Editing Sigmoid function (section)

== Applications ==
[[File:Gohana inverted S-curve.png|thumb|right|320px|Inverted logistic S-curve to model the relation between wheat yield and soil salinity]]

Many natural processes, such as those of complex system [[learning curve]]s, exhibit a progression from small beginnings that accelerates and approaches a climax over time.<ref>{{cite web |author1=Laurens Speelman, Yuki Numata |title=Harnessing the Power of S-Curves |url=https://rmi.org/insight/harnessing-the-power-of-s-curves/ |website=RMI |publisher=[[RMI (energy organization)|Rocky Mountain Institute]] |date=2022}}</ref> When a specific mathematical model is lacking, a sigmoid function is often used.<ref name="Gibbs_2000" />

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 [[logistic function#In agriculture: modeling crop response|modeling crop response in agriculture]].

In [[artificial neural network]]s, sometimes non-smooth functions are used instead for efficiency; these are known as [[hard sigmoid]]s.

In [[audio signal processing]], sigmoid functions are used as [[waveshaper]] [[transfer function]]s to emulate the sound of [[analog circuitry]] [[clipping (audio)|clipping]].<ref name="Smith_2010" />

In [[biochemistry]] and [[pharmacology]], the [[Hill equation (biochemistry)|Hill]] and [[Hill–Langmuir equation]]s 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 curve]]s 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 [[Unum type 3|type III Unums]].<ref name="Gustafson-Yonemoto_2017" />

An hierarchy of sigmoid growth models with increasing complexity (number of parameters) was built<ref name="app-kleshtanova2023">{{cite journal
 | author = Kleshtanova, Viktoria and Ivanov, Vassil V and Hodzhaoglu, Feyzim and Prieto, Jose Emilio and Tonchev, Vesselin
 | title = Heterogeneous Substrates Modify Non-Classical Nucleation Pathways: Reanalysis of Kinetic Data from the Electrodeposition of Mercury on Platinum Using Hierarchy of Sigmoid Growth Models
 | journal = Crystals
 | volume = 13
 | number = 12
 | pages = 1690
 | year = 2023
 | publisher = MDPI
 | doi = 10.3390/cryst13121690
| doi-access = free
 | bibcode = 2023Cryst..13.1690K
 }}</ref> with the primary goal to re-analyze kinetic data, the so called N-t curves, from heterogeneous [[nucleation]] experiments,<ref name="app-Markov_1976">{{cite journal
 | author = Markov, I. and Stoycheva, E.
 | title = Saturation Nucleus Density in the Electrodeposition of Metals onto Inert Electrodes II. Experimental
 | journal = Thin Solid Films
 | volume = 35
 | number = 1
 | pages = 21–35
 | year = 1976
 | publisher = Elsevier
 | doi = 10.1016/0040-6090(76)90109-7
}}</ref> in [[electrochemistry]]. The hierarchy includes at present three models, with 1, 2 and 3 parameters, if not counting the maximal number of nuclei N<sub>max</sub>, respectively—a tanh<sup>2</sup> based model called α<sub>21</sub><ref name="app-Ivanov_2023">{{cite journal
 | author = Ivanov, V.V. and Tielemann, C. and Avramova, K. and Reinsch, S. and Tonchev, V.
 | title = Modelling Crystallization: When the Normal Growth Velocity Depends on the Supersaturation
 | journal = Journal of Physics and Chemistry of Solids
 | volume = 181
 | pages = 111542
 | year = 2023
 | publisher = Elsevier
 | doi = 10.1016/j.jpcs.2022.111542
| doi-broken-date = 28 January 2025
 }}</ref> originally devised to describe diffusion-limited crystal growth (not aggregation!) in 2D, the Johnson-Mehl-Avrami-Kolmogorov (JMAKn) model,<ref name="app-Fanfoni_1998">{{cite journal
 | author = Fanfoni, M. and Tomellini, M.
 | title = The Johnson-Mehl-Avrami-Kohnogorov Model: A Brief Review
 | journal = Il Nuovo Cimento D
 | volume = 20
 | pages = 1171–1182
 | year = 1998
 | publisher = Springer
 | doi = 10.1007/s002690050098
}}</ref> and the Richards model.<ref name="app-Tjorve_2010">{{cite journal
 | author = Tjørve, E. and Tjørve, K.M.C.
 | title = A Unified Approach to the Richards-Model Family for Use in Growth Analyses: Why We Need Only Two Model Forms
 | journal = Journal of Theoretical Biology
 | volume = 267
 | number = 3
 | pages = 417–425
 | year = 2010
 | publisher = Elsevier
 | doi = 10.1016/j.jtbi.2010.02.027
| pmid = 20176032
 }}</ref> 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.<ref name="app-kleshtanova2023"/>