13,] 1 2e+00 1.9161 2.0211 1.9946 10000 255.98 470.24 176.30 337.80 2840.2
## [14,] 2 2e+00 2.9886 3.0199 2.9836 10000 255.91 721.92 176.16 526.47 2844.3
## [15,] 4 2e+00 4.9052 5.0275 4.9783 10000 256.19 1137.59 176.85 867.47 2848.8
## [16,] 8 2e+00 8.9506 9.0482 8.9707 10000 256.61 2042.35 176.85 1582.89 2850.5
## [17,] 16 2e+00 16.5316 17.1257 16.8396 10000 255.42 3720.44 175.64 2903.65 2835.8
## [18,] 32 2e+00 31.8944 33.4905 32.8890 9965 256.75 7074.87 176.20 5619.66 2852.3
## [19,] 1 4e+00 1.8816 2.0085 1.8253 10000 256.22 440.62 160.70 302.38 2845.5
## [20,] 2 4e+00 2.7680 2.9329 2.6728 10000 256.20 608.46 160.76 444.98 2846.7
## [21,] 4 4e+00 4.7211 4.8095 4.3736 10000 255.93 1007.97 160.53 757.86 2846.5
## [22,] 8 4e+00 8.4540 8.5647 7.7899 10000 255.96 1765.85 160.60 1357.74 2846.4
## [23,] 16 4e+00 15.5473 16.1004 14.6217 10000 255.49 3241.52 160.30 2492.29 2839.5
## [24,] 32 4e+00 30.0134 31.2622 28.1783 9972 257.05 6115.98 160.48 4816.58 2853.4
## varlocalD VRlocalD
## [1,] 41629 1.9304
## [2,] 82656 2.8408
## [3,] 165929 4.7218
## [4,] 334375 8.4475
## [5,] 652945 15.8283
## [6,] 1300729 29.7413
## [7,] 39704 1.8888
## [8,] 79503 2.7847
## [9,] 155408 4.4825
## [10,] 313076 7.9939
## [11,] 622576 14.8864
## [12,] 1231872 28.5515
## [13,] 35758 1.8010
## [14,] 73344 2.6382
## [15,] 142657 4.1762
## [16,] 293555 7.5281
## [17,] 568685 13.7777
## [18,] 1135388 26.2181
## [19,] 29197 1.6516
## [20,] 57743 2.2875
## [21,] 120780 3.6934
## [22,] 244349 6.4497
## [23,] 477183 11.6941
## [24,] 931136 21.6652## mu scale varration chatF chatW Nsim N varN n varn localD varlocalD
## [1,] 1 1e-04 0.58924 2.0292 2.0278 10000 256 0 180.54 106.380 2846.1 25755
## [2,] 2 1e-04 0.89261 3.0374 3.0393 10000 256 0 180.63 161.231 2846.6 40384
## [3,] 4 1e-04 1.47404 5.0449 5.0525 10000 256 0 180.70 266.359 2849.0 63596
## [4,] 8 1e-04 2.65284 9.0685 9.0925 10000 256 0 180.93 479.967 2850.2 104436
## [5,] 16 1e-04 4.98912 17.1378 17.0805 10000 256 0 179.92 897.636 2840.5 188417
## [6,] 32 1e-04 9.67104 33.2156 33.2628 10000 256 0 180.62 1746.787 2853.4 343799
## [7,] 1 1e+00 0.55110 2.0102 2.0092 10000 256 0 180.31 99.367 2846.8 23167
## [8,] 2 1e+00 0.77483 3.0217 3.0199 10000 256 0 180.21 139.632 2843.1 35139
## [9,] 4 1e+00 1.26138 5.0446 5.0453 10000 256 0 180.40 227.557 2846.8 50756
## [10,] 8 1e+00 2.17174 9.0731 9.0736 10000 256 0 180.22 391.394 2847.0 77807
## [11,] 16 1e+00 4.10780 17.1478 17.1136 10000 256 0 179.59 737.721 2839.0 128838
## [12,] 32 1e+00 7.99488 33.1655 33.0066 10000 256 0 179.83 1437.712 2847.1 219923
## [13,] 1 2e+00 0.48443 1.9249 1.9154 10000 256 0 179.05 86.737 2846.7 20602
## [14,] 2 2e+00 0.70217 2.8855 2.8755 10000 256 0 179.13 125.779 2847.9 31113
## [15,] 4 2e+00 1.12666 4.8686 4.8486 10000 256 0 178.92 201.577 2844.0 42535
## [16,] 8 2e+00 1.92840 8.8594 8.7881 10000 256 0 178.03 343.316 2847.9 58624
## [17,] 16 2e+00 3.52786 16.8071 16.6270 10000 256 0 177.70 626.892 2847.2 87609
## [18,] 32 2e+00 7.09988 32.8090 32.1553 10000 256 0 176.59 1253.801 2844.7 141560
## [19,] 1 4e+00 0.47417 1.8839 1.8539 10000 256 0 176.15 83.525 2844.5 17642
## [20,] 2 4e+00 0.64272 2.7892 2.7427 10000 256 0 175.91 113.063 2846.1 25688
## [21,] 4 4e+00 0.97093 4.6870 4.5955 10000 256 0 175.48 170.383 2846.9 33945
## [22,] 8 4e+00 1.73923 8.6250 8.3802 10000 256 0 173.81 302.298 2843.6 44001
## [23,] 16 4e+00 3.09755 16.3679 15.7848 10000 256 0 172.01 532.822 2848.1 54732
## [24,] 32 4e+00 6.45844 31.7910 29.9547 10000 256 0 167.96 1084.766 2841.6 76268
## VRlocalD
## [1,] 1.5745
## [2,] 1.9005
## [3,] 2.4157
## [4,] 3.3229
## [5,] 5.2196
## [6,] 8.6300
## [7,] 1.5165
## [8,] 1.7855
## [9,] 2.1317
## [10,] 2.7346
## [11,] 3.8884
## [12,] 5.9024
## [13,] 1.4594
## [14,] 1.6931
## [15,] 1.9502
## [16,] 2.3061
## [17,] 2.9528
## [18,] 4.1609
## [19,] 1.3940
## [20,] 1.5730
## [21,] 1.7568
## [22,] 1.9832
## [23,] 2.2192
## [24,] 2.7067## mu scale n nvalid estimate SE.estimate RSE trueD RB COV
## [1,] 1 1e-04 180.76 1000 2850.5 213.22 0.075200 2844.4 -0.0021367 0.8360
## [2,] 2 1e-04 179.36 1000 2826.3 212.06 0.075678 2844.4 0.0063846 0.7290
## [3,] 4 1e-04 178.85 1000 2824.1 211.90 0.076214 2844.4 0.0071628 0.5870
## [4,] 8 1e-04 179.54 1000 2840.1 212.21 0.076903 2844.4 0.0015156 0.4560
## [5,] 16 1e-04 181.53 1000 2881.9 213.72 0.077795 2844.4 -0.0131552 0.3790
## [6,] 32 1e-04 184.82 987 2965.3 217.53 0.081205 2844.4 -0.0424851 0.2766
## chatF varration a SEa RBa RSEa COVa pCVn
## [1,] 1.9739 1.9227 0.063423 0.00055790 0.00161431 0.0088051 0.83100 0.98613
## [2,] 2.9824 2.9929 0.063480 0.00055960 0.00071431 0.0088282 0.75900 0.98615
## [3,] 4.9483 5.4891 0.063358 0.00057051 0.00264306 0.0090273 0.64600 0.98555
## [4,] 8.8896 9.8519 0.063238 0.00058835 0.00452230 0.0093613 0.49700 0.98433
## [5,] 16.7772 16.2780 0.063035 0.00062604 0.00773051 0.0101110 0.36300 0.98102
## [6,] 32.7160 32.0965 0.062445 0.00071643 0.01701753 0.0121070 0.27052 0.97237## mu scale n nvalid estimate SE.estimate RSE trueD RB COV chatF
## [1,] 1 2 179.14 1000 2822.8 212.28 0.075306 2844.4 0.0075961 0.996 1.9135
## [2,] 2 2 178.23 1000 2808.2 211.69 0.075548 2844.4 0.0127534 0.974 2.8475
## [3,] 4 2 178.35 1000 2814.1 212.08 0.075618 2844.4 0.0106765 0.922 4.7972
## [4,] 8 2 179.01 1000 2831.7 213.04 0.075642 2844.4 0.0044680 0.828 8.8168
## [5,] 16 2 177.55 1000 2816.7 212.84 0.076367 2844.4 0.0097519 0.689 16.6758
## [6,] 32 2 175.51 1000 2815.5 214.98 0.077974 2844.4 0.0101690 0.529 32.1971
## varration a SEa RBa RSEa COVa pCVn
## [1,] 0.49980 0.063469 0.00055324 0.00088721 0.0087248 0.858 0.98643
## [2,] 0.77999 0.063482 0.00055583 0.00068254 0.0087684 0.756 0.98631
## [3,] 1.18750 0.063405 0.00056545 0.00190234 0.0089411 0.614 0.98562
## [4,] 1.91385 0.063260 0.00057844 0.00417795 0.0091875 0.482 0.98453
## [5,] 3.59820 0.063125 0.00060059 0.00630514 0.0096317 0.344 0.98222
## [6,] 6.79155 0.062534 0.00067588 0.01560910 0.0112573 0.286 0.97376