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A sigmoid function is a Mathematical Function that produces a '''sigmoid curve''' — a curve having an "S" shape. Often, ''sigmoid function'' refers to the special case of the Logistic Function shown at right and defined by the formula: : MEMBERS OF THE SIGMOID FAMILY In general, a sigmoid function is Real -valued and Differentiable , having a non- Negative or non- Positive first Derivative , one Local Minimum , and one Local Maximum . Besides the logistic function, sigmoid functions include the ordinary Arc-tangent , the Hyperbolic Tangent , and the Error Function . The Integral of any smooth, positive, "bump-shaped" function will be sigmoidal, thus the Cumulative Distribution Function s for many common Probability Distribution s are sigmoidal. The logistic sigmoid function is related to the hyperbolic tangent, e.g., by : SIGMOID FUNCTIONS IN NEURAL NETWORKS Sigmoid functions are often used in Neural Network s to introduce Nonlinearity in the model and/or to make sure that certain signals remain 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 . A reason for its popularity in neural networks is because the sigmoid function satisfies this property: : This simple polynomial relationship between the derivative and itself is computationally easy to perform. SEE ALSO |
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