dmixture {ForestFit}

Computing probability density function of the well-known mixture models

Description

Computes probability density function (pdf) of the mixture model. The general form for the pdf of the mixture model is given by

f(x,{\Theta}) = \sum_{j=1}^{K}\omega_j f_j(x,\theta_j),

where \Theta=(\theta_1,\dots,\theta_K)^T, is the whole parameter vector, \theta_j for j=1,\dots,K is the parameter space of the j-th component, i.e. \theta_j=(\alpha_j,\beta_j)^{T}, f_j(.,\theta_j) is the pdf of the j-th component, and known constant K is the number of components. The vector of mixing parameters is given by \omega=(\omega_1,\dots,\omega_K)^T where \omega_js sum to one, i.e., \sum_{j=1}^{K}\omega_j=1. Parameters \alpha_j and \beta_j are the shape and scale parameters of the j-th component or both are the shape parameters. In the latter case, the parameters \alpha and \beta are called the first and second shape parameters, respectively. We note that the constants \omega_js sum to one, i.e. \sum_{j=1}^{K}\omega_j=1. The families considered for each component include Birnbaum-Saunders, Burr type XII, Chen, F, Frechet, Gamma, Gompertz, Log-normal, Log-logistic, Lomax, skew-normal, and Weibull with pdf given by the following.

• Birnbaum-Saunders

  f(x,\theta)=\frac{\sqrt{\frac{x}{\beta}}+\sqrt{\frac{\beta}{x}}}{2\alpha x}\phi \Biggl( \frac{\sqrt{\frac{x}{\beta}}-\sqrt{\frac{\beta}{x}}}{\alpha}\Biggr),

• Burr XII

  f(x,\theta)=\alpha \beta x^{\alpha-1} \Bigl(1+x^{\alpha}\Bigr)^{-\beta-1},

• Chen

  f(x,\theta)=\alpha \beta x^{\alpha}\exp\bigl(x^\alpha\bigr) \exp\Bigl\{-\beta \exp\bigl(x^\alpha\bigr)+\beta\Bigr\},

• F

  f(x,\theta)=\frac{\Gamma\Bigl(\frac{\alpha+\beta}{2}\Bigl)}{\Gamma\bigl(\frac{\alpha}{2}\bigl) \Gamma\bigl(\frac{\beta}{2}\bigl)}\Bigl( \frac{\alpha}{\beta}\Bigl)^{\frac{\alpha}{2}} x^{\frac{\alpha}{2}-1}\Big(1+\frac{\alpha}{\beta}x\Big)^{-\frac{\alpha+\beta}{2}},

• Frechet

  f(x,\theta)=\frac{\alpha}{ \beta} \Bigl( \frac {x}{\beta}\Bigr) ^{-\alpha-1}\exp\Bigl\{ -\Bigl( \frac {x}{\beta}\Bigr)^{-\alpha} \Bigr\},

• gamma

  f(x,\theta)=\bigl[ \beta^\alpha \Gamma(\alpha)\bigr]^{-1} x^{\alpha-1} \exp\Bigl( -\frac {x}{\beta}\Bigr),

• Gompertz

  f(x,\theta)=\beta\exp\bigl(\alpha x\bigr) \exp\Biggl\{\frac{\beta \exp\bigl(\alpha x\bigr)-1}{\alpha} \Biggr\},

• log-logistic

  f(x,\theta)=\frac{ \alpha}{ \beta^{\alpha}} x^{\alpha-1} \left[ \Bigl( \frac {x}{\beta}\Bigr)^\alpha +1\right]^{-2},

• log-normal

  f(x,\theta)=\bigl(\sqrt{2\pi} \beta x \bigr)^{-1}\exp\biggl\{ -\frac {1}{2}\left( \frac {\log x-\alpha}{\beta}\right) ^2\biggr\},

• Lomax

  f(x,\theta)=\frac{\alpha \beta}{(1+\alpha x)^{\beta+1}},

• skew-normal

  f(x,\theta)=2\phi\Bigl(\frac{x-\alpha}{\beta}\Bigr)\Phi\Bigl(\lambda\frac{x-\alpha}{\beta}\Bigr),

• Weibull

  f(x,\theta)=\frac {\alpha}{\beta} \Bigl( \frac {x}{\beta} \Bigr)^{\alpha - 1}\exp\Bigl\{ -\Bigl( \frac {x}{\beta}\Bigr)^\alpha \Bigr\},

where \theta=(\alpha,\beta). In the skew-normal case, \phi(.) and \Phi(.) are the density and distribution functions of the standard normal distribution, respectively.

Usage

dmixture(data, g, K, param)

Arguments

data

Vector of observations.

g

Name of the family including "birnbaum-saunders", "burrxii", "chen", "f", "Frechet", "gamma", "gompetrz", "log-normal", "log-logistic", "lomax", "skew-normal", and "weibull".

K

Number of components.

param

Vector of the \omega, \alpha, \beta, and \lambda.

Details

For the skew-normal case, \alpha, \beta, and \lambda are the location, scale, and skewness parameters, respectively.

Value

A vector of the same length as data, giving the pdf of the mixture model of families computed at data.

Author(s)

Mahdi Teimouri

Examples

data<-seq(0,20,0.1)
K<-2
weight<-c(0.6,0.4)
alpha<-c(1,2)
beta<-c(2,1)
param<-c(weight,alpha,beta)
dmixture(data, "weibull", K, param)