Mixed Poisson Models

Description

The package provides functions, which support to fit parameters of different mixed Poisson models using the Expectation-Maximization (EM) algorithm of estimation, cf. (Ghitany et al., 2012, pp. 6848). In the model the assumptions are: conditional N|\theta is of distribution N|\theta \sim POIS(\lambda\theta), parameter \theta is a random variable distributed according to the density function f_{\theta}(\cdot), E[\theta]=1 and \lambda=\exp(\mathbf{x}_{i}'\mathbf{\boldsymbol \beta}) – the regression component. The E-step is carried out through the numerical integration using Laquerre quadrature. The M-step estimates the parameters \beta using GLM Poisson with pseudo values from E-step and mixing parameters using optimize function.

Details

Author(s)

Alicja Wolny-Dominiak and Michal Trzesiok

Maintainer: <alicja.wolny-dominiak@ue.katowice.pl>

References

Karlis, D. (2005). EM algorithm for mixed Poisson and other discrete distributions. Astin Bulletin, 35(01), 3-24. Ghitany, M. E., Karlis, D., Al-Mutairi, D. K., & Al-Awadhi, F. A. (2012). An EM algorithm for multivariate mixed Poisson regression models and its application. Applied Mathematical Sciences, 6(137), 6843-6856.

Gamma density

Description

The function returns the vector of values of density function for of Gamma distribution with one parameter \gamma.

Usage

Gamma.density(theta, gamma.par)

Arguments

Details

The pdf of Gamma is of the form f_\theta(\theta)=\frac{\gamma^\gamma}{\Gamma(\gamma)}\theta^{\gamma-1}\exp(-\gamma\theta)

Value

Author(s)

Michal Trzesiok

Examples

Gamma.density(c(2,3,5,4,6,7,4), 5)

Estimation of delta parameter of inverse-Gaussian distribution

Description

The function estimates the value of the parameter delta using \texttt{optimize}.

Usage

est.delta(t)

Arguments

Details

The form of the distribution is as in the function \texttt{ll.invGauss}

Value

Author(s)

Michal Trzesiok

Examples

est.delta(t=c(3,8))

Estimation of gamma parameter of Gamma distribution

Description

The function estimates the value of the parameter gamma using \texttt{optimize}.

Usage

est.gamma(t)

Arguments

Details

The form of the distribution is as in the function \texttt{ll.gamma}

Value

Author(s)

Michal Trzesiok

Examples

est.gamma(t=c(3,8))

Estimation of nu parameter of log-normal distribution

Description

The function estimates the value of the parameter nu using \texttt{optimize}.

Usage

est.nu(t)

Arguments

Details

The form of the distribution is as in the function \texttt{ll.lognorm}

Value

Author(s)

Michal Trzesiok

Examples

est.nu(t=c(3,8))

inverse-Gaussian Density

Description

The function returns the vector of values of density function for of inverse-Gaussian distribution with one parameter \delta.

Usage

invGauss.density(theta, delta)

Arguments

Details

The pdf of inverse-Gaussian is of the form f_\theta(\theta)=\frac{\delta}{2\pi}\exp(\delta^2)\theta^{-\frac{3}{2}} \exp(-\frac{\delta^2}{2}(\frac{1}{\theta}+\theta))

Value

Author(s)

Michal Trzesiok

Examples

invGauss.density(c(2,3,5,4,6,7,6), 5)

Estimation of Lambda in M-step – Expectation-Maximization (EM) algorithm

Description

The function fits the GLM Poisson with given offset.

Usage

lambda_m_step(variable, X, offset)

Arguments

Details

It fits the GLM Poisson, where variable \sim 1 and the ofsset is given as the vector of the variable's length. The results are used in M-step of EM algorithm, cf. [Karlis, 2012] pp. 6850.

Value

Author(s)

Alicja Wolny–Dominiak, Michal Trzesiok

Examples

set.seed(1234)
variable=rpois(50,4)
X=as.matrix(rep(1, length(variable)))
t=pseudo_values(variable, mixing=c("invGauss"), lambda=4, delta=1, n=100)
lambda_m_step(variable, X, offset=t$pseudo_values)

Estimation of starting lambda in Expectation-Maximization (EM) algorithm

Description

The function fits the GLM Poisson without regressors.

Usage

lambda_start(variable, X)

Arguments

Details

It fits the GLM Poisson, where variable \sim 1. The results are taken as the starting value of EM algorithm.

Value

Author(s)

Alicja Wolny–Dominiak, Michal Trzesiok

Examples

set.seed(1234)
variable=rpois(50,4)
X=as.matrix(rep(1, length(variable)))
t=pseudo_values(variable, mixing=c("invGauss"), lambda=4, delta=1, n=100)
lambda_m_step(variable, X, offset=t$pseudo_values)

Gamma Log-likelihood

Description

The function returns the value of log-likelihood function for of Gamma distribution with one parameter \gamma.

Usage

ll.gamma(gamma.par, t)

Arguments

Details

The pdf of Gamma is of the form f_\theta(\theta)=\frac{\gamma^\gamma}{\Gamma(\gamma)}\theta^{\gamma-1}\exp(-\gamma\theta)

Value

Author(s)

Michal Trzesiok

Examples

ll.gamma(1, c(3,8))

Inverse-Gaussian Log-likelihood

Description

The function returns the value of log-likelihood function for of inverse-Gaussian distribution with one parameter \delta.

Usage

ll.invGauss(delta, t)

Arguments

Details

The pdf of inverse-Gaussian is of the form f_\theta(\theta)=\frac{\delta}{2\pi}\exp(\delta^2)\theta^{-\frac{3}{2}} \exp(-\frac{\delta^2}{2}(\frac{1}{\theta}+\theta))

Value

Author(s)

Michal Trzesiok

Examples

ll.invGauss(1, c(3,8))

Log-normal Log-likelihood

Description

The function returns the value of log-likelihood function of log-normal distribution with one parameter \nu.

Usage

ll.lognorm(nu, t)

Arguments

Details

The pdf of log-normal is of the form f_\theta(\theta)=\frac{1}{\sqrt{2\pi\nu\theta}}\exp[-\frac{(\log(\theta)+\frac{\nu^2}{2})^2}{2\nu^2}]

Value

Author(s)

Michal Trzesiok

Examples

ll.lognorm(1, c(3,8))

Log-normal Density

Description

The function returns the vector of values of density function for of log-normal distribution with one parameter \nu.

Usage

lognorm.density(theta, nu)

Arguments

Details

The pdf of log-normal is of the form f_\theta(\theta)=\frac{1}{\sqrt{2\pi\nu\theta}}\exp[-\frac{(\log(\theta)+\frac{\nu^2}{2})^2}{2\nu^2}]

Value

Author(s)

Michal Trzesiok

Examples

lognorm.density(c(2,3,5,4,6,7,6), 5)

Poisson-Gamma Distribution (Negative-Binomial)

Description

The function fits a mixed Poisson distribution, in which the random parameter follows Gamma distribution (the negative-binomial distribution). As teh method of estimation Expectation-maximization algorithm is used. In M-step the analytical formulas taken from [Karlis, 2005] are applied.

Usage

pg.dist(variable, alpha.start, beta.start, epsylon)

Arguments

Details

This function provides estimated parameters of the model N|\lambda \sim Poisson(\lambda) where \lambda parameter is also a random variable follows Gamma distribution with hiperparameters \alpha, \beta. The pdf of Gamma is of the form f_\lambda(\lambda)=\frac{\lambda^{\alpha-1}\exp(-\beta\lambda)\beta^\lambda}{\Gamma(\alpha)} .

Value

References

Karlis, D. (2005). EM algorithm for mixed Poisson and other discrete distributions. Astin bulletin, 35(01), 3-24.

Examples

library(MASS)
pGamma1 = pg.dist(variable=quine$Days)
print(pGamma1)

Poisson-Lindley Distribution

Description

The function fits a mixed Poisson distribution, in which the random parameter follows Lindley distribution. As teh method of estimation Expectation-maximization algorithm is used.

Usage

pl.dist(variable, p.start, epsylon)

Arguments

Details

This function provides estimated parameters of the model N|\lambda \sim Poisson(\lambda) where \lambda parameter is also a random variable follows Lindley distribution with hiperparameter p. The pdf of Lindley is of the form f_\lambda(\lambda)=\frac{p^2}{p+1}(\lambda+1)\exp(-\lambda p) .

Value

References

Karlis, D. (2005). EM algorithm for mixed Poisson and other discrete distributions. Astin bulletin, 35(01), 3-24.

Examples

library(MASS)
pLindley = pl.dist(variable=quine$Days)
print(pLindley)

Pseudo values – Expectation-Maximization (EM) algorithm

Description

The function returns the pseudo values t_i defined as the conditional expectation E[\theta_i|k_1,...,k_n], where k_1,...,k_n are realizations of the count variable N.

Usage

pseudo_values(variable, mixing, lambda, gamma.par, nu, delta, n)

Arguments

Details

The function calculates the vector of pseudo values t_i=E[\theta_i|k_1,...,k_n] in E-step of EM algorithm. It applies the numerical integration using laguerre.quadrature in the nominator and the denominator of the formula

The proper parameter \gamma, \nu, \delta should be chosen according to the mixing distribution.

Value

Author(s)

Alicja Wolny–Dominiak, Michal Trzesiok

Examples

variable=rpois(30,4)
pseudo_values(variable, mixing="Gamma", lambda=4, gamma.par=0.7, n=100)