Package 'MixedPoisson'

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

Package:	MixedPoisson
Type:	Package
Version:	1.0
Date:	2015-07-13
License:	GPL-2

Author(s)

Alicja Wolny-Dominiak and Michal Trzesiok

Maintainer: <[email protected]>

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.

---

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

t	the vector of values

Details

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

Value

nu	the estimates of $\nu$
ll.delta.max	the value of loglikehood

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

t	the vector of values

Details

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

Value

gamma	the estimates of $\gamma$
ll.gamma.max	the value of loglikehood

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

t	the vector of values

Details

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

Value

nu	the estimates of $\nu$
ll.nu.max	the value of loglikehood

Author(s)

Michal Trzesiok

Examples

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

---

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

theta	the vector of values
gamma.par	the parameter of Gamma distribution

Details

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

Value

Gamma.density(theta, nu)

the density – the vector of values

Author(s)

Michal Trzesiok

Examples

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

---

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

theta	the vector of values
delta	the parameter of inverse-Gaussian distribution

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

invGauss.density(theta, delta)

the density – the vector of values

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

variable	the vector of numbers
X	model matrix of the form $X=model.matrix(~regressor)$. In the model without regressor the $X$ sould be defined as $X=as.matrix(rep(1, length(variable)))$
offset	offset in GLM Poisson

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

lambda	$\hat \lambda=\hat \beta X$
beta	regressor parameters
glm	output of $\texttt{glm}$

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

variable	the vector of numbers
X	model matrix of the form $X=model.matrix(~regressor)$. In the model without regressor the $X$ sould be defined as $X=as.matrix(rep(1, length(variable)))$

Details

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

Value

lambda	$\hat \lambda=\hat \beta X$
beta	regressor parameters
glm	output of $\texttt{glm}$

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

gamma.par	$\gamma$ parameter
t	the vector of values

Details

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

Value

ll.gamma

the 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

delta	$\delta$ parameter
t	the vector of values

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

ll.invGauss

the 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

nu	$\nu$ parameter
t	the vector of values

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

ll.lognorm

the 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

theta	the vector of values
nu	the parameter of log-normal distribution

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

lognorm.density(theta, nu)

the density – the vector of values

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

variable	The count variable.
alpha.start	The starting value of the parameter alpha. Default to 1.
beta.start	The starting value of the parameter beta. Default to 0.3
epsylon	Default to epsylon = 10^(-8)

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

alpha	the parameter of mixing Gamma distribution
beta	the parameter of mixing Gamma distribution
theta	the value 1/beta
n.iter	the number of steps in EM algorithm

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

variable	The count variable.
p.start	The starting value of p parameter. Default to 0.1.
epsylon	Default to epsylon = 10^(-8)

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

p	the parameter of mixing Lindley distribution
n.iter	the number of steps in EM algorithm

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

variable	the vector of numbers
mixing	the name of mixing distribution – "Gamma", "lognorm", "invGauss"
lambda	$\lambda$ parameter in mixed Poisson model
gamma.par	$\gamma$ parameter in Gamma mixing distribution
nu	$\nu$ parameter in log-normal mixing distribution
delta	$\delta$ parameter in inverse-Gaussian mixing distribution
n	The integer value for the Laguerre quadrature. Default to 100

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

pseudo_values	pseudo values $t_1,...,t_n$
nominator	nominator in the formula
denominator	denominator in the formula

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)

---