LaplacesDemon (version 16.1.1)

dist.Normal.Mixture: Mixture of Normal Distributions

Description

These functions provide the density, cumulative, and random generation for the mixture of univariate normal distributions with probability \(p\), mean \(\mu\) and standard deviation \(\sigma\).

Usage

dnormm(x, p, mu, sigma, log=FALSE)
pnormm(q, p, mu, sigma, lower.tail=TRUE, log.p=FALSE)
rnormm(n, p, mu, sigma)

Arguments

x,q

This is vector of values at which the density will be evaluated.

p

This is a vector of length \(M\) of probabilities for \(M\) components. The sum of the vector must be one.

n

This is the number of observations, which must be a positive integer that has length 1.

mu

This is a vector of length \(M\) that is the mean parameter \(\mu\).

sigma

This is a vector of length \(M\) that is the standard deviation parameter \(\sigma\), which must be positive.

lower.tail

Logical. This defaults to TRUE.

log,log.p

Logical. If TRUE, then probabilities \(p\) are given as \(\log(p)\).

Value

dnormm gives the density, pnormm returns the CDF, and rnormm generates random deviates.

Details

• Application: Continuous Univariate

• Density: \(p(\theta) = \sum p_i \mathcal{N}(\mu_i, \sigma^2_i)\)

• Inventor: Unknown

• Notation 1: \(\theta \sim \mathcal{N}(\mu, \sigma^2)\)

• Notation 2: \(p(\theta) = \mathcal{N}(\theta | \mu, \sigma^2)\)

• Parameter 1: mean parameters \(\mu\)

• Parameter 2: standard deviation parameters \(\sigma > 0\)

• Mean: \(E(\theta) = \sum p_i \mu_i\)

• Variance: \(var(\theta) = \sum p_i \sigma^{0.5}_i\)

• Mode:

A mixture distribution is a probability distribution that is a combination of other probability distributions, and each distribution is called a mixture component, or component. A probability (or weight) exists for each component, and these probabilities sum to one. A mixture distribution (though not these functions here in particular) may contain mixture components in which each component is a different probability distribution. Mixture distributions are very flexible, and are often used to represent a complex distribution with an unknown form. When the number of mixture components is unknown, Bayesian inference is the only sensible approach to estimation.

A normal mixture, or Gaussian mixture, distribution is a combination of normal probability distributions.

See Also

ddirichlet and dnorm.

Examples

# NOT RUN {
library(LaplacesDemon)
p <- c(0.3,0.3,0.4)
mu <- c(-5, 1, 5)
sigma <- c(1,2,1)
x <- seq(from=-10, to=10, by=0.1)
plot(x, dnormm(x, p, mu, sigma, log=FALSE), type="l") #Density
plot(x, pnormm(x, p, mu, sigma), type="l") #CDF
plot(density(rnormm(10000, p, mu, sigma))) #Random Deviates
# }