Lambda {secrdesign}

Expected Detections

Description

Compute the expected number of detections as a function of location (Lambda), and the expected total numbers of individuals n, recaptures r and movements m for a population sampled with an array of detectors (Enrm) or the number of individuals detected at two or more detectors (En2).

Usage

Lambda(traps, mask, detectpar, noccasions, detectfn = c("HHN", "HHR", "HEX", 
    "HAN", "HCG", 'HN', 'HR', 'EX'))
Enrm(D, ...)

minnrRSE(D, ..., CF = 1.0, distribution = c("poisson","binomial"))

En2(D, traps, mask, detectpar, noccasions, detectfn = c("HHN", "HHR", "HEX", 
    "HAN", "HCG", "HN", "HR", "EX"))

Qpm(D, traps, mask, detectpar, noccasions, detectfn = c("HHN", "HHR", "HEX", 
    "HAN", "HCG", "HN", "HR", "EX"))

Arguments

traps

traps object

mask

mask object

detectpar

a named list giving a value for each parameter of detection function

noccasions

integer number of sampling occasions

detectfn

integer code or character string for shape of detection function – see detectfn

D

population density animals / hectare; may be scalar or vector of length nrow(mask)

...

arguments passed to Lambda

CF

numeric correction factor

distribution

character distribution of n

Details

The detector attribute of traps may be ‘multi’, ‘proximity’ or ‘count’. It is assumed that detectpar and detector type do not differ among occasions.

The calculation is based on an additive hazard model. If detectfn is not a hazard function (‘HHN’, ‘HEX’, ‘HHR’, ‘HAN’ and ‘HCG’) then an attempt is made to approximate one of the hazard functions (HN -> HHN, HR -> HHR, EX -> HEX). The default is ‘HHN’.

For hazard function \lambda(d) and S occasions, we define \Lambda(x) = \sum_s \sum_k \lambda(d_k(x)).

Formulae for expected counts are given in secrdesign-Enrm.pdf.

minnrRSE has mostly the same inputs as Enrm but returns sqrt(CF/min(n,r)). The correction factor CF may be used to adjust for systematic bias (e.g., for a line of detectors CF = 1.4 may be appropriate). The default distribution = 'poisson' is for Poisson-distributed N and n. To adjust the prediction for fixed N (binomial n) use distribution = 'binomial' (see ../doc/secrdesign-tools.pdf Appendix 2).

From 2.7.0, the first argument of minnrRSE may also be the output from GAoptim.

En2 is defined for detectors ‘multi’, ‘proximity’ and ‘count’.

Qpm returns the optimisation criteria Q_p and Q_{p_m} of Dupont et al. (2021), defined only for ‘proximity’ and ‘count’ detectors. The criteria are mask-dependent, and En2 is generally preferred. For ‘proximity’ and ‘count’ detectors the following expressions give the same result:

En2(D, trp, msk, dp)

Qpm(D, trp, msk, dp) * maskarea(msk) * D

given constant density ‘D’, detectors ‘trp’, mask ‘msk’ and detection parameters ‘dp’.

Value

Lambda – mask object with covariates ‘Lambda’ (\Lambda(x)), ‘sumpk’ and ‘sumq2’ (intermediate values for computation of expected counts - see ../doc/expectedcounts.pdf)

Enrm – numeric vector of length 3, the values of E(n), E(r) and E(m)

minnrRSE – rule-of-thumb RSE(D-hat) Efford and Boulanger (2019)

En2 – numeric vector comprising the values E(n) and E(number of animals detected at 2 or more sites)

Qpm – numeric vector comprising the criteria Q_p and Q_{p_m} of Dupont et al. (2021)

References

Dupont, G., Royle, J. A., Nawaz, M. A. and Sutherland, C. (2021) Optimal sampling design for spatial capture–recapture. Ecology 102 e03262.

Efford, M. G., and Boulanger, J. (2019) Fast evaluation of study designs for spatially explicit capture–recapture. Methods in Ecology and Evolution, 10, 1529–1535. DOI: 10.1111/2041-210X.13239

See Also

getdetectpar, optimalSpacing, scenarioSummary, GAoptim

Examples

tr <- traps(captdata)
detector(tr) <- "multi"
msk <- make.mask(tr, buffer = 100, type = 'trapbuffer')

L <- Lambda(tr, msk, list(lambda0 = 0.2, sigma = 20), 5)
nrm <- Enrm(D = 5, tr, msk, list(lambda0 = 0.2, sigma = 20), 5)
nrm

En2(D = 5, tr, msk, list(lambda0 = 0.2, sigma = 20), 5)

plot(L, cov = "Lambda", dots = FALSE)
plot(tr, add = TRUE)
mtext(side = 3,  paste(paste(names(nrm), round(nrm,1)), collapse = ", "))