EnvStats: LognormalMix – R documentation

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LognormalMix

Mixture of Two Lognormal Distributions

Description

Density, distribution function, quantile function, and random generation for a mixture of two lognormal distribution with parameters meanlog1, sdlog1, meanlog2, sdlog2, and p.mix.

Usage

dlnormMix(x, meanlog1 = 0, sdlog1 = 1, meanlog2 = 0, sdlog2 = 1, p.mix = 0.5)
  plnormMix(q, meanlog1 = 0, sdlog1 = 1, meanlog2 = 0, sdlog2 = 1, p.mix = 0.5) 
  qlnormMix(p, meanlog1 = 0, sdlog1 = 1, meanlog2 = 0, sdlog2 = 1, p.mix = 0.5) 
  rlnormMix(n, meanlog1 = 0, sdlog1 = 1, meanlog2 = 0, sdlog2 = 1, p.mix = 0.5)

Arguments

`x`	vector of quantiles.
`q`	vector of quantiles.
`p`	vector of probabilities between 0 and 1.
`n`	sample size. If `length(n)` is larger than 1, then `length(n)` random values are returned.
`meanlog1`	vector of means of the first lognormal random variable on the log scale. The default is `meanlog1=0`.
`sdlog1`	vector of standard deviations of the first lognormal random variable on the log scale. The default is `sdlog1=1`.
`meanlog2`	vector of means of the second lognormal random variable on the log scale. The default is `meanlog2=0`.
`sdlog2`	vector of standard deviations of the second lognormal random variable on the log scale. The default is `sdlog2=1`.
`p.mix`	vector of probabilities between 0 and 1 indicating the mixing proportion. For `rlnormMix` this must be a single, non-missing number.

Details

Let f(x; μ, σ) denote the density of a lognormal random variable with parameters meanlog=μ and sdlog=σ. The density, g, of a lognormal mixture random variable with parameters meanlog1=μ_1, sdlog1=σ_1, meanlog2=μ_2, sdlog2=σ_2, and p.mix=p is given by:

g(x; μ_1, σ_1, μ_2, σ_2, p) = (1 - p) f(x; μ_1, σ_1) + p f(x; μ_2, σ_2)

Value

dlnormMix gives the density, plnormMix gives the distribution function, qlnormMix gives the quantile function, and rlnormMix generates random deviates.

Note

A lognormal mixture distribution is often used to model positive-valued data that appear to be “contaminated”; that is, most of the values appear to come from a single lognormal distribution, but a few “outliers” are apparent. In this case, the value of meanlog2 would be larger than the value of meanlog1, and the mixing proportion p.mix would be fairly close to 0 (e.g., p.mix=0.1). The value of the second standard deviation (sdlog2) may or may not be the same as the value for the first (sdlog1).

Author(s)

Steven P. Millard (EnvStats@ProbStatInfo.com)

References

Gilliom, R.J., and D.R. Helsel. (1986). Estimation of Distributional Parameters for Censored Trace Level Water Quality Data: 1. Estimation Techniques. Water Resources Research 22, 135-146.

Johnson, N. L., S. Kotz, and A.W. Kemp. (1992). Univariate Discrete Distributions. Second Edition. John Wiley and Sons, New York, pp.53-54, and Chapter 8.

Johnson, N. L., S. Kotz, and N. Balakrishnan. (1994). Continuous Univariate Distributions, Volume 1. Second Edition. John Wiley and Sons, New York.

Examples

# Density of a lognormal mixture with parameters meanlog1=0, sdlog1=1, 
  # meanlog2=2, sdlog2=3, p.mix=0.5, evaluated at 1.5: 

  dlnormMix(1.5, meanlog1 = 0, sdlog1 = 1, meanlog2 = 2, sdlog2 = 3, p.mix = 0.5) 
  #[1] 0.1609746

  #----------

  # The cdf of a lognormal mixture with parameters meanlog1=0, sdlog1=1, 
  # meanlog2=2, sdlog2=3, p.mix=0.2, evaluated at 4: 

  plnormMix(4, 0, 1, 2, 3, 0.2) 
  #[1] 0.8175281

  #----------

  # The median of a lognormal mixture with parameters meanlog1=0, sdlog1=1, 
  # meanlog2=2, sdlog2=3, p.mix=0.2: 

  qlnormMix(0.5, 0, 1, 2, 3, 0.2) 
  #[1] 1.156891

  #----------

  # Random sample of 3 observations from a lognormal mixture with 
  # parameters meanlog1=0, sdlog1=1, meanlog2=3, sdlog2=4, p.mix=0.2. 
  # (Note: the call to set.seed simply allows you to reproduce this example.)

  set.seed(20) 
  rlnormMix(3, 0, 1, 2, 3, 0.2) 
  #[1] 0.08975283 1.07591103 7.85482514

EnvStats

Package for Environmental Statistics, Including US EPA Guidance

v2.4.0

GPL (>= 3)

Authors

Steven P. Millard [aut], Alexander Kowarik [ctb, cre] (<https://orcid.org/0000-0001-8598-4130>)

Initial release

2020-10-20