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Logistic curve

The sigmoid curve is the curve whose formula is the sigmoid function
<math>P = \frac{1}{1 + e^{-t}},</math>
so-called because of its sigmoid shape. The sigmoid function is the solution of the first-order non-linear differential equation
<math>\frac{dP}{dt}=P(1-P),</math>
the continuous version of the logistic map. If P represents population size and t represents time, then the somewhat more general equation
<math>\frac{dP}{dt}=kP(C-P),</math>
where k is a constant proportional to the growth rate and C is a carrying capacity, expresses the fact that the rate of population growth is jointly proportional to the present population size and the amount by which that size falls short of the carrying capacity. The sigmoid function is the inverse of the logit function.

The sigmoid curve shows early exponential growth which slows to linear growth then decelerates until it reaches a saturation level at y = 1.

The conversion from the log-likelihood ratio of two alternatives to a probability takes the form of a sigmoid curve.

Members of the family of curves with obtained by linear scaling and translation of the sigmoid curve are called logistic curves, and are found in a range of fields, from biology to economics.

See also: Hubbert curve, Logistic regression, Generalised logistic curve, log-likelihood ratio

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