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Encyclopedia > Logistic function
Logistic curve, specifically the sigmoid function
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Logistic curve, specifically the sigmoid function

A logistic function or logistic curve models the S-curve of growth of some set P. The initial stage of growth is approximately exponential; then, as competition arises, the growth slows, and at maturity, growth stops. plot of the logistic curve Generated using gnuplot with set nokey, set terminal png, and set size 0. ... plot of the logistic curve Generated using gnuplot with set nokey, set terminal png, and set size 0. ...


As shown below, the untrammeled growth can be modelled as a rate term +rKP (a percentage of P). But then, as the population grows, some members of P (modelled as rP2) interfere with each other in competition for some critical resource (which can be called the bottleneck, modelled by K). This competition diminishes the growth rate, until the set P ceases to grow (this is called maturity).


A logistic function is defined by the mathematical formula:

P(t) = afrac{1 + m e^{-t/tau}}{1 + n e^{-t/tau}} !

for real parameters a, m, n, and τ. These functions find applications in a range of fields, from biology to economics. Biology is the branch of science dealing with the study of life. ... Buyers bargain for good prices while sellers put forth their best front in Chichicastenango Market, Guatemala. ...


For example, in the development of an embryo, a fertilized ovum splits, and the cell count grows: 1, 2, 4, 8, 16, 32, 64, etc. This is exponential growth. But the fetus can grow only as large as the uterus can hold; thus other factors start slowing down the increase in the cell count, and the rate of growth slows (but the baby is still growing, of course). After a suitable time, the child is born and keeps growing. Ultimately, the cell count is stable; the person's height is constant; the growth has stopped, at maturity. Embryos (and one tadpole) of the wrinkled frog (Rana rugosa). ... Fetus at eight weeks Foetus redirects here. ...


Concentration of reactants and products in autocatalytical reactions follow the logistic function. A single chemical reaction is said to have undergone autocatalysis, or be autocatalytic, if the reaction product is itself the catalyst for that reaction. ...


In such examples, relationships between variables are modelled. In addition, an important logistic function is the Rasch model, which is a general stochastic measurement model. This model is used as a basis for measurement rather than for modelling relationships between variables for which measurements have already been obtained (as in the preceding example). In particular, the Rasch model forms a basis for maximum likelihood estimation of the locations of objects to be measured on a continuum, based on collections of categorical data. For example, the model can be applied in order to estimate the abilities of persons on a continuum based on assessment responses that have been categorized as correct and incorrect. Rasch models are probabilistic measurement models which find their application primarily in psychological and attainment assessment, and are being increasingly used in other areas, including the health profession. ... Stochastic, from the Greek stochos or goal, means of, relating to, or characterized by conjecture; conjectural; random. ... Various meters Measurement is the process of estimating the ratio of a magnitude of a quantity to a unit of the same type. ... Maximum likelihood estimation (MLE) is a popular statistical method used to make inferences about parameters of the underlying probability distribution of a given data set. ... Look up continuum in Wiktionary, the free dictionary. ...

Contents


The Verhulst equation

A typical application of the logistic equation is a common model of population growth, which states that: Countries by population growth rate Population growth is changing of the amount of population over time. ...

  • the rate of reproduction is proportional to the existing population, all else being equal
  • the rate of reproduction is proportional to the amount of available resources, all else being equal. Thus the second term models the competition for available resources, which tends to limit the 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: Graph of a differential equation In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ...

frac{dP}{dt}=rPleft(1 - frac{P}{K}right) qquad mbox{(1)}, !

where the constant r defines the growth rate and K is the carrying capacity. The general solution to this equation is a logistic function. 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. The solution to the equation (with P0 being the initial population) is As population density increases, birth rates decrease and death rates increase. ... The word ecology is often used in common parlance as a synonym for the natural environment or environmentalism. ... In biology, a species is the basic unit of biodiversity. ... In ecology, r-selection (note: lower case r) relates to the selection of traits (in organisms) that allow success in unstable or unpredictable environments. ... In ecology, K-selection (note : upper case K) relates to the selection of traits (in organisms) that allow success in stable or predictable environments. ... Natural selection is the process by which individual organisms with favorable traits are more likely to survive and reproduce. ... A life cycle includes the major sexual stages of a species, especially in regard to its ploidy. ...

P(t) = frac{K P_0 e^{rt}}{K + P_0 left( e^{rt} - 1right)}

where

lim_{ttoinfty} P(t) = K.,

Sigmoid function

Main article: sigmoid function

The special case of the logistic function with a = 1,m = 0,n = 1,τ = 1, namely The logistic curve A sigmoid function is a mathematical function that produces a sigmoid curve — a curve having an S shape. ...

P(t) = frac{1}{1 + e^{-t}}!

is called sigmoid function or sigmoid curve. The name is due to the sigmoid shape of its graph. This function is also called the standard logistic function and is often encountered in many technical domains, especially in artificial neural networks as a transfer function, probability, statistics, biomathematics, and economics. Sigmoid generally means resembling the letter S or the lower-case Greek letter sigma (ς). ... An artificial neural network (ANN), also called a simulated neural network (SNN) or commonly just neural network (NN) is an interconnected group of artificial neurons that uses a mathematical or computational model for information processing based on a connectionist approach to computation. ... A transfer function is a mathematical representation of the relation between the input and output of a linear time-invariant system. ... The word probability derives from the Latin probare (to prove, or to test). ... A graph of a bell curve in a normal distribution showing statistics used in educational assessment, comparing various grading methods. ... Mathematical biology or biomathematics is an interdisciplinary field of academic study which models natural, biological processes using mathematical techniques. ... Buyers bargain for good prices while sellers put forth their best front in Chichicastenango Market, Guatemala. ...


Properties of the sigmoid function

The (standard) sigmoid function is the solution of the first-order non-linear differential equation Graph of a differential equation In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ...

frac{dP}{dt}=P(1-P), quadmbox{(2)}!

with boundary condition P(0) = 1 / 2. Equation (2) is the continuous version of the logistic map. The logistic map is a polynomial mapping, often cited as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations. ...


The sigmoid curve shows early exponential growth for negative t, which slows to linear growth of slope 1/4 near t = 0, then approaches y = 1 with an exponentially decaying gap. In mathematics, a quantity that grows exponentially (or geometrically) is one that grows at a rate proportional to its size. ...


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; the conversion from the log-likelihood ratio of two alternatives also takes the form of a sigmoid curve. In mathematics, especially as applied in statistics, the logit (pronounced with a long o and a soft g, IPA ) of a number p between 0 and 1 is Plot of logit in the range 0 to 1, base is e (The base of the logarithm function used here is of... In probability theory and statistics the odds in favor of an event or a proposition are the quantity p / (1 − p), where p is the probability of the event or proposition. ... The word probability derives from the Latin probare (to prove, or to test). ... A likelihood-ratio test is a statistical test relying on a test statistic computed by taking the ratio of the maximum value of the likelihood function under the constraint of the null hypothesis to the maximum with that constraint relaxed. ...


History

The Verhulst equation, (1), was first published by Pierre F. Verhulst in 1838 after he had read Thomas Malthus' An Essay on the Principle of Population. Pierre François Verhulst (October 28, 1804 - February 15, 1849, Brussels, Belgium) was a mathematician and a doctor in number theory from the University of Ghent in 1825. ... | Jöns Jakob Berzelius, discoverer of protein 1838 was a common year starting on Monday (see link for calendar). ... The Rev. ...


Verhulst derived his equation logistique (logistic equation) to describe the self-limiting growth of a biological population. The equation is also sometimes called the Verhulst-Pearl equation following its rediscovery in 1920. Alfred J. Lotka derived the equation again in 1925, calling it the law of population growth. Biology is the branch of science dealing with the study of life. ... 1920 (MCMXX) was a leap year starting on Thursday (link will take you to calendar) // Events January January 3 - Babe Ruth is traded by the Boston Red Sox to the New York Yankees for $125,000, the largest sum ever paid for a player at that time. ... Alfred James Lotka (March 2, 1880 - December 5, 1949) was a US mathematician and statistician, most famous for his work in population dynamics. ... 1925 (MCMXXV) was a common year starting on Thursday (link will take you to calendar). ...


Critics

Despite its persistent popularity, the logistic function has been heavily criticised in the field of population dynamics. One such critic is demographer, and Professor of Population, Joel E. Cohen (How Many People Can The Earth Support, 1995). Cohen explains that Verhulst attempted to fit a logistic curve based on the logistic function to 3 separate censuses of the population of the United States of America in order to predict future growth. All 3 sets of predictions failed. Population dynamics is the study of marginal and long-term changes in the numbers, individual weights and age composition of individuals in one or several populations, and biological and environmental processes influencing those changes. ... The logistic function or logistic curve is defined by the mathematical formula: for real parameters a, m, n, and . ... A census is the process of obtaining information about every member of a population (not necessarily a human population). ...


In 1924, Professor Ray Pearl and Lowell J. Reed used Verhulst's model to predict an upper limit of 2 billion for the world population. This was passed in 1930. A later attempt by Pearl and an associate Sophia Gould in 1936 then estimated an upper limit of 2.6 billion. This was passed in 1955. Raymond Pearl (b. ... The world population is the total number of humans alive on the planet Earth at a given time. ...


These criticisms are echoed by Professor Peter Turchin (Complex Population Dynamics, 2003), who nonetheless concludes that it provides a useful framework for single-species dynamics and contributes to models for multispecies interactions.


Nevertheless, the logistics curve has been a staple of models both mathematical and sociological, for instance the transformation theory of George Land, which uses the concept of the S-curve to prescribe appropriate business behaviour in various stages of a technology's growth. Transformation theory, first explained by George Land (also George Ainsworth Land) (1927-) is a description of the structure of change in natural systems. ...


See also

The generalised logistic (or Richards) curve is a widely used and flexible function for growth modelling. ... The Hubbert curve, named after the geophysicist M. King Hubbert, is the derivative of the logistic curve. ... In probability theory and statistics, the logistic distribution is a continuous probability distribution. ... The logistic map is a polynomial mapping, often cited as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations. ... It has been suggested that Logit be merged into this article or section. ... A likelihood-ratio test is a statistical test relying on a test statistic computed by taking the ratio of the maximum value of the likelihood function under the constraint of the null hypothesis to the maximum with that constraint relaxed. ... It has been suggested that this article or section be merged with Malthus. ... A Gompertz curve is a type of mathematical model for a time series, where growth is slowest at the start and end of a time period. ...

References

  • Kingsland, S. E. (1995) Modeling nature ISBN 0226437280

External links


  Results from FactBites:
 
NationMaster - Encyclopedia: Sigmoid function (1388 words)
Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output).
Logistic curve, specifically the sigmoid function A logistic function or logistic curve models the S-curve of growth of some set P. The initial stage of growth is approximately exponential; then, as competition arises, the growth slows, and at maturity, growth stops.
Sigmoid functions are often used in neural networks to introduce nonlinearity in the model and/or to clamp signals to within a specified range.
  More results at FactBites »

 
 

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