CMP: The Conway-Maxwell-Poisson Distribution

In opisthokonta/goalmodel: Prediction Models For Goals Scored

The Conway-Maxwell-Poisson Distribution

Description

Functions to compute the probability (dCMP) and cumulative probability (pCMP) for the Conway-Maxwell-Poisson distribution, with rate parameter lambda and dispersion parameter upsilon. There is also a function for computing the expectation (eCMP) and for allowing a re-parameterization by the mean paramater mu (lambdaCMP).

Usage

dCMP(x, lambda, upsilon, log = FALSE, error = 0.01)

pCMP(q, lambda, upsilon, lower_tail = TRUE, error = 0.01)

qCMP(p, lambda, upsilon, lower_tail = TRUE, error = 0.01)

eCMP(lambda, upsilon, method = "sum", error = 0.01)

lambdaCMP(mu, upsilon, method = "sum", error = 0.01)

Arguments

x	vector of non-negative integers.
lambda	vector of non-negative rate parameters.
upsilon	vector of non-negative dispersion parameters.
log	logical; if TRUE, probabilities p are given as log(p).
error	numeric. Upper bound for the error in computing the normalizing constant.
q	vector of probabilities.
lower_tail	logical; if TRUE (default), probabilities are P[X ≤ x], otherwise, P[X > x].
p	vector of quantiles.
method	character; either 'fast' or 'sum'. See 'details'.
mu	vector of non-negative mean parameters.

Details

When upsilon is greater than 1, the variance is lower than the expectation, while it is larger than expectation when upsilon is in the 0-1 range. The variance equals expectation when upsilon = 1, and then the distribution is the same as the Poisson distribution.

There is no general closed form for the expectation for the CMP distribution, and so it must be approximated. Two methods are available for this. The 'fast' method uses the following approximation in Shmueli et al (2005) for the relationship between the parameters and the expectation:

mu = E(x) ~ lambda^(1/upsilon) - ((upsilon - 1) / (2*upsilon))

The 'sum' method relies on using a truncated sum from 0 to K, where K is determined using the error argument, as in Shmueli et al (2005). For the expectation the sum(x*p(x)) from x = (0, Inf) is approximated. For lambda, the equation sum((mu-x) * p(x)) = 0 is solved (see Huang 2017), with the left hand side sum is approximated.

References

• Shmueli et al (2005) A useful distribution for fitting discrete data: revival of the Conway–Maxwell–Poisson distribution

• Huang (2017) Mean-parametrized Conway–Maxwell–Poisson regression models for dispersed counts

See Also

upsilon.ml for a function for estimating the dispersion parameter.

opisthokonta/goalmodel documentation built on April 3, 2024, 1:32 a.m.