sads (version 0.6.3)

trueLL-methods: True likelihood for continuous variables

Description

Calculates the corrected likelihood for independent observations of a continuous variable that follows a given (truncated) density function, given a measurement precision.

Methods

signature(object = "fitsad", dist = "missing", coef = "missing", trunc = "missing", dec.places = "ANY", breaks="ANY", ...)

log-likelihood value of an abundance distribution model object fitted with function fitsad.

signature(object = "fitrad", dist = "missing", coef = "missing", trunc = "missing", dec.places = "ANY", breaks="ANY", ...)

log-likelihood value of an abundance distribution model object fitted with function fitrad.

signature(x = "numeric", dist = "character", coef = "list", trunc = "ANY", dec.places = "ANY", breaks="ANY", ...)

log-likelihood value of a vector of abundances object, fitted to a continuous distribution named by dist with parameters defined in coef.

signature(object = "histogram", dist = "character", coef = "list", trunc = "missing", dec.places = "ANY", breaks="ANY", ...)

log-likelihood value of the abundances given by histogram object, fitted to a continuous distribution named by dist with parameters defined in coef.

Arguments

object

vector or histogram of observed values, or an object of fitsad-class or fitrad-class

dist

character; root name of the continuous density distribution of the variable - e.g., lnorm for the lognormal distribution; gamma for the gamma distribution.

coef

named list; values of the coefficients of the continuous density distribution, in the same order and with the same names as in the probability function defined by the argument dist.

trunc

real number trunc <= min(x); value from which the density distribution is truncated. Currently only lower-tail truncation (or right-truncation) is implemented. If this argument is omitted the whole distribution is used (default).

dec.places

positive integer; number of decimal places used in the measurement of the observed values. Observed values will be rounded to this number of decimals. This argument defines the measurement precision, see details. If neither dec.places nor breaks is given, this defaults to zero.

breaks

A numeric vector giving the breakpoints between count classes. Either dec.places or breaks can be provided, but not both.

...

named arguments to be passed to the probability function defined by the argument dens.

Author

Paulo I. Prado and Andre Chalom

Details

The (log)likelihood function is often defined as any function proportional to the (sum) product of (log) probability density of the observations. In its original formulation, however, the likelihood is proportional to the product of probabilities, not probabilities densities (Fisher 1922, stressed by Lindsey 1999). For continuous variables, these probabilities are calculated through integration of the probability density function in an interval. For observed values of continuous variables, this interval is defined by the measurement precision. The likelihood function is thus any function proportional to

prod ( integral_(x-D)^(x+D) f(x) dx )

where x is the observed value, f(x) the density function and D is half the measurement precision:

D = 10^(-'dec.places')/2

For continuous variables aggregated in discrete classes, such as frequency tables of histograms, the probability of a given observation is

prod ( integral_L^U f(x) dx )

where L and U are the lower and upper limits of the classes, as defined by argument breaks.

Because products of probabilities quickly tends to zero, probabilities are usually converted to their logs and then summed to give the log-likelihood function.

References

Fisher, R.A. 1922. On the mathematical foundations of theoretical statistics. Phil. Trans. R. Soc. Lond. A 222: 309--368.

Lindsey, J.K. 1999. Some statistical heresies. The Statistician 48(1): 1--40.

See Also

logLik in the package MASS and logLik-methods in package bbmle.

Examples

##Random sample of a lognormal distribution with precision = 0.1
x <- round(rlnorm(500, meanlog = 3, sdlog = 0.5), 1)
## Log-likelihood given by fitdistr
library(MASS)
logLik(fitdistr(x, "lognormal"))
## Which is the sum of log of densities
sum( dlnorm(x, meanlog=mean(log(x)), sdlog=sd(log(x)), log=TRUE) )
## Correct log-likelihood
trueLL(x, "lnorm", coef=list(meanlog=mean(log(x)), sdlog=sd(log(x))), dec.places=1)
# Alternative invocation
trueLL(fitlnorm(x))

## Data in classes
xoc <- octav(x)
xc <- as.numeric(as.character(xoc$octave))
xb <- 2^(c(min(xc)-1, xc))
xh <- hist(x, breaks=xb, plot=FALSE)
xll <- trueLL(xh, dist="lnorm", coef = list(meanlog=mean(log(x)), sd=sd(log(x))))
xp <- diff(plnorm(xh$breaks, mean(log(x)), sd(log(x))))
xll2 <- sum( rep(log(xp), xh$counts))
all.equal(xll, xll2) # should be TRUE