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Neural network tutorial: The back-propagation algorithm (Part 1)
Neural network tutorial: The back-propagation algorithm (Part 1)
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Organizational Learning Tool: The Sigmoid Curve
Organizational Learning Tool: The Sigmoid Curve
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Julia Programming : The Sigmoid Function Programming Exercise
Julia Programming : The Sigmoid Function Programming Exercise
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Intro to Neural Networks
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Contrast Enhancement of Color Images using Tunable Sigmoid Function.wmv
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The Gompertz Sigmoid Function and Its Derivative
The Gompertz Sigmoid Function and Its Derivative
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The Gompertz Sigmoid Function and Its Derivative
The Gompertz Sigmoid Function and Its Derivative
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Equivalence of two activation functions in hidden layer: example
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Mathematical Biology. 19: Sigmoidal Functions, Multisite Systems
Mathematical Biology. 19: Sigmoidal Functions, Multisite Systems
::2014/02/25::
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The Sigmoid Curve
The Sigmoid Curve
::2012/11/11::
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16
Plot 5 of 6 - Continuous A* - Obstacle Created Using the Product of Sigmoid Functions
Plot 5 of 6 - Continuous A* - Obstacle Created Using the Product of Sigmoid Functions
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Retro Sigmoid Vestibular Nerve Section for Meniere
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Neural Network Part 2
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Spiral Weaving Time
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::2013/08/24::
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Jump Your Sigmoid Curve
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25
Plot 6 of 6 - Continuous A* - Simple Maze
Plot 6 of 6 - Continuous A* - Simple Maze
::2008/11/12::
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26
5.3.2 Draw and label a graph showing a sigmoid (S-shaped) population growth curve
5.3.2 Draw and label a graph showing a sigmoid (S-shaped) population growth curve
::2013/04/05::
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Comparison of Six Sigmoid Growth Curve Models
Comparison of Six Sigmoid Growth Curve Models
::2013/12/09::
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Sigmoid Microbial Survival Curves
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Company Introduction: About Sigmoid Consulting Group
Company Introduction: About Sigmoid Consulting Group
::2013/10/15::
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Sigmoid 1
Sigmoid 1
::2014/01/22::
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You don
You don't know what you don't know: the sigmoid curve
::2010/03/11::
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32
14 - 5 - OPTIONAL VIDEO RBMs are infinite sigmoid belief nets [17 mins]
14 - 5 - OPTIONAL VIDEO RBMs are infinite sigmoid belief nets [17 mins]
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Developing neural network in MATLAB method2 nntool] [fitting tool]
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Hand Assisted Lap Sigmoid Colectomy
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Laparoscopische sigmoid resectie
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2. Bob Buford explains the Sigmoid Curve
2. Bob Buford explains the Sigmoid Curve
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Logistic Sigmoid Market Model
Logistic Sigmoid Market Model
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38
Developing neural network in MATLAB method1 command window] [fitting tool]
Developing neural network in MATLAB method1 command window] [fitting tool]
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TableCurve2D 012 Sigmoid
TableCurve2D 012 Sigmoid
::2013/08/16::
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Neural Network Tutorial - Ch. 7.1 More transfer functions (Part 1)
Neural Network Tutorial - Ch. 7.1 More transfer functions (Part 1)
::2012/01/30::
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41
sigmoid sme
sigmoid sme
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The sigmoid growth curve
The sigmoid growth curve
::2011/05/18::
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43
A Neural Network Learning his name
A Neural Network Learning his name
::2014/04/30::
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44
Introduction to Sigmoid: 1A
Introduction to Sigmoid: 1A
::2008/02/21::
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45
Forex ANN Neural Network Close Price Prediction
Forex ANN Neural Network Close Price Prediction
::2014/04/26::
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Introduction to Sigmoid: 1C Retrieving a Model
Introduction to Sigmoid: 1C Retrieving a Model
::2008/02/21::
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47
5.1 Loss Functions | 5 Support Vector Machines | Pattern Recognition Class 2012
5.1 Loss Functions | 5 Support Vector Machines | Pattern Recognition Class 2012
::2012/11/22::
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48
EC50 and IC50 Determination in Excel
EC50 and IC50 Determination in Excel
::2013/07/10::
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49
Introduction to Sigmoid: 1D Viewing a Model
Introduction to Sigmoid: 1D Viewing a Model
::2008/02/21::
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Intorduction to Sigmoid: 1E Simulation
Intorduction to Sigmoid: 1E Simulation
::2008/02/21::
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RESULTS [51 .. 101]
From Wikipedia, the free encyclopedia
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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[edit]

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[edit]

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[edit]

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.

See also[edit]

References[edit]

  1. ^ Han, Jun; Morag, Claudio (1995). "The influence of the sigmoid function parameters on the speed of backpropagation learning". In Mira, José; Sandoval, Francisco. From Natural to Artificial Neural Computation. pp. 195–201. 
  2. ^ Gibbs, M.N. (Nov 2000). "Variational Gaussian process classifiers". IEEE Transactions on Neural Networks 11 (6): 1458–1464. doi:10.1109/72.883477. 
  • Mitchell, Tom M. (1997). Machine Learning. WCB–McGraw–Hill. ISBN 0-07-042807-7. . In particular see "Chapter 4: Artificial Neural Networks" (in particular pp. 96–97) where Mitchell uses the word "logistic function" and the "sigmoid function" synonymously – this function he also calls the "squashing function" – and the sigmoid (aka logistic) function is used to compress the outputs of the "neurons" in multi-layer neural nets.
  • Humphrys, Mark. "Continuous output, the sigmoid function".  Properties of the sigmoid, including how it can shift along axes and how its domain may be transformed.
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