Sigmoid

Sigmoid(x)

The Sigmoid function is

[math]\displaystyle{ Sigmoid(x) = {1\over{1+\exp(-x)}} }[/math]

The Sigmoid function goes by several other names including the logistic function, the inverse logit function, and the expit function.

There are other functions that are also sigmoidal in shape, most notably the ArcTan and Tanh functions. These other sigmoidal fucntions differ in their asymptotic values. The Sigmoid(x) function goes to 0 as «x» goes to [math]\displaystyle{ -\infty }[/math] and to 1 as «x» goes to [math]\displaystyle{ +\infty }[/math].

The inverse of the Sigmoid function is the Logit function (aka log-odds function). Hence, Sigmoid returns the probability value that corresponds to a given log-odds.

Library

Advanced Math Functions

Uses

Since the Logit function is the link function in generalized linear regression that results in logistic regression, the Sigmoid function is used to apply the coefficients of logistic regression to make predictions. So if c are the coefficients returned by the LogisticRegression function, where c is a vector indexed by «K», and x is a new data point (also indexed by «K»), then the predicted probability for x is computed using:

Sigmoid(Sum(c*x, K))

Notes

dSigmoid

You can use the following User-Defined Function to compute the derivative of the Sigmoid function:

Function dSigmoid(x) := 0.25 / Cosh(90*x/Pi)^2

Relation to Tanh

Sigmoid() varies from 0 to 1. A related function is Tanh(), which goes from -1 to 1, but is on a different x-axis scale. The following equivalence holds (recall that in Analytica, the Tanh() function expects its parameter to be in degrees).

2*Sigmoid(x)-1 = Tanh(x*90/Pi)

History

Sigmoid was introduced as a built-in in Analytica 4.5, superseding the earlier InvLogit function that was part of the add-on Generalized Regression library.