dgumbel: calcuate the density of Gumbel copula

In MonkeyLB/TDR: Estimate the reproducibility of sequencing-based experiment

Description

compute the density of Gumbel copula given the two individual variables, i.g. sequencing read counts.

Usage

dgumbel(u, v, alpha)

Arguments

u	a vector containing cumulative densities for one replicate data.
v	a vector containing cumulative densities for the other replicate data.
alpha	the single paramter for the Gumbel copula. It has a range of (0, 1], perfect dependence is archieved if α approximates 0, while α = 0 implies no dependence.

Details

The formula for the distribution of bivariate Clayton copula is:

C(u, v | α) = exp(-((-log(u))^α+(-log(v))^α)^(1/α))

The formula for the corresponding density is:

c(u, v | α) = C(u, v | α)*(u*v)^(-1)*((-log(u))^α+ (-log(v))^α)^(-2+2/α)*(log(u)*log(v))^(α-1)* (1+(α-1)*((-log(u))^α+(-log(v))^α)^(-1/α))

where 0 < α ≤ 0 in our application.

Value

dgumbel calculates the density of Gumbel copula given the cumulative densities of two random variables and the parameter of Gumbel copula. The empirical cumulative densities of the two random variables could be obtained using the function empdist(). Invalid arguments will result in value NaN.

References

Nelsen, R. (2006). An Introduction to Copula, Second Edition, Springer. G. G. Venter (2001). Tails of copulas. In Proceedings ASTIN Washington, USA, pages 68-113.

Examples

data(Chipseq_TF)
x1 <- Chipseq_TF[,1]
x2 <- Chipseq_TF[,2]
U=empdist(x1, x2)
u <- U[,1]
v <- U[,2]
alpha <- 2
dgumbel(u, v, alpha)

MonkeyLB/TDR documentation built on May 7, 2019, 4:59 p.m.