Editing Log-normal distribution (section)

=== Alternative parameterizations ===
In addition to the characterization by <math>\mu, \sigma</math> or <math>\mu^*, \sigma^*</math>, here are multiple ways how the log-normal distribution can be parameterized. [[ProbOnto]], the knowledge base and ontology of [[probability distribution]]s<ref>{{cite web | url = http://www.probonto.org | title = ProbOnto |access-date = 1 July 2017}}</ref><ref>{{cite journal | pmid = 27153608 | doi = 10.1093/bioinformatics/btw170 | pmc = 5013898 | volume = 32 | issue = 17 | pages = 2719–2721 | title = ProbOnto: ontology and knowledge base of probability distributions | year = 2016 | journal = Bioinformatics | last1 = Swat | first1 = MJ | last2 = Grenon | first2 = P | last3 = Wimalaratne | first3 = S}}</ref> lists seven such forms: [[File:LogNormal17.jpg|thumb|400px|Overview of parameterizations of the log-normal distributions.]]

* {{math|LogNormal1(''μ'',''σ'')}} with [[mean]], {{math|''μ''}}, and [[standard deviation]], {{math|''σ''}}, both on the log-scale <ref name="Forbes">Forbes et al. Probability Distributions (2011), John Wiley & Sons, Inc.</ref> <math display="block">P(x;\boldsymbol\mu,\boldsymbol\sigma) = \frac{1}{x \sigma \sqrt{2 \pi}} \exp\left[-\frac{(\ln x - \mu)^2}{2 \sigma^2}\right]</math>
* {{math|LogNormal2(''μ'',''υ'')}} with mean, {{math|''μ''}}, and variance, {{math|''υ''}}, both on the log-scale <math display="block">P(x;\boldsymbol\mu,\boldsymbol {v}) = \frac{1}{x \sqrt{v} \sqrt{2 \pi}} \exp\left[-\frac{(\ln x - \mu)^2}{2 v}\right]</math>
* {{math|LogNormal3(''m'',''σ'')}} with [[median]], {{math|''m''}}, on the natural scale and standard deviation, {{math|''σ''}}, on the log-scale<ref name="Forbes" /> <math display="block">P(x;\boldsymbol m,\boldsymbol \sigma) =\frac{1}{x \sigma \sqrt{2 \pi}} \exp\left[-\frac{\ln^2(x/m)}{2 \sigma^2}\right]</math>
* {{math|LogNormal4(''m'',cv)}} with median, {{math|''m''}}, and [[coefficient of variation]], {{math|cv}}, both on the natural scale <math display="block">P(x;\boldsymbol m,\boldsymbol {cv}) = \frac{1}{x \sqrt{\ln(cv^2+1)} \sqrt{2 \pi}} \exp\left[-\frac{\ln^2(x/m)}{2\ln(cv^2+1)}\right]</math>
* {{math|LogNormal5(''μ'',''τ'')}} with mean, {{math|''μ''}}, and [[Precision (statistics)|precision]], {{math|''τ''}}, both on the log-scale<ref>Lunn, D. (2012). The BUGS book: a practical introduction to Bayesian analysis. Texts in statistical science. CRC Press.</ref> <math display="block">P(x;\boldsymbol\mu,\boldsymbol \tau) = \sqrt{\frac{\tau}{2 \pi}} \frac{1}{x} \exp\left[-\frac{\tau}{2}(\ln x-\mu)^2\right]</math>
* {{math|LogNormal6(''m'',''σ<sub>g</sub>'')}} with median, {{math|''m''}}, and [[geometric standard deviation]], {{math|''σ<sub>g</sub>''}}, both on the natural scale<ref>{{cite journal | last1 = Limpert | first1 = E. | last2 = Stahel | first2 = W. A. | last3 = Abbt | first3 = M. | year = 2001 | title = Log-normal distributions across the sciences: Keys and clues | journal = BioScience | volume = 51 | issue = 5 | pages = 341–352 | doi = 10.1641/0006-3568(2001)051[0341:LNDATS]2.0.CO;2 | doi-access = free }}</ref> <math display="block">
P(x;\boldsymbol m,\boldsymbol {\sigma_g}) = \frac{1}{x \sqrt{2 \pi} \, \ln\sigma_g} \exp\left[-\frac{\ln^2(x/m)}{2 \ln^2(\sigma_g)}\right]</math>
* {{math|LogNormal7(''μ<sub>N</sub>'',''σ<sub>N</sub>'')}} with mean, {{math|''μ<sub>N</sub>''}}, and standard deviation, {{math|''σ<sub>N</sub>''}}, both on the natural scale<ref>{{cite journal | last1 = Nyberg | first1 = J. | display-authors = etal | year = 2012 | title = PopED – An extended, parallelized, population optimal design tool | journal = Comput Methods Programs Biomed | volume = 108 | issue = 2 | pages = 789–805 | doi = 10.1016/j.cmpb.2012.05.005 | pmid = 22640817 }}</ref> <math display="block">P(x;\boldsymbol {\mu_N},\boldsymbol {\sigma_N}) = \frac{1}{x \sqrt{2 \pi \ln\left(1+\sigma_N^2/\mu_N^2\right)}} \exp\left[-\frac{\left( \ln x - \ln\frac{\mu_N}{\sqrt{1 + \sigma_N^2/\mu_N^2}}\right)^2}{2 \ln\left(1 + \frac{\sigma_N^2}{\mu_N^2}\right)}\right]</math>

==== Examples for re-parameterization ====
Consider the situation when one would like to run a model using two different optimal design tools, for example PFIM<ref>{{cite journal | last1 = Retout | first1 = S | last2 = Duffull | first2 = S | last3 = Mentré | first3 = F | year = 2001 | title = Development and implementation of the population Fisher information matrix for the evaluation of population pharmacokinetic designs | journal = Comp Meth Pro Biomed | volume = 65 | issue = 2 | pages = 141–151 | doi = 10.1016/S0169-2607(00)00117-6 | pmid = 11275334 }}</ref> and PopED.<ref>The PopED Development Team (2016). PopED Manual, Release version 2.13. Technical report, Uppsala University.</ref> The former supports the LN2, the latter LN7 parameterization, respectively. Therefore, the re-parameterization is required, otherwise the two tools would produce different results.

For the transition <math>\operatorname{LN2}(\mu, v) \to \operatorname{LN7}(\mu_N, \sigma_N)</math> following formulas hold <math display="inline">\mu_N = \exp(\mu+v/2) </math> and <math display="inline">\sigma_N = \exp(\mu+v/2)\sqrt{\exp(v)-1}</math>.

For the transition <math>\operatorname{LN7}(\mu_N, \sigma_N) \to \operatorname{LN2}(\mu, v)</math> following formulas hold <math display="inline">\mu = \ln \mu_N - \frac{1}{2} v </math> and <math display="inline"> v = \ln(1+\sigma_N^2/\mu_N^2)</math>.

All remaining re-parameterisation formulas can be found in the specification document on the project website.<ref name="probontoWebsite">ProbOnto website, URL: http://probonto.org</ref>