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of lambda bell shaped function m13 integration of the sum of lambda bell shaped functions m14 integration of the product of two lambda bell shaped functions m15 integration of the difference of two shifted sigmoid functions m16 integration of the product of two shifted sigmoid functions m17 convolution of sigmoid functions m18 integration of the product of lambda and sigmoid function m19 solutions of ordinary differential equations m20 solutions of partial differential equation pde m21 solutions of functional differential equation fde m22 sum of a sigmoid function and some derivatives m23 combination of sigmoid functions its derivative and integral m24 filtering sigmoid functions m25 special cases of gauss hypergeometric functions m26 feedback closed loop systems m27 recursive functions m28 recursive time delayed feed forward loops m29 solutions of stochastic differential equation m30 chaotic sigmoid functions consult reference 1 for more details definition edit 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 13 14 properties edit 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 displaystyle 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 edit some sigmoid functions compared in the drawing all functions are normalized in such a way that their slope at the origin is 1 logistic function f x 1 1 e x displaystyle f x frac 1 1 e x hyperbolic tangent shifted and scaled version of the logistic function above f x tanh x e x e x e x e x displaystyle f x tanh x frac e x e x e x e x arctangent function f x arctan x displaystyle f x arctan x gudermannian function f x gd x 0 x d t cosh t 2 arctan tanh x 2 displaystyle 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 erf x 2 π 0 x e t 2 d t displaystyle f x operatorname erf x frac 2 sqrt pi int _ 0 x e t 2 dt generalised logistic function f x 1 e x α α 0 displaystyle f x left 1 e x right alpha quad alpha 0 smoothstep function f x 0 1 1 u 2 n d u 1 0 x 1 u 2 n d u x 1 sgn x x 1 n z 1 displaystyle 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 leq 1 operatorname sgn x x geq 1 end cases quad n in mathbb z geq 1 some algebraic functions for example f x x 1 x 2 displaystyle f x frac x sqrt 1 x 2 and in a more general form 15 f x x 1 x k 1 k displaystyle f x frac x left 1 x k right 1 k up to shifts and scaling many sigmoids are special cases of f x φ φ x β α displaystyle f x varphi varphi x beta alpha where φ x λ 1 λ x 1 λ λ 0 e x λ 0 displaystyle varphi x lambda begin cases 1 lambda x 1 lambda lambda neq 0 e x lambda 0 end cases is the inverse of the negative box cox transformation and α 1 displaystyle alpha 1 and β 1 displaystyle beta 1 are shape parameters 16 smooth transition function 17 normalized to 1 1 f x 2 1 e 2 m x 1 x 2 1 x 1 sgn x x 1 tanh m x 1 x 2 x 1 sgn x x 1 displaystyle begin aligned f x begin cases displaystyle frac 2 1 e 2m frac x 1 x 2 1 x 1 operatorname sgn x x geq 1 end cases begin cases displaystyle tanh left m frac x 1 x 2 right x 1 operatorname sgn x x geq 1 end cases end aligned using the hyperbolic tangent mentioned above here m displaystyle m is a free parameter encoding the slope at x 0 displaystyle x 0 which must be greater than or equal to 3 displaystyle 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 1 displaystyle x leq 1 and at 1 for all x 1 displaystyle x geq 1 nonetheless it is smooth infinitely differentiable c displaystyle c infty everywhere including at x 1 displaystyle x pm 1 applications edit inverted logistic s curve to model the relation between wheat yield and soil salinity 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 18 when a specific mathematical model is lacking a sigmoid function is often used 19 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 non smooth functions are sometimes used 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 20 in digital signal processing in general sigmoid functions due to their higher order of continuity have much faster asymptotic rolloff in the frequency domain than a heavyside step function and therefore are useful to smooth discontinuities before sampling to reduce aliasing this is for example used to generate square waves in many kinds of digital synthesizer in biochemistry and pharmacology the hill and hill langmuir equations are sigmoid functions in driving changing lanes is accomplished smoothly by following a sigmoid trajectory in computer graphics and real time rendering sigmoid functions are used to blend colors or geometry between two values producing smooth transitions 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 21 a hierarchy of sigmoid growth models with increasing complexity number of parameters was built 22 with the primary goal to re analyze kinetic data the so called n t curves from heterogeneous nucleation experiments 23 in electrochemistry the hierarchy includes at present three models with 1 2 and 3 parameters if not counting the maximal number of nuclei n max respectively a tanh 2 based model called α 21 24 originally devised to describe diffusion limited crystal growth not aggregation in 2d the johnson mehl avrami kolmogorov jmak model 25 and the richards model 26 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 22 see also edit wikimedia commons has media related to sigmoid functions step function linear combination of indicator functions of real intervals sign function function returning minus 1 zero or plus 1 heaviside step function indicator function of positive numbers logistic regression statistical model for a binary dependent variable logit function in statistics softplus function smoothed ramp function pages displaying short descriptions of redirect targets softmax function smooth approximation of one hot arg max swish function mathematical activation function in data analysis weibull distribution continuous probability distribution fermi dirac statistics statistical description for the behavior of fermions references edit 1 2 3 grebenc andrej august 2025 foundations for a general theory of sigmoid functions modelling with 30 methods intelligent systems conference cham springer nature switzerland pp 481 511 doi 10 1007 978 3 031 99958 1 elliot d l 1993 a better activation function for artificial neural networks technical report college park md institute for systems research university of maryland tr 93 8 richards f j 1959 a flexible growth function for empirical use journal of experimental botany 10 2 290 300 doi 10 1093 jxb 10 2 290 verhulst p f 1838 notice sur la loi que la population suit dans son accroissement correspondance mathématique et physique 10 113 121 laplace p s 1774 mémoire sur la probabilité des causes par les évènements mémoires de l académie royale des sciences présentés par divers savans 6 621 656 rayleigh l 1880 on the resultant of a large number of vibrations of the same pitch and of arbitrary phase philosophical magazine 5th series 10 73 78 cauchy a l 1853 memoire sur les resultats moyens d un tres grand nombre des observations comptes rendus hebdomadaires des séances de l académie des sciences 37 381 385 fisk p r 1961 the graduation of income distributions econometrica 29 2 171 185 doi 10 2307 1909287 gauss c f 1809 theoria motvs corporvm coelestivm in sectionibvs conicis solem ambientivm hamburg perthes et besser glaisher james whitbread lee 1871 on a class of definite integrals london edinburgh and dublin philosophical magazine and journal of science 4th series 42 277 294 302 doi 10 1080 14786447108640568 galton f 1874 on men of science their nature and their nurture proceedings of the royal institution of great britain 7 227 236 grebenc andrej 2026 chaotic sigmoid functions to be published han jun morag claudio 1995 the influence of the sigmoid function parameters on the speed of backpropagation learning in mira josé sandoval francisco eds from natural to artificial neural computation lecture notes in computer science vol 930 pp 195 201 doi 10 1007 3 540 59497 3_175 isbn 978 3 540 59497 0 ling yibei he bin december 1993 entropic analysis of biological growth models ieee transactions on biomedical engineering 40 12 1193 2000 doi 10 1109 10 250574 pmid 8125495 dunning andrew j kensler jennifer coudeville laurent bailleux fabrice 2015 12 28 some extensions in continuous methods for immunological correlates of protection bmc medical research methodology 15 107 107 doi 10 1186 s12874 015 0096 9 pmc 4692073 pmid 26707389 grex growth curve explorer github 2022 07 09 archived from the original on 2022 08 25 retrieved 2022 08 25 epsilondelta 2022 08 16 smooth transition function in one dimension smooth transition function series part 1 13 29 14 04 via www youtube com laurens speelman yuki numata 2022 harnessing the power of s curves rmi rocky mountain institute gibbs mark n mackay d november 2000 variational gaussian process classifiers ieee transactions on neural networks 11 6 1458 1464 doi 10 1109 72 883477 pmid 18249869 s2cid 14456885 smith julius o 2010 physical audio signal processing 2010 ed w3k publishing isbn 978 0 9745607 2 4 archived from the original on 2022 07 14 retrieved 2020 03 28 gustafson john l yonemoto isaac 2017 06 12 beating floating point at its own game posit arithmetic pdf archived pdf from the original on 2022 07 14 retrieved 2019 12 28 1 2 kleshtanova viktoria and ivanov vassil v and hodzhaoglu feyzim and prieto jose emilio and tonchev vesselin 2023 heterogeneous substrates modify non classical nucleation pathways reanalysis of kinetic data from the electrodeposition of mercury on platinum using a hierarchy of sigmoid growth models crystals 13 12 mdpi 1690 doi 10 3390 cryst13121690 hdl 10261 341589 cite journal cs1 maint multiple names authors list link 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 ivanov v v and tielemann c and avramova k and reinsch s and tonchev v 2023 modeling crystallization when the normal growth velocity depends on the supersaturation journal of physics and chemistry of solids 181 111542 elsevier doi 10 1016 j jpcs 2023 111542 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 further reading edit mitchell tom m 1997 machine learning wcb mcgraw hill isbn 978 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 retrieved 2022 07 14 nb properties of the sigmoid including how it can shift along axes and how its domain may be transformed v t e artificial intelligence ai history timeline glossary lists algorithms companies institutions projects software open source proprietary concepts automated reasoning automated planning constraint satisfaction knowledge representation parameter hyperparameter loss functions regression bias variance tradeoff double descent overfitting clustering gradient descent sgd quasi newton method conjugate gradient method backpropagation attention convolution normalization batchnorm activation softmax sigmoid rectifier gating weight initialization regularization datasets augmentation prompt engineering reinforcement learning q learning sarsa imitation policy gradient diffusion latent diffusion model autoregression adversary rag uncanny valley llm post training rlhf self supervised learning reflection recursive self improvement hallucination word embedding vibe coding blended ai open source ai sovereign ai symbolic ai neuro symbolic ai situated approach actor critic algorithm applications automated theorem proving general game playing machine learning in context learning artificial neural network deep learning language model large nmt reasoning model context protocol intelligent agent ai agent artificial human companion humanity s last exam lethal autonomous weapons laws generative ai weak ai hypothetical artificial general intelligence agi artificial superintelligence asi agent2agent protocol physical ai implementations audio visual alexnet wavenet human image synth...
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