Fit the N-mixture model of Royle (2004)

pcount(formula, data, K, mixture=c("P", "NB", "ZIP"),
starts, method="BFGS", se=TRUE, engine=c("C", "R", "TMB"), threads=1, ...)

## Arguments

formula Double right-hand side formula describing covariates of detection and abundance, in that order an unmarkedFramePCount object supplying data to the model. Integer upper index of integration for N-mixture. This should be set high enough so that it does not affect the parameter estimates. Note that computation time will increase with K. character specifying mixture: "P", "NB", or "ZIP". vector of starting values Optimization method used by optim. logical specifying whether or not to compute standard errors. Either "C", "R", or "TMB" to use fast C++ code, native R code, or TMB (required for random effects) during the optimization. Set the number of threads to use for optimization in C++, if OpenMP is available on your system. Increasing the number of threads may speed up optimization in some cases by running the likelihood calculation in parallel. If threads=1 (the default), OpenMP is disabled. Additional arguments to optim, such as lower and upper bounds

## Details

This function fits N-mixture model of Royle (2004) to spatially replicated count data.

See unmarkedFramePCount for a description of how to format data for pcount.

This function fits the latent N-mixture model for point count data (Royle 2004, Kery et al 2005).

The latent abundance distribution, $$f(N | \mathbf{\theta})$$ can be set as a Poisson, negative binomial, or zero-inflated Poisson random variable, depending on the setting of the mixture argument, mixture = "P", mixture = "NB", mixture = "ZIP" respectively. For the first two distributions, the mean of $$N_i$$ is $$\lambda_i$$. If $$N_i \sim NB$$, then an additional parameter, $$\alpha$$, describes dispersion (lower $$\alpha$$ implies higher variance). For the ZIP distribution, the mean is $$\lambda_i(1-\psi)$$, where psi is the zero-inflation parameter.

The detection process is modeled as binomial: $$y_{ij} \sim Binomial(N_i, p_{ij})$$.

Covariates of $$\lambda_i$$ use the log link and covariates of $$p_{ij}$$ use the logit link.

## Value

unmarkedFit object describing the model fit.

## Author

Ian Fiske and Richard Chandler

unmarkedFramePCount, pcountOpen, ranef, parboot

Royle, J. A. (2004) N-Mixture Models for Estimating Population Size from Spatially Replicated Counts. Biometrics 60, pp. 108--105.

Kery, M., Royle, J. A., and Schmid, H. (2005) Modeling Avaian Abundance from Replicated Counts Using Binomial Mixture Models. Ecological Applications 15(4), pp. 1450--1461.

Johnson, N.L, A.W. Kemp, and S. Kotz. (2005) Univariate Discrete Distributions, 3rd ed. Wiley.

## Examples


if (FALSE) {

# Simulate data
set.seed(35)
nSites <- 100
nVisits <- 3
x <- rnorm(nSites)               # a covariate
beta0 <- 0
beta1 <- 1
lambda <- exp(beta0 + beta1*x)   # expected counts at each site
N <- rpois(nSites, lambda)       # latent abundance
y <- matrix(NA, nSites, nVisits)
p <- c(0.3, 0.6, 0.8)            # detection prob for each visit
for(j in 1:nVisits) {
y[,j] <- rbinom(nSites, N, p[j])
}

# Organize data
visitMat <- matrix(as.character(1:nVisits), nSites, nVisits, byrow=TRUE)

umf <- unmarkedFramePCount(y=y, siteCovs=data.frame(x=x),
obsCovs=list(visit=visitMat))
summary(umf)

# Fit a model
fm1 <- pcount(~visit-1 ~ x, umf, K=50)
fm1

plogis(coef(fm1, type="det")) # Should be close to p

# Empirical Bayes estimation of random effects
(fm1re <- ranef(fm1))
plot(fm1re, subset=site %in% 1:25, xlim=c(-1,40))
sum(bup(fm1re))         # Estimated population size
sum(N)                  # Actual population size

# Real data
data(mallard)
mallardUMF <- unmarkedFramePCount(mallard.y, siteCovs = mallard.site,
obsCovs = mallard.obs)
(fm.mallard <- pcount(~ ivel+ date + I(date^2) ~ length + elev + forest, mallardUMF, K=30))
(fm.mallard.nb <- pcount(~ date + I(date^2) ~ length + elev, mixture = "NB", mallardUMF, K=30))

}