Sigmoid function

A sigmoid function is a bounded, differentiable, real function that is defined for all real input values and has a positive derivative at each point. == Properties ==

In general, a sigmoid function is monotonic, and has a first derivative which is bell shaped. Conversely, the integral of any continuous, non-negative, bell-shaped function (with one local maximum and no local minimum, unless degenerate) will be sigmoidal. Thus the cumulative distribution functions for many common probability distributions are sigmoidal. One such example is the error function, which is related to the cumulative distribution function of a normal distribution; another is the arctan function, which is related to the cumulative distribution function of a Cauchy distribution. A sigmoid function is constrained by a pair of horizontal asymptotes as x \rightarrow \pm \infty. A sigmoid function is convex for values less than a particular point, and it is concave for values greater than that point: in many of the examples here, that point is 0. == Examples ==

• Logistic function f(x) = \frac{1}{1 + e^{-x}} • Hyperbolic tangent (shifted and scaled version of the logistic function, above) f(x) = \tanh x = \frac{e^x-e^{-x}}{e^x+e^{-x}} • Arctangent function f(x) = \arctan x • Gudermannian function f(x) = \operatorname{gd}(x) = \int_0^x \frac{dt}{\cosh t} = 2\arctan\left(\tanh\left(\frac{x}{2}\right)\right) • Error function f(x) = \operatorname{erf}(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} \, dt • Generalised logistic function f(x) = \left(1 + e^{-x} \right)^{-\alpha}, \quad \alpha > 0 • Smoothstep function f(x) = \begin{cases} {\displaystyle \left( \int_0^1 \left(1 - u^2\right)^N du \right)^{-1} \int_0^x \left( 1 - u^2 \right)^N \ du}, & |x| \le 1 \\ \\ \sgn(x) & |x| \ge 1 \\ \end{cases} \quad N \in \mathbb{Z} \ge 1 • Some algebraic functions, for example f(x) = \frac{x}{\sqrt{1+x^2}} • and in a more general form normalized to (−1,1): f(x) = \begin{cases} {\displaystyle 2\frac{e^{\frac{1}{u}}}{e^{\frac{1}{u}}+e^{\frac{-1}{1+u}}} - 1}, u=\frac{x+1}{-2}, & |x| AManWithNoPlan simplified below --> \begin{align}f(x) &= \begin{cases} {\displaystyle \frac{2}{1+e^{-2m\frac{x}{1-x^2}}} - 1}, & |x| using the hyperbolic tangent mentioned above. Here, m is a free parameter encoding the slope at x=0, which must be greater than or equal to \sqrt{3} because any smaller value will result in a function with multiple inflection points, which is therefore not a true sigmoid. This function is unusual because it actually attains the limiting values of −1 and 1 within a finite range, meaning that its value is constant at −1 for all x \leq -1 and at 1 for all x \geq 1. Nonetheless, it is smooth (infinitely differentiable, C^\infty) everywhere, including at x = \pm 1. == Applications ==

Many natural processes, such as those of complex systems learning curves, exhibit a progression from small beginnings that accelerates and approaches a climax over time. When a specific mathematical model is lacking, a sigmoid function is often used. with the primary goal to re-analyze kinetic data, the so-called N-t curves, from heterogeneous nucleation experiments, in electrochemistry. The hierarchy includes at present three models, with 1, 2, and 3 parameters, if not counting the maximal number of nuclei Nmax, respectively—a tanh2 based model called α21 originally devised to describe diffusion-limited crystal growth (not aggregation!) in 2D, the Johnson–Mehl–Avrami–Kolmogorov (JMAK) model, and the Richards model. It was shown that for the concrete purpose, even the simplest model works, and thus it was implied that the experiments revisited are an example of two-step nucleation with the first step being the growth of the metastable phase in which the nuclei of the stable phase form. == See also ==