Log-normal distribution
Template:Short description {{#invoke:infobox3cols|infoboxTemplate | bodyclass = ib-prob-dist | templatestyles = Infobox probability distribution/styles.css | bodystyle = {{#if:|width: {{{box_width}}};}}
| title = Log-normal distribution | subheader =
| image1 = {{#if: Plot of the Lognormal PDF
Identical parameter <math> \mu </math> but differing parameters <math>\sigma</math>|
{{#invoke:InfoboxImage|InfoboxImage|image=Plot of the Lognormal PDF
Identical parameter <math> \mu </math> but differing parameters <math>\sigma</math>|alt=}}}}
| caption1 =
| image2 = {{#if: Plot of the Lognormal CDF
<math> \mu = 0 </math>|
{{#invoke:InfoboxImage|InfoboxImage|image=Plot of the Lognormal CDF
<math> \mu = 0 </math>|alt=}}}}
| caption2 =
| label1 = Notation | data1 = <math> \operatorname{Lognormal}\left( \mu,\,\sigma^2 \right) </math>
| label2 = Parameters | data2 = {{#if: | | Template:Plainlist }} | data2a = Template:Plainlist | data2b =
| label3 = Support | data3 = {{#if: | | <math> x \in ( 0, +\infty ) </math> }} | data3a = <math> x \in ( 0, +\infty ) </math> | data3b =
| label4 = {{#switch: continuous
| discrete | mass = PMF | multivariate | continuous | density = PDF | #default = Template:Error }}
| data4 = {{#if: | | <math> \frac{ 1 }{ x \sigma \sqrt{2\pi } } \exp\left( - \frac{ \left( \ln x - \mu \right)^2}{ 2 \sigma^2 } \right)</math> }} | data4a = <math> \frac{ 1 }{ x \sigma \sqrt{2\pi } } \exp\left( - \frac{ \left( \ln x - \mu \right)^2}{ 2 \sigma^2 } \right)</math> | data4b =
| label5 = CDF | data5 = {{#if: | | <math>\begin{align} &\frac{ 1 }{2}\left[1 + \operatorname{erf}\left( \frac{ \ln x - \mu }{\sigma\sqrt{2 }} | data5a = <math>\begin{align} &\frac{ 1 }{2}\left[1 + \operatorname{erf}\left( \frac{ \ln x - \mu }{\sigma\sqrt{2 | data5b =
| label6 = Quantile | data6 =
| label7 = Mean | data7 = {{#if: | | }} | data7a = | data7b =
| label8 = Median | data8 = {{#if: | | }} | data8a = | data8b =
| label9 = Mode | data9 = {{#if: | | }} | data9a = | data9b =
| label10 = {{#switch: continuous
| discrete | continuous | mass | density = Variance | multivariate = Variance | #default = Template:Error }}
| data10 = {{#if: | | }} | data10a = | data10b =
| label11 = MAD | data11 = {{#if: | | }} | data11a = | data11b =
| label12 = AAD | data12 = {{#if: | | }} | data12a = | data12b =
| label13 = Skewness | data13 = {{#if: | | }} | data13a = | data13b =
| label14 = Excess kurtosis | data14 = {{#if: | | }} | data14a = | data14b =
| label15 = Entropy | data15 = {{#if: | | }} | data15a = | data15b =
| label16 = MGF | data16 = {{#if: | | }} | data16a = | data16b =
| label17 = CF | data17 = {{#if: | | }} | data17a = | data17b =
| label18 = PGF | data18 = {{#if: | | }} | data18a = | data18b =
| label19 = Fisher information | data19 = {{#if: | | }} | data19a = | data19b =
| label20 = Kullback–Leibler divergence | data20 =
| label21 = Jensen-Shannon divergence | data21 =
| label22 = Method of moments | data22 = {{#if: | | }} | data22a = | data22b =
| label23 = Expected shortfall | data23 =
}}{{#invoke:Check for unknown parameters|check|unknown=Template:Main other|preview=Page using Template:Infobox probability distribution with unknown parameter "_VALUE_"|ignoreblank=y| box_width | bPOE | cdf | cdf_caption | cdf_image | cdf_image_alt | cdf2 | cf | cf2 | char | char2 | entropy | entropy2 | ES | fisher | fisher2 | intro | JSDdiv | KLDiv | kurtosis | kurtosis2 | mad | mad2 | aad | aad2 | mean | mean2 | median | median2 | mgf | mgf2 | mode | mode2 | moments | moments2 | name | notation | parameters | parameters2 | pdf | pdf_caption | pdf_image | pdf_image_alt | pdf2 | pgf | pgf2 | quantile | skewness | skewness2 | support | support2 | type | variance | variance2 }} \right)\right] \\[1ex] &= \Phi{\left(\frac{\ln x -\mu}{\sigma} \right)} \end{align}</math>
| quantile = <math>\begin{align}
&\exp\left( \mu + \sqrt{2\sigma^2}\operatorname{erf}^{-1}(2 p - 1) \right) \\[1ex] & = \exp(\mu + \sigma \Phi^{-1}(p)) \end{align}</math>
| mean = <math> \exp\left( \mu + \frac{\sigma^2}{2} \right) </math>
| median = <math> \exp( \mu ) </math>
| mode = <math> \exp\left( \mu - \sigma^2 \right) </math>
| variance = <math> \left[ \exp(\sigma^2) - 1 \right] \exp\left( 2 \mu + \sigma^2\right ) </math>
| skewness = <math> \left[ \exp\left( \sigma^2 \right) + 2 \right] \sqrt{\exp(\sigma^2) - 1 }</math>
| kurtosis = <math> \exp\left( 4 \sigma^2 \right) + 2 \exp\left( 3 \sigma^2 \right) + 3 \exp\left( 2\sigma^2 \right) - 6 </math>
| entropy = <math> \log_2 \left( \sqrt{2\pi e} \, \sigma e^{ \mu } \right) </math>
| mgf = defined only for numbers with a Template:Nowrap real part, see text
| char = representation <math> \sum_{n=0}^{\infty} \frac{ {\left(i t\right)}^n }{ n! }e^{ n \mu + n^2 \sigma^2/2} </math> is asymptotically divergent, but adequate for most numerical purposes
| fisher = <math> \frac{1}{\sigma^2} \begin{pmatrix} 1 & 0 \\ 0 & 2 \end{pmatrix} </math>
| moments = <math> \mu = \ln \operatorname{E}[X] - \frac{1}{2} \ln\left( \frac{ \operatorname{Var}[X] }{ \operatorname{E}[X]^2 } + 1 \right),</math>
<math> \sigma = \sqrt{ \ln \left( \frac{ \operatorname{Var}[X] }{ \operatorname{E}[X]^2 } + 1 \right) } </math>
| ES = <math>\begin{align}
&\frac{ e^{ \mu + \frac{ \sigma^2 }{2}} }{ 2p } \left[ 1 + \operatorname{erf} \left( \frac{ \sigma }{ \sqrt{2} } + \operatorname{erf}^{-1}(2p-1) \right) \right] \\[0.5ex]
&= \frac{e^{ \mu + \frac{ \sigma^2 }{2}}}{1-p} \left[1 - \Phi(\Phi^{-1}(p) - \sigma)\right]
\end{align}</math><ref name="norton">Template:Cite journal </ref>
}}
In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable Template:Mvar is log-normally distributed, then Template:Math has a normal distribution.<ref name=":1">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref name=":2">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> Equivalently, if Template:Mvar has a normal distribution, then the exponential function of Template:Mvar, Template:Math, has a log-normal distribution. A random variable which is log-normally distributed takes only positive real values. It is a convenient and useful model for measurements in exact and engineering sciences, as well as medicine, economics and other topics (e.g., energies, concentrations, lengths, prices of financial instruments, and other metrics).
The distribution is occasionally referred to as the Galton distribution or Galton's distribution, after Francis Galton.<ref name="JKB"/> The log-normal distribution has also been associated with other names, such as McAlister, Gibrat and Cobb–Douglas.<ref name="JKB"/>
A log-normal process is the statistical realization of the multiplicative product of many independent random variables, each of which is positive. This is justified by considering the central limit theorem in the log domain (sometimes called Gibrat's law). The log-normal distribution is the maximum entropy probability distribution for a random variate Template:Mvar—for which the mean and variance of Template:Math are specified.<ref>Template:Cite journal</ref>
DefinitionsEdit
Generation and parametersEdit
Let <math> Z </math> be a standard normal variable, and let <math>\mu</math> and <math>\sigma</math> be two real numbers, with Template:Nowrap Then, the distribution of the random variable
<math display="block"> X = e^{\mu + \sigma Z} </math>
is called the log-normal distribution with parameters <math>\mu</math> and Template:Nowrap These are the expected value (or mean) and standard deviation of the variable's natural logarithm, Template:Nowrap not the expectation and standard deviation of <math> X </math> itself.
This relationship is true regardless of the base of the logarithmic or exponential function: If <math>\log_a X </math> is normally distributed, then so is <math>\log_b X </math> for any two positive numbers Template:Nowrap Likewise, if <math> e^Y </math> is log-normally distributed, then so is Template:Nowrap where Template:Nowrap
In order to produce a distribution with desired mean <math>\mu_X</math> and variance Template:Nowrap one uses <math> \mu = \ln \frac{ \mu_X^2 }{ \sqrt{ \mu_X^2 + \sigma_X^2 } } </math> and Template:Nowrap
Alternatively, the "multiplicative" or "geometric" parameters <math> \mu^* = e^\mu </math> and <math> \sigma^* = e^\sigma </math> can be used. They have a more direct interpretation: <math> \mu^* </math> is the median of the distribution, and <math> \sigma^* </math> is useful for determining "scatter" intervals, see below.
Probability density functionEdit
A positive random variable <math> X </math> is log-normally distributed (i.e., Template:Nowrap if the natural logarithm of <math> X </math> is normally distributed with mean <math> \mu</math> and variance Template:Nowrap
<math display="block"> \ln X \sim \mathcal{N}(\mu,\sigma^2)</math>
Let <math> \Phi </math> and <math> \varphi </math> be respectively the cumulative probability distribution function and the probability density function of the <math> \mathcal{N}( 0, 1 ) </math> standard normal distribution, then we have that<ref name=":1"/><ref name="JKB"/> the probability density function of the log-normal distribution is given by:
<math display="block">\begin{align} f_X(x) & = \frac{d}{dx} \Pr\nolimits_X\left[ X \le x \right] \\[6pt] & = \frac{d}{dx} \Pr\nolimits_X\left[ \ln X \le \ln x \right] \\[6pt] & = \frac{d}{dx} \Phi{\left( \frac{ \ln x -\mu }{ \sigma } \right)} \\[6pt] & = \varphi{\left( \frac{\ln x - \mu} \sigma \right)} \frac{d}{dx} \left( \frac{ \ln x - \mu }{ \sigma }\right) \\[6pt] & = \varphi{\left( \frac{ \ln x - \mu }{ \sigma } \right)} \frac{ 1 }{ \sigma x } \\[6pt] & = \frac{ 1 }{ x \sigma\sqrt{2 \pi } } \exp\left( -\frac{ (\ln x-\mu)^2 }{2 \sigma^2} \right) ~. \end{align}</math>
Cumulative distribution functionEdit
The cumulative distribution function is
<math display="block"> F_X(x) = \Phi{\left( \frac{\ln x - \mu} \sigma \right)} </math>
where <math> \Phi </math> is the cumulative distribution function of the standard normal distribution (i.e., Template:Nowrap
This may also be expressed as follows:<ref name=":1" />
<math display="block"> \frac{1}{2} \left[ 1 + \operatorname{erf} \left(\frac{\ln x - \mu}{\sigma\sqrt{2}}\right) \right] = \frac12 \operatorname{erfc} \left(-\frac{\ln x - \mu}{\sigma\sqrt{2}}\right) </math>
where Template:Big is the complementary error function.
Multivariate log-normalEdit
If <math>\boldsymbol X \sim \mathcal{N}(\boldsymbol\mu,\,\boldsymbol\Sigma)</math> is a multivariate normal distribution, then <math>Y_i = \exp(X_i)</math> has a multivariate log-normal distribution.<ref>Template:Cite conference</ref><ref>Template:Cite conference</ref> The exponential is applied element-wise to the random vector <math>\boldsymbol X</math>. The mean of <math>\boldsymbol Y</math> is
<math display="block">\operatorname{E}[\boldsymbol Y]_i = e^{\mu_i + \frac{1}{2} \Sigma_{ii}} ,</math>
and its covariance matrix is
<math display="block">\operatorname{Var}[\boldsymbol Y]_{ij} = e^{\mu_i + \mu_j + \frac{1}{2}(\Sigma_{ii} + \Sigma_{jj}) } \left( e^{\Sigma_{ij}} - 1\right) . </math>
Since the multivariate log-normal distribution is not widely used, the rest of this entry only deals with the univariate distribution.
Characteristic function and moment generating functionEdit
All moments of the log-normal distribution exist and
<math display="block">\operatorname{E}[X^n] = e^{n\mu+n^2\sigma^2/2}</math>
This can be derived by letting <math display="inline">z = \tfrac{\ln x - \mu}{\sigma} - n \sigma</math> within the integral. However, the log-normal distribution is not determined by its moments.<ref name="Heyde">Template:Citation</ref> This implies that it cannot have a defined moment generating function in a neighborhood of zero.<ref>Template:Cite book</ref> Indeed, the expected value <math>\operatorname{E}[e^{t X}]</math> is not defined for any positive value of the argument <math>t</math>, since the defining integral diverges.
The characteristic function <math>\operatorname{E}[e^{i t X}]</math> is defined for real values of Template:Mvar, but is not defined for any complex value of Template:Mvar that has a negative imaginary part, and hence the characteristic function is not analytic at the origin. Consequently, the characteristic function of the log-normal distribution cannot be represented as an infinite convergent series.<ref name="Holgate">Template:Cite journal</ref> In particular, its Taylor formal series diverges:
<math display="block">\sum_{n=0}^\infty \frac{{\left(it\right)}^n}{n!} e^{n\mu + n^2\sigma^2/2}</math>
However, a number of alternative divergent series representations have been obtained.<ref name="Holgate" /><ref name="Barakat">Template:Cite journal</ref><ref name="Barouch">Template:Cite journal</ref><ref name="Leipnik">Template:Cite journal</ref>
A closed-form formula for the characteristic function <math>\varphi(t)</math> with <math>t</math> in the domain of convergence is not known. A relatively simple approximating formula is available in closed form, and is given by<ref name="Asmussen">S. Asmussen, J.L. Jensen, L. Rojas-Nandayapa (2016). "On the Laplace transform of the Lognormal distribution", Methodology and Computing in Applied Probability 18 (2), 441-458. Thiele report 6 (13).</ref>
<math display="block">\varphi(t) \approx \frac{\exp\left(-\frac{W^2(-it\sigma^2e^\mu) + 2W(-it\sigma^2e^\mu)}{2\sigma^2} \right)}{\sqrt{1 + W{\left(-it\sigma^2e^\mu\right)}}}</math>
where <math>W</math> is the Lambert W function. This approximation is derived via an asymptotic method, but it stays sharp all over the domain of convergence of <math>\varphi</math>.
PropertiesEdit
Probability in different domainsEdit
The probability content of a log-normal distribution in any arbitrary domain can be computed to desired precision by first transforming the variable to normal, then numerically integrating using the ray-trace method.<ref name="Das">Template:Cite journal</ref> (Matlab code)
Probabilities of functions of a log-normal variableEdit
Since the probability of a log-normal can be computed in any domain, this means that the cdf (and consequently pdf and inverse cdf) of any function of a log-normal variable can also be computed.<ref name="Das"/> (Matlab code)
Geometric or multiplicative momentsEdit
The geometric or multiplicative mean of the log-normal distribution is <math>\operatorname{GM}[X] = e^\mu = \mu^*</math>. It equals the median. The geometric or multiplicative standard deviation is <math>\operatorname{GSD}[X] = e^{\sigma} = \sigma^*</math>.<ref name="ReferenceA">Template:Cite journal</ref><ref>Template:Cite journal</ref>
By analogy with the arithmetic statistics, one can define a geometric variance, <math>\operatorname{GVar}[X] = e^{\sigma^2}</math>, and a geometric coefficient of variation,<ref name="ReferenceA" /> <math>\operatorname{GCV}[X] = e^{\sigma} - 1</math>, has been proposed. This term was intended to be analogous to the coefficient of variation, for describing multiplicative variation in log-normal data, but this definition of GCV has no theoretical basis as an estimate of <math>\operatorname{CV}</math> itself (see also Coefficient of variation).
Note that the geometric mean is smaller than the arithmetic mean. This is due to the AM–GM inequality and is a consequence of the logarithm being a concave function. In fact,<ref name="Acoustic Stimuli Revisited 2016">Template:Cite journal</ref>
<math display="block">\operatorname{E}[X] = e^{\mu + \frac12 \sigma^2} = e^{\mu} \cdot \sqrt{e^{\sigma^2}} = \operatorname{GM}[X] \cdot \sqrt{\operatorname{GVar}[X]}.</math>
In finance, the term <math>e^{-\sigma^2/2}</math> is sometimes interpreted as a convexity correction. From the point of view of stochastic calculus, this is the same correction term as in Itō's lemma for geometric Brownian motion.
Arithmetic momentsEdit
For any real or complex number Template:Mvar, the Template:Mvar-th moment of a log-normally distributed variable Template:Mvar is given by<ref name="JKB"/> <math display="block">\operatorname{E}[X^n] = e^{n\mu + \frac{1}{2}n^2\sigma^2}.</math>
Specifically, the arithmetic mean, expected square, arithmetic variance, and arithmetic standard deviation of a log-normally distributed variable Template:Mvar are respectively given by:<ref name=":1" />
<math display="block">\begin{align}
\operatorname{E}[X] & = e^{\mu + \tfrac{1}{2}\sigma^2}, \\[4pt]
\operatorname{E}[X^2] & = e^{2\mu + 2\sigma^2}, \\[4pt]
\operatorname{Var}[X] & = \operatorname{E}[X^2] - \operatorname{E}[X]^2
= {\left(\operatorname{E}[X]\right)}^2 \left(e^{\sigma^2} - 1\right) \\[2pt]
&= e^{2\mu + \sigma^2} \left(e^{\sigma^2} - 1\right), \\[4pt]
\operatorname{SD}[X] & = \sqrt{\operatorname{Var}[X]}
= \operatorname{E}[X] \sqrt{e^{\sigma^2} - 1} \\[2pt]
&= e^{\mu + \tfrac{1}{2}\sigma^2} \sqrt{e^{\sigma^2} - 1},
\end{align}</math>
The arithmetic coefficient of variation <math>\operatorname{CV}[X]</math> is the ratio <math>\tfrac{\operatorname{SD}[X]}{\operatorname{E}[X]}</math>. For a log-normal distribution it is equal to<ref name=":2" /> <math display="block">\operatorname{CV}[X] = \sqrt{e^{\sigma^2} - 1}.</math> This estimate is sometimes referred to as the "geometric CV" (GCV),<ref>Sawant, S.; Mohan, N. (2011) "FAQ: Issues with Efficacy Analysis of Clinical Trial Data Using SAS" Template:Webarchive, PharmaSUG2011, Paper PO08</ref><ref>Template:Cite journal</ref> due to its use of the geometric variance. Contrary to the arithmetic standard deviation, the arithmetic coefficient of variation is independent of the arithmetic mean.
The parameters Template:Math and Template:Math can be obtained, if the arithmetic mean and the arithmetic variance are known:
<math display="block">\begin{align} \mu &= \ln \frac{\operatorname{E}[X]^2}{\sqrt{\operatorname{E}[X^2]}} = \ln \frac{\operatorname{E}[X]^2}{\sqrt{\operatorname{Var}[X] + \operatorname{E}[X]^2}}, \\[1ex]
\sigma^2 &= \ln \frac{\operatorname{E}[X^2]}{\operatorname{E}[X]^2}
= \ln \left(1 + \frac{\operatorname{Var}[X]}{\operatorname{E}[X]^2}\right).
\end{align}</math>
A probability distribution is not uniquely determined by the moments Template:Math for Template:Math. That is, there exist other distributions with the same set of moments.<ref name="JKB"/> In fact, there is a whole family of distributions with the same moments as the log-normal distribution.Template:Citation needed
Mode, median, quantilesEdit
The mode is the point of global maximum of the probability density function. In particular, by solving the equation <math>(\ln f)'=0</math>, we get that:
<math display="block">\operatorname{Mode}[X] = e^{\mu - \sigma^2}.</math>
Since the log-transformed variable <math>Y = \ln X</math> has a normal distribution, and quantiles are preserved under monotonic transformations, the quantiles of <math>X</math> are
<math display="block">q_X(\alpha) = \exp\left[\mu + \sigma q_\Phi(\alpha)\right] = \mu^* (\sigma^*)^{q_\Phi(\alpha)},</math>
where <math>q_\Phi(\alpha)</math> is the quantile of the standard normal distribution.
Specifically, the median of a log-normal distribution is equal to its multiplicative mean,<ref>Template:Cite book print edition. Online eBook Template:ISBN</ref>
<math display="block">\operatorname{Med}[X] = e^\mu = \mu^* ~.</math>
Partial expectationEdit
The partial expectation of a random variable <math>X</math> with respect to a threshold <math>k</math> is defined as
<math display="block"> g(k) = \int_k^\infty x \, f_X(x \mid X > k)\, dx . </math>
Alternatively, by using the definition of conditional expectation, it can be written as <math>g(k) = \operatorname{E}[X\mid X>k] \Pr(X>k)</math>. For a log-normal random variable, the partial expectation is given by:
<math display="block">\begin{align} g(k) &= \int_k^\infty x f_X(x \mid X > k)\, dx \\[1ex] &= e^{\mu+\tfrac{1}{2} \sigma^2}\, \Phi{\left(\frac{\mu-\ln k}{\sigma} - \sigma\right)} \end{align} </math>
where <math>\Phi</math> is the normal cumulative distribution function. The derivation of the formula is provided in the Talk page. The partial expectation formula has applications in insurance and economics, it is used in solving the partial differential equation leading to the Black–Scholes formula.
Conditional expectationEdit
The conditional expectation of a log-normal random variable <math>X</math>—with respect to a threshold <math>k</math>—is its partial expectation divided by the cumulative probability of being in that range:
<math display="block">\begin{align} \operatorname{E}[X\mid X<k] & = e^{\mu +\frac{\sigma^2}{2}} \cdot \frac{\Phi {\left[\frac{\ln k - \mu}{\sigma} - \sigma \right]}}{\Phi {\left[\frac{\ln k-\mu}{\sigma} \right]}} \\[8pt] \operatorname{E}[X \mid X \geq k] &= e^{\mu +\frac{\sigma^2}{2}} \cdot \frac{\Phi {\left[\frac{\mu - \ln k}{\sigma} + \sigma \right]}}{1 - \Phi {\left[\frac{\ln k -\mu}{\sigma}\right]}} \\[8pt] \operatorname{E}[X\mid X\in [k_1,k_2]] &= e^{\mu +\frac{\sigma^2}{2}} \cdot \frac{
\Phi{\left[\frac{\ln k_2 - \mu}{\sigma} - \sigma \right]} - \Phi{\left[\frac{\ln k_1 - \mu}{\sigma} - \sigma\right]}
}{
\Phi \left[\frac{\ln k_2 - \mu}{\sigma}\right]-\Phi \left[\frac{\ln k_1 - \mu}{\sigma}\right]
} \end{align}</math>
Alternative parameterizationsEdit
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 distributions<ref>{{#invoke:citation/CS1|citation |CitationClass=web
}}</ref><ref>Template:Cite journal</ref> lists seven such forms:
- Template:Math with mean, Template:Math, and standard deviation, Template:Math, 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>
- Template:Math with mean, Template:Math, and variance, Template:Math, 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>
- Template:Math with median, Template:Math, on the natural scale and standard deviation, Template:Math, 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>
- Template:Math with median, Template:Math, and coefficient of variation, Template:Math, 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>
- Template:Math with mean, Template:Math, and precision, Template:Math, 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>
- Template:Math with median, Template:Math, and geometric standard deviation, Template:Math, both on the natural scale<ref>Template:Cite journal</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>
- Template:Math with mean, Template:Math, and standard deviation, Template:Math, both on the natural scale<ref>Template:Cite journal</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-parameterizationEdit
Consider the situation when one would like to run a model using two different optimal design tools, for example PFIM<ref>Template:Cite journal</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>
Multiple, reciprocal, powerEdit
- Multiplication by a constant: If <math>X \sim \operatorname{Lognormal}(\mu, \sigma^2)</math> then <math>a X \sim \operatorname{Lognormal}( \mu + \ln a, \sigma^2)</math> for <math> a > 0. </math>
- Reciprocal: If <math>X \sim \operatorname{Lognormal}(\mu, \sigma^2)</math> then <math>\tfrac{1}{X} \sim \operatorname{Lognormal}(-\mu, \sigma^2).</math>
- Power: If <math>X \sim \operatorname{Lognormal}(\mu, \sigma^2)</math> then <math>X^a \sim \operatorname{Lognormal}(a\mu, a^2 \sigma^2)</math> for <math>a \neq 0.</math>
Multiplication and division of independent, log-normal random variablesEdit
If two independent, log-normal variables <math>X_1</math> and <math>X_2</math> are multiplied [divided], the product [ratio] is again log-normal, with parameters <math>\mu = \mu_1 + \mu_2</math> Template:Nowrap and Template:Nowrap where Template:Nowrap
More generally, if <math>X_j \sim \operatorname{Lognormal} (\mu_j, \sigma_j^2)</math> are <math>n</math> independent, log-normally distributed variables, then <math display="inline">Y = \prod_{j=1}^n X_j \sim \operatorname{Lognormal} \Big( \sum_{j=1}^n\mu_j, \sum_{j=1}^n \sigma_j^2 \Big).</math>
Multiplicative central limit theoremEdit
The geometric or multiplicative mean of <math>n</math> independent, identically distributed, positive random variables <math>X_i</math> shows, for <math>n \to \infty</math>, approximately a log-normal distribution with parameters <math>\mu = \operatorname{E}[\ln X_i]</math> and <math>\sigma^2 = \operatorname{var}[\ln X_i ]/n</math>, assuming <math>\sigma^2</math> is finite.
In fact, the random variables do not have to be identically distributed. It is enough for the distributions of <math>\ln X_i</math> to all have finite variance and satisfy the other conditions of any of the many variants of the central limit theorem.
This is commonly known as Gibrat's law.
Heavy-tailness of the Log-NormalEdit
Whether a Log-Normal can be considered or not a true heavy-tail distribution is still debated. The main reason is that its variance is always finite, differently from what happen with certain Pareto distributions, for instance. However a recent study has shown how it is possible to create a Log-Normal distribution with infinite variance using Robinson Non-Standard Analysis.<ref>Template:Cite journal</ref>
OtherEdit
A set of data that arises from the log-normal distribution has a symmetric Lorenz curve (see also Lorenz asymmetry coefficient).<ref name="EcolgyArticle">Template:Cite journal</ref>
The harmonic <math>H</math>, geometric <math>G</math> and arithmetic <math>A</math> means of this distribution are related;<ref name="Rossman1990">Template:Cite journal</ref> such relation is given by
<math display="block">H = \frac{G^2} A.</math>
Log-normal distributions are infinitely divisible,<ref name="OlofThorin1978LNInfDivi"/> but they are not stable distributions, which can be easily drawn from.<ref name="Gao"/>
Related distributionsEdit
- If <math>X \sim \mathcal{N}(\mu, \sigma^2)</math> is a normal distribution, then <math>\exp(X) \sim \operatorname{Lognormal}(\mu, \sigma^2).</math>
- If <math>X \sim \operatorname{Lognormal}(\mu, \sigma^2)</math> is distributed log-normally, then <math>\ln X \sim \mathcal{N}(\mu, \sigma^2)</math> is a normal random variable.
- Let <math>X_j \sim \operatorname{Lognormal}(\mu_j, \sigma_j^2)</math> be independent log-normally distributed variables with possibly varying <math>\sigma</math> and <math>\mu</math> parameters, and <math display="inline">Y = \sum_{j = 1}^n X_j</math>. The distribution of <math>Y</math> has no closed-form expression, but can be reasonably approximated by another log-normal distribution <math>Z</math> at the right tail.<ref name="Asmussen2">Template:Cite journal</ref> Its probability density function at the neighborhood of 0 has been characterized<ref name = Gao/> and it does not resemble any log-normal distribution. A commonly used approximation due to L.F. Fenton (but previously stated by R.I. Wilkinson and mathematically justified by Marlow<ref name="Marlow">Template:Cite journal</ref>) is obtained by matching the mean and variance of another log-normal distribution: <math display="block">\begin{align}
\sigma^2_Z &= \ln\!\left[ \frac{\sum_j e^{2\mu_j+\sigma_j^2} \left(e^{\sigma_j^2} - 1\right)}{{\left(\sum_j e^{\mu_j + \sigma_j^2/2}\right)}^2} + 1\right], \\[1ex]
\mu_Z &= \ln\!\left[ \sum_j e^{\mu_j+\sigma_j^2/2} \right] - \frac{\sigma^2_Z}{2}.
\end{align}</math> In the case that all <math>X_j</math> have the same variance parameter Template:Nowrap these formulas simplify to <math display="block">\begin{align}
\sigma^2_Z &= \ln\!\left[ \left(e^{\sigma^2} - 1\right) \frac{\sum_j e^{2\mu_j}}{{\left(\sum_j e^{\mu_j}\right)}^2} + 1\right], \\[1ex]
\mu_Z &= \ln\!\left[ \sum_j e^{\mu_j} \right] + \frac{\sigma^2}{2} - \frac{\sigma^2_Z}{2}.
\end{align}</math> For a more accurate approximation, one can use the Monte Carlo method to estimate the cumulative distribution function, the pdf and the right tail.<ref name="BotLec2017">Template:Cite conference </ref><ref name="AGL2016">Template:Cite arXiv </ref>
The sum of correlated log-normally distributed random variables can also be approximated by a log-normal distributionTemplate:Citation needed <math display="block">\begin{align} S_+ &= \operatorname{E}\left[\sum_i X_i \right]
= \sum_i \operatorname{E}[X_i]
= \sum_i e^{\mu_i + \sigma_i^2/2}
\\[2ex] \sigma^2_{Z} &= \frac{1}{S_+^2} \, \sum_{i,j} \operatorname{cor}_{ij} \sigma_i \sigma_j \operatorname{E}[X_i] \operatorname{E}[X_j] \\[1ex]
&= \frac{1}{S_+^2} \, \sum_{i,j} \operatorname{cor}_{ij} \sigma_i \sigma_j e^{\mu_i+\sigma_i^2/2} e^{\mu_j+\sigma_j^2/2}
\\[2ex]
\mu_Z &= \ln S_+ - \sigma_{Z}^2/2
\end{align}</math>
- If <math>X \sim \operatorname{Lognormal}(\mu, \sigma^2)</math> then <math>X+c</math> is said to have a Three-parameter log-normal distribution with support Template:Nowrap<ref name="Sangal1970">Template:Cite journal</ref> Template:Nowrap Template:Nowrap
- The log-normal distribution is a special case of the semi-bounded Johnson's SU-distribution.<ref name="Johnson1949">Template:Cite journal</ref>
- If <math>X\mid Y \sim \operatorname{Rayleigh}(Y)</math> with <math> Y \sim \operatorname{Lognormal}(\mu, \sigma^2)</math>, then <math> X \sim \operatorname{Suzuki}(\mu, \sigma)</math> (Suzuki distribution).
- A substitute for the log-normal whose integral can be expressed in terms of more elementary functions<ref>Template:Cite journal</ref> can be obtained based on the logistic distribution to get an approximation for the CDF <math display="block"> F(x;\mu,\sigma) = \left[\left(\frac{e^\mu}{x}\right)^{\pi/(\sigma \sqrt{3})} + 1\right]^{-1}.</math> This is a log-logistic distribution.
Statistical inference
Estimation of parameters
Maximum likelihood estimator
For determining the maximum likelihood estimators of the log-normal distribution parameters Template:Math and Template:Math, we can use the same procedure as for the normal distribution. Note that <math display="block">L(\mu, \sigma) = \prod_{i=1}^n \frac 1 {x_i} \varphi_{\mu,\sigma} (\ln x_i),</math> where <math>\varphi</math> is the density function of the normal distribution <math>\mathcal N(\mu,\sigma^2)</math>. Therefore, the log-likelihood function is <math display="block"> \ell (\mu,\sigma \mid x_1, x_2, \ldots, x_n) = - \sum _i \ln x_i + \ell_N (\mu, \sigma \mid \ln x_1, \ln x_2, \dots, \ln x_n).</math>
Since the first term is constant with regard to μ and σ, both logarithmic likelihood functions, <math>\ell</math> and <math>\ell_N</math>, reach their maximum with the same <math>\mu</math> and <math>\sigma</math>. Hence, the maximum likelihood estimators are identical to those for a normal distribution for the observations <math>\ln x_1, \ln x_2, \dots, \ln x_n)</math>, <math display="block">\widehat \mu = \frac {\sum_i \ln x_i}{n}, \qquad \widehat \sigma^2 = \frac {\sum_i {\left( \ln x_i - \widehat \mu \right)}^2} {n}.</math>
For finite n, the estimator for <math>\mu</math> is unbiased, but the one for <math>\sigma</math> is biased. As for the normal distribution, an unbiased estimator for <math>\sigma</math> can be obtained by replacing the denominator n by n−1 in the equation for <math>\widehat\sigma^2</math>.
From this, the MLE for the expectancy of x is:<ref>Shen, Wei-Hsiung. "Estimation of parameters of a lognormal distribution." Taiwanese Journal of Mathematics 2.2 (1998): 243–250. pdf </ref> <math> \widehat{\theta}_\text{MLE} = \widehat{\operatorname{E}[X]}_\text{MLE} = e^{\hat \mu + {\hat{\sigma}^2}/{2}} </math>
Method of momentsEdit
When the individual values <math>x_1, x_2, \ldots, x_n</math> are not available, but the sample's mean <math>\bar x</math> and standard deviation s is, then the method of moments can be used. The corresponding parameters are determined by the following formulas, obtained from solving the equations for the expectation <math>\operatorname{E}[X]</math> and variance <math>\operatorname{Var}[X]</math> for <math>\mu</math> and <math>\sigma</math>:<ref>Henry (https://math.stackexchange.com/users/6460/henry), Method of moments estimator for lognormal distribution, URL (version: 2022-01-12): https://math.stackexchange.com/q/4355343</ref> <math display="block"> \begin{align} \mu &= \ln \frac{ \bar x} {\sqrt{1+\widehat\sigma^2/\bar x^2} } , \\[1ex] \sigma^2 &= \ln\left(1 + {\widehat\sigma^2} / \bar x^2 \right). \end{align}</math>
Other estimatorsEdit
Other estimators also exist, such as Finney's UMVUE estimator,<ref>Finney, D. J. "On the distribution of a variate whose logarithm is normally distributed." Supplement to the Journal of the Royal Statistical Society 7.2 (1941): 155–161.</ref> the "Approximately Minimum Mean Squared Error Estimator", the "Approximately Unbiased Estimator" and "Minimax Estimator",<ref>Longford, Nicholas T. "Inference with the lognormal distribution." Journal of Statistical Planning and Inference 139.7 (2009): 2329–2340.</ref> also "A Conditional Mean Squared Error Estimator",<ref>Zellner, Arnold. "Bayesian and non-Bayesian analysis of the log-normal distribution and log-normal regression." Journal of the American Statistical Association 66.334 (1971): 327–330.</ref> and other variations as well.<ref>Tang, Qi. "Comparison of different methods for estimating log-normal means". MS thesis. East Tennessee State University, 2014. linkhttps://dc.etsu.edu/cgi/viewcontent.cgi?article=3728&context=etd#page=12.13 pdf]</ref><ref>Kwon, Yeil. "An alternative method for estimating lognormal means." Communications for Statistical Applications and Methods 28.4 (2021): 351–368. link</ref>
Interval estimatesEdit
The most efficient way to obtain interval estimates when analyzing log-normally distributed data consists of applying the well-known methods based on the normal distribution to logarithmically transformed data and then to back-transform results if appropriate.
Prediction intervalsEdit
A basic example is given by prediction intervals: For the normal distribution, the interval <math>[\mu-\sigma,\mu+\sigma]</math> contains approximately two thirds (68%) of the probability (or of a large sample), and <math>[\mu-2\sigma,\mu+2\sigma]</math> contain 95%. Therefore, for a log-normal distribution,
- <math>[\mu^*/\sigma^*,\mu^*\cdot\sigma^*]=[\mu^* {}^\times\!\!/ \sigma^*]</math> contains 2/3, and
- <math>[\mu^*/(\sigma^*)^2,\mu^*\cdot(\sigma^*)^2] = [\mu^* {}^\times\!\!/ (\sigma^*)^2]</math> contains 95% of the probability. Using estimated parameters, then approximately the same percentages of the data should be contained in these intervals.
Confidence interval for eμEdit
Using the principle, note that a confidence interval for <math>\mu</math> is <math>[\widehat\mu \pm q \cdot \widehat\mathop{se}]</math>, where <math>\mathop{se} = \widehat\sigma / \sqrt{n}</math> is the standard error and q is the 97.5% quantile of a t distribution with n-1 degrees of freedom. Back-transformation leads to a confidence interval for <math>\mu^* = e^\mu</math> (the median), is: <math display="block">[\widehat\mu^* {}^\times\!\!/ (\operatorname{sem}^*)^q]</math> with <math>\operatorname{sem}^*=(\widehat\sigma^*)^{1/\sqrt{n}}</math>
Confidence interval for Template:MathEdit
The literature discusses several options for calculating the confidence interval for <math>\mu</math> (the mean of the log-normal distribution). These include bootstrap as well as various other methods.<ref name = "Olsson2005">Olsson, Ulf. "Confidence intervals for the mean of a log-normal distribution." Journal of Statistics Education 13.1 (2005).pdf html</ref><ref>user10525, How do I calculate a confidence interval for the mean of a log-normal data set?, URL (version: 2022-12-18): https://stats.stackexchange.com/q/33395</ref>
The Cox MethodTemplate:Efn proposes to plug-in the estimators <math display="block">\widehat \mu = \frac {\sum_i \ln x_i}{n}, \qquad S^2 = \frac {\sum_i \left( \ln x_i - \widehat \mu \right)^2} {n-1}</math>
and use them to construct approximate confidence intervals in the following way: <math>\mathrm{CI}(\operatorname{E}(X)) : \exp\left(\hat \mu + \frac{S^2}{2} \pm z_{1-\frac{\alpha}{2}} \sqrt{\frac{S^2}{n} + \frac{S^4}{2(n-1)}} \right)</math>
Template:Hidden begin We know that Template:Nowrap</math>.}} Also, <math>\widehat \mu</math> is a normal distribution with parameters: <math>\widehat \mu \sim N\left(\mu, \frac{\sigma^2}{n}\right)</math>
<math>S^2</math> has a chi-squared distribution, which is approximately normally distributed (via CLT), with parameters: Template:Nowrap Hence, Template:Nowrap
Since the sample mean and variance are independent, and the sum of normally distributed variables is also normal, we get that: <math>\widehat \mu + \frac{S^2}{2} \dot \sim N\left(\mu + \frac{\sigma^2}{2}, \frac{\sigma^2}{n} + \frac{\sigma^4}{2(n-1)}\right)</math> Based on the above, standard confidence intervals for <math>\mu + \frac{\sigma^2}{2}</math> can be constructed (using a Pivotal quantity) as: <math>\hat \mu + \frac{S^2}{2} \pm z_{1-\frac{\alpha}{2}} \sqrt{\frac{S^2}{n} + \frac{S^4}{2(n-1)} } </math> And since confidence intervals are preserved for monotonic transformations, we get that: <math>\mathrm{CI}\left(\operatorname{E}[X] = e^{\mu + \frac{\sigma^2}{2}}\right): \exp\left(\hat \mu + \frac{S^2}{2} \pm z_{1-\frac{\alpha}{2}} \sqrt{\frac{S^2}{n} + \frac{S^4}{2(n-1)}} \right)</math>
As desired.
Olsson 2005, proposed a "modified Cox method" by replacing <math>z_{1-\frac{\alpha}{2}}</math> with <math>t_{n-1, 1-\frac{\alpha}{2}}</math>, which seemed to provide better coverage results for small sample sizes.<ref name = "Olsson2005" />Template:Rp
Confidence interval for comparing two log normalsEdit
Comparing two log-normal distributions can often be of interest, for example, from a treatment and control group (e.g., in an A/B test). We have samples from two independent log-normal distributions with parameters <math>(\mu_1, \sigma_1^2)</math> and <math>(\mu_2, \sigma_2^2)</math>, with sample sizes <math>n_1</math> and <math>n_2</math> respectively.
Comparing the medians of the two can easily be done by taking the log from each and then constructing straightforward confidence intervals and transforming it back to the exponential scale.
<math display="block">\mathrm{CI}(e^{\mu_1-\mu_2}): \exp\left(\hat \mu_1 - \hat \mu_2 \pm z_{1-\frac{\alpha}{2}} \sqrt{\frac{S_1^2}{n} + \frac{S_2^2}{n} } \right)</math>
These CI are what's often used in epidemiology for calculation the CI for relative-risk and odds-ratio.<ref>Confidence Intervals for Risk Ratios and Odds Ratios</ref> The way it is done there is that we have two approximately Normal distributions (e.g., p1 and p2, for RR), and we wish to calculate their ratio.Template:Efn
However, the ratio of the expectations (means) of the two samples might also be of interest, while requiring more work to develop. The ratio of their means is:
<math display="block">\frac{\operatorname{E}(X_1)}{\operatorname{E}(X_2)} = \frac{e^{\mu_1 + \sigma_1^2 / 2}}{e^{\mu_2 + \sigma_2^2 /2}} = e^{(\mu_1 - \mu_2) + \frac{1}{2} \left(\sigma_1^2 - \sigma_2^2\right)}</math>
Plugin in the estimators to each of these parameters yields also a log normal distribution, which means that the Cox Method, discussed above, could similarly be used for this use-case:
<math display="block">\mathrm{CI}\left( \frac{\operatorname{E}(X_1)}{\operatorname{E}(X_2)} = \frac{e^{\mu_1 + \sigma_1^2 / 2}}{e^{\mu_2 + \sigma_2^2 / 2}} \right): \exp\left(\left(\hat \mu_1 - \hat \mu_2 + \tfrac{1}{2}S_1^2 - \tfrac{1}{2}S_2^2\right) \pm z_{1-\frac{\alpha}{2}} \sqrt{ \frac{S_1^2}{n_1} + \frac{S_2^2}{n_2} + \frac{S_1^4}{2(n_1-1)} + \frac{S_2^4}{2(n_2-1)} } \right)</math>
To construct a confidence interval for this ratio, we first note that <math>\hat \mu_1 - \hat \mu_2</math> follows a normal distribution, and that both <math>S_1^2</math> and <math>S_2^2</math> has a chi-squared distribution, which is approximately normally distributed (via CLT, with the relevant parameters).
This means that <math display="block">(\hat \mu_1 - \hat \mu_2 + \frac{1}{2}S_1^2 - \frac{1}{2}S_2^2) \sim N\left((\mu_1 - \mu_2) + \frac{1}{2}(\sigma_1^2 - \sigma_2^2), \frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2} + \frac{\sigma_1^4}{2(n_1-1)} + \frac{\sigma_2^4}{2(n_2-1)} \right)</math>
Based on the above, standard confidence intervals can be constructed (using a Pivotal quantity) as: <math>(\hat \mu_1 - \hat \mu_2 + \frac{1}{2}S_1^2 - \frac{1}{2}S_2^2) \pm z_{1-\frac{\alpha}{2}} \sqrt{ \frac{S_1^2}{n_1} + \frac{S_2^2}{n_2} + \frac{S_1^4}{2(n_1-1)} + \frac{S_2^4}{2(n_2-1)} } </math> And since confidence intervals are preserved for monotonic transformations, we get that: <math>CI\left( \frac{\operatorname{E}(X_1)}{\operatorname{E}(X_2)} = \frac{e^{\mu_1 + \frac{\sigma_1^2}{2}}}{e^{\mu_2 + \frac{\sigma_2^2}{2}}} \right):e^{\left((\hat \mu_1 - \hat \mu_2 + \frac{1}{2}S_1^2 - \frac{1}{2}S_2^2) \pm z_{1-\frac{\alpha}{2}} \sqrt{ \frac{S_1^2}{n_1} + \frac{S_2^2}{n_2} + \frac{S_1^4}{2(n_1-1)} + \frac{S_2^4}{2(n_2-1)} } \right)}</math>
As desired.
It's worth noting that naively using the MLE in the ratio of the two expectations to create a ratio estimator will lead to a consistent, yet biased, point-estimation (we use the fact that the estimator of the ratio is a log normal distribution):Template:Efn \end{align} </math>}}Template:Citation needed
<math display="block">\begin{align} \operatorname{E}\left[ \frac{\widehat \operatorname{E}(X_1)}{\widehat \operatorname{E}(X_2)} \right] &= \operatorname{E}\left[\exp\left(\left(\widehat \mu_1 - \widehat \mu_2\right) + \tfrac{1}{2} \left(S_1^2 - S_2^2\right)\right)\right] \\ &\approx \exp\left[{(\mu_1 - \mu_2) + \frac{1}{2}(\sigma_1^2 - \sigma_2^2) + \frac{1}{2}\left( \frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2} + \frac{\sigma_1^4}{2(n_1-1)} + \frac{\sigma_2^4}{2(n_2-1)} \right) }\right] \end{align} </math>
Extremal principle of entropy to fix the free parameter σEdit
In applications, <math>\sigma</math> is a parameter to be determined. For growing processes balanced by production and dissipation, the use of an extremal principle of Shannon entropy shows that<ref name="bai">Template:Cite journal</ref> <math display="block">\sigma = \frac 1 \sqrt{6} </math>
This value can then be used to give some scaling relation between the inflexion point and maximum point of the log-normal distribution.<ref name = bai/> This relationship is determined by the base of natural logarithm, <math>e = 2.718\ldots</math>, and exhibits some geometrical similarity to the minimal surface energy principle. These scaling relations are useful for predicting a number of growth processes (epidemic spreading, droplet splashing, population growth, swirling rate of the bathtub vortex, distribution of language characters, velocity profile of turbulences, etc.). For example, the log-normal function with such <math>\sigma</math> fits well with the size of secondarily produced droplets during droplet impact <ref name="wu"/> and the spreading of an epidemic disease.<ref name="Wang">Template:Cite journal</ref>
The value <math display="inline">\sigma = 1 \big/ \sqrt{6}</math> is used to provide a probabilistic solution for the Drake equation.<ref name="Bloetscher">Template:Cite journal</ref>
Occurrence and applicationsEdit
The log-normal distribution is important in the description of natural phenomena. Many natural growth processes are driven by the accumulation of many small percentage changes which become additive on a log scale. Under appropriate regularity conditions, the distribution of the resulting accumulated changes will be increasingly well approximated by a log-normal, as noted in the section above on "Multiplicative Central Limit Theorem". This is also known as Gibrat's law, after Robert Gibrat (1904–1980) who formulated it for companies.<ref>Template:Cite journal</ref> If the rate of accumulation of these small changes does not vary over time, growth becomes independent of size. Even if this assumption is not true, the size distributions at any age of things that grow over time tends to be log-normal.Template:Citation needed Consequently, reference ranges for measurements in healthy individuals are more accurately estimated by assuming a log-normal distribution than by assuming a symmetric distribution about the mean.Template:Citation needed
A second justification is based on the observation that fundamental natural laws imply multiplications and divisions of positive variables. Examples are the simple gravitation law connecting masses and distance with the resulting force, or the formula for equilibrium concentrations of chemicals in a solution that connects concentrations of educts and products. Assuming log-normal distributions of the variables involved leads to consistent models in these cases.
Specific examples are given in the following subsections.<ref name=":0">Template:Cite journal</ref> contains a review and table of log-normal distributions from geology, biology, medicine, food, ecology, and other areas.<ref name=":4" /> is a review article on log-normal distributions in neuroscience, with annotated bibliography.
Human behaviorEdit
- The length of comments posted in Internet discussion forums follows a log-normal distribution.<ref name=":3">Template:Cite journal</ref>
- Users' dwell time on online articles (jokes, news etc.) follows a log-normal distribution.<ref>Template:Cite conference</ref>
- The length of chess games tends to follow a log-normal distribution.<ref>{{#invoke:citation/CS1|citation
|CitationClass=web }}</ref>
- Onset durations of acoustic comparison stimuli that are matched to a standard stimulus follow a log-normal distribution.<ref name="Acoustic Stimuli Revisited 2016"/>
Biology and medicineEdit
- Measures of size of living tissue (length, skin area, weight).<ref>Template:Cite book</ref>
- Incubation period of diseases.<ref>Sartwell, Philip E. "The distribution of incubation periods of infectious disease." American journal of hygiene 51 (1950): 310–318.</ref>
- Diameters of banana leaf spots, powdery mildew on barley.<ref name=":0" />
- For highly communicable epidemics, such as SARS in 2003, if public intervention control policies are involved, the number of hospitalized cases is shown to satisfy the log-normal distribution with no free parameters if an entropy is assumed and the standard deviation is determined by the principle of maximum rate of entropy production.<ref>Template:Cite journal</ref>
- The length of inert appendages (hair, claws, nails, teeth) of biological specimens, in the direction of growth.Template:Citation needed
- The normalised RNA-Seq readcount for any genomic region can be well approximated by log-normal distribution.
- The PacBio sequencing read length follows a log-normal distribution.<ref>Template:Cite journal</ref>
- Certain physiological measurements, such as blood pressure of adult humans (after separation on male/female subpopulations).<ref>Template:Cite journal</ref>
- Several pharmacokinetic variables, such as Cmax, elimination half-life and the elimination rate constant.<ref>Template:Cite journal</ref>
- In neuroscience, the distribution of firing rates across a population of neurons is often approximately log-normal. This has been first observed in the cortex and striatum <ref>Template:Cite conference</ref> and later in hippocampus and entorhinal cortex,<ref>Template:Cite journal</ref> and elsewhere in the brain.<ref name=":4">Template:Cite journal</ref><ref>Template:Cite journal</ref> Also, intrinsic gain distributions and synaptic weight distributions appear to be log-normal<ref>Template:Cite journal</ref> as well.
- Neuron densities in the cerebral cortex, due to the noisy cell division process during neurodevelopment.<ref>Template:Cite journal</ref>
- In operating-rooms management, the distribution of surgery duration.
- In the size of avalanches of fractures in the cytoskeleton of living cells, showing log-normal distributions, with significantly higher size in cancer cells than healthy ones.<ref>Template:Cite journal</ref>
ChemistryEdit
- Particle size distributions and molar mass distributions.
- The concentration of rare elements in minerals.<ref>Template:Cite journal</ref>
- Diameters of crystals in ice cream, oil drops in mayonnaise, pores in cocoa press cake.<ref name=":0" />
Physical sciencesEdit
- In hydrology, the log-normal distribution is used to analyze extreme values of such variables as monthly and annual maximum values of daily rainfall and river discharge volumes.<ref>Template:Cite book</ref>
- The image on the right illustrates an example of fitting the log-normal distribution to ranked annually maximum one-day rainfalls showing also the 90% confidence belt based on the binomial distribution.<ref>CumFreq, free software for distribution fitting</ref>
- The rainfall data are represented by plotting positions as part of a cumulative frequency analysis.
- In physical oceanography, the sizes of icebergs in the midwinter Southern Atlantic Ocean were found to follow a log-normal size distribution. The iceberg sizes, measured visually and by radar from the F.S. Polarstern in 1986, were thought to be controlled by wave action in heavy seas causing them to flex and break.<ref>Template:Cite journal</ref>
- In atmospheric science, log-normal distributions (or distributions made by combining multiple log-normal functions) have been used to characterize both measurements and models of the sizes and concentrations of many different types of particles, from volcanic ash, to clouds and rain, to airborne microbes.<ref>Template:Cite journal</ref><ref>Template:Cite journal</ref><ref>Template:Cite journal</ref><ref>Template:Cite journal</ref> The log-normal distribution is strictly empirical, so more physically-based distributions have been adopted to better understand processes controlling size distributions of particles such as volcanic ash.<ref>Template:Cite journal</ref>
Social sciences and demographicsEdit
- In economics, there is evidence that the income of 97–99% of the population is distributed log-normally.<ref>Clementi, Fabio; Gallegati, Mauro (2005) "Pareto's law of income distribution: Evidence for Germany, the United Kingdom, and the United States", EconWPA</ref> (The distribution of higher-income individuals follows a Pareto distribution).<ref>Template:Cite conference</ref>
- If an income distribution follows a log-normal distribution with standard deviation <math>\sigma</math>, then the Gini coefficient, commonly use to evaluate income inequality, can be computed as <math>G = \operatorname{erf}\left(\frac{\sigma }{2 }\right)</math> where <math>\operatorname{erf}</math> is the error function, since <math> G = 2 \Phi{\left(\frac{\sigma }{\sqrt{2}}\right)} - 1</math>, where <math>\Phi(x)</math> is the cumulative distribution function of a standard normal distribution.
- In finance, in particular the Black–Scholes model, changes in the logarithm of exchange rates, price indices, and stock market indices are assumed normal<ref>Template:Cite journal</ref> (these variables behave like compound interest, not like simple interest, and so are multiplicative). However, some mathematicians such as Benoit Mandelbrot have argued <ref>Template:Cite book</ref> that log-Lévy distributions, which possesses heavy tails would be a more appropriate model, in particular for the analysis for stock market crashes. Indeed, stock price distributions typically exhibit a fat tail.<ref>Bunchen, P., Advanced Option Pricing, University of Sydney coursebook, 2007</ref> The fat tailed distribution of changes during stock market crashes invalidate the assumptions of the central limit theorem.
- In scientometrics, the number of citations to journal articles and patents follows a discrete log-normal distribution.<ref>Template:Cite journal</ref><ref>Template:Cite journal</ref>
- City sizes (population) satisfy Gibrat's Law.<ref>Template:Cite journal</ref> The growth process of city sizes is proportionate and invariant with respect to size. From the central limit theorem therefore, the log of city size is normally distributed.
- The number of sexual partners appears to be best described by a log-normal distribution.<ref>Template:Cite journal</ref>
TechnologyEdit
- In reliability analysis, the log-normal distribution is often used to model times to repair a maintainable system.<ref>Template:Cite book</ref>
- In wireless communication, "the local-mean power expressed in logarithmic values, such as dB or neper, has a normal (i.e., Gaussian) distribution."<ref>{{#invoke:citation/CS1|citation
|CitationClass=web }}</ref> Also, the random obstruction of radio signals due to large buildings and hills, called shadowing, is often modeled as a log-normal distribution.
- Particle size distributions produced by comminution with random impacts, such as in ball milling.<ref>Template:Cite journal</ref>
- The file size distribution of publicly available audio and video data files (MIME types) follows a log-normal distribution over five orders of magnitude.<ref>
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- File sizes of 140 million files on personal computers running the Windows OS, collected in 1999.<ref>Template:Cite journal</ref><ref name=":3" />
- Sizes of text-based emails (1990s) and multimedia-based emails (2000s).<ref name=":3" />
- In computer networks and Internet traffic analysis, log-normal is shown as a good statistical model to represent the amount of traffic per unit time. This has been shown by applying a robust statistical approach on a large groups of real Internet traces. In this context, the log-normal distribution has shown a good performance in two main use cases: (1) predicting the proportion of time traffic will exceed a given level (for service level agreement or link capacity estimation) i.e. link dimensioning based on bandwidth provisioning and (2) predicting 95th percentile pricing.<ref>Template:Cite arXiv</ref>
- in physical testing when the test produces a time-to-failure of an item under specified conditions, the data is often best analyzed using a lognormal distribution.<ref>ASTM D3654, Standard Test Method for Shear Adhesion on Pressure-Sensitive Tapesw</ref><ref>ASTM D4577, Standard Test Method for Compression Resistance of a container Under Constant Load>\</ref>
See alsoEdit
- Heavy-tailed distribution
- Log-distance path loss model
- Modified lognormal power-law distribution
- Fading
NotesEdit
ReferencesEdit
Further readingEdit
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