Open-population distance-sampling family

The unmarked distsampOpen model: a Dail-Madsen open N-mixture (as dyn_abun()) observed by distance sampling at each primary period – the open-population counterpart of the single-season gdistremoval(). Initial abundance N_1 ~ Poisson(lambda), then N_t = Binomial(N_{t-1}, omega) + Poisson(gamma) (apparent survival omega + recruitment gamma); at each primary period the detected birds are distance-sampled into bins.

Usage

distsamp_open(
  transect = c("line", "point"),
  cutpoints = NULL,
  K_max = NULL,
  mixture = c("poisson", "negbin", "zip", "zinb"),
  dynamics = c("constant", "notrend", "trend", "autoreg", "ricker", "gompertz")
)

Arguments

transect: Transect geometry: "line" (default) or "point".
cutpoints: Distance-bin edges, length dim(y)[2] + 1, strictly increasing and starting at >= 0.
K_max: Truncation for the latent abundance HMM. Defaults to 3 * max(period total) + 40.
mixture: Initial-abundance mixing distribution. "poisson" (default), "negbin" (negative binomial, Var(N_1) = lambda + lambda^2 / r, with an overdispersion r estimated jointly and reported as log_r), "zip" (zero-inflated Poisson) or "zinb" (zero-inflated negative binomial). A structural-zero site is never occupied across any primary period, so all its band counts are zero; the observed per-site marginal is the two-component mixture omega * 1{all zero} + (1 - omega) * L_open, with L_open the exact open-population distance marginal and omega an intercept-only structural-zero probability reported as zi_logit (distinct from omega, the survival arm). The negative-binomial size / zero-inflation is layered over the same forward-HMM marginal; the Poisson path is unchanged.
dynamics: Population-dynamics form for the transition N_t | N_{t-1} (following unmarked::distsampOpen()): "constant" (default) is N_t = Binomial(N_{t-1}, omega) + Poisson(gamma); "notrend" ties recruitment so the expected abundance is stationary (gamma = (1 - omega) * lambda, no free gamma arm); "trend" is an exponential-growth process N_t ~ Poisson(N_{t-1} * gamma) with no survival term (omega unused); "autoreg" adds density-dependent recruitment Poisson(N_{t-1} * gamma) on top of survival; "ricker" and "gompertz" are the density-regulated recruitment forms with an estimated carrying capacity K (reported as K, on the log scale, modelled through the omega = ~ ... formula slot) and a growth rate reported as r (modelled through the gamma = ~ ... slot). "constant" / "notrend" fit with the exact analytic gradient; the density-dependent forms (trend / autoreg / ricker / gompertz) use the exact forward-HMM marginal with a numeric gradient.

Value

A tobs_family object.

Details

The distance-band allocation is conditional on the period total detected, so it factors out of the abundance HMM (the gdistremoval() trick): the marginal is the dyn_abun() forward recursion with the detection probability set to the overall distance detection pdist, plus the per-period band multinomials.

Four site-level arms: log abundance lambda (the tobs() formula), log distance scale sigma (detection), logit survival omega (omega = ~ ...), and log recruitment gamma (gamma = ~ ...), the last two intercept-only by default.

Inputs

y is a 3D array [n_sites x n_bins x n_seasons] of per-distance-band counts at each primary period (secondary occasions absorbed into the period total).

References

Dail, D., Madsen, L. (2011). Models for estimating abundance from repeated counts of an open metapopulation. Biometrics 67, 577-587. Sollmann, R., Gardner, B., Chandler, R. B., Royle, J. A., Sillett, T. S. (2015). An open-population hierarchical distance sampling model. Ecology 96, 325-331.

Examples

f <- distsamp_open(transect = "line", cutpoints = c(0, 10, 20, 30, 40),
                   dynamics = "notrend")
f