| Title: | Functions for the Logitnormal Distribution |
|---|---|
| Description: | Density, distribution, quantile and random generation function for the logitnormal distribution. Estimation of the mode and the first two moments. Estimation of distribution parameters. |
| Authors: | Thomas Wutzler |
| Maintainer: | Thomas Wutzler <[email protected]> |
| License: | GPL-2 |
| Version: | 0.8.39.9000 |
| Built: | 2024-10-29 05:32:26 UTC |
| Source: | https://github.com/bgctw/logitnorm |
Utilities for the logitnormal distribution in R
Density, distribution, quantile and random generation function.
Estimation of the mode and the first two moments.
Estimation of distribution parameters from observations.
The package provides the main distribution functions:
density dlogitnorm,
distribution plogitnorm,
quantile qlogitnorm, and
random generation function rlogitnorm.
Transformation functions
Moments and mode
Expected value and variance: momentsLogitnorm
Mode: modeLogitnorm
Estimating parameters
from mode and upper quantile: twCoefLogitnormMLE
from mode and constraint to be unimodal and maximally flat: twCoefLogitnormMLEFlat
from median and upper quantile: twCoefLogitnorm
from expected value, i.e. mean and upper quantile: twCoefLogitnormE
from a confidence interval which is symmetric at normal scale: twCoefLogitnormCi
from prescribed quantiles: twCoefLogitnormN
Have a look at the package vignettes.
Thomas Wutzler
Frederic, P. & Lad, F. (2008) Two Moments of the Logitnormal Distribution. Communications in Statistics-Simulation and Computation, 37, 1263-1269
Density function of logitnormal distribution
dlogitnorm(x, mu = 0, sigma = 1, log = FALSE, ...)dlogitnorm(x, mu = 0, sigma = 1, log = FALSE, ...)
x |
vector of quantiles |
mu |
scale distribution parameter |
sigma |
location distribution parameter |
log |
if TRUE, the log-density is returned |
... |
further arguments passed to |
density function: dlogitnorm
distribution function: plogitnorm
quantile function: qlogitnorm
random generation function: rlogitnorm
The function is only defined in interval (0,1), but the density returns 0 outside the support region.
Thomas Wutzler
Mode of the logitnormal distribution by numerical optimization
modeLogitnorm(mu, sigma, tol = invlogit(mu)/1000)modeLogitnorm(mu, sigma, tol = invlogit(mu)/1000)
mu |
parameter mu |
sigma |
parameter sigma |
tol |
precisions of the estimate |
Thomas Wutzler
First two moments of the logitnormal distribution by numerical integration
momentsLogitnorm(mu, sigma, abs.tol = 0, ...)momentsLogitnorm(mu, sigma, abs.tol = 0, ...)
mu |
parameter mu |
sigma |
parameter sigma |
abs.tol |
changing default to |
... |
further parameters to the |
named numeric vector with components
mean: expected value, i.e. first moment
var: variance, i.e. second moment
Thomas Wutzler
(res <- momentsLogitnorm(4,1)) (res <- momentsLogitnorm(5,0.1))(res <- momentsLogitnorm(4,1)) (res <- momentsLogitnorm(5,0.1))
Distribution function for logitnormal distribution
plogitnorm(q, mu = 0, sigma = 1, ...)plogitnorm(q, mu = 0, sigma = 1, ...)
q |
vector of quantiles |
mu |
location distribution parameter |
sigma |
scale distribution parameter |
... |
further arguments to pnorm |
Thomas Wutzler
Quantiles of logitnormal distribution.
qlogitnorm(p, mu = 0, sigma = 1, ...)qlogitnorm(p, mu = 0, sigma = 1, ...)
p |
vector of probabilities |
mu |
location distribution parameter |
sigma |
scale distribution parameter |
... |
further arguments to plogis |
Thomas Wutzler
Random number generation for logitnormal distribution
rlogitnorm(n, mu = 0, sigma = 1, ...)rlogitnorm(n, mu = 0, sigma = 1, ...)
n |
number of observations |
mu |
distribution parameter |
sigma |
distribution parameter |
... |
arguments to |
Thomas Wutzler
Estimating coefficients of logitnormal distribution from median and upper quantile
twCoefLogitnorm(median, quant, perc = 0.975)twCoefLogitnorm(median, quant, perc = 0.975)
median |
numeric vector: the median of the density function |
quant |
numeric vector: the upper quantile value |
perc |
numeric vector: the probability for which the quantile was specified |
numeric matrix with columns c("mu","sigma")
rows correspond to rows in median, quant, and perc
Thomas Wutzler
# estimate the parameters, with median at 0.7 and upper quantile at 0.9 med = 0.7; upper = 0.9 med = 0.2; upper = 0.4 (theta <- twCoefLogitnorm(med,upper)) x <- seq(0,1,length.out = 41)[-c(1,41)] # plotting grid px <- plogitnorm(x,mu = theta[1],sigma = theta[2]) #percentiles function plot(px~x); abline(v = c(med,upper),col = "gray"); abline(h = c(0.5,0.975),col = "gray") dx <- dlogitnorm(x,mu = theta[1],sigma = theta[2]) #density function plot(dx~x); abline(v = c(med,upper),col = "gray") # vectorized (theta <- twCoefLogitnorm(seq(0.4,0.8,by = 0.1),0.9)) .tmp.f <- function(){ # xr = rlogitnorm(1e5, mu = theta["mu"], sigma = theta["sigma"]) # median(xr) invlogit(theta["mu"]) qlogitnorm(0.975, mu = theta["mu"], sigma = theta["sigma"]) }# estimate the parameters, with median at 0.7 and upper quantile at 0.9 med = 0.7; upper = 0.9 med = 0.2; upper = 0.4 (theta <- twCoefLogitnorm(med,upper)) x <- seq(0,1,length.out = 41)[-c(1,41)] # plotting grid px <- plogitnorm(x,mu = theta[1],sigma = theta[2]) #percentiles function plot(px~x); abline(v = c(med,upper),col = "gray"); abline(h = c(0.5,0.975),col = "gray") dx <- dlogitnorm(x,mu = theta[1],sigma = theta[2]) #density function plot(dx~x); abline(v = c(med,upper),col = "gray") # vectorized (theta <- twCoefLogitnorm(seq(0.4,0.8,by = 0.1),0.9)) .tmp.f <- function(){ # xr = rlogitnorm(1e5, mu = theta["mu"], sigma = theta["sigma"]) # median(xr) invlogit(theta["mu"]) qlogitnorm(0.975, mu = theta["mu"], sigma = theta["sigma"]) }
Calculates mu and sigma of the logitnormal distribution from lower and upper quantile, i.e. confidence interval.
twCoefLogitnormCi(lower, upper, perc = 0.975, sigmaFac = qnorm(perc), isTransScale = FALSE)twCoefLogitnormCi(lower, upper, perc = 0.975, sigmaFac = qnorm(perc), isTransScale = FALSE)
lower |
value at the lower quantile, i.e. practical minimum |
upper |
value at the upper quantile, i.e. practical maximum |
perc |
numeric vector: the probability for which the quantile was specified |
sigmaFac |
sigmaFac = 2 is 95% sigmaFac = 2.6 is 99% interval |
isTransScale |
if true lower and upper are already on logit scale |
named numeric vector: mu and sigma parameter of the logitnormal distribution.
Thomas Wutzler
mu = 2 sd = c(1,0.8) p = 0.99 lower <- l <- qlogitnorm(1 - p, mu, sd ) # p-confidence interval upper <- u <- qlogitnorm(p, mu, sd ) # p-confidence interval cf <- twCoefLogitnormCi(lower,upper, perc = p) all.equal( cf[,"mu"] , c(mu,mu) ) all.equal( cf[,"sigma"] , sd )mu = 2 sd = c(1,0.8) p = 0.99 lower <- l <- qlogitnorm(1 - p, mu, sd ) # p-confidence interval upper <- u <- qlogitnorm(p, mu, sd ) # p-confidence interval cf <- twCoefLogitnormCi(lower,upper, perc = p) all.equal( cf[,"mu"] , c(mu,mu) ) all.equal( cf[,"sigma"] , sd )
Estimating coefficients of logitnormal distribution from expected value, i.e. mean, and upper quantile.
twCoefLogitnormE(mean, quant, perc = c(0.975), method = "BFGS", theta0 = c(mu = 0, sigma = 1), returnDetails = FALSE, ...)twCoefLogitnormE(mean, quant, perc = c(0.975), method = "BFGS", theta0 = c(mu = 0, sigma = 1), returnDetails = FALSE, ...)
mean |
the expected value of the density function |
quant |
the quantile values |
perc |
the probabilities for which the quantiles were specified |
method |
method of optimization (see |
theta0 |
starting parameters |
returnDetails |
if TRUE, the full output of optim is returned with attribute resOptim |
... |
further arguments to optim |
named numeric matrix with estimated parameters of the logitnormal
distribution.
colnames: c("mu","sigma")
Thomas Wutzler
# estimate the parameters (thetaE <- twCoefLogitnormE(0.7,0.9)) x <- seq(0,1,length.out = 41)[-c(1,41)] # plotting grid px <- plogitnorm(x,mu = thetaE[1],sigma = thetaE[2]) #percentiles function plot(px~x); abline(v = c(0.7,0.9),col = "gray"); abline(h = c(0.5,0.975),col = "gray") dx <- dlogitnorm(x,mu = thetaE[1],sigma = thetaE[2]) #density function plot(dx~x); abline(v = c(0.7,0.9),col = "gray") z <- rlogitnorm(1e5, mu = thetaE[1],sigma = thetaE[2]) mean(z) # about 0.7 # vectorized (theta <- twCoefLogitnormE(mean = seq(0.4,0.8,by = 0.1),quant = 0.9))# estimate the parameters (thetaE <- twCoefLogitnormE(0.7,0.9)) x <- seq(0,1,length.out = 41)[-c(1,41)] # plotting grid px <- plogitnorm(x,mu = thetaE[1],sigma = thetaE[2]) #percentiles function plot(px~x); abline(v = c(0.7,0.9),col = "gray"); abline(h = c(0.5,0.975),col = "gray") dx <- dlogitnorm(x,mu = thetaE[1],sigma = thetaE[2]) #density function plot(dx~x); abline(v = c(0.7,0.9),col = "gray") z <- rlogitnorm(1e5, mu = thetaE[1],sigma = thetaE[2]) mean(z) # about 0.7 # vectorized (theta <- twCoefLogitnormE(mean = seq(0.4,0.8,by = 0.1),quant = 0.9))
Estimating coefficients of logitnormal distribution from mode and upper quantile
twCoefLogitnormMLE(mle, quant, perc = 0.999)twCoefLogitnormMLE(mle, quant, perc = 0.999)
mle |
numeric vector: the mode of the density function |
quant |
numeric vector: the upper quantile value |
perc |
numeric vector: the probability for which the quantile was specified |
numeric matrix with columns c("mu","sigma")
rows correspond to rows in mle, quant, and perc
Thomas Wutzler
# estimate the parameters, with mode 0.7 and upper quantile 0.9 mode = 0.7; upper = 0.9 (theta <- twCoefLogitnormMLE(mode,upper)) x <- seq(0,1,length.out = 41)[-c(1,41)] # plotting grid px <- plogitnorm(x,mu = theta[1],sigma = theta[2]) #percentiles function plot(px~x); abline(v = c(mode,upper),col = "gray"); abline(h = c(0.999),col = "gray") dx <- dlogitnorm(x,mu = theta[1],sigma = theta[2]) #density function plot(dx~x); abline(v = c(mode,upper),col = "gray") # vectorized (theta <- twCoefLogitnormMLE(mle = seq(0.4,0.8,by = 0.1),quant = upper)) # flat (theta <- twCoefLogitnormMLEFlat(mode))# estimate the parameters, with mode 0.7 and upper quantile 0.9 mode = 0.7; upper = 0.9 (theta <- twCoefLogitnormMLE(mode,upper)) x <- seq(0,1,length.out = 41)[-c(1,41)] # plotting grid px <- plogitnorm(x,mu = theta[1],sigma = theta[2]) #percentiles function plot(px~x); abline(v = c(mode,upper),col = "gray"); abline(h = c(0.999),col = "gray") dx <- dlogitnorm(x,mu = theta[1],sigma = theta[2]) #density function plot(dx~x); abline(v = c(mode,upper),col = "gray") # vectorized (theta <- twCoefLogitnormMLE(mle = seq(0.4,0.8,by = 0.1),quant = upper)) # flat (theta <- twCoefLogitnormMLEFlat(mode))
Estimating coefficients of a maximally flat unimodal logitnormal distribution given the mode
twCoefLogitnormMLEFlat(mle)twCoefLogitnormMLEFlat(mle)
mle |
numeric vector: the mode of the density function |
Thomas Wutzler
Estimating coefficients from a vector of quantiles and percentiles (non-vectorized).
twCoefLogitnormN(quant, perc = c(0.5, 0.975), method = "BFGS", theta0 = c(mu = 0, sigma = 1), returnDetails = FALSE, ...)twCoefLogitnormN(quant, perc = c(0.5, 0.975), method = "BFGS", theta0 = c(mu = 0, sigma = 1), returnDetails = FALSE, ...)
quant |
the quantile values |
perc |
the probabilities for which the quantiles were specified |
method |
method of optimization (see |
theta0 |
starting parameters |
returnDetails |
if TRUE, the full output of optim is returned instead of only entry par |
... |
further parameters passed to optim, e.g. |
named numeric vector with estimated parameters of the logitnormal distribution.
names: c("mu","sigma")
Thomas Wutzler
# experiment of re-estimation the parameters from generated observations thetaTrue <- c(mu = 0.8, sigma = 0.7) obsTrue <- rlogitnorm(thetaTrue["mu"],thetaTrue["sigma"], n = 500) obs <- obsTrue + rnorm(100, sd = 0.05) # some observation uncertainty plot(density(obsTrue),col = "blue"); lines(density(obs)) # re-estimate parameters based on the quantiles of the observations (theta <- twCoefLogitnorm( median(obs), quantile(obs,probs = 0.9), perc = 0.9)) # add line of estimated distribution x <- seq(0,1,length.out = 41)[-c(1,41)] # plotting grid dx <- dlogitnorm(x,mu = theta[1],sigma = theta[2]) lines( dx ~ x, col = "orange")# experiment of re-estimation the parameters from generated observations thetaTrue <- c(mu = 0.8, sigma = 0.7) obsTrue <- rlogitnorm(thetaTrue["mu"],thetaTrue["sigma"], n = 500) obs <- obsTrue + rnorm(100, sd = 0.05) # some observation uncertainty plot(density(obsTrue),col = "blue"); lines(density(obs)) # re-estimate parameters based on the quantiles of the observations (theta <- twCoefLogitnorm( median(obs), quantile(obs,probs = 0.9), perc = 0.9)) # add line of estimated distribution x <- seq(0,1,length.out = 41)[-c(1,41)] # plotting grid dx <- dlogitnorm(x,mu = theta[1],sigma = theta[2]) lines( dx ~ x, col = "orange")
Estimating coefficients of logitnormal distribution from mode and given mu
twSigmaLogitnorm(mle, mu = 0)twSigmaLogitnorm(mle, mu = 0)
mle |
numeric vector: the mode of the density function |
mu |
for mu = 0 the distribution will be the flattest case (maybe bimodal) |
For a mostly flat unimodal distribution use twCoefLogitnormMLE(mle,0)
numeric matrix with columns c("mu","sigma")
rows correspond to rows in mle and mu
Thomas Wutzler
mle <- 0.8 (theta <- twSigmaLogitnorm(mle)) # x <- seq(0,1,length.out = 41)[-c(1,41)] # plotting grid px <- plogitnorm(x,mu = theta[1],sigma = theta[2]) #percentiles function plot(px~x); abline(v = c(mle),col = "gray") dx <- dlogitnorm(x,mu = theta[1],sigma = theta[2]) #density function plot(dx~x); abline(v = c(mle),col = "gray") # vectorized (theta <- twSigmaLogitnorm(mle = seq(0.401,0.8,by = 0.1)))mle <- 0.8 (theta <- twSigmaLogitnorm(mle)) # x <- seq(0,1,length.out = 41)[-c(1,41)] # plotting grid px <- plogitnorm(x,mu = theta[1],sigma = theta[2]) #percentiles function plot(px~x); abline(v = c(mle),col = "gray") dx <- dlogitnorm(x,mu = theta[1],sigma = theta[2]) #density function plot(dx~x); abline(v = c(mle),col = "gray") # vectorized (theta <- twSigmaLogitnorm(mle = seq(0.401,0.8,by = 0.1)))