Many natural processes, including those of complex system learning curves, exhibit a progression from small beginnings that accelerates and approaches a climax over time. When a detailed description is lacking, a sigmoid function is often used. A sigmoid curve is produced by a mathematical function having an "S" shape. Often, sigmoid function refers to the special case of the logistic function shown at right and defined by the formula
Another example is the Gompertz curve. It is used in modeling systems that saturate at large values of t. Another example is the ogee curve as used in the spillway of some dams. A wide variety of sigmoid functions have been used as the activation function of artificial neurons, including the logistic function and tanh(x).
In general, a sigmoid function is real-valued and differentiable, having either a non-negative or non-positive first derivative which is bell shaped. There are also a pair of horizontal asymptotes as . The logistic functions are sigmoidal and are characterized as the solutions of the differential equation
Besides the logistic function, sigmoid functions include the ordinary arctangent, the hyperbolic tangent, and the error function, but also the generalised logistic function and algebraic functions like .
The integral of any smooth, positive, "bump-shaped" function will be sigmoidal, thus the cumulative distribution functions for many common probability distributions are sigmoidal. The most famous such example is the error function.
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