R: Mixture of Normal Distributions

dist.Normal.Mixture {LaplacesDemon} | R Documentation

## 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).

### Details

- Application: Continuous Univariate

- Density: p(theta) = sum p[i] N(mu[i], sigma[i]^2)

- Inventor: Unknown

- Notation 1: theta ~ N(mu, sigma^2)

- Notation 2: p(theta) = 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[i]^(0.5)

- 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.

### Value

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

### Author(s)

Statisticat, LLC. software@bayesian-inference.com

### See Also

`ddirichlet` and `dnorm`.

### Examples

    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

[Package LaplacesDemon version 16.1.1 Index]