Description

Estimate the expected and realised populations in a region, using a fitted spatially explicit capture–recapture model. Density is assumed to follow an inhomogeneous Poisson process in two dimensions. Expected N is the volume under a fitted density surface; realised N is the number of individuals within the region for the current realisation of the process (cf Johnson et al. 2010; see Note).

Usage

Arguments

Details

If the density surface of the fitted model is flat (i.e. object$model$D == ~1 or object$CL == TRUE) then E(N) is simply the density multiplied by the area of region, and the standard error is also a simple product. In the conditional likelihood case, the density and standard error are obtained by first calling derived.

If, on the other hand, the density has been modelled then the density surface is predicted at each point in region and E(N) is obtained by discrete summation. Pixel size may have a minor effect on the result - check by varying spacing. Sampling variance is determined by the delta method, using a numerical approximation to the gradient of E(N) with respect to each beta parameter.

The region may be defined as a mask object (if omitted, the mask component of object will be used). Alternatively, region may be a SpatialPolygonsDataFrame object (see package sp), and a raster mask will be constructed on the fly using the specified spacing. See make.mask for an example importing a shapefile to a SpatialPolygonsDataFrame.

Note: The option of specifying a polygon rather than a mask for region does not work if the density model in object uses spatial covariates: these must be passed in a mask.

Group-specific N has yet to be implemented.

Population size is adjusted automatically for the number of clusters in ‘mashed’ models (see mash). However, the population size reported is that associated with a single cluster unless regionmask is specified.

pooled.RN = TRUE handles the special case of a multi-session model in which the region of interest spans several patches (i.e., sampling in each session is localised within region. This is not yet fully implemented.

Setting ncores = NULL uses the existing value from the environment variable RCPP_PARALLEL_NUM_THREADS (see setNumThreads).

Use par.region.N to apply region.N in parallel to several models.

Value

If se.N = FALSE, the numeric value of expected population size, otherwise, a dataframe with rows ‘E.N’ and ‘R.N’, and columns as below.

For multiple sessions, the value is a list with one component per session, each component as above.

If keep.region = TRUE then the mask object for the region is saved as the attribute ‘region’ (see Examples).

The area in hectares of the region is saved as attribute ‘regionarea’.

Note

The estimates of expected and realised N are generally very similar, or identical, but realised N usually has lower estimated variance, especially if the n detected animals comprise a large fraction.

Realised N is given by R(N) = n + integral_over_B (1 - p.(X))D(X) dX (the second term represents undetected animals). This definition strictly holds only when region B is at least as large as the region of integration used to fit the model; only with this condition can we be sure all n detected animals have centres within B. The sampling variance of R(N), technically a mean square prediction error (Johnson et al. 2010), is approximated by summing the expected Poisson variance of the true number of undetected animals and a delta-method estimate of its sampling variance, obtained as for E(N).

By default, a shortcut is used to compute the sampling variance of realised N. With this option (RN.method = ‘poisson’) the sampling variance is the sampling variance of E(N) minus the estimate of E(N) (representing Poisson process variance). This has been found to give reliable confidence intervals in simulations (Efford and Fewster 2013).

If RN.method is neither ‘MSPE’ nor ‘poisson’ (ignoring case) then the estimate of expected N is also used for realised N, and the ‘poisson’ shortcut variance is used.

Johnson et al. (2010) use the notation μ(B) for expected N and N(B) for realised N in region B.

In our case, the relative SE (CV) of μ(B) is the same as that for the estimated density D if D has been estimated using the Poisson distribution option in secr.fit or derived(). If D has been estimated with the binomial distribution option, its relative SE for simple models will be the same as that of N(B), assuming that B is the full extent of the original mask.

References

Borchers, D. L. and Efford, M. G. (2008) Spatially explicit maximum likelihood methods for capture–recapture studies. Biometrics 64, 377–385.

Efford, M. G. and Fewster, R. M. (2013) Estimating population size by spatially explicit capture–recapture. Oikos 122, 918–928.

Johnson, D. S., Laake, J. L. and Ver Hoef, J. M. (2010) A model-based approach for making ecological inference from distance sampling data. Biometrics 66, 310–318.

See Also

secr.fit, derived, make.mask, expected.n, closedN

Examples