dmixture: Computing probability density function of the well-known...

In ForestFit: Statistical Modelling for Plant Size Distributions

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,{Θ}) = ∑_{j=1}^{K}ω_j f_j(x,θ_j),

where Θ=(θ_1,…,θ_K)^T, is the whole parameter vector, θ_j for j=1,…,K is the parameter space of the j-th component, i.e. θ_j=(α_j,β_j)^{T}, f_j(.,θ_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 ω=(ω_1,…,ω_K)^T where ω_js sum to one, i.e., ∑_{j=1}^{K}ω_j=1. Parameters α_j and β_j are the shape and scale parameters of the j-th component or both are the shape parameters. In the latter case, the parameters α and β are called the first and second shape parameters, respectively. We note that the constants ω_js sum to one, i.e. ∑_{j=1}^{K}ω_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,θ)=\frac{√{\frac{x}{β}}+√{\frac{β}{x}}}{2α x}φ \Biggl( \frac{√{\frac{x}{β}}-√{\frac{β}{x}}}{α}\Biggr),

• Burr XII

  f(x,θ)=α β x^{α-1} \Bigl(1+x^{α}\Bigr)^{-β-1},

• Chen

  f(x,θ)=α β x^{α}\exp\bigl(x^α\bigr) \exp\Bigl\{-β \exp\bigl(x^α\bigr)+β\Bigr\},

• F

  f(x,θ)=\frac{Γ\Bigl(\frac{α+β}{2}\Bigl)}{Γ\bigl(\frac{α}{2}\bigl) Γ\bigl(\frac{β}{2}\bigl)}\Bigl( \frac{α}{β}\Bigl)^{\frac{α}{2}} x^{\frac{α}{2}-1}\Big(1+\frac{α}{β}x\Big)^{-\frac{α+β}{2}},

• Frechet

  f(x,θ)=\frac{α}{ β} \Bigl( \frac {x}{β}\Bigr) ^{-α-1}\exp\Bigl\{ -\Bigl( \frac {x}{β}\Bigr)^{-α} \Bigr\},

• gamma

  f(x,θ)=\bigl[ β^α Γ(α)\bigr]^{-1} x^{α-1} \exp\Bigl( -\frac {x}{β}\Bigr),

• Gompertz

  f(x,θ)=β\exp\bigl(α x\bigr) \exp\Biggl\{\frac{β \exp\bigl(α x\bigr)-1}{α} \Biggr\},

• log-logistic

  f(x,θ)=\frac{ α}{ β^{α}} x^{α-1} ≤ft[ \Bigl( \frac {x}{β}\Bigr)^α +1\right]^{-2},

• log-normal

  f(x,θ)=\bigl(√{2π} β x \bigr)^{-1}\exp\biggl\{ -\frac {1}{2}≤ft( \frac {\log x-α}{β}\right) ^2\biggr\},

• Lomax

  f(x,θ)=\frac{α β}{(1+α x)^{β+1}},

• skew-normal

  f(x,θ)=2φ\Bigl(\frac{x-α}{β}\Bigr)Φ\Bigl(λ\frac{x-α}{β}\Bigr),

• Weibull

  f(x,θ)=\frac {α}{β} \Bigl( \frac {x}{β} \Bigr)^{α - 1}\exp\Bigl\{ -\Bigl( \frac {x}{β}\Bigr)^α \Bigr\},

where θ=(α,β). In the skew-normal case, φ(.) and Φ(.) 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 ω, α, β, and λ.

Details

For the skew-normal case, α, β, and λ 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)

ForestFit documentation built on March 7, 2023, 8:27 p.m.