R: Maximum Likelihood Estimation for Bivariate Poisson Lognormal...

MLE for bivariate Poisson lognormal distribution {poilog} | R Documentation

## Maximum Likelihood Estimation for Bivariate Poisson Lognormal Distribution

### Description

`bipoilogMLE` fits the bivariate Poisson lognormal distribution to data

### Usage

    bipoilogMLE(n1, n2 = NULL,
                startVals = c(mu1=1, mu2=1, sig1=2, sig2=2, rho=0.5),
                nboot = 0, zTrunc = TRUE, file = NULL,
                method = "BFGS", control = list(maxit=1000))

### Arguments

`n1` | a vector or a matrix with two columns of pairwise counts of observed individuals for each species
`n2` | if n1 is not given as a matrix, a vector of counts with same ordering of species as in argument n1
`startVals` | starting values of parameters
`nboot` | number of parametric bootstraps, defaults to zero
`zTrunc` | logical; if TRUE (default) the zero-truncated distribution is fitted
`file` | text file to hold copies of bootstrap estimates
`method` | method to use during optimization, see details
`control` | a list of control parameters for the optimization routine, see details

### Details

The function estimates the parameters `mu1`, `sig1`, `mu2`, `sig2` and `rho`. In cases of incomplete sampling the estimates of `mu1` and `mu2` will be confounded with the sampling intensities (see `rbipoilog`). Assuming sampling intensities ν1 and ν2, the estimates of the means are \code{mu1}+ \ln nu1 and \code{mu2}+ \ln nu2. Parameters `sig1`, `sig2` and `rho` can be estimated without any knowledge of sampling intensities. The parameters must be given starting values for the optimization procedure (default starting values are used if starting values are not specified in the function call).

A zero-truncated distribution (see `dbipoilog`) is assumed by default (`zTrunc = TRUE`). In cases where the number of zeros is known the `zTrunc` argument should be set to `FALSE`.

The function uses the optimization procedures in `optim` to obtain the maximum likelihood estimate. The `method` and `control` arguments are passed to `optim`, see the help page for this function for additional methods and control parameters.

The approximate fraction of species revealed by each sample is estimated as 1 minus the zero term of the univariate Poisson lognormal distribution: 1-q(0;\code{mu1},\code{sig1}) and 1-q(0;\code{mu2},\code{sig2}).

Parametric bootstrapping could be time consuming for large data sets. If argument `file` is specified, e.g. `file =`‘`C:\\myboots.txt`’, the matrix with bootstrap estimates are copied into a tab-seperated text-file providing extra backup. Bootstrapping is done by simulating new sets of observations conditioned on the observed number of species (see `rbipoilog`).

### Value

`par` | Maximum likelihood estimates of the parameters
`p` | Approximate fraction of species revealed by the samples for sample 1 and 2 respectively
`logLval` | Log likelihood of the data given the estimated parameters
`gof` | Godness of fit measure obtained by checking the rank of logLval against logLval's obtained during the bootstrap procedure, (`gof` < 0.05) or (`gof` > 0.95) indicates lack of fit
`boot` | A data frame containing the bootstrap replicates of parameters and logLval

### Author(s)

Vidar Grøtan vidar.grotan@bio.ntnu.no, Steinar Engen

### References

Engen, S., R. Lande, T. Walla and P. J. DeVries. 2002. Analyzing spatial structure of communities using the two-dimensional Poisson lognormal species abundance model. American Naturalist 160, 60-73.

### See Also

`optim`, `dbipoilog`, `rbipoilog`

### Examples

    ## simulate observations
    xy  <- rbipoilog(S=30,mu1=1,mu2=1,sig1=2,sig2=2,rho=0.5)

    ## obtain estimates of parameters
    est <- bipoilogMLE(xy)

    ## similar, but now with bootstrapping
    ## Not run: est <- bipoilogMLE(xy,nboot=10)

    ## change start values and request tracing information
    ## from optimization procedure
    est <- bipoilogMLE(xy,startVals=c(2,2,4,4,0.3),
           control=list(maxit=1000,trace=1, REPORT=1))

    ## effect of sampling intensity
    xy  <- rbipoilog(S=100,mu1=1,mu2=1,sig1=2,sig2=2,rho=0.5,nu1=0.5,nu2=0.5)
    est <- bipoilogMLE(xy)
    ## the expected estimates of mu1 and mu2 are now 1-log(0.5) = 0.3 (approximately)

[Package poilog version 0.4 Index]