Sigmoid activation function

If the activation function is the sigmoid function given in equation (10.56), then its derivative is... 

In case of MLP, the activation function has to be monotonous and differentiable (because of Eq. 6.122). Frequently used is the sigmoid function... 

Fig. 6.20. Sigmoid activation function with diverse values of the steepness parameter y...

Sigmoidal activation functions generate an output signal that is related uniquely to the size of the input signal, so, unlike both binary and capped linear activation functions, they provide an unambiguous transfer of information from the input side of a node to its output. 

We also experimented with a sigmoidal activation function to transform the values o to the range [0, 1]. 

The sigmoidal activation function produced better results. If o.(x, y) is almost zero, i.e. local space average color and the color of the current pixel are similar, then the output will be 0.5, i.e. gray. For small deviations around the average color, the output will vary linearly. The output will be saturated at 1 for large positive deviations. It will approach 0... 

This value is used to scale o" before it is sent through a sigmoidal activation function. 

Therefore, if the computed value for a channel is greater than zero, then it will be scaled to one. If the computed value for a channel is lesser than zero, it will be scaled to minus one. If both reflectances are equivalent, then the output will be zero. The output is constant over all channels. The sigmoidal activation function does not change the computed color. In any case, the output is constant over all channels. Therefore, the patch will appear to be achromatic if this algorithm is used. The results obtained by this algorithm do not agree with the results obtained by Helson. 

An activation function with limits on the amplitude of the output of a neuron. The amplitude range is usually given in a closed interval [0,1] or [-1,1]. Activation function y>(-) defines the output yu of a neuron (see Eq. 3.65) in terms of the activation potential Vk (see Eq. 3.64). Typical activation functions include the unit step change and sigmoid functions. 

Sigmoid Function. This S-shaped function is by far the most common form of activation function used. A t3rpical expression is... 

After the network propagates from the input layer to the output layer, the error between the desired output and actual output will be back-propagated to the previous layer. In the hidden layers, the error for each node is computed by the weighted sum of errors in the next layer s nodes. In a three-layered network, the next layer means the output layer. The activation function is usually a sigmoid function with the weights modified according to (2) or (3). 

Activation function Every neuron has its own activation function and generally only two activation functions are used in a particular NN. Neurons in the input layer use the identity function as the activation function. That is, the output of an input neuron equals its input. The activation functions of hidden and output layers can be differentiable and non-linear in theory. Several well-behaved (bounded, monotonically increasing and differentiable) activation functions are commonly used in practice, including (1) the sigmoid function f X) = (1 + exp(-A)) (2) the hyperbolic tangent function f X) = (exp(A) - exp(-A))/ (exp(A) + exp(-A)) (3) the sine or cosine function f(X) = sin(A) or f X) = cos(A) (4) the linear function f X) = X (5) the radial basis function. Among them, the sigmoid function is the most popular, while the radial basis function is only used for radial basis function networks. 

Here QfD is the H x m matrix of weights (cOrk) in the hidden layer, (D represents a vector of weights for a single node and P is a bias value associated with each node. The function is a sigmoidal activation function, typically of the form ... 

As a note of interest, Qin McAvoy (1992) have shown that NNPLS models can be collapsed to multilayer perceptron architectures. In this case it was therefore possible to represent the best NNPLS model in the form of a single layer neural network with 29 hidden nodes using tan-sigmoidal activation functions and an output layer of 146 nodes with purely linear functions. 

The output is calculated in two steps first, the input and output signals are delayed to different degrees. Second a nonlinear aetivation fimetion /( ) (here a static neural network) estimates the output. In (Nelles 2001) a sigmoid fimetion is proposed for the nonlinear activation function, which is used in this eontext. Other fimetions for nonlinear dynamie modeling e.g. Ham-merstein models, Wiener models, neural or wavelet network are also possible. 

The activation function F of the output neurons can be any monotone, nondecreasing differentiable function. Sigmoid or logistic functions are usually used. 

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