A common example of a sigmoid function is thelogistic function, which is defined by the formula[1]
Other sigmoid functions are given in theExamples section. In some fields, most notably in the context ofartificial neural networks, the term "sigmoid function" is used as a synonym for "logistic function".
Special cases of the sigmoid function include theGompertz curve (used in modeling systems that saturate at large values ofx) and theogee curve (used in thespillway of somedams). Sigmoid functions have domain of allreal numbers, with return (response) value commonlymonotonically increasing but could be decreasing. Sigmoid functions most often show a return value (y axis) in the range 0 to 1. Another commonly used range is from −1 to 1.
There is also theHeaviside step function, which instantaneously transitions between 0 and 1.
A sigmoid function is abounded,differentiable, real function that is defined for all real input values and has a positive derivative at each point.[1][2]
A sigmoid function isconvex for values less than a particular point, and it isconcave for values greater than that point: in many of the examples here, that point is 0.
Up to shifts and scaling, many sigmoids are special cases of where is the inverse of the negativeBox–Cox transformation, and and are shape parameters.[4]
using the hyperbolic tangent mentioned above. Here, is a free parameter encoding the slope at, which must be greater than or equal to 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 and at 1 for all. Nonetheless, it issmooth (infinitely differentiable,)everywhere, including at.
Inverted logistic S-curve to model the relation between wheat yield and soil salinity
Many natural processes, such as those of complex systemlearning curves, exhibit a progression from small beginnings that accelerates and approaches a climax over time.[6] When a specific mathematical model is lacking, a sigmoid function is often used.[7]
Examples of the application of the logistic S-curve to the response of crop yield (wheat) to both the soil salinity and depth towater table in the soil are shown inmodeling crop response in agriculture.
InDigital signal processing in general, sigmoid functions, due to their higher order of continuity, have much faster asymptotic rolloff in thefrequency domain than a Heavyside step function, and therefore are useful to smoothen discontinuities before sampling to reduce aliasing. This is, for example, used to generate square waves in many kinds ofDigital synthesizer.
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 thepH scale.
The logistic function can be calculated efficiently by utilizingtype III Unums.[9]
An hierarchy of sigmoid growth models with increasing complexity (number of parameters) was built[10] with the primary goal to re-analyze kinetic data, the so called N-t curves, from heterogeneousnucleation experiments,[11] inelectrochemistry. 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[12] originally devised to describe diffusion-limited crystal growth (not aggregation!) in 2D, theJohnson–Mehl–Avrami–Kolmogorov (JMAK) model,[13] and theRichards model.[14] 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.[10]
^Markov, I. and Stoycheva, E. (1976). "Saturation Nucleus Density in the Electrodeposition of Metals onto Inert Electrodes II. Experimental".Thin Solid Films.35 (1). Elsevier:21–35.doi:10.1016/0040-6090(76)90237-6.{{cite journal}}: CS1 maint: multiple names: authors list (link)
^Fanfoni, M. and Tomellini, M. (1998). "The Johnson-Mehl-Avrami-Kolmogorov Model: A Brief Review".Il Nuovo Cimento D.20. Springer:1171–1182.doi:10.1007/BF03185527.{{cite journal}}: CS1 maint: multiple names: authors list (link)
^Tjørve, E. and Tjørve, K.M.C. (2010). "A Unified Approach to the Richards-Model Family for Use in Growth Analyses: Why We Need Only Two Model Forms".Journal of Theoretical Biology.267 (3). Elsevier:417–425.doi:10.1016/j.jtbi.2010.09.008.{{cite journal}}: CS1 maint: multiple names: authors list (link)
Mitchell, Tom M. (1997).Machine Learning. WCBMcGraw–Hill.ISBN978-0-07-042807-2.. (NB. In particular see "Chapter 4: Artificial Neural Networks" (in particular pp. 96–97) where Mitchell uses the word "logistic function" and the "sigmoid function" synonymously – this function he also calls the "squashing function" – and the sigmoid (aka logistic) function is used to compress the outputs of the "neurons" in multi-layer neural nets.)
Humphrys, Mark."Continuous output, the sigmoid function".Archived from the original on 2022-07-14. Retrieved2022-07-14. (NB. Properties of the sigmoid, including how it can shift along axes and how its domain may be transformed.)
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