# Logistic Sigmoid Function

A Logistic Sigmoid Function is a sigmoid exponential probability function of the form [math]\displaystyle{ f(t,A,B,C) \equiv (C + Ae^{-Bt})^{-1} }[/math].

**AKA:**Inverse Logit.**Context:**- It can (typically) be a member of a logistic statistical model family
- It can (often) be represented as [math]\displaystyle{ f(x) = (1 + e^{-x})^{-1} }[/math].
- It can be a solution to a simple First-Order Non-Linear Differential Equation.
- It can show early exponential growth for negative
*t*, linear growth rof slope 1/4 near*t*= 0, then approaches*y*=1 with an exponentially decaying gap. - It can be a Fitted Logistic Function (produced by a logistic function fitting system that maximizes maximum likelihood).
- It can range from being a Binomial Logistic Function to being a Multinomial Logistic Function.
- It can be an inverse function of a Logit Function.

**Example(s):**- [math]\displaystyle{ f(x) = (3.4 + 1.22e^{-5.4x})^{-1} }[/math].
- a Fitted Logistic Function, such as
- a Neural Logistic Activation Function.
- …

**Counter-Example(s):****See:**Double Logistic Function, Euler's Number, Logistic Regression Model.

## References

### 2018

- (Wikipedia, 2018) ⇒ https://en.wikipedia.org/wiki/logistic_function Retrieved:2018-2-4.
- A
**logistic function**or**logistic curve**is a common "S" shape (sigmoid curve), with equation: : [math]\displaystyle{ f(x) = \frac{L}{1 + e^{-k(x-x_0)}} }[/math] where*e*= the natural logarithm base (also known as Euler's number),*x*_{0}= the*x*-value of the sigmoid's midpoint,*L*= the curve's maximum value, and*k*= the steepness of the curve. For values of*x*in the domain of real numbers from −∞ to +∞, the S-curve shown on the right is obtained (with the graph of*f*approaching*L*as*x*approaches +∞ and approaching zero as*x*approaches −∞). The function was named in 1844–1845 by Pierre François Verhulst, who studied it in relation to population growth. The initial stage of growth is approximately exponential; then, as saturation begins, the growth slows, and at maturity, growth stops.

- The logistic function finds applications in a range of fields, including artificial neural networks, biology (especially ecology), biomathematics, chemistry, demography, economics, geoscience, mathematical psychology, probability, sociology, political science, linguistics, and statistics.

- A

### 2018

- (Wikipedia, 2018) ⇒ https://en.wikipedia.org/wiki/logistic_function#Mathematical_properties Retrieved:2018-2-4.
- The
*standard logistic function*is the logistic function with parameters (*k*= 1,*x*_{0}= 0,*L*= 1) which yields : [math]\displaystyle{ \begin{align} f(x) &= \frac{1}{1 + e^{-x}} \\ &= \frac{e^x}{1 + e^x} \\ &= \tfrac12 + \tfrac12 \tanh(\tfrac{x}{2}) \\ \end{align} }[/math] In practice, due to the nature of the exponential function*e*^{−x}, it is often sufficient to compute the standard logistic function for*x*over a small range of real numbers such as a range contained in [−6, +6].The logistic function has the symmetry property that: : [math]\displaystyle{ 1-f(x) = f(-x). }[/math] Thus, [math]\displaystyle{ x \mapsto f(x) - 1/2 }[/math] is an odd function.

The logistic function is an offset and scaled hyperbolic tangent function : [math]\displaystyle{ f(x) = \tfrac12 + \tfrac12 \tanh(\tfrac{x}{2}) }[/math] or : [math]\displaystyle{ \tanh(x) = 2 \, f(2x) - 1 }[/math] .

This follows from : [math]\displaystyle{ \begin{align} \tanh(x) & = \frac{e^x - e^{-x}}{e^x + e^{-x}} = \frac{e^x \cdot \left(1 - e^{-2x}\right)}{e^x \cdot \left(1 + e^{-2x}\right)} \\[6pt] & = f(2x) - \frac{e^{-2x}}{1+e^{-2x}} = f(2x) - \frac{e^{-2x} + 1 - 1}{1+e^{-2x}} = 2f(2x)-1. \end{align} }[/math]

- The

### 2013

- (Wikipedia, 2013) ⇒ http://en.wikipedia.org/wiki/Logistic_function
- A
**logistic function**or logistic curve is a common sigmoid function, given its name (in reference to its S-shape) in 1844 or 1845 by Pierre François Verhulst who studied it in relation to population growth. A generalized logistic curve can model the "S-shaped" behaviour (abbreviated S-curve) of growth of some population*P*. The initial stage of growth is approximately exponential; then, as saturation begins, the growth slows, and at maturity, growth stops. ...

- A

### 2009

- http://en.wiktionary.org/wiki/logistic_function
- (mathematics) A function, the result of the division of two exponential functions, that gives rise to the logistic curve.