Binned distance-sampling family

Latent abundance N_i ~ Poisson(lambda_i) (or negative binomial) in a covered region, observed through a half-normal or hazard-rate detection function over distance bins. With B bins the detected counts are multinomial over (bin 1, ..., bin B, undetected) with cell probabilities pi_b = integral_bin g(x; sigma) f(x) dx and 1 - sum_b pi_b, where f(x) is the distance density (uniform for a line transect, proportional to distance for a point transect). The latent N is summed out in closed form (truncation K_max), so the fit is a direct Laplace approximation (no EM), with a NUTS path over the same marginal; the per-bin detection integrals are evaluated by Gauss-Legendre quadrature.

Usage

distance(
  key = c("halfnorm", "hazard"),
  transect = c("line", "point"),
  cutpoints = NULL,
  K_max = NULL,
  mixture = c("poisson", "negbin")
)

Arguments

key

detection-function key. "halfnorm" (default) or "hazard" (the hazard-rate shape b is estimated as a scalar, reported as log_shape).

transect

"line" (default; perpendicular distances uniform) or "point" (radial distances, density proportional to distance).

cutpoints

numeric distance-bin edges, length n_bins + 1 (0 = c_0 < c_1 < ... < c_B). Required.

K_max

upper bound for the latent-abundance marginal sum (the exact integration over N is truncated at K_max). NULL (default) lets the engine pick max(y) + 100 (matching unmarked::pcount()); raise it if a fit warns that the posterior over N puts mass on the boundary.

mixture

latent-abundance distribution, both fitted via tulpa's closed-form marginal Laplace engine. "poisson" is the default; "negbin" (negative binomial, Var(N) = lambda + lambda^2 / r) adds an overdispersion parameter r. Non-spatially the log size log_r is estimated jointly with the coefficients and reported with an SE; on the areal-spatial path (method = "nested_laplace") r is integrated over the outer hyperparameter grid and reported as a posterior mean / sd. Named mixture (after unmarked::pcount()) to avoid collision with the model-type family argument of tobs().

Value

A tobs_family object.

Details

The tobs() formula is the abundance (log lambda) model; detection is the site-level detection-scale (log sigma) model. The response y is an n_sites x n_bins integer matrix of per-bin detected counts. The bin edges and transect geometry travel with the family: distance(cutpoints = ...).

References

Buckland, S. T., Anderson, D. R., Burnham, K. P., Laake, J. L., Borchers, D. L., Thomas, L. (2001). Introduction to Distance Sampling. Oxford. Royle, J. A., Dawson, D. K., Bates, S. (2004). Modeling abundance effects in distance sampling. Ecology 85, 1591-1597.

Examples

# \donttest{
sim <- simulate_distance(N = 200, key = "halfnorm", transect = "line",
                         n_abund_covs = 1, n_sigma_covs = 1, seed = 1)
fit <- tobs(~ abund_cov1, data = sim$data,
            family = distance(key = "halfnorm", transect = "line",
                              cutpoints = sim$cutpoints),
            detection = ~ sigma_cov1, y = sim$y, method = "laplace")
summary(fit)
# }