LognormalMixAlt {EnvStats} R Documentation

## Mixture of Two Lognormal Distributions (Alternative Parameterization)

### Description

Density, distribution function, quantile function, and random generation for a mixture of two lognormal distribution with parameters mean1, cv1, mean2, cv2, and p.mix.

### Usage

  dlnormMixAlt(x, mean1 = exp(1/2), cv1 = sqrt(exp(1) - 1),
mean2 = exp(1/2), cv2 = sqrt(exp(1) - 1), p.mix = 0.5)
plnormMixAlt(q, mean1 = exp(1/2), cv1 = sqrt(exp(1) - 1),
mean2 = exp(1/2), cv2 = sqrt(exp(1) - 1), p.mix = 0.5)
qlnormMixAlt(p, mean1 = exp(1/2), cv1 = sqrt(exp(1) - 1),
mean2 = exp(1/2), cv2 = sqrt(exp(1) - 1), p.mix = 0.5)
rlnormMixAlt(n, mean1 = exp(1/2), cv1 = sqrt(exp(1) - 1),
mean2 = exp(1/2), cv2 = sqrt(exp(1) - 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. mean1 vector of means of the first lognormal random variable. The default is meanlog1=sqrt(exp(1) - 1). cv1 vector of coefficient of variations of the first lognormal random variable. The default is sdlog1=sqrt(exp(1) - 1). mean2 vector of means of the second lognormal random variable. The default is mean2=sqrt(exp(1) - 1). cv2 vector of coefficient of variations of the second lognormal random variable. The default is sdlog2=sqrt(exp(1) - 1). p.mix vector of probabilities between 0 and 1 indicating the mixing proportion. For rlnormMixAlt this must be a single, non-missing number.

### Details

Let f(x; \eta, \theta) denote the density of a lognormal random variable with parameters mean=\eta and cv=\theta. The density, g, of a lognormal mixture random variable with parameters mean1=\eta_1, cv1=\theta_1, mean2=\eta_2, cv2=\theta_2, and p.mix=p is given by:

g(x; \eta_1, \theta_1, \eta_2, \theta_2, p) = (1 - p) f(x; \eta_1, \theta_1) + p f(x; \eta_2, \theta_2)

The default values for mean1 and cv1 correspond to a lognormal distribution with parameters meanlog=0 and sdlog=1. Similarly for the default values of mean2 and cv2.

### Value

dlnormMixAlt gives the density, plnormMixAlt gives the distribution function, qlnormMixAlt gives the quantile function, and rlnormMixAlt 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 mean2 would be larger than the value of mean1, and the mixing proportion p.mix would be fairly close to 0 (e.g., p.mix=0.1).

### 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 mean=2, cv1=3,
# mean2=4, cv2=5, p.mix=0.5, evaluated at 1.5:

dlnormMixAlt(1.5, mean1 = 2, cv1 = 3, mean2 = 4, cv2 = 5, p.mix = 0.5)
#[1] 0.1436045

#----------

# The cdf of a lognormal mixture with parameters mean=2, cv1=3,
# mean2=4, cv2=5, p.mix=0.5, evaluated at 1.5:

plnormMixAlt(1.5, mean1 = 2, cv1 = 3, mean2 = 4, cv2 = 5, p.mix = 0.5)
#[1] 0.6778064

#----------

# The median of a lognormal mixture with parameters mean=2, cv1=3,
# mean2=4, cv2=5, p.mix=0.5:

qlnormMixAlt(0.5, 2, 3, 4, 5, 0.5)
#[1] 0.6978355

#----------

# Random sample of 3 observations from a lognormal mixture with
# parameters mean1=2, cv1=3, mean2=4, cv2=5, p.mix=0.5.
# (Note: the call to set.seed simply allows you to reproduce this example.)

set.seed(20)
rlnormMixAlt(3, 2, 3, 4, 5, 0.5)
#[1]  0.70672151 14.43226313  0.05521329


[Package EnvStats version 2.8.1 Index]