Logistic distribution

Logistic
Probability density function
Cumulative distribution function
Parameters	μ location (real) s > 0 scale (real)
Support	x ∈ (−∞, ∞)
PDF	${\frac {e^{-{\frac {x-\mu }{s}}}}{s\left(1+e^{-{\frac {x-\mu }{s}}}\right)^{2}}}\!$
CDF	${\frac {1}{1+e^{-{\frac {x-\mu }{s}}}}}\!$
Mean	$\mu$
Median	$\mu$
Mode	$\mu$
Variance	${\tfrac {s^{2}\pi ^{2}}{3}}$
Skewness	$0$
Ex. kurtosis	$1.2$
Entropy	$\ln(s)+2=\ln(\sigma )+1.404576,$ where σ is the standard deviation.
MGF	$e^{{\mu t}}\operatorname {B}(1-st,1+st)$ for st ∈ (−1, 1) Beta function
CF	$e^{{it\mu }}{\frac {\pi st}{\sinh(\pi st)}}$

In probability theory and statistics, the logistic distribution is a continuous probability distribution. Its cumulative distribution function is the logistic function, which appears in logistic regression and feedforward neural networks. It resembles the normal distribution in shape but has heavier tails (higher kurtosis). The Tukey lambda distribution can be considered a generalization of the logistic distribution since it adds a shape parameter, λ, and corresponds to the logistic distribution when λ is zero.

Specification

Probability density function

The probability density function (pdf) of the logistic distribution is given by:

f(x;\mu ,s)={\frac {e^{\frac {x-\mu }{s}}}{s\left(1+e^{\frac {x-\mu }{s}}\right)^{2}}}={\frac {1}{4s}}\operatorname {sech} ^{2}\!\left({\frac {x-\mu }{2s}}\right).

Because the pdf can be expressed in terms of the square of the hyperbolic secant function "sech", it is sometimes referred to as the sech-square(d) distribution.^[1]

Cumulative distribution function

The logistic distribution receives its name from its cumulative distribution function (cdf), which is an instance of the family of logistic functions. The cumulative distribution function of the logistic distribution is also a scaled version of the hyperbolic tangent.

F(x;\mu ,s)={\frac {1}{1+e^{{-{\frac {x-\mu }{s}}}}}}={\frac 12}+{\frac 12}\;\operatorname {tanh}\!\left({\frac {x-\mu }{2s}}\right).

In this equation, x is the random variable, μ is the mean, and s is a scale parameter proportional to the standard deviation.

Quantile function

The inverse cumulative distribution function (quantile function) of the logistic distribution is a generalization of the logit function. Its derivative is called the quantile density function. They are defined as follows:

Q(p;\mu ,s)=\mu +s\,\ln \left({\frac {p}{1-p}}\right).

Q'(p;s)={\frac {s}{p(1-p)}}.

Alternative parameterization

An alternative parameterization of the logistic distribution can be derived by expressing the scale parameter, $s$ , in terms of the standard deviation, $\sigma$ , using the substitution $s\,=\,q\,\sigma$ , where $q\,=\,{\sqrt {3}}/{\pi }\,=\,0.551328895\ldots$ . The alternative forms of the above functions are reasonably straightforward.

Applications

The logistic distribution—and the S-shaped pattern of its cumulative distribution function (the logistic function) and quantile function (the logit function)—have been extensively used in many different areas.

Logistic regression

One of the most common applications is in logistic regression, which is used for modeling categorical dependent variables (e.g., yes-no choices or a choice of 3 or 4 possibilities), much as standard linear regression is used for modeling continuous variables (e.g., income or population). Specifically, logistic regression models can be phrased as latent variable models with error variables following a logistic distribution. This phrasing is common in the theory of discrete choice models, where the logistic distribution plays the same role in logistic regression as the normal distribution does in probit regression. Indeed, the logistic and normal distributions have a quite similar shape. However, the logistic distribution has heavier tails, which often increases the robustness of analyses based on it compared with using the normal distribution.

Physics

The pdf of this distribution has the same functional form as the derivative of the Fermi function. In the theory of electron properties in semiconductors and metals, this derivative sets the relative weight of the various electron energies in their contributions to electron transport. Those energy levels whose energies are closest to the distribution's "mean" (Fermi level) dominate processes such as electronic conduction, with some smearing induced by temperature.^[2]^:34 Note however that the pertinent probability distribution in Fermi–Dirac statistics is actually a simple Bernoulli distribution, with the probability factor given by the Fermi function.

The logistic distribution arises as limit distribution of a finite-velocity damped random motion described by a telegraph process in which the random times between consecutive velocity changes have independent exponential distributions with linearly increasing parameters.^[3]

Hydrology

Fitted cumulative logistic distribution to October rainfalls using CumFreq, see also Distribution fitting

In hydrology the distribution of long duration river discharge and rainfall (e.g., monthly and yearly totals, consisting of the sum of 30 respectively 360 daily values) is often thought to be almost normal according to the central limit theorem.^[4] The normal distribution, however, needs a numeric approximation. As the logistic distribution, which can be solved analytically, is similar to the normal distribution, it can be used instead. The blue picture illustrates an example of fitting the logistic distribution to ranked October rainfalls—that are almost normally distributed—and it shows the 90% confidence belt based on the binomial distribution. The rainfall data are represented by plotting positions as part of the cumulative frequency analysis.

Chess ratings

Both the United States Chess Federation and FIDE have switched their formulas for calculating chess ratings from the normal distribution to the logistic distribution; see Elo rating system.

Related distributions

Logistic distribution mimics the Sech distribution.
If X ~ Logistic(μ, β) then kX + loc ~ Logistic(kμ+loc, kβ).
If X ~ U(0, 1) then μ + β(log(X)−log(1−X)) ~ Logistic(μ, β).
If X, Y ~ Gumbel(α, β) then X−Y ~ Logistic(0, β).
If X, Y ~ GEV(α, β, 0) then X−Y ~ Logistic(0, β).
If X ~ Gumbel(α, β) and Y ~ GEV(α, β, 0) then X+Y ~ Logistic(2α, β).
If X ~ Logistic(α, β) then exp(X) ~ LogLogistic $\left(e^{\alpha },{\frac {1}{\beta }}\right)$ , and exp(X−k) ~ ShiftedLogLogistic $\left(e^{\alpha },{\frac {1}{\beta }},k\right)$ .
If X ~ Exponential(1) then

\mu +\beta \log \left(e^{{X}}-1\right)\sim {\mathrm {Logistic}}(\mu ,\beta ).

If X, Y ~ Exponential(1) then

\mu -\beta \log \left({\frac {X}{Y}}\right)\sim {\mathrm {Logistic}}(\mu ,\beta ).

Derivations

Higher order moments

The n-th order central moment can be expressed in terms of the quantile function:

\operatorname {E}[(X-\mu )^{n}]=\int _{{-\infty }}^{\infty }(x-\mu )^{n}dF(x)=\int _{0}^{1}{\big (}Q(p)-\mu {\big )}^{n}dp=s^{n}\int _{0}^{1}\left[\ln \!\left({\frac {p}{1-p}}\right)\right]^{n}dp.

This integral is well-known^[5] and can be expressed in terms of Bernoulli numbers:

\operatorname {E}[(X-\mu )^{n}]=s^{n}\pi ^{n}(2^{n}-2)\cdot |B_{n}|.

Notes

↑ Johnson, Kotz & Balakrishnan (1995, p.116).
↑ Davies, John H. (1998). The Physics of Low-dimensional Semiconductors: An Introduction. Cambridge University Press. ISBN 9780521484916.
↑ A. Di Crescenzo, B. Martinucci (2010) "A damped telegraph random process with logistic stationary distribution", J. Appl. Prob., vol. 47, pp. 84–96.
↑ Ritzema (ed.), H.P. (1994). Frequency and Regression Analysis (PDF). Chapter 6 in: Drainage Principles and Applications, Publication 16, International Institute for Land Reclamation and Improvement (ILRI), Wageningen, The Netherlands. pp. 175–224. ISBN 90-70754-33-9.
↑ A001896

References

John S. deCani & Robert A. Stine (1986). "A note on deriving the information matrix for a logistic distribution". The American Statistician. American Statistical Association. 40: 220–222. doi:10.2307/2684541.
N., Balakrishnan (1992). Handbook of the Logistic Distribution. Marcel Dekker, New York. ISBN 0-8247-8587-8.
Johnson, N. L., Kotz, S., Balakrishnan N. (1995). Continuous Univariate Distributions. Vol. 2 (2nd ed.). ISBN 0-471-58494-0.

Modis, Theodore (1992) Predictions: Society's Telltale Signature Reveals the Past and Forecasts the Future, Simon & Schuster, New York. ISBN 0-671-75917-5

External links

Wikimedia Commons has media related to Logistic distribution.

Probability distributions

List

Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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