## Geometric series distribution

### Description

Density, distribution function, quantile function and random generation for the Geometric Series distribution, with parameter k.

### Usage

dgs( x, k, S, log = FALSE )
pgs( q, k, S, lower.tail = TRUE, log.p = FALSE )
qgs( p, k, S, lower.tail = TRUE, log.p = FALSE )
rgs( n, k, 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. n number of random values to return. k positive real, 0 < k < 1; geometric series coefficient; the ratio between the abundances of i-th and (i+1)-th species. q vector of (non-negative integer) quantiles. In the context of species abundance distributions, a vector of abundance ranks of species in a sample. p vector of probabilities. S positive integer 0 < S < Inf, number of elements in a collection. In the context of species abundance distributions, the number of species in a sample. 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 Geometric series distribution gives the probability (or expected proportion of occurrences) of the i-th most abundant element in a collection:

p(i) = C * k * (1-k)^(i-1)

where C is a normalization constant which makes the summation of p(i) over S equals to one:

C = 1/(1 - (1-k)^S)

where S is the number of species in the sample.

Therefore, [dpq]gs can be used as rank-abundance model for species ranks in a sample or biological community see fitrad-class.

### Value

dgs gives the (log) density and pgs gives the (log) distribution function of ranks, and qgs gives the corresponding quantile function.

### Note

The Geometric series is NOT the same as geometric distribution. In the context of community ecology, the first can be used as a rank-abundance model and the former as a species-abundance model. See fitsad and fitrad and vignettes of sads package.

### References

Doi, H. and Mori, T. 2012. The discovery of species-abundance distribution in an ecological community. Oikos 122: 179–182.

May, R.M. 1975. Patterns of Species Abundance and Diversity. In Cody, M.L. and Diamond, J.M. (Eds) Ecology and Evolution of Communities. Harvard University Press. pp 81–120.

fitgs, fitrad to fit the Geometric series as a rank-abundance model.

### Examples

x <- 1:25
PDF <- dgs(x=x, k=0.1, S=25)
CDF <- pgs(q=x, k=0.1, S=25)
par(mfrow=c(1,2))
plot(x,CDF, ylab="Cumulative Probability", type="b",
main="Geometric series distribution, CDF")
plot(x,PDF, ylab="Probability, log-scale", type="h",
main="Geometric series distribution, PDF", log="y")
par(mfrow=c(1,1))

## quantile is the inverse of CDF
all.equal(qgs(CDF, k=0.1, S=25), x)