Sigmoid function

A sigmoid function is any <a href="/facts/Mathematical_function/CdReEKCh">mathematical function</a> whose <a href="/facts/Graph_of_a_function/htYX0BGX">graph</a> has a characteristic S-shaped or sigmoid curve.
A common example of a sigmoid function is the <a href="/facts/Logistic_function/IaQ254t7">logistic function</a>, which is defined by the formula

σ
        (
        x
        )
        =
        
          
            1
            
              1
              +
              
                e
                
                  −
                  x
                
              
            
          
        
        =
        
          
            
              e
              
                x
              
            
            
              1
              +
              
                e
                
                  x
                
              
            
          
        
        =
        1
        −
        σ
        (
        −
        x
        )
        .
      
    
    {\displaystyle \sigma (x)={\frac {1}{1+e^{-x}}}={\frac {e^{x}}{1+e^{x}}}=1-\sigma (-x).}

Other sigmoid functions are given in the Examples section. In some fields, most notably in the context of <a href="/facts/Artificial_neural_network/6V1jMlkx">artificial neural networks</a>, the term "sigmoid function" is used as a synonym for "logistic function".
Special cases of the sigmoid function include the <a href="/facts/Gompertz_curve/9nYLlKvx">Gompertz curve</a> (used in modeling systems that saturate at large values of x) and the <a href="/facts/Ogee_curve/317SgIMu">ogee curve</a> (used in the <a href="/facts/Spillway/PBPNrfc2">spillway</a> of some <a href="/facts/Dam/BroJFhEI">dams</a>). Sigmoid functions have domain of all <a href="/facts/Real_number/R02gw5Pb">real numbers</a>, with return (response) value commonly <a href="/facts/Monotonically_increasing/MjQK0YL2">monotonically increasing</a> but could be decreasing. Sigmoid functions most often show a return value (y axis) in the range 0 to 1. Another commonly used range is from −1 to 1.
A wide variety of sigmoid functions including the logistic and <a href="/facts/Hyperbolic_tangent/Y4Cb5S35">hyperbolic tangent</a> functions have been used as the <a href="/facts/Activation_function/S4NImL6L">activation function</a> of <a href="/facts/Artificial_neuron/TSH6nsgg">artificial neurons</a>. Sigmoid curves are also common in statistics as <a href="/facts/Cumulative_distribution_function/WaKU8tp4">cumulative distribution functions</a> (which go from 0 to 1), such as the integrals of the <a href="/facts/Logistic_density/QlQNt0uC">logistic density</a>, the <a href="/facts/Normal_density/UapjjPyQ">normal density</a>, and <a href="/facts/Student%2527s_t-distribution/DeT1SDqH">Student's t probability density functions</a>. The logistic sigmoid function is invertible, and its inverse is the <a href="/facts/Logit/5674C2DL">logit</a> function.

Sigmoid function open-in-new

Sigmoid function