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LogNormal distribution

The LogNormal distribution can be characterized by the exponent of its parameters:

  • exp(μ): the median
  • exp(σ): the multiplicative standard deviation $\sigma^*$.

Function σstar returns the multiplicative standard deviation.

A distribution can be specified by taking the log of median and $\sigma^*$

d = LogNormal(log(2), log(1.2))
σstar(d) == 1.2

Alternatively the distribution can be specified by its mean and either

  • Multiplicative standard deviation,$\sigma^*$, using type AbstractΣstar
  • Standard deviation at log-scale, $\sigma$, or
  • relative error, $cv$. 
d = fit(LogNormal, 2, Σstar(1.2))
(mean(d), σstar(d)) == (2, 1.2)
d = fit_mean_Σ(LogNormal, 2, 1.2)
(mean(d),  d.σ) == (2, 1.2)
d = fit_mean_relerror(LogNormal, 2, 0.2)
(mean(d), std(d)/mean(d)) .≈ (2, 0.2)

ScaledLogNormal: LogNormal(-x)

To support the use case of a distribution of strictly negative values, the fitting of a mirrored LogNormal on -x is supported.

There is a type-alias ScaledLogNormal = LocationScale{T, Continuous, LogNormal{T}} where T, denoting a scaled and shifted LogNormal distribution.

There are fitting function dispatched by this type that fit such a mirrored distribution.

d = fit_mean_Σ(ScaledLogNormal, -1, log(1.1))
(mean(d), σstar(d)) .≈ (-1.0, 1.1)
d = fit(ScaledLogNormal, -1.0, @qp_ll(-1.32), Val(:mode))
(mode(d), quantile(d, 0.025)) .≈ (-1.0, -1.32)

Note the usage of lower quantile for the mirrored distribution, here.

Detailed API

DistributionFits.σstar — Method
σstar(d)

Get the multiplicative standard deviation of LogNormal distribution d.

Arguments

  • d: The type of distribution to fit

Examples

d = LogNormal(2,log(1.2))
σstar(d) == 1.2
source
StatsAPI.fit — Method
fit(D, mean, σstar::AbstractΣstar)
fit_mean_Σ(D, mean, σ::Real)

Fit a statistical distribution of type D to mean and multiplicative standard deviation, σstar, or scale parameter at log-scale: σ.

Arguments

  • D: The type of distribution to fit
  • mean: The moments of the distribution
  • σstar::AbstractΣstar: The multiplicative standard deviation
  • σ: The standard-deviation parameter at log-scale

The first version uses type AbstractΣstar to distinguish from other methods of function fit.

Examples

d = fit(LogNormal, 2, Σstar(1.1));
(mean(d), σstar(d)) == (2, 1.1)
source
DistributionFits.AbstractΣstar — Type
Σstar <: AbstractΣstar

Represent the multiplicative standard deviation of a LogNormal distribution.

Supports dispatch of fit. Invoking the type as a function returns its single value.

Examples

a = Σstar(4.2)
a() == 4.2
source
DistributionFits.fit_mean_relerror — Function
fit_mean_relerror(D, mean, relerror)

Fit a distribution of type D to mean and relative error.

Arguments

  • D: The type of distribution to fit
  • mean: The first moment of the distribution
  • relerror: The relative error, i.e. the coefficient of variation

Examples

d = fit_mean_relerror(LogNormal, 10.0, 0.03);
(mean(d), std(d)/mean(d)) .≈ (10.0, 0.03)
source