Logistic Distribution - Applications

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. 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.

Other applications:

• Biology – to describe how species populations grow in competition
• Epidemiology – to describe the spreading of epidemics
• Psychology – to describe learning
• Technology – to describe how new technologies diffuse and substitute for each other
• Marketing – the diffusion of new-product sales
• Energy – the diffusion and substitution of primary energy sources, as in the Hubbert curve
• Hydrology - 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. 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.
• Physics - the cdf of this distribution describes a Fermi gas and more specifically the number of electrons within a metal that can be expected to occupy a given quantum state. Its range is between 0 and 1, reflecting the Pauli exclusion principle. The value is given as a function of the kinetic energy corresponding to that state and is parametrized by the Fermi energy and also the temperature (and Boltzmann constant). By changing the sign in front of the "1" in the denominator, one goes from Fermi–Dirac statistics to Bose–Einstein statistics. In this case, the expected number of particles (bosons) in a given state can exceed unity, which is indeed the case for systems such as lasers.

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.