polonoUC: The Poisson Lognormal Distribution

In VGAM: Vector Generalized Linear and Additive Models

The Poisson Lognormal Distribution

Description

Density, distribution function and random generation for the Poisson lognormal distribution.

Usage

dpolono(x, meanlog = 0, sdlog = 1, bigx = 170, ...)
ppolono(q, meanlog = 0, sdlog = 1,
        isOne = 1 - sqrt( .Machine$double.eps ), ...)
rpolono(n, meanlog = 0, sdlog = 1)

Arguments

x, q	vector of quantiles.
n	number of observations. If length(n) > 1 then the length is taken to be the number required.
meanlog, sdlog	the mean and standard deviation of the normal distribution (on the log scale). They match the arguments in Lognormal.
bigx	Numeric. This argument is for handling large values of x and/or when integrate fails. A first order Taylor series approximation [Equation (7) of Bulmer (1974)] is used at values of x that are greater or equal to this argument. For bigx = 10, he showed that the approximation has a relative error less than 0.001 for values of meanlog and sdlog “likely to be encountered in practice”. The argument can be assigned Inf in which case the approximation is not used.
isOne	Used to test whether the cumulative probabilities have effectively reached unity.
...	Arguments passed into integrate.

Details

The Poisson lognormal distribution is similar to the negative binomial in that it can be motivated by a Poisson distribution whose mean parameter comes from a right skewed distribution (gamma for the negative binomial and lognormal for the Poisson lognormal distribution).

Value

dpolono gives the density, ppolono gives the distribution function, and rpolono generates random deviates.

Note

By default, dpolono involves numerical integration that is performed using integrate. Consequently, computations are very slow and numerical problems may occur (if so then the use of ... may be needed). Alternatively, for extreme values of x, meanlog, sdlog, etc., the use of bigx = Inf avoids the call to integrate, however the answer may be a little inaccurate.

For the maximum likelihood estimation of the 2 parameters a VGAM family function called polono(), say, has not been written yet.

Author(s)

T. W. Yee. Some anonymous soul kindly wrote ppolono() and improved the original dpolono().

References

Bulmer, M. G. (1974). On fitting the Poisson lognormal distribution to species-abundance data. Biometrics, 30, 101–110.

See Also

lognormal, poissonff, negbinomial.

Examples

meanlog <- 0.5; sdlog <- 0.5; yy <- 0:19
sum(proby <- dpolono(yy, m = meanlog, sd = sdlog))  # Should be 1
max(abs(cumsum(proby) - ppolono(yy, m = meanlog, sd = sdlog)))  # 0?

## Not run:  opar = par(no.readonly = TRUE)
par(mfrow = c(2, 2))
plot(yy, proby, type = "h", col = "blue", ylab = "P[Y=y]", log = "",
     main = paste0("Poisson lognormal(m = ", meanlog,
                  ", sdl = ", sdlog, ")"))

y <- 0:190  # More extreme values; use the approxn & plot on a log scale
(sum(proby <- dpolono(y, m = meanlog, sd = sdlog, bigx = 100)))  # 1?
plot(y, proby, type = "h", col = "blue", ylab = "P[Y=y] (log)", log = "y",
     main = paste0("Poisson lognormal(m = ", meanlog,
                  ", sdl = ", sdlog, ")"))  # Note the kink at bigx

# Random number generation
table(y <- rpolono(n = 1000, m = meanlog, sd = sdlog))
hist(y, breaks = ((-1):max(y))+0.5, prob = TRUE, border = "blue")
par(opar) 
## End(Not run)

VGAM documentation built on Sept. 18, 2024, 9:09 a.m.