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):
  • a Hyperbolic Tangent (tanh) Function.
  • a Gaussian Probability Function.
  • a Decision Tree.
• 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),
    • x₀ = 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.

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, x₀ = 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]

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. ...

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.

---