Logistic curve
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The sigmoid curve is the curve whose formula is the sigmoid function. Members of the family of curves 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.

The sigmoid function

The sigmoid function is

so-called because of its sigmoid shape. The sigmoid function is the solution of the first-order non-linear differential equation

the continuous version of the logistic map. If P represents population size and t represents time, then the somewhat more general equation

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.

History

The Verhulst equation, (1), was first published by Pierre F. Verhulst in 1838, after he had read Thomas Malthus' Essay on the Principle of Population. Verhulst derived his logistique equation (logistic equation) to describe the self-limiting growth of a biological population. The equation is also sometimes called the Verhulst-Pearl equation following its rediscovery in 1920. A. J. Lotka derived the equation again in 1925, calling it the law of population growth.

See also

• Hubbert curve
• Logistic regression
• Generalised logistic curve
• log-likelihood ratio

Bibliography

• Kingsland, S. E. (1985) Modeling nature ISBN 0226437280

External links

• http://www.dartmouth.edu/~math3f98/csc98/chap5/CSC.USAPop5.html
• http://www.ento.vt.edu/~sharov/PopEcol/lec5/logist.html
• http://www.nlreg.com/aids.htm
• http://luna.cas.usf.edu/~mbrannic/files/regression/Logistic.html

The original article can be found at: http://en.wikipedia.org/wiki/Logistic_curve.
All text is available under the terms of the GNU Free Documentation License.