Sigmoid function: Meaning (information, definition, explanation, facts)

logistic function.

The sigmoid function or sigmoid curve is the mathematical function defined by the formula

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.

The terms sigmoid function or sigmoid curve are also used in the general sense of any real function of a real variable with a sigmoid ("S"-shaped) graph: i.e. a differentiable function whose first derivative is non-negative and has a single local maximum at a finite argument. Besides the standard function above and its examples of sigmoid functions (in this general sense) are the ordinary arc-tangent, the hyperbolic tangent, and the error function.

Properties

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

with boundary condition P(0) = 1 / 2. 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.

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.