A logistic function or logistic curve is a common sigmoid function, given its name in 1844 or 1845 by Pierre François Verhulst who studied it in relation to population growth. A generalized logistic curve can model the "S-shaped" behaviour (abbreviated S-curve) of growth of some population P. The initial stage of growth is approximately exponential; then, as saturation begins, the growth slows, and at maturity, growth stops.
A simple logistic function may be defined by the formula
where the variable P might be considered to denote a population, where e is Euler's number and the variable t might be thought of as time. For values of t in the range of real numbers from −∞ to +∞, the S-curve shown is obtained. In practice, due to the nature of the exponential function e−t, it is sufficient to compute t over a small range of real numbers such as .
The logistic function finds applications in a range of fields, including artificial neural networks, biology, biomathematics, demography, economics, chemistry, mathematical psychology, probability, sociology, political science, and statistics. It has an easily calculated derivative:
It also has the property that
Thus, the function is odd.
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