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Logistic function

logistic function, logistic function equation
A logistic function or logistic curve is a common "S" shape sigmoid curve, with equation:

f x = L 1 + e − k x − x 0 ^


  • e = the natural logarithm base also known as Euler's number,
  • x0 = the x-value of the sigmoid's midpoint,
  • L = the curve's maximum value, and
  • k = the steepness of the curve

For values of x in the range of real numbers from −∞ to +∞, the S-curve shown on the right is obtained with the graph of f approaching L as x approaches +∞ and approaching zero as x approaches −∞

The function was named in 1844–1845 by Pierre François Verhulst, who studied it in relation to population growth The initial stage of growth is approximately exponential; then, as saturation begins, the growth slows, and at maturity, growth stops

The logistic function finds applications in a range of fields, including artificial neural networks, biology especially ecology, biomathematics, chemistry, demography, economics, geoscience, mathematical psychology, probability, sociology, political science, linguistics, and statistics


  • 1 Mathematical properties
    • 11 Derivative
    • 12 Logistic differential equation
    • 13 Rotational symmetry about 0, ½
  • 2 Applications
    • 21 In ecology: modeling population growth
      • 211 Time-varying carrying capacity
    • 22 In statistics and machine learning
      • 221 Logistic regression
      • 222 Neural networks
    • 23 In medicine: modeling of growth of tumors
    • 24 In chemistry: reaction models
    • 25 In physics: Fermi distribution
    • 26 In linguistics: language change
    • 27 In economics and sociology: diffusion of innovations
  • 3 See also
  • 4 Notes
  • 5 References
  • 6 External links

Mathematical properties

The standard logistic function is the logistic function with parameters k = 1, x0 = 0, L = 1 which yields

f x = 1 1 + e − x

In practice, due to the nature of the exponential function e−x, it is often sufficient to compute the standard logistic function for x over a small range of real numbers such as a range contained in


The standard logistic function has an easily calculated derivative:

d d x f x = e − x 1 + e − x 2 = 1 1 + e − x e − x 1 + e − x = f x 1 − f x fx=^=\left\right\left\right=fx1-fx

It also has the property that

1 − f x = f − x

Thus, x ↦ f x − 1 / 2 is an odd function

Logistic differential equation

The standard logistic function is the solution of the simple first-order non-linear ordinary differential equation

d d x f x = f x 1 − f x fx=fx1-fx

with boundary condition f0 = 1/2 This equation is the continuous version of the logistic map

The qualitative behavior is easily understood in terms of the phase line: the derivative is null when function is unit and the derivative is positive for f between 0 and 1, and negative for f above 1 or less than 0 though negative populations do not generally accord with a physical model This yields an unstable equilibrium at 0, and a stable equilibrium at 1, and thus for any function value greater than zero and less than unit, it grows to unit

The above equation can be rewritten in the following steps:

d d x f x = f x 1 − f x fx=fx1-fx d y d x = y 1 − y =y1-y d y d x = y − y 2 =y-y^ d y d x − y = − y 2 -y=-y^

Which is a special case of the Bernoulli differential equation and has the following solution:

f x = e x e x + C +C

Choosing the constant of integration C = 1 gives the other well-known form of the definition of the logistic curve

f x = e x e x + 1 = 1 1 + e − x +1=

More quantitatively, as can be seen from the analytical solution, the logistic curve shows early exponential growth for negative argument, which slows to linear growth of slope 1/4 for an argument near zero, then approaches one with an exponentially decaying gap

The logistic function is the inverse of the natural logit function and so can be used to convert the logarithm of odds into a probability In mathematical notation the logistic function is sometimes written as expit in the same form as logit The conversion from the log-likelihood ratio of two alternatives also takes the form of a logistic curve

The logistic sigmoid function is related to the hyperbolic tangent, Ap by

2 f x = 1 + tanh ⁡ x 2 \right


tanh ⁡ x = 2 f 2 x − 1

The latter relationship follows from

tanh ⁡ x = e x − e − x e x + e − x = e x ⋅ 1 − e − 2 x e x ⋅ 1 + e − 2 x = f 2 x − e − 2 x 1 + e − 2 x = f 2 x − e − 2 x + 1 − 1 1 + e − 2 x = 2 f 2 x − 1 -e^+e^=\cdot \left1-e^\right\cdot \left1+e^\right=f2x-=f2x-+1-1=2\,f2\,x-1

The hyperbolic tangent relationship leads to another form for the logistic function's derivative:

d d x f x = 1 4 sech 2 ⁡ x 2 fx=\operatorname ^\left\right ,

which ties the logistic function into the Logistic distribution

Rotational symmetry about 0, ½

The sum of the logistic function and its reflection about the vertical axis, f −x is

1 1 + e − x + 1 1 + e − − x = 1 + e x + 1 + e − x 1 + e − x 1 + e x = 2 + e x + e − x 1 + e x + e − x + e x − x = 2 + e x + e − x 2 + e x + e − x = 1 +=+1+e^1+e^=+e^+e^+e^=+e^+e^=1

The logistic function is thus rotationally symmetrical about the point 0, 1/2


In ecology: modeling population growth

Pierre-François Verhulst 1804–1849

A typical application of the logistic equation is a common model of population growth see also population dynamics, originally due to Pierre-François Verhulst in 1838, where the rate of reproduction is proportional to both the existing population and the amount of available resources, all else being equal The Verhulst equation was published after Verhulst had read Thomas Malthus' An Essay on the Principle of Population Verhulst derived his logistic equation to describe the self-limiting growth of a biological population The equation was rediscovered in 1911 by A G McKendrick for the growth of bacteria in broth and experimentally tested using a technique for nonlinear parameter estimation The equation is also sometimes called the Verhulst-Pearl equation following its rediscovery in 1920 by Raymond Pearl 1879–1940 and Lowell Reed 1888–1966 of the Johns Hopkins University Another scientist, Alfred J Lotka derived the equation again in 1925, calling it the law of population growth

Letting P represent population size N is often used in ecology instead and t represent time, this model is formalized by the differential equation:

d P d t = r P ⋅ 1 − P K =rP\cdot \left1-\right

where the constant r defines the growth rate and K is the carrying capacity

In the equation, the early, unimpeded growth rate is modeled by the first term +rP The value of the rate r represents the proportional increase of the population P in one unit of time Later, as the population grows, the modulus of the second term which multiplied out is −rP2/K becomes almost as large as the first, as some members of the population P interfere with each other by competing for some critical resource, such as food or living space This antagonistic effect is called the bottleneck, and is modeled by the value of the parameter K The competition diminishes the combined growth rate, until the value of P ceases to grow this is called maturity of the population The solution to the equation with P 0 being the initial population is

P t = K P 0 e r t K + P 0 e r t − 1 e^\lefte^-1\right


lim t → ∞ P t = K Pt=K

Which is to say that K is the limiting value of P: the highest value that the population can reach given infinite time or come close to reaching in finite time It is important to stress that the carrying capacity is asymptotically reached independently of the initial value P0 > 0, also in case that P0 > K

In ecology, species are sometimes referred to as r-strategist or K-strategist depending upon the selective processes that have shaped their life history strategies Choosing the variable dimensions so that n measures the population in units of carrying capacity, and τ measures time in units of 1/r, gives the dimensionless differential equation

d n d τ = n 1 − n =n1-n

Time-varying carrying capacity

Since the environmental conditions influence the carrying capacity, as a consequence it can be time-varying: Kt > 0, leading to the following mathematical model:

d P d t = r P ⋅ 1 − P K t =rP\cdot \left1-\right

A particularly important case is that of carrying capacity that varies periodically with period T:

K t + T = K t

It can be shown that in such a case, independently from the initial value P0 > 0, Pt will tend to a unique periodic solution Pt, whose period is T

A typical value of T is one year: In such case Kt may reflect periodical variations of weather conditions

Another interesting generalization is to consider that the carrying capacity Kt is a function of the population at an earlier time, capturing a delay in the way population modifies its environment This leads to a logistic delay equation, which has a very rich behavior, with bistability in some parameter range, as well as a monotonic decay to zero, smooth exponential growth, punctuated unlimited growth ie, multiple S-shapes, punctuated growth or alternation to a stationary level, oscillatory approach to a stationary level, sustainable oscillations, finite-time singularities as well as finite-time death

In statistics and machine learning

Logistic functions are used in several roles in statistics For example, they are the cumulative distribution function of the logistic family of distributions, and they are, a bit simplified, used to model the chance a chess player has to beat his opponent in the Elo rating system More specific examples now follow

Logistic regression

Main article: Logistic regression

Logistic functions are used in logistic regression to model how the probability p of an event may be affected by one or more explanatory variables: an example would be to have the model

p = P a + b x

where x is the explanatory variable and a and b are model parameters to be fitted

Logistic regression and other log-linear models are also commonly used in machine learning A generalisation of the logistic function to multiple inputs is the softmax activation function, used in multinomial logistic regression

Another application of the logistic function is in the Rasch model, used in item response theory In particular, the Rasch model forms a basis for maximum likelihood estimation of the locations of objects or persons on a continuum, based on collections of categorical data, for example the abilities of persons on a continuum based on responses that have been categorized as correct and incorrect

Neural networks

Logistic functions are often used in neural networks to introduce nonlinearity in the model and/or to clamp signals to within a specified range A popular neural net element computes a linear combination of its input signals, and applies a bounded logistic function to the result; this model can be seen as a "smoothed" variant of the classical threshold neuron

A common choice for the activation or "squashing" functions, used to clip for large magnitudes to keep the response of the neural network bounded is

g h = 1 1 + e − 2 β h

which is a logistic function These relationships result in simplified implementations of artificial neural networks with artificial neurons Practitioners caution that sigmoidal functions which are antisymmetric about the origin eg the hyperbolic tangent lead to faster convergence when training networks with backpropagation

The logistic function is itself the derivative of another proposed activation function, the softplus

In medicine: modeling of growth of tumors

See also: Gompertz curve § Growth of tumors

Another application of logistic curve is in medicine, where the logistic differential equation is used to model the growth of tumors This application can be considered an extension of the above-mentioned use in the framework of ecology see also the Generalized logistic curve, allowing for more parameters Denoting with Xt the size of the tumor at time t, its dynamics are governed by:

X ′ = r 1 − X K X =r\left1-\rightX

which is of the type:

X ′ = F X X , F ′ X ≤ 0 =F\leftX\rightX,F^X\leq 0

where FX is the proliferation rate of the tumor

If a chemotherapy is started with a log-kill effect, the equation may be revised to be

X ′ = r 1 − X K X − c t X =r\left1-\rightX-ctX ,

where ct is the therapy-induced death rate In the idealized case of very long therapy, ct can be modeled as a periodic function of period T or in case of continuous infusion therapy as a constant function, and one has that

1 T ∫ 0 T c t d t > r → lim t → + ∞ x t = 0 \int _^>r\rightarrow \lim _xt=0

ie if the average therapy-induced death rate is greater than the baseline proliferation rate then there is the eradication of the disease Of course, this is an oversimplified model of both the growth and the therapy eg it does not take into account the phenomenon of clonal resistance

In chemistry: reaction models

The concentration of reactants and products in autocatalytic reactions follow the logistic function

In physics: Fermi distribution

The logistic function determines the statistical distribution of fermions over the energy states of a system in thermal equilibrium In particular, it is the distribution of the probabilities that each possible energy level is occupied by a fermion, according to Fermi–Dirac statistics

In linguistics: language change

In linguistics, the logistic function can be used to model language change: an innovation that is at first marginal begins to spread more quickly with time, and then more slowly as it becomes more universally adopted

In economics and sociology: diffusion of innovations

The logistic function can be used to illustrate the progress of the diffusion of an innovation through its life cycle

In The Laws of Imitation 1890, Gabriel Tarde describes the rise and spread of new ideas through imitative chains In particular, Tarde identifies three main stages through which innovations spread: the first one corresponds to the difficult beginnings, during which the idea has to struggle within a hostile environment full of opposing habits and beliefs; the second one corresponds to the properly exponential take-off of the idea, with f x = 2 x ; finally, the third stage is logarithmic, with f x = log ⁡ x , and corresponds to the time when the impulse of the idea gradually slows down while, simultaneously new opponent ideas appear The ensuing situation halts or stabilizes the progress of the innovation, which approaches an asymptote

In the history of economy, when new products are introduced there is an intense amount of research and development which leads to dramatic improvements in quality and reductions in cost This leads to a period of rapid industry growth Some of the more famous examples are: railroads, incandescent light bulbs, electrification, cars and air travel Eventually, dramatic improvement and cost reduction opportunities are exhausted, the product or process are in widespread use with few remaining potential new customers, and markets become saturated

Logistic analysis was used in papers by several researchers at the International Institute of Applied Systems Analysis IIASA These papers deal with the diffusion of various innovations, infrastructures and energy source substitutions and the role of work in the economy as well as with the long economic cycle Long economic cycles were investigated by Robert Ayres 1989 Cesare Marchetti published on long economic cycles and on diffusion of innovations Arnulf Grübler’s book 1990 gives a detailed account of the diffusion of infrastructures including canals, railroads, highways and airlines, showing that their diffusion followed logistic shaped curves

Carlota Perez used a logistic curve to illustrate the long Kondratiev business cycle with the following labels: beginning of a technological era as irruption, the ascent as frenzy, the rapid build out as synergy and the completion as maturity

See also

  • Diffusion of innovations
  • Generalised logistic curve
  • Gompertz curve
  • Heaviside step function
  • Hubbert curve
  • Logistic distribution
  • Logistic map
  • Logistic regression
  • Logistic smooth-transmission model
  • Logit
  • Log-likelihood ratio
  • Malthusian growth model
  • Population dynamics
  • r/K selection theory
  • Shifted Gompertz distribution
  • Tipping point sociology
  • Rectifier neural networks


  1. ^ Verhulst, Pierre-François 1838 "Notice sur la loi que la population poursuit dans son accroissement" PDF Correspondance mathématique et physique 10: 113–121 Retrieved 3 December 2014 
  2. ^ Verhulst, Pierre-François 1845 "Recherches mathématiques sur la loi d'accroissement de la population" Nouveaux Mémoires de l'Académie Royale des Sciences et Belles-Lettres de Bruxelles 18: 1–42 Retrieved 2013-02-18 
  3. ^ expit documentation for R's clusterPower package
  4. ^ Raul Rojas Neural Networks - A Systematic Introduction PDF Retrieved 15 October 2016 
  5. ^ A G McKendricka; M Kesava Paia1 January 1912 "XLV—The Rate of Multiplication of Micro-organisms: A Mathematical Study" Proceedings of the Royal Society of Edinburgh 31: 649–653 doi:101017/S0370164600025426 
  6. ^ Raymond Pearl and Lowell Reed June 1920 "On the Rate of Growth of the Population of the United States" PDF Proc of the National Academy of Sciences 6 6 p 275 
  7. ^ Yukalov, V I; Yukalova, E P; Sornette, D 2009 "Punctuated evolution due to delayed carrying capacity" Physica D: Nonlinear Phenomena 238 17: 1752 doi:101016/jphysd200905011 
  8. ^ Gershenfeld 1999, p150
  9. ^ LeCun, Y; Bottou, L; Orr, G; Muller, K 1998 Orr, G; Muller, K, eds Efficient BackProp PDF Neural Networks: Tricks of the trade Springer ISBN 3-540-65311-2 
  10. ^ Bod, Hay, Jennedy eds 2003, pp 147–156
  11. ^ Ayres, Robert 1989 "Technological Transformations and Long Waves" PDF 
  12. ^ Marchetti, Cesare 1996 "Pervasive Long Waves: Is Society Cyclotymic" PDF 
  13. ^ Marchetti, Cesare 1988 "Kondratiev Revisited-After One Cycle" PDF 
  14. ^ Grübler, Arnulf 1990 The Rise and Fall of Infrastructures: Dynamics of Evolution and Technological Change in Transport PDF Heidelberg and New York: Physica-Verlag 
  15. ^ Perez, Carlota 2002 Technological Revolutions and Financial Capital: The Dynamics of Bubbles and Golden Ages UK: Edward Elgar Publishing Limited ISBN 1-84376-331-1 


  • Jannedy, Stefanie; Bod, Rens; Hay, Jennifer 2003 Probabilistic Linguistics Cambridge, Massachusetts: MIT Press ISBN 0-262-52338-8 
  • Gershenfeld, Neil A 1999 The Nature of Mathematical Modeling Cambridge, UK: Cambridge University Press ISBN 978-0-521-57095-4 
  • Kingsland, Sharon E 1995 Modeling nature: episodes in the history of population ecology Chicago: University of Chicago Press ISBN 0-226-43728-0 
  • Weisstein, Eric W "Logistic Equation" MathWorld 

External links

  • LJ Linacre, Why logistic ogive and not autocatalytic curve, accessed 2009-09-12
  • http://lunacasusfedu/~mbrannic/files/regression/Logistichtml
  • Weisstein, Eric W "Sigmoid Function" MathWorld 
  • Online experiments with JSXGraph
  • Esses are everywhere
  • Seeing the s-curve is everything

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