Description Usage Arguments Details Value Author(s) References See Also Examples

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

and `s`

.

1 2 3 4 |

`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]. |

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.

`dzipf`

gives the (log) density, `pzipf`

gives the (log)
distribution function, `qzipf`

gives the quantile function.

Paulo I Prado prado@ib.usp.br and Murilo Dantas Miranda.

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.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 | ```
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
``` |

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