## Neural network tutorial: The back-propagation algorithm (Part 1)

Channel: nqramjets   |   2012/01/07
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Neural network tutorial: The back-propagation algorithm (Part 1)
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The Sigmoid Curve
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Intro to Neural Networks
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Finding a Logistic Function
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Julia Programming : The Sigmoid Function Programming Exercise
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Normalised Tunable Sigmoid Function 2.0
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Contrast Enhancement of Color Images using Tunable Sigmoid Function.wmv
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Normalised Tunable Sigmoid Function Demo in Unity3D
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The Gompertz Sigmoid Function and Its Derivative
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The Gompertz Sigmoid Function and Its Derivative
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Sigmoid function displacement time servo control
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1.6 Sigmoid Emax model - Hill factor
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SIGMOID 0000
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2012-04-10_13.53.16 Sigmoid [HD]
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Función Sigmoide y codificación en Java
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The S curve, before pedals...
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Comm 9. The S Curve - Hardy
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Logistic Sigmoid Market Model
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6 the s curve
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Plot 5 of 6 - Continuous A* - Obstacle Created Using the Product of Sigmoid Functions
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Sigmoid Microbial Survival Curves
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Intorduction to Sigmoid: 1E Simulation
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2. Bob Buford explains the Sigmoid Curve
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Pioneer S curve 2
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Equivalence of two activation functions in hidden layer: example
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Fun at Scurv's Curve
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AKD_BASIC_S-Curve-Smoothing
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A _Small_Study_08- 30to 09-01-2013
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The Sigmoid Curve
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Blops Search Sesh w/Sigmoid and BikeRider
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Sigmoid Argonaut - Brittle Brail
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Fusion Summit 2011: 04 Safe Sigmoid Access (Hubertus Feussner)
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Axial sigmoids PA, AP, Butterfly.wmv
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Retro Sigmoid Vestibular Nerve Section for Meniere's Disease
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3D Modeling system for polar mesh: sigmoidal morphing & dynamic gravity (drag: 01)
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Ignite Columbus 2 - Joshua Scott - Jump!
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Neural Network Part 2
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S-Curves
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Colon Resection
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Plot 6 of 6 - Continuous A* - Simple Maze
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Logistic regression
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5.1 Loss Functions | 5 Support Vector Machines | Pattern Recognition Class 2012
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You don't know what you don't know: the sigmoid curve
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decision feedback equalizer
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Polypectomy of Colon Sigmoid Polyp
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S curves n normal curves
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Spiral Weaving Time
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IMG 0343
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RESULTS [51 .. 101]
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, including 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.

## References

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.