1
Sigmoid function
::2014/08/16::
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Julia Programming : The Sigmoid Function Programming Exercise
::2014/03/02::
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Organizational Learning Tool: The Sigmoid Curve
::2013/10/15::
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The Sigmoid Curve
::2012/11/11::
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Intro to Neural Networks
::2013/06/10::
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Neural network tutorial: The back-propagation algorithm (Part 1)
::2012/01/07::
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Contrast Enhancement of Color Images using Tunable Sigmoid Function.wmv
::2011/03/16::
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Sigmoid function displacement time servo control
::2009/05/30::
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Normalised Tunable Sigmoid Function Demo in Unity3D
::2013/06/26::
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The Gompertz Sigmoid Function and Its Derivative
::2009/07/16::
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Normalised Tunable Sigmoid Function 2.0
::2013/11/24::
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The Gompertz Sigmoid Function and Its Derivative
::2010/04/13::
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Mod-08 Lec-26 Multilayer Feedforward Neural networks with Sigmoidal activation functions;
::2013/12/02::
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::2013/05/08::
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Impact of Bias on the Sigmoid Activation function
::2014/10/19::
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Mathematical Biology. 19: Sigmoidal Functions, Multisite Systems
::2014/02/25::
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Retro Sigmoid Vestibular Nerve Section for Meniere's Disease
::2009/07/02::
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Curve Fitting in Excel
::2013/06/26::
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Polypectomy of Colon Sigmoid Polyp
::2012/10/04::
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Logistic function
::2014/08/13::
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3. Learning Sigmoid Belief Nets
::2013/11/09::
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Inferior Mesenteric Artery: Easy Anatomy Mnemonic Tutorial- Branches: colic, sigmoid, rectal
::2013/07/29::
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5.3.2 Draw and label a graph showing a sigmoid (S-shaped) population growth curve
::2013/04/04::
24
Neural Network Part 2
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Plot 5 of 6 - Continuous A* - Obstacle Created Using the Product of Sigmoid Functions
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Logistic regression
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You don't know what you don't know: the sigmoid curve
::2010/03/11::
28
Spiral Weaving Time
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Company Introduction: About Sigmoid Consulting Group
::2013/10/15::
30
Developing neural network in MATLAB method2 nntool] [fitting tool]
::2013/07/23::
31
14 - 5 - OPTIONAL VIDEO RBMs are infinite sigmoid belief nets [17 mins]
::2013/03/29::
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1.6 Sigmoid Emax model - Hill factor
::2012/03/12::
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Developing neural network in MATLAB method1 command window] [fitting tool]
::2013/07/23::
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2. Bob Buford explains the Sigmoid Curve
::2012/02/22::
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Equivalence of two activation functions in hidden layer: example
::2014/01/04::
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The sigmoid growth curve
::2011/05/18::
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::2014/04/24::
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Plot 6 of 6 - Continuous A* - Simple Maze
::2008/11/12::
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Comparison of Six Sigmoid Growth Curve Models
::2013/12/09::
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::2010/10/21::
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Comparison of Six Sigmoid Growth Curve Models
::2014/06/18::
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Colon Resection
::2010/12/13::
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The Sigmoid Curve (Positive Psychology)
::2012/10/04::
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5.1 Loss Functions | 5 Support Vector Machines | Pattern Recognition Class 2012
::2012/11/22::
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Neural Network Tutorial - Ch. 7.1 More transfer functions (Part 1)
::2012/01/30::
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Apple core lesion of sigmoid colon, suggesting colon cancer
::2011/09/15::
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sigmoid sme
::2007/08/06::
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CSCI4477-R11-classify-housing-SVM
::2013/02/22::
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Forex ANN Neural Network Close Price Prediction
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Plot of the error function

A sigmoid function is a mathematical function having an "S" shape (sigmoid curve). Often, sigmoid function refers to the special case of the logistic function shown in the first figure and defined by the formula

$S(t) = \frac{1}{1 + e^{-t}}.$

Other examples of similar shapes include the Gompertz curve (used in modeling systems that saturate at large values of t) and the ogee curve (used in the spillway of some dams). A wide variety of sigmoid functions have been used as the activation function of artificial neurons, including the logistic and hyperbolic tangent functions. Sigmoid curves are also common in statistics as cumulative distribution functions, such as the integrals of the logistic distribution, the normal distribution, and Student's t probability density functions.

## Definition

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.[1]

## Properties

In general, a sigmoid function is real-valued and differentiable, having either a non-negative or non-positive first derivative[citation needed] which is bell shaped. There are also a pair of horizontal asymptotes as $t \rightarrow \pm \infty$. The differential equation $\tfrac{d}{dt} S(t) = c_1 S(t) \left( c_2 - S(t) \right)$, with the inclusion of a boundary condition providing a third degree of freedom, $c_3$, provides a class of functions of this type.

## Examples

Some sigmoid functions compared. In the drawing all functions are normalized in such a way that their slope at the origin is 1.

Many natural processes, such as those of complex system learning curves, exhibit a progression from small beginnings that accelerates and approaches a climax over time. When a detailed description is lacking, a sigmoid function is often used[2] .

Besides the logistic function, sigmoid functions include the ordinary arctangent, the hyperbolic tangent, the Gudermannian function, and the error function, but also the generalised logistic function and algebraic functions like $f(x)=\tfrac{x}{\sqrt{1+x^2}}$.

The integral of any smooth, positive, "bump-shaped" function will be sigmoidal, thus the cumulative distribution functions for many common probability distributions are sigmoidal. The most famous such example is the error function, which is related to the cumulative distribution function (CDF) of a normal distribution.