- Conway-Maxwell-Poisson distribution
median =No closed form
mode =Not listed
skewness =Not listed
kurtosis =Not listed
entropy =Not listed
mgf =Not listed
char =Not listed
probability theoryand statistics, the Poisson distributionis the most common distribution used to model count data. However, due to the Poisson distribution's reliance on a single parameter, the mean, data exhibiting overdispersionor underdispersionunder the Poisson model must be modeled under alternative distributions to account for these statistically significant deviations. Typically, the negative binomial distributionis used to model data with over-dispersion, however the Conway-Maxwell-Poisson (CMP) distribution provides an improved, yet relatively unknown, alternative. In particular, the CMP distribution is suitable for over and under-dispersed data, and is a member of the exponential family, which has many favorable properties.
The CMP distribution was originally proposed by Conway and Maxwell in 1962 [Citation|last1=Conway| first1=R. W.| last2=Maxwell| first2= W. L.| year=1962| title=A queuing model with state dependent service rates|journal= Journal of Industrial Engineering| volume=12| pages=132–136] as a solution to handling queueing systems with state-dependent service rates. The probabilistic and statistical properties of the distribution were published by Shmueli et al. (2005) [Shmueli G., Minka T., Kadane J.B., Borle S., and Boatwright, P.B. "A useful distribution for fitting discrete data: revival of the Conway-Maxwell-Poisson distribution." Journal of the Royal Statistical Society: Series C (Applied Statistics) 54.1 (2005): 127-142. [http://dx.doi.org/10.1111/j.1467-9876.2005.00474.x] ] .
CMP is defined to be the distribution with
probability mass function
:for "x" = 0,1,2,... , and ≥ 0,where
The function serves as a normalization constant so the probability mass function sums to one. Note that does not have a closed form.
The additional parameter which does not appear in the
Poisson distributionallows for adjustment of the rate of decay. This rate of decay is a non-linear decrease in ratios of successive probabilities, specifically
When , the CMP distribution becomes the standard
Poisson distributionand as , the distribution approaches a Bernoulli distributionwith parameter . When the CMP distribution reduces to a geometric distributionwith probability of success .
For the CMP distribution, moments can be found through the recursive formula
There are a few methods of estimating the parameters of the CMP distribution from the data. Two methods will be discussed, the "quick and crude method" and the "accurate and intensive method".
Quick and crude method: weighted least squares
The "quick and crude method" provides a simple, efficient method to derive rough estimates of the parameters of the CMP distribution and determine if the distribution would be an appropriate model. Following the use of this method, an alternative method should be employed to compute more accurate estimates of the parameters if the model is deemed appropriate.
This method uses the relationship of successive probabilities as discussed above. By taking logarithms of both sides of this equation, the following linear relationship arises
where denotes . When estimating the parameters, the probabilities can be replaced by the
relative frequenciesof and . To determine if the CMP distribution is an appropriate model, these values should be plotted against for all ratios without zero counts. If the data appear to be linear, then the model is likely to be a good fit.
Once the appropriateness of the model is determined, the parameters can be estimated by fitting a regression of on . However, the basic assumption of
homoscedasticityis violated, so a weighted least squaresregression must be used. The inverse weight matrix will have the variances of each ratio on the diagonal with the one-step covariances on the first off-diagonal, both given below.
Accurate and intensive method: maximum likelihood
where and . Maximizing the likelihood yields the following two equations
which do not have an analytic solution.
maximum likelihoodestimates are approximated numerically by the Newton-Raphson method. In each iteration, the expectations, variances, and covariance of and are approximated by using the estimates for and from the previous iteration in the expression
This is continued until convergence of and .
Generalized Linear Model
The basic Conway-Maxwell Poisson (COM-Poisson) distribution discussed above has also been used as the basis for a
generalized linear model(GLM) using a Bayesian formulation. Guikema and Coffelt (2008) [Guikema, S.D. and J.P. Coffelt (2008) “A Flexible Count Data Regression Model for Risk Analysis,” Risk Analysis, Vol. 28, No. 1, pp. 213-223. [http://dx.doi.org/10.1111/j.1539-6924.2008.01014.x] ] developed a dual-link GLM based on the Conway-Maxwell Poisson distribution, and Lord et al. (2008) [Lord, D., S.D. Guikema, and S. Geedipally (2008) "Application of the Conway-Maxwell-Poisson Generalized Linear Model for Analyzing Motor Vehicle Crashes," Accident Analysis & Prevention, Vol. 40, No. 3, pp. 1123-1134. [http://dx.doi.org/10.1016/j.aap.2007.12.003] ] used this model evaluate traffic accident data. The COM-Poisson GLM developed by Guikema and Coffelt (2008) is based on a reformulation of the COM-Poisson distribution above, replacing with . The integral part of is then the mode of the distribution. Guikema and Coffelt (2008) and Lord et al. (2008) used a full Bayesian estimation approach with MCMC sampling implemented in WinBugswith non-informative priors for the regression parameters. This approach is computationally expensive, but it yields the full posterior distributions for the regression parameters and allows expert knowledge to be incorporated through the use of informative priors.
Galit Shmueli (University of Maryland) and Kimberly Sellers (Georgetown University) developed a classical GLM formulation for a CMP regression which generalizes Poisson regression and logistic regression. They take advantage of the exponential family properties of the CMP distribution to obtain elegant model estimation (via maximum-likelihood), inference, and interpretation. This approach requires substantially less computational time than the Bayesian approach of Guikema and Coffelt (2008), at the cost of not allowing expert knowledge to be incorporated into the model. The Sellers-Shmueli formulation yields standard errors for the regression parameters (via the Fisher Information matrix) compared to the full posterior distributions obtainable via the Guikema and Coffelt (2008) Bayesian formulation. It also devises a statistical test for testing for the level of dispersion compared to a Poisson model. Code for fitting a CMP-regression and testing for dispersion is available at http://www9.georgetown.edu/faculty/kfs7/research.
The two GLM frameworks developed for the COM-Poisson distribution significantly extend the usefulness of this distribution for data analysis problems.
* [http://cran.r-project.org/web/packages/compoisson/index.html Conway-Maxwell-Poisson distribution package for R (compoisson) by Jeffrey Dunn, part of Comprehensive R Archive Network (CRAN)]
* [http://alumni.media.mit.edu/~tpminka/software/compoisson Conway-Maxwell-Poisson distribution package for R (compoisson) by Tom Minka, third party package]
Wikimedia Foundation. 2010.
Look at other dictionaries:
Conway–Maxwell–Poisson distribution — Conway–Maxwell–Poisson parameters: support: pmf: cdf … Wikipedia
Maxwell–Boltzmann distribution — Maxwell–Boltzmann Probability density function Cumulative distribution function parameters … Wikipedia
Maxwell speed distribution — Classically, an ideal gas molecules bounce around with somewhat arbitrary velocities, never interacting with each other. In reality, however, an ideal gas is subjected to intermolecular forces. It is to be noted that the aforementioned classical… … Wikipedia
Compound Poisson distribution — In probability theory, a compound Poisson distribution is the probability distribution of the sum of a Poisson distributed number of independent identically distributed random variables. In the simplest cases, the result can be either a… … Wikipedia
Maxwell'sche Geschwindigkeitsverteilung — Maxwell Boltzmann Verteilung Parameter Definitionsbereich Wahrscheinlichkeitsdichte … Deutsch Wikipedia
Maxwell-Boltzmann-Geschwindigkeitsverteilung — Maxwell Boltzmann Verteilung Parameter Definitionsbereich Wahrscheinlichkeitsdichte … Deutsch Wikipedia
Maxwell-Verteilung — Maxwell Boltzmann Verteilung Parameter Definitionsbereich Wahrscheinlichkeitsdichte … Deutsch Wikipedia
Maxwell-Boltzmann-Verteilung — Parameter Definitionsbereich Wahrscheinlichkeitsdichte … Deutsch Wikipedia
Siméon Denis Poisson — Infobox Scientist box width = 300px name = Siméon Poisson image size = 200px caption = Siméon Denis Poisson (1781 1840) birth date = 21 June 1781 birth place = Pithiviers, France death date = 25 April 1840 death place = Sceaux, France residence … Wikipedia
Normal distribution — This article is about the univariate normal distribution. For normally distributed vectors, see Multivariate normal distribution. Probability density function The red line is the standard normal distribution Cumulative distribution function … Wikipedia