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Translate means and standard errors/deviations of lognormal distributions between log and linear space.

Source: R/uncertainty.R
lognormal-moments.Rd

logToLin makes the transformation from the moments of a lognormal distribution in log space to the moments of the same distribution in linear space.

Usage

logToLin(meanlog, sdlog, mslist)

linToLog(meanlin, sdlin, mslist)

mixedToLog(meanlin, sdlog, meanlog, sdlin)

Arguments

meanlog

The mean of the distribution on the log scale, with the same meaning as the meanlog argument to stats::rlnorm.

sdlog

The standard deviation of the distribution on the log scale, with the same meaning as the sdlog argument to stats::rlnorm.

mslist

Optionally, a list containing the meanlog and sdlog as the first and second elements. If this argument is specified, the first two arguments should be left missing.

meanlin

The mean of the lognormal distribution on the linear scale.

sdlin

The standard deviation of the lognormal distribution on the linear scale.

Details

Terminology

Let L be a sample from a lognormal distribution in linear space; for example, L=rlnorm(1000) could be a set of predicted flux rates in kg/day. Then let N be the corresponding set of predictions in log space, as they emerge from a model with a logged term on the left-hand side; for example, the output of predict(lm(log(L_obs) ~ ...)) where L_obs is the set of observed flux rates used to calibrate the lm model. N is related to L by N=log(L) or L=exp(N), but the parameters of their distributions are more complicated.

(It could also be that L is not a set of predicted flux rates but simply the uncertainty distribution of a single flux rate prediction. The same principles apply.)

Given the above definition,

  • meanlin = mean(L) = mean(exp(N)) = exp(mean(N) + sd(N)^2/2)

  • sdlin = sd(L) = sd(exp(N)) = sqrt((exp(sd(N)^2) - 1) * exp(2*mean(N) + sd(N)^2))

  • meanlog = mean(N) = mean(log(L)) = log(mean(L)^2/sqrt(sd(L)^2 + mean(L)^2))

  • sdlog = sd(N) = sd(log(L)) = sqrt(log(1 + sd(L)^2/mean(L)^2))

Note that meanlin does NOT simply equal exp(mean(N)), and sdlin does NOT equal exp(sd(N)); this function exists to apply those more complicated expressions on the right in the above list. Specifically, this function helps you convert from meanlog=mean(N), sdlog=sd(N) to an appropriate mean and sd for L in linear space.

Examples

# Explore with draws from a lognormally distributed sample
library(ggplot2)
logparams <- list(meanlog = 1, sdlog = 0.5)
loglin_data <- data.frame(rlnorm=rlnorm(1000, meanlog=logparams$meanlog, sdlog=logparams$sdlog))
ggplot(loglin_data, aes(x=log(rlnorm))) + geom_density() +
  geom_vline(xintercept=logparams$meanlog, color="red") +
  geom_vline(xintercept=logparams$meanlog+c(-1,1)*logparams$sdlog, color="blue")

linparams <- logToLin(ms=logparams)
ggplot(loglin_data, aes(x=rlnorm)) + geom_density() +
  geom_vline(xintercept=linparams$meanlin, color="red") +
  geom_vline(xintercept=linparams$meanlin+c(-1,1)*linparams$sdlin, color="blue")

#
# logToLin
logToLin(meanlog=1, sdlog=0.5)
#>    meanlin    sdlin
#> 1 3.080217 1.641572
logToLin(mslist=list(meanlog=1, sdlog=0.5))
#>    meanlin    sdlin
#> 1 3.080217 1.641572
logToLin(ms=linToLog(meanlin=3.080217, sdlin=1.641572))
#>    meanlin    sdlin
#> 1 3.080217 1.641572
#
# linToLog
linToLog(meanlin=1, sdlin=0.5)
#>      meanlog     sdlog
#> 1 -0.1115718 0.4723807
linToLog(mslist=list(meanlin=1, sdlin=0.5))
#>      meanlog     sdlog
#> 1 -0.1115718 0.4723807
linToLog(ms=logToLin(meanlog=-0.1115718, sdlog=0.4723807))
#>      meanlog     sdlog
#> 1 -0.1115718 0.4723807
#
# mixedToLog
linparams <- data.frame(meanlin=1, sdlin=0.5)
logparams <- linToLog(ms=linparams)
mixedToLog(meanlin=linparams$meanlin, sdlog=logparams$sdlog)
#>      meanlog     sdlog
#> 1 -0.1115718 0.4723807
if (FALSE) mixedToLog(meanlog=logparams$meanlog, sdlin=linparams$sdlin) # \dontrun{}
logToLin(ms=mixedToLog(meanlin=linparams$meanlin, sdlog=logparams$sdlog))
#>   meanlin sdlin
#> 1       1   0.5