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A sigmoid function is a mathematical function that produces a sigmoid curve — a curve having an "S" shape. Often, sigmoid function refers to the special case of the logistic function shown at right and defined by the formula plot of the logistic curve Generated using gnuplot with set nokey, set terminal png, and set size 0. ...
The logistic function or logistic curve is defined by the mathematical formula: for real parameters a, m, n, and . ...
Mathematics is commonly defined as the study of patterns of structure, change, and space; more informally, one might say it is the study of figures and numbers. Mathematical knowledge is constantly growing, through research and application, but mathematics itself is not usually considered a natural science. ...
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). A...
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. ...
Derivative of the sigmoid function
The derivative of the sigmoid function can be approximated by
Members of the sigmoid family In general, a sigmoid function is real-valued and differentiable, having either a non-negative or non-positive first derivative and exactly one inflection point. In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...
In mathematics, the derivative of a function is one of the two central concepts of calculus. ...
A negative number is a number that is less than zero, such as −3. ...
A negative number is a number that is less than zero, such as −3. ...
For a non-technical overview of the subject, see Calculus. ...
Plot of y = x3 with inflection point of (0,0). ...
Besides the logistic function, sigmoid functions include the ordinary arc-tangent, the hyperbolic tangent, and the error function, but also algebraic functions like  . The integral of any smooth, positive, "bump-shaped" function will be sigmoidal, thus the cumulative distribution functions for many common probability distributions are sigmoidal. 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. ...
All of the trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. Trigonometric functions: , , , , , In mathematics, the trigonometric functions (also called circular functions) are functions of an angle; they are important when studying triangles and modeling periodic phenomena, among many other...
In mathematics, the hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. ...
Plot of the error function In mathematics, the error function (also called the Gauss error function) is a non-elementary function which occurs in probability, statistics and partial differential equations. ...
In calculus, the integral of a function is an extension of the concept of a sum. ...
In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the random variable X takes on a value less than...
In mathematics and statistics, a probability distribution is a function of the probabilities of a mutually exclusive and exhaustive set of events. ...
The logistic sigmoid function is related to the hyperbolic tangent, e.g. by
Sigmoid functions in neural networks Sigmoid 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 sigmoid function to the result; this model can be seen as a "smoothed" variant of the classical threshold neuron. // See also Artificial neural network. ...
In mathematics, nonlinear systems represent systems whose behavior is not expressible as a sum of the behaviors of its descriptors. ...
A clamp meter (clamp-on ammeter) is a type of ammeter which measures electrical current without the need to disconnect the wiring through which the current is flowing. ...
In mathematics, the range of a function is the set of all output values produced by that function. ...
An artificial neuron (also called a node or Nv neuron or Binary neuron or McCulloch-Pitts neuron) is an abstraction of biological neurons and the basic unit in an artificial neural network. ...
In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. ...
The perceptron is a type of artificial neural network invented in 1957 at the Cornell Aeronautical Laboratory by Frank Rosenblatt. ...
A reason for its popularity in neural networks is because the sigmoid function satisfies the differential equation
- y' = y(1 − y)
The right hand side is a low order 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 sigmoid 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. An artificial neural network (ANN), often just called a neural network (NN), is an interconnected group of artificial neurons that uses a mathematical model or computational model for information processing based on a connectionist approach to computation. ...
An artificial neuron (also called a node or Nv neuron or Binary neuron or McCulloch-Pitts neuron) is an abstraction of biological neurons and the basic unit in an artificial neural network. ...
Double sigmoid function The double sigmoid is a function similar to the sigmoid function with numerous applications. Its general formula is: Image File history File links No higher resolution available. ...
Image File history File links No higher resolution available. ...
 where d is its centre and s is the steepness factor.
It is based on the Gaussian curve and graphically it is similar to two identical sigmoids bonded together at the point x = d. Probability density function of Gaussian distribution (bell curve). ...
One of its applications is non-linear normalization of a sample, as it has the property of eliminating outliers. Broadly, normalization (also spelled normalisation) is any process that makes something more normal, which typically means conforming to some regularity or rule, or returning from some state of abnormality. ...
Figure 1. ...
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