Sigmoid curve

The logistic function or logistic curve is defined by the mathematical formula:

<math>P(t) = a\frac{1 + m e^{-t/\tau}}{1 + n e^{-t/\tau}}<math>

for real parameters a, m, n, and <math>\tau<math>.

Applications

These functions are found in a range of fields, from biology to economics. For example, a common model of population growth states that:

• the rate of reproduction is proportional to the existing population, all else being equal
• the rate of reproduction is proportional to the amount of available resources, all else being equal. Thus the second term models the competition for available resources, which tends to limit the population growth.

Letting P represent population size and t represent time, this model is formalized by the differential equation:

<math>\frac{dP}{dt}=kP(C-P) \qquad \mbox{(1)},<math>

where the constant k defines the growth rate and C is the carrying capacity. The solution to this equation is a logistic function.

The logistic function is the inverse of the logit function and so can be used to convert the logarithm of odds into a probability; the conversion from the log-likelihood ratio of two alternatives also 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. Alfred J. Lotka derived the equation again in 1925, calling it the law of population growth.

Sigmoid function

The sigmoid function or sigmoid curve is a special case of the logistic function (for parameters <math> a=1, m=0, n=1, \tau=1 <math>):

<math>P(t) = \frac{1}{1 + e^{-t}}<math>

Its name is due to the sigmoid shape of its graph. This function is also called the standard logistic function and is often encountered in many technical domains, especially probability, statistics, biomathematics, and economics. See also sigmoid curve, for related uses of this term.

Properties of the sigmoid function

The (standard) sigmoid function is the solution of the first-order non-linear differential equation

<math>\frac{dP}{dt}=P(1-P), \quad\mbox{(1)}<math>

with boundary condition <math>P(0)=1/2<math>. Equation (1) is the continuous version of the logistic map.

The sigmoid curve shows early exponential growth for negative t, which slows to linear growth of slope 1/4 near t = 0, then approaches y = 1 with an exponentially decaying gap.

The sigmoid function is the inverse of the logit function.

Sigmoid functions in neural nets

In a neural network, a sigmoid function is often used to introduce nonlinearity in the model and/or to make sure that certain signals remains within a specified range. A popular neural net element computes a linear combination of its input signals, and applies a bounded sigmoid function to the result; this model can be seen as a "smoothed" variant of the classical threshold neuron.

See also

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

References

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