Sigmoid function
Special cases of the sigmoid function include the Gompertz curve (used in modeling systems that saturate at large values of x) and the ogee curve (used in the spillway of some dams).
Sigmoid functions have domain of all real numbers, with return (response) value commonly monotonically increasing but could be decreasing.
Sigmoid functions most often show a return value (y axis) in the range 0 to 1.
A sigmoid function is a bounded, differentiable, real function that is defined for all real input values and has a non-negative derivative at each point[1] [2] and exactly one inflection point.
In general, a sigmoid function is monotonic, and has a first derivative which is bell shaped.
A sigmoid function is constrained by a pair of horizontal asymptotes as
because any smaller value will result in a function with multiple inflection points, which is therefore not a true sigmoid.
Many natural processes, such as those of complex system 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.
[6] The van Genuchten–Gupta model is based on an inverted S-curve and applied to the response of crop yield to soil salinity.
Examples of the application of the logistic S-curve to the response of crop yield (wheat) to both the soil salinity and depth to water table in the soil are shown in modeling crop response in agriculture.
In artificial neural networks, sometimes non-smooth functions are used instead for efficiency; these are known as hard sigmoids.
In audio signal processing, sigmoid functions are used as waveshaper transfer functions to emulate the sound of analog circuitry clipping.
[7] In biochemistry and pharmacology, the Hill and Hill–Langmuir equations are sigmoid functions.
In computer graphics and real-time rendering, some of the sigmoid functions are used to blend colors or geometry between two values, smoothly and without visible seams or discontinuities.
Titration curves between strong acids and strong bases have a sigmoid shape due to the logarithmic nature of the pH scale.
The logistic function can be calculated efficiently by utilizing type III Unums.
[8] An hierarchy of sigmoid growth models with increasing complexity (number of parameters) was built[9] with the primary goal to re-analyze kinetic data, the so called N-t curves, from heterogeneous nucleation experiments,[10] 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[11] originally devised to describe diffusion-limited crystal growth (not aggregation!)
[13] 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.
