Sigmoid function

A sigmoid function gooseneck or S- function is a mathematical function with an S- shaped graph. Often, the term sigmoid logistic function is based on the special case determined by the equation

Will be described. This is Euler's number. This particular sigmoid function is therefore a scaled and shifted hyperbolic tangent function and has yielded corresponding symmetries.

The inverse of this function is:

Sigmoid functions in general

In general, a sigmoid function is a bounded and differentiable real function with a consistently positive or consistently negative first derivative and exactly one turning point.

Except the logistic function includes the amount of the arc-tangent sigmoid, the hyperbolic tangent and the error function, all of which are transcendental, but also simple algebraic functions, such as. The integral of any continuous, positive function with a "mountain" (more precisely, with exactly one local maximum and no local minimum, eg the Gaussian bell curve ) is also a sigmoid function. Therefore, many cumulative distribution functions are sigmoidal.

Sigmoid functions in neural networks

Sigmoid functions are often used in artificial neural networks as the activation function, as the use of differentiable functions using learning mechanisms, such as the back propagation algorithm allows. As the activating function of an artificial neuron, the sigmoid function is applied to the sum of the weighted input values ​​to obtain the output of the neuron.

The sigmoid function is mainly due to their simple differentiability preferably used as the activation function, because it applies to the logistic function:

For the derivative of the hyperbolic tangent sigmoid function applies: