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Logistic distribution

 
Statistics Dictionary: logistic distribution
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A continuous distribution with probability density function f given by




,
where α is the mean of the distribution and β is a positive parameter. The distribution has variance
⅓β2 π2
.



Logistic distribution. The distributions shown are for α=0. The variance of these distributions is ⅓β2π2.


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Wikipedia: Logistic distribution
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Logistic
Probability density function
Standard logistic PDF
Cumulative distribution function
Standard logistic CDF
Parameters \mu\, location (real)
s>0\, scale (real)
Support x \in (-\infty; +\infty)\!
Probability density function (pdf) \frac{e^{-(x-\mu)/s}} {s\left(1+e^{-(x-\mu)/s}\right)^2}\!
Cumulative distribution function (cdf) \frac{1}{1+e^{-(x-\mu)/s}}\!
Mean \mu\,
Median \mu\,
Mode \mu\,
Variance \frac{\pi^2}{3} s^2\!
Skewness 0\,
Excess kurtosis 6/5\,
Entropy \ln(s)+2\,
Moment-generating function (mgf) e^{\mu\,t}\,\mathrm{B}(1-s\,t,\;1+s\,t)\!
for |s\,t|<1\!, Beta function
Characteristic function e^{i \mu t}\,\mathrm{B}(1-ist,\;1+ist)\,
for |ist|<1\,

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).

Contents

Specification

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:

F(x; \mu,s) = \frac{1}{1+e^{-(x-\mu)/s}} \!
= \frac12 + \frac12 \;\operatorname{tanh}\!\left(\frac{x-\mu}{2\,s}\right).

Probability density function

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

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

See also: hyperbolic secant distribution

Quantile function

The inverse cumulative distribution function of the logistic distribution is F − 1, a generalization of the logit function, defined as follows:

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

Alternative parameterization

An alternative parameterization of the logistic distribution can be derived using the substitution \sigma^2 = \pi^2\,s^2/3. This yields the following density function:

g(x;\mu,\sigma) = f(x;\mu,\sigma\sqrt{3}/\pi) = \frac{\pi}{\sigma\,4\sqrt{3}} \,\operatorname{sech}^2\!\left(\frac{\pi}{2 \sqrt{3}} \,\frac{x-\mu}{\sigma}\right).


Applications

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.

The logistic distribution and the S-shaped pattern that results from it have been extensively used in many different areas the most important of which include:

♦ Biology - to describe how species populations grow in competition[1]

♦ Epidemiology - to describe the spreading of epidemics[2]

♦ Psychology - to describe learning[3]

♦ Technology - to describe how new technologies diffuse and substitute for each other[4]

♦ Market - the diffusion of new-product sales[5]

♦ Energy - the diffusion and substitution of primary energy sources[6]

Wiki letter w.svg
 Please help improve this article by expanding it. Further information might be found on the talk page. (January 2007)

Related distributions

If log(X) has a logistic distribution then X has a log-logistic distribution and X – a has a shifted log-logistic distribution.

Derivations

Expected Value

E[X]=\int_{-\infty}^{\infty} {\frac{xe^{-(x-\mu)/s}} {s\left(1+e^{-(x-\mu)/s}\right)^2}} \! dx = \int_{-\infty}^{\infty} \frac{x}{4\,s} \;\operatorname{sech}^2\!\left(\frac{x-\mu}{2\,s}\right)dx


Substitute: u=\frac{(x-\mu)}{2s}, du=\frac{1}{2s} dx


E[X]=\int_{-\infty}^{\infty} \frac{2\,s\,u+\mu}{2} \;\operatorname{sech}^2\!\left(u\right)du
E[X]=s\int_{-\infty}^{\infty} u \;\operatorname{sech}^2\!\left(u\right)du + \frac{\mu}{2} \int_{-\infty}^{\infty} \;\operatorname{sech}^2\!\left(u\right)du


Note the odd function: \int_{-\infty}^{\infty} u \;\operatorname{sech}^2\!\left(u\right)du = 0


E[X]=\frac{\mu}{2} \int_{-\infty}^{\infty} \;\operatorname{sech}^2\!\left(u\right)du = \frac{\mu}{2}\,2 = \mu

See also

Notes

  1. ^ P. F. Verhulst, "Recherches mathématiques sur la loi d'accroissement de la population", Nouveaux Mémoirs de l'Académie Royale des Sciences et des Belles-Lettres de Bruxelles, vol. 18 (1845); Alfred J. Lotka, Elements of Physical Biology, (Baltimore, MD: Williams & Wilkins Co., 1925).
  2. ^ Theodore Modis, Predictions: Society's Telltale Signature Reveals the Past and Forecasts the Future, Simon & Schuster, New York, 1992, pp 97-105.
  3. ^ Theodore Modis, Predictions: Society's Telltale Signature Reveals the Past and Forecasts the Future, Simon & Schuster, New York, 1992, Chapter 2.
  4. ^ J. C. Fisher and R. H. Pry , "A Simple Substitution Model of Technological Change", Technological Forecasting & Social Change, vol. 3, no. 1 (1971).
  5. ^ Theodore Modis, Conquering Uncertainty, McGraw-Hill, New York, 1998, Chapter 1.
  6. ^ Cesare Marchetti, "Primary Energy Substitution Models: On the Interaction between Energy and Society", Technological Forecasting & Social Change, vol. 10, (1977).

References

  • 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. ed.). ISBN 0-471-58494-0. 
v • d • e
Probability distributions
 
Continuous univariate supported on the whole real line (−∞, ∞)

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Statistics Dictionary. A Dictionary of Statistics. Second edition revised. Copyright © Oxford University Press, 2008. All rights reserved.  Read more
Wikipedia. This article is licensed under the Creative Commons Attribution/Share-Alike License. It uses material from the Wikipedia article "Logistic distribution".  Read more