Utilities for the logitnormal distribution in R

Description

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.

Details

The package provides the main distribution functions:

• density dlogitnorm,

• distribution plogitnorm,

• quantile qlogitnorm, and

• random generation function rlogitnorm.

Transformation functions

• (0,1) -> (-Inf,Inf): logit

• (-Inf,Inf) -> (0,1): invlogit

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.

Author(s)

Thomas Wutzler

References

Frederic, P. & Lad, F. (2008) Two Moments of the Logitnormal Distribution. Communications in Statistics-Simulation and Computation, 37, 1263-1269

dlogitnorm

Description

Density function of logitnormal distribution

Usage

dlogitnorm(x, mu = 0, sigma = 1, log = FALSE, 
    ...)

Arguments

Details

Logitnorm distribution

• 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.

Author(s)

Thomas Wutzler

See Also

logitnorm

invlogit

Description

Transforming (-Inf,Inf) to original scale (0,1)

Usage

invlogit(q, ...)

Arguments

Details

function f(z) = \frac{e^{z}}{e^{z} + 1} \! = \frac{1}{1 + e^{-z}} \!

Author(s)

Thomas Wutzler

See Also

logit

logitnorm

logit

Description

Transforming (0,1) to normal scale (-Inf Inf)

Usage

logit(p, ...)

Arguments

Details

function logit(p) = log \left( \frac{p}{1-p} \right) = log(p) - log(1-p)

Author(s)

Thomas Wutzler

See Also

invlogit

logitnorm

modeLogitnorm

Description

Mode of the logitnormal distribution by numerical optimization

Usage

modeLogitnorm(mu, sigma, tol = invlogit(mu)/1000)

Arguments

Author(s)

Thomas Wutzler

See Also

logitnorm

momentsLogitnorm

Description

First two moments of the logitnormal distribution by numerical integration

Usage

momentsLogitnorm(mu, sigma, abs.tol = 0, 
    ...)

Arguments

Value

named numeric vector with components

• mean: expected value, i.e. first moment

• var: variance, i.e. second moment

Author(s)

Thomas Wutzler

Examples

(res <- momentsLogitnorm(4,1))
(res <- momentsLogitnorm(5,0.1))

plogitnorm

Description

Distribution function for logitnormal distribution

Usage

plogitnorm(q, mu = 0, sigma = 1, ...)

Arguments

Author(s)

Thomas Wutzler

See Also

logitnorm

qlogitnorm

Description

Quantiles of logitnormal distribution.

Usage

qlogitnorm(p, mu = 0, sigma = 1, ...)

Arguments

Author(s)

Thomas Wutzler

See Also

logitnorm

rlogitnorm

Description

Random number generation for logitnormal distribution

Usage

rlogitnorm(n, mu = 0, sigma = 1, ...)

Arguments

Author(s)

Thomas Wutzler

See Also

logitnorm

twCoefLogitnorm

Description

Estimating coefficients of logitnormal distribution from median and upper quantile

Usage

twCoefLogitnorm(median, quant, perc = 0.975)

Arguments

Value

numeric matrix with columns c("mu","sigma") rows correspond to rows in median, quant, and perc

Author(s)

Thomas Wutzler

See Also

logitnorm

Examples

# 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"])
}

twCoefLogitnormCi

Description

Calculates mu and sigma of the logitnormal distribution from lower and upper quantile, i.e. confidence interval.

Usage

twCoefLogitnormCi(lower, upper, perc = 0.975, 
    sigmaFac = qnorm(perc), isTransScale = FALSE)

Arguments

Value

named numeric vector: mu and sigma parameter of the logitnormal distribution.

Author(s)

Thomas Wutzler

See Also

logitnorm

Examples

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 )

twCoefLogitnormE

Description

Estimating coefficients of logitnormal distribution from expected value, i.e. mean, and upper quantile.

Usage

twCoefLogitnormE(mean, quant, perc = c(0.975), 
    method = "BFGS", theta0 = c(mu = 0, sigma = 1), 
    returnDetails = FALSE, ...)

Arguments

Value

named numeric matrix with estimated parameters of the logitnormal distribution. colnames: c("mu","sigma")

Author(s)

Thomas Wutzler

See Also

logitnorm

Examples

# 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))

twCoefLogitnormMLE

Description

Estimating coefficients of logitnormal distribution from mode and upper quantile

Usage

twCoefLogitnormMLE(mle, quant, perc = 0.999)

Arguments

Value

numeric matrix with columns c("mu","sigma") rows correspond to rows in mle, quant, and perc

Author(s)

Thomas Wutzler

See Also

logitnorm

Examples

# 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))

twCoefLogitnormMLEFlat

Description

Estimating coefficients of a maximally flat unimodal logitnormal distribution given the mode

Usage

twCoefLogitnormMLEFlat(mle)

Arguments

Author(s)

Thomas Wutzler

twCoefLogitnormN

Description

Estimating coefficients from a vector of quantiles and percentiles (non-vectorized).

Usage

twCoefLogitnormN(quant, perc = c(0.5, 0.975), 
    method = "BFGS", theta0 = c(mu = 0, sigma = 1), 
    returnDetails = FALSE, ...)

Arguments

Value

named numeric vector with estimated parameters of the logitnormal distribution. names: c("mu","sigma")

Author(s)

Thomas Wutzler

See Also

logitnorm

Examples

# 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")

twSigmaLogitnorm

Description

Estimating coefficients of logitnormal distribution from mode and given mu

Usage

twSigmaLogitnorm(mle, mu = 0)

Arguments

Details

For a mostly flat unimodal distribution use twCoefLogitnormMLE(mle,0)

Value

numeric matrix with columns c("mu","sigma") rows correspond to rows in mle and mu

Author(s)

Thomas Wutzler

See Also

logitnorm

Examples

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)))