logistic curve

Logistic function

A logistic function or logistic curve is the most common sigmoid curve. It models the S-curve of growth of some set P. The initial stage of growth is approximately exponential; then, as saturation begins, the growth slows, and at maturity, growth stops.

A logistic function is defined by the mathematical formula:

Z(theta) = frac{e^{theta}}{1 + e^{theta}} !

The logistic function finds applications in a range of fields, including biology, economics, chemistry, and statistics.

Logistic sigmoid function

The sigmoid function with a = 1, m = 0, n = 1, τ = 1, namely

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

is called the logistic sigmoid function or sometimes the standard 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, mathematical psychology and economics.

Properties

The logistic (standard) sigmoid function is the solution of the first-order non-linear differential equation

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 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.

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 logistic curve.

The logistic sigmoid function is related to the hyperbolic tangent, e.g. by

2 , P(t) = 1 + tanh left(frac{t}{2} right).

In 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 reason for its popularity in neural networks is because the logistic function satisfies the differential equation

y' = y(1-y).,

The right hand side is a low-degree polynomial. Furthermore, the polynomial has factors y and 1 − y, both of which are simple to compute. Given y = sig(t) at a particular t, the derivative of the logistic function at that t can be obtained by multiplying the two factors together. These relationships result in simplified implementations of artificial neural networks with artificial neurons.

In statistics

Logistic functions are used in several roles in statistics. Firstly, they are the cumulative distribution function of the logistic family of distributions. Secondly they 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 + bx),

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

An important 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.

In Ecology: modeling population growth

A typical application of the logistic equation is a common model of population growth, which states that:

  • 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:

frac{dP}{dt}=rPleft(1 - frac{P}{K}right)

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

Interpreting the equation shown above: 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 second term, which multiplied out is −rP2/K, becomes larger than 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 modelled 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).

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 P_0 being the initial population) is

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

where

lim_{ttoinfty} P(t) = 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 P(0) > 0, also in case that P(0) > K.

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. Verhulst derived his 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.

Time-varying carrying capacity

Since the environmental conditions influences the carrying capacity, as a consequence it can be time-varying: K(t) > 0, leading to the following mathematical model:

frac{dP}{dt}=rPleft(1 - frac{P}{K(t)}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 P(0) > 0, P(t) will tend to a unique periodic solution P*(t), whose period is T.

A typical value of T is one year: in such case K(t) reflects periodical variations of the climate.

In medicine: modeling of 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. Denoting with X(t) the size of the tumor at time t, its dynamics is governed by:

X^{prime}=rleft(1 - frac{X}{K}right)X

which is of the type:

X^{prime}=Fleft(Xright)X

where F(X) 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^{prime}=rleft(1 - frac{X}{K}right)X - c(t)X,

where c(t) is the therapy-induced death rate. In the idealized case of very long therapy, c(t) 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

frac{1}{T}int_{0}^{T}{c(t), dt} > r Rightarrow lim_{t rightarrow +infty }x(t)=0

i.e. 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 over-simplified model of both the growth and the therapy (e.g. it does not take into account the phenomenon of clonal resistance).

In chemistry: reaction models

Concentration of reactants and products in autocatalytical reactions follow the logistic function.

Double logistic function

The double logistic is a function similar to the logistic function with numerous applications. Its general formula is:

y = mbox{sign}(x-d) , Bigg(1-expbigg(-bigg(frac{x-d}{s}bigg)^2bigg)Bigg),
where d is its centre and s is the steepness factor. Here "sign" represents the sign function.

It is based on the Gaussian curve and graphically it is similar to two identical logistic sigmoids bonded together at the point x = d.

One of its applications is non-linear normalization of a sample, as it has the property of eliminating outliers.

See also

References

  • Kingsland, S. E. (1995) Modeling nature ISBN 0-226-43728-0

External links

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