Logistic curve

A logistic function or logistic curve is a common "S" shape (sigmoid curve), with equation:

where

For values of x in the range of real numbers from −∞ to +∞, the S-curve shown on the right is obtained (with the graph of f approaching L as x approaches +∞ and approaching zero as x approaches −∞).

The function was named in 1844–1845 by Pierre François Verhulst, who studied it in relation to population growth. The initial stage of growth is approximately exponential; then, as saturation begins, the growth slows, and at maturity, growth stops.

The logistic function finds applications in a range of fields, including artificial neural networks, biology (especially ecology), biomathematics, chemistry, demography, economics, geoscience, mathematical psychology, probability, sociology, political science, linguistics, and statistics.

The standard logistic function is the logistic function with parameters (k = 1, x₀ = 0, L = 1) which yields

In practice, due to the nature of the exponential function e^−x, it is often sufficient to compute the standard logistic function for x over a small range of real numbers such as a range contained in [−6, +6].

The standard logistic function has an easily calculated derivative:

${\displaystyle f(x)={\frac {1}{1+e^{-x}}}={\frac {e^{x}}{1+e^{x}}}}$

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