## Zipf distribution

### Description

Density, distribution function, quantile function and random generation for Zipf distribution with parameters `N` and `s`.

### Usage

```dzipf( x, N, s, log=FALSE)
pzipf( q, N, s, lower.tail=TRUE, log.p=FALSE)
qzipf( p, N, s, lower.tail = TRUE, log.p = FALSE)
rzipf( n, N, s)
```

### Arguments

 `x` vector of (non-negative integer) quantiles. In the context of species abundance distributions, this is a vector of abundance ranks of species in a sample. `q` vector of (non-negative integer) quantiles. In the context of species abundance distributions, a vector of abundance ranks of species in a sample. `n` number of random values to return. `p` vector of probabilities. `N` positive integer 0 < N < Inf, total number of elements of a collection. In the context of species abundance distributions, usually the number of species in a sample. `s` positive real s > 0; Zipf's exponent `log, log.p` logical; if TRUE, probabilities p are given as log(p). `lower.tail` logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x].

### Details

The Zipf distribution describes the probability or frequency of occurrence of a given element from a set of `N` elements. According to Zipf's law, this probability is inversely proportional to a power `s` of the frequency rank of the element in the set. The density function is

p(x) = ((x+v)^(-s)) / sum(((1:N)+v)^(-s))

Since p(x) is proportional to a power of `x`, the Zipf distribution is a power distribution. The Zeta distribution is a special case at the limit N -> Inf.

The Zipf distribution has a wide range of applications (Li 2011). One of its best known applications is describing the probability of occurrence of a given word that has a ranking `x` in a corpus with a total of `N` words. It can also be used to describe the probability of the abundance rank of a given species in a sample or assemblage of `N` species.

### Value

`dzipf` gives the (log) density, `pzipf` gives the (log) distribution function, `qzipf` gives the quantile function.

### References

Johnson N. L., Kemp, A. W. and Kotz S. (2005) Univariate Discrete Distributions, 3rd edition, Hoboken, New Jersey: Wiley. Section 11.2.20.

Li, W. (2002) Zipf's Law everywhere. Glottometrics 5:14-21

Zipf's Law. http://en.wikipedia.org/wiki/Zipf's_law.

`dzipf` and `rzipf` and related functions in zipfR package; `Zeta` for zeta distribution in VGAM package. `fitzipf` to fit Zipf distribution as a rank-abundance model.

### Examples

```x <- 1:20
PDF <- dzipf(x=x, N=100, s=2)
CDF <- pzipf(q=x, N=100, s=2)
par(mfrow=c(1,2))
plot(x,CDF, ylab="Cumulative Probability", type="b",
main="Zipf distribution, CDF")
plot(x,PDF, ylab="Probability", type="h",
main="Zipf distribution, PDF")
par(mfrow=c(1,1))

## quantile is the inverse of CDF
all.equal( qzipf(CDF, N=100, s=2), x) # should be TRUE

## Zipf distribution is discrete hence
all.equal( sum(dzipf(1:10, N=10, s=2)), pzipf(10, N=10, s=2)) # should be TRUE
```