Inprobability theory, alog-normal (orlognormal)distribution is a continuousprobability distribution of arandom variable whoselogarithm isnormally distributed. Thus, if the random variableX is log-normally distributed, thenY = lnX has a normal distribution.[2][3] Equivalently, ifY has a normal distribution, then theexponential function ofY,X = exp(Y), 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 andengineering sciences, as well asmedicine,economics and other topics (e.g., energies, concentrations, lengths, prices of financial instruments, and other metrics).
The distribution is occasionally referred to as theGalton distribution orGalton's distribution, afterFrancis Galton.[4] The log-normal distribution has also been associated with other names, such asMcAlister,Gibrat andCobb–Douglas.[4]
Let be astandard normal variable, and let and be two real numbers, with. Then, the distribution of the random variable
is called the log-normal distribution with parameters and. These are theexpected value (ormean) andstandard deviation of the variable's naturallogarithm,,not the expectation and standard deviation of itself.
The cumulative distribution function and the corresponding probability density function of a log-normal distribution plotted on a semi-log plot. The median value is equal to 1, and the modal value (the peak of the probability density function) is equal to 1/e. In this case, 1/e corresponds to roughly the 16th percentile, 1 corresponds to the median (50th percentile), and e corresponds to roughly the 84th percentile.A log-normal distribution on a semi-log plot with a median of 1, whose ~84th percentile value is 10 times the median, whose ~16th percentile value is 1/10th the median, and whose modal value (~.005) is at roughly the 1st percentile. The peak density of values around the mode is high, but the distribution would possess a tall, narrow peak, and a long right tail on a linear-linear plot.
This relationship is true regardless of the base of the logarithmic or exponential function: If is normally distributed, then so is for any two positive numbers. Likewise, if is log-normally distributed, then so is, where.
In order to produce a distribution with desired mean and variance, one uses and.
Alternatively, the "multiplicative" or "geometric" parameters and can be used. They have a more direct interpretation: is themedian of the distribution, and is useful for determining "scatter" intervals, see below.
A positive random variable is log-normally distributed (i.e.,), if the natural logarithm of is normally distributed with mean and variance:
Let and be respectively the cumulative probability distribution function and the probability density function of the standard normal distribution, then we have that[2][4] theprobability density function of the log-normal distribution is given by:
If is amultivariate normal distribution, then has a multivariate log-normal distribution.[6][7] The exponential is applied element-wise to the random vector. The mean of is
All moments of the log-normal distribution exist and
This can be derived by letting within the integral. However, the log-normal distribution is not determined by its moments.[8] This implies that it cannot have a defined moment generating function in a neighborhood of zero.[9] Indeed, the expected value is not defined for any positive value of the argument, since the defining integral diverges.
Thecharacteristic function is defined for real values oft, but is not defined for any complex value oft that has a negative imaginary part, and hence the characteristic function is notanalytic at the origin. Consequently, the characteristic function of the log-normal distribution cannot be represented as an infinite convergent series.[10] In particular, its Taylorformal series diverges:
A closed-form formula for the characteristic function with in the domain of convergence is not known. A relatively simple approximating formula is available in closed form, and is given by[14]
where is theLambert W function. This approximation is derived via an asymptotic method, but it stays sharp all over the domain of convergence of.
By analogy with the arithmetic statistics, one can define a geometric variance,, and ageometric coefficient of variation,[15], has been proposed. This term was intended to beanalogous 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 itself (see alsoCoefficient of variation).
Note that the geometric mean is smaller than the arithmetic mean. This is due to theAM–GM inequality and is a consequence of the logarithm being aconcave function. In fact,[17]
For any real or complex numbern, then-thmoment of a log-normally distributed variableX is given by[4]
Specifically, the arithmetic mean, expected square, arithmetic variance, and arithmetic standard deviation of a log-normally distributed variableX are respectively given by:[2]
The arithmeticcoefficient of variation is the ratio. For a log-normal distribution it is equal to[3]This estimate is sometimes referred to as the "geometric CV" (GCV),[18][19] 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 parametersμ andσ can be obtained, if the arithmetic mean and the arithmetic variance are known:
A probability distribution is not uniquely determined by the momentsE[Xn] = enμ +1/2n2σ2 forn ≥ 1. That is, there exist other distributions with the same set of moments.[4] In fact, there is a whole family of distributions with the same moments as the log-normal distribution.[citation needed]
The partial expectation of a random variable with respect to a threshold is defined as
Alternatively, by using the definition ofconditional expectation, it can be written as. For a log-normal random variable, the partial expectation is given by:
The conditional expectation of a log-normal random variable—with respect to a threshold—is its partial expectation divided by the cumulative probability of being in that range:
In addition to the characterization by or, here are multiple ways how the log-normal distribution can be parameterized.ProbOnto, the knowledge base and ontology ofprobability distributions[21][22] lists seven such forms:
Overview of parameterizations of the log-normal distributions.
Consider the situation when one would like to run a model using two different optimal design tools, for example PFIM[27] and PopED.[28] 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 following formulas hold and.
For the transition following formulas hold and.
All remaining re-parameterisation formulas can be found in the specification document on the project website.[29]
The geometric or multiplicative mean of independent, identically distributed, positive random variables shows, for, approximately a log-normal distribution with parameters and, assuming is finite.
In fact, the random variables do not have to be identically distributed. It is enough for the distributions of to all have finite variance and satisfy the other conditions of any of the many variants of thecentral limit theorem.
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.[30]
If is distributed log-normally, then is a normal random variable.
Let be independent log-normally distributed variables with possibly varying and parameters, and. The distribution of has no closed-form expression, but can be reasonably approximated by another log-normal distribution at the right tail.[35] Its probability density function at the neighborhood of 0 has been characterized[34] 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[36]) is obtained by matching the mean and variance of another log-normal distribution: In the case that all have the same variance parameter, these formulas simplify to
For a more accurate approximation, one can use theMonte Carlo method to estimate the cumulative distribution function, the pdf and the right tail.[37][38] The cdf and pdf of the sum of correlated log-normally distributed random variables can also be approximated by Monte Carlo simulation.[39]
If then is said to have aThree-parameter log-normal distribution with support.[40],.
A substitute for the log-normal whose integral can be expressed in terms of more elementary functions[42] can be obtained based on thelogistic distribution to get an approximation for theCDF This is alog-logistic distribution.
For determining themaximum likelihood estimators of the log-normal distribution parametersμ andσ, we can use thesame procedure as for thenormal distribution. Note thatwhere is the density function of the normal distribution. Therefore, the log-likelihood function is
Since the first term is constant with regard toμ andσ, both logarithmic likelihood functions, and, reach their maximum with the same and. Hence, the maximum likelihood estimators are identical to those for a normal distribution for the observations,
For finiten, the estimator for is unbiased, but the one for is biased. As for the normal distribution, an unbiased estimator for can be obtained by replacing the denominatorn byn−1 in the equation for.
From this, the MLE for the expectancy of x is:[43]
When the individual values are not available, but the sample's mean andstandard deviations is, then themethod of moments can be used. The corresponding parameters are determined by the following formulas, obtained from solving the equations for the expectation and variance for and:[44]
Other estimators also exist, such as Finney'sUMVUE estimator,[45] the "Approximately Minimum Mean Squared Error Estimator", the "Approximately Unbiased Estimator" and "Minimax Estimator",[46] also "A Conditional Mean Squared Error Estimator",[47] and other variations as well.[48][49]
The most efficient way to obtaininterval 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.
A basic example is given byprediction intervals: For the normal distribution, the interval contains approximately two thirds (68%) of the probability (or of a large sample), and contain 95%. Therefore, for a log-normal distribution,
contains 2/3, and
contains 95% of the probability. Using estimated parameters, then approximately the same percentages of the data should be contained in these intervals.
Using the principle, note that aconfidence interval for is, where is the standard error andq is the 97.5% quantile of at distribution withn-1 degrees of freedom. Back-transformation leads to a confidence interval for (the median), is: with
The literature discusses several options for calculating theconfidence interval for (the mean of the log-normal distribution). These includebootstrap as well as various other methods.[50][51]
The Cox Method[a] proposes to plug-in the estimators
Since the sample mean and variance are independent, and the sum of normally distributed variables isalso normal, we get that:Based on the above, standardconfidence intervals for can be constructed (using aPivotal quantity) as:And since confidence intervals are preserved for monotonic transformations, we get that:
As desired.
Olsson 2005, proposed a "modified Cox method" by replacing with, which seemed to provide better coverage results for small sample sizes.[50]: Section 3.4
Comparing two log-normal distributions can often be of interest, for example, from a treatment and control group (e.g., in anA/B test). We have samples from two independent log-normal distributions with parameters and, with sample sizes and 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.
These CI are what are often used in epidemiology for calculation of the CI forrelative-risk andodds-ratio.[54] 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.[b]
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:
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:
[Proof]
To construct a confidence interval for this ratio, we first note that follows a normal distribution, and that both and has achi-squared distribution, which isapproximately normally distributed (viaCLT, with the relevantparameters).
This means that
Based on the above, standardconfidence intervals can be constructed (using aPivotal quantity) as:And since confidence intervals are preserved for monotonic transformations, we get that:
As desired.
It is worth noting that naively using theMLE in the ratio of the two expectations to create aratio estimator will lead to aconsistent, yet biased, point-estimation (we use the fact that the estimator of the ratio is a log normal distribution):[c][citation needed]
Extremal principle of entropy to fix the free parameterσ
In applications, 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[55]
This value can then be used to give some scaling relation between the inflexion point and maximum point of the log-normal distribution.[55] This relationship is determined by the base of natural logarithm,, 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 fits well with the size of secondarily produced droplets during droplet impact[56] and the spreading of an epidemic disease.[57]
The value is used to provide a probabilistic solution for the Drake equation.[58]
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 asGibrat's law, after Robert Gibrat (1904–1980) who formulated it for companies.[59] 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.[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.[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.[60] contains a review and table of log-normal distributions from geology, biology, medicine, food, ecology, and other areas.[61] is a review article on log-normal distributions in neuroscience, with annotated bibliography.
Diameters of banana leaf spots, powdery mildew on barley.[60]
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 ofentropy production.[67]
The length of inert appendages (hair, claws, nails, teeth) of biological specimens, in the direction of growth.[citation needed]
The normalised RNA-Seq readcount for any genomic region can be well approximated by log-normal distribution.
ThePacBio sequencing read length follows a log-normal distribution.[68]
Certain physiological measurements, such as blood pressure of adult humans (after separation on male/female subpopulations).[69]
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[71] and later in hippocampus and entorhinal cortex,[72] and elsewhere in the brain.[61][73] Also, intrinsic gain distributions and synaptic weight distributions appear to be log-normal[74] as well.
Neuron densities in the cerebral cortex, due to the noisy cell division process during neurodevelopment.[75]
In operating-rooms management, the distribution ofsurgery 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.[76]
Inhydrology, 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.[78]
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 thebinomial distribution.
Inphysical 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.[79]
Inatmospheric 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.[80][81][82][83] 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.[84]
Ineconomics, there is evidence that theincome of 97–99% of the population is distributed log-normally.[85] (The distribution of higher-income individuals follows aPareto distribution).[86]
If an income distribution follows a log-normal distribution with standard deviation, then theGini coefficient, commonly used to evaluate income inequality, can be computed as where is theerror function, since, where is the cumulative distribution function of a standard normal distribution.
Infinance, in particular theBlack–Scholes model, changes in thelogarithm of exchange rates, price indices, and stock market indices are assumed normal[87] (these variables behave like compound interest, not like simple interest, and so are multiplicative). However, some mathematicians such asBenoit Mandelbrot have argued[88] thatlog-Lévy distributions, which possessheavy tails, would be a more appropriate model, in particular for the analysis forstock market crashes. Indeed, stock price distributions typically exhibit afat tail.[89] The fat tailed distribution of changes during stock market crashes invalidate the assumptions of thecentral limit theorem.
Inscientometrics, the number of citations to journal articles and patents follows a discrete log-normal distribution.[90][91]
City sizes (population) satisfy Gibrat's Law.[92] The growth process of city sizes is proportionate and invariant with respect to size. From thecentral 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.[93]
Inreliability analysis, the log-normal distribution is often used to model times to repair a maintainable system.[94]
Inwireless communication, "the local-mean power expressed in logarithmic values, such as dB or neper, has a normal (i.e., Gaussian) distribution."[95] Also, the random obstruction of radio signals due to large buildings and hills, calledshadowing, is often modeled as a log-normal distribution.
Particle size distributions produced by comminution with random impacts, such as inball milling.[96]
File sizes of 140 million files on personal computers running the Windows OS, collected in 1999.[98][62]
Sizes of text-based emails (1990s) and multimedia-based emails (2000s).[62]
In computer networks andInternet 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.[99]
inphysical 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.[100][101]
^The Cox Method was quoted as "personal communication" in Land, 1971,[52] and was also given in Zhou and Gao (1997)[53] and Olsson 2005[50]: Section 3.3
^The issue is that we do not know how to do it directly, so we take their logs, and then use thedelta method to say that their logs is itself (approximately) normal. This trick allows us to pretend that their exp was log normal, and use that approximation to build the CI. Notice that in the RR case, the median and the mean in the base distribution (i.e., before taking the log), is actually identical (since they are originally normal, and not log normal).For example, and Hence, building a CI based on the log and then back-transform will give us. So while we expect the CI to be for the median, in this case, it is actually also for the mean in the original distribution.i.e., if the original was log-normal, we would expect that. But in practice, we KNOW that. Hence, the approximation we have is in the second step (of the delta method), but the CI are actually for the expectation (not just the median). This is because we are starting from a base distribution that is normal, and then using another approximation after the log again to normal. This means that a big approximation part of the CI is from the delta method.
^The formula can found by just treating the estimated means and variances as approximately normal, which indicates the terms is itself a log-normal, enabling us to quickly get the expectation. The bias can be partially minimized by using:
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