R: The Logarithmic Series Distribution

logseries {vcdExtra} | R Documentation

## The Logarithmic Series Distribution

### Description

The logarithmic series distribution is a long-tailed distribution introduced by Fisher etal. (1943) in connection with data on the abundance of individuals classified by species.

These functions provide the density, distribution function, quantile function and random generation for the logarithmic series distribution with parameter `prob`.

### Usage

    dlogseries(x, prob = 0.5, log = FALSE)

    plogseries(q, prob = 0.5, lower.tail = TRUE, log.p = FALSE)

    qlogseries(p, prob = 0.5, lower.tail = TRUE, log.p = FALSE, max.value = 10000)

    rlogseries(n, prob = 0.5)

### Arguments

`x, q` | vector of quantiles representing the number of events.
`prob` | parameter for the distribution, `0 < prob < 1`
`log, log.p` | ogical; if TRUE, probabilities `p` are given as `log(p)`
`lower.tail` | logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x].
`p` | vector of probabilities
`max.value` | maximum value returned by `qlogseries`
`n` | number of observations for `rlogseries`

### Details

The logarithmic series distribution with `prob` = p has density

p ( x ) = α p^x / x

for x = 1, 2, …, where α= -1 / \log(1 - p) and 0 < p <1. Note that counts `x==2` cannot occur.

### Value

`dlogseries` gives the density, `plogseries` gives the distribution function, `qlogseries` gives the quantile function, and `rlogseries` generates random deviates.

### Author(s)

Michael Friendly, using original code modified from the `gmlss.dist` package by Mikis Stasinopoulos.

### References

http://en.wikipedia.org/wiki/Logarithmic_distribution

Fisher, R. A. and Corbet, A. S. and Williams, C. B. (1943). The relation between the number of species and the number of individuals Journal of Animal Ecology, 12, 42-58.

### See Also

`Distributions`, ~~~

### Examples

    XL <-expand.grid(x=1:5, p=c(0.33, 0.66, 0.99))
    lgs.df <- data.frame(XL, prob=dlogseries(XL[,"x"], XL[,"p"]))
    lgs.df$p = factor(lgs.df$p)
    str(lgs.df)

    require(lattice)
    mycol <- palette()[2:4]
    xyplot( prob ~ x, data=lgs.df, groups=p,
    	xlab=list('Number of events (k)', cex=1.25),
    	ylab=list('Probability',  cex=1.25),
    	type='b', pch=15:17, lwd=2, cex=1.25, col=mycol,
    	key = list(
    					title = 'p',
    					points = list(pch=15:17, col=mycol, cex=1.25),
    					lines = list(lwd=2, col=mycol),
    					text = list(levels(lgs.df$p)),
    					x=0.9, y=0.98, corner=c(x=1, y=1)
    					)
    	)

    # random numbers
    hist(rlogseries(200, prob=.4), xlab='x')
    hist(rlogseries(200, prob=.8), xlab='x')

[Package vcdExtra version 0.7-1 Index]