# Logistic function

The logistic function is a sigmoid function that maps the real numbers to the unit interval $$(0, 1)$$ using the formula $$\displaystyle f(x) = \frac{1}{1 + e^{-x}}$$.

More generally, there exists a family of logistic functions that can be written as $$\displaystyle f(x) = \frac{M}{1 + \alpha^{c(x_0 - x)}}$$, where:

• $$M$$ is the upper bound of the function (in which case the function maps to the interval $$(0, M)$$ instead). When $$M = 1$$, the logistic function is usually being used to calculate some probability or proportion of a total.

• $$x_0$$ is the inflection point of the curve, or the value of $$x$$ where the function’s growth stops speeding up and starts slowing down.

• $$\alpha$$ is a variable controlling the steepness of the curve.

• $$c$$ is a scaling factor for the distance.

## Applications

• The logistic function is used to model growth rates of populations in an ecosystem with a limited carrying capacity.

• The inverse logistic function (with $$\alpha = 2$$) is used to convert a probability to log-odds form for use in Bayes’ rule.

• The logistic function (with $$\alpha = 10$$ and $$c = 1/400$$) is used to calculate the expected probability of a player winning given a specific difference in rating in the Elo rating system.

Parents:

• Mathematics

Mathematics is the study of numbers and other ideal objects that can be described by axioms.