Sigmoid function

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

The ANN model had four neurones in the input layer one for each operating variable and one for the bias. The output was selected to be cumulative mass distribution thirteen neurones were used to represent it. A sigmoid functional... 

The concentration Cz is a sigmoid function ( growth curve ) of time. Presumably some of product Z must be present at r = 0 in order to initiate the reaction. 

If k is much larger than k", Eq. (6-64) takes the form of Eq. (6-61) for the fraction Fhs thus we may expect the experimental rate constant to be a sigmoid function of pH. If k" is larger than k, the / -pH plot should resemble the Fs-pH plot. Equation (6-64) is a very important relationship for the description of pH effects on reaction rates. Most sigmoid pH-rate profiles can be quantitatively accounted for with its use. Relatively minor modifications [such as the addition of rate terms first-order in H or OH to Eq. (6-63)] can often extend the description over the entire pH range. 

The general principle behind most commonly used back-propagation learning methods is the delta rule, by which an objective function involving squares of the output errors from the network is minimized. The delta rule requires that the sigmoidal function used at each neuron be continuously differentiable. This methods identifies an error associated with each neuron for each iteration involving a cause-effect pattern. Therefore, the error for each neuron in the output layer can be represented as ... 

The form of the stochastic transfer function p x) is shown in figure 10.7. Notice that the steepness of the function near a - 0 depends entirely on T. Notice also that this form approaches that of a simple threshold function as T —> 0, so that the deterministic Hopfield net may be recovered by taking the zero temperature limit of the stochastic system. While there are a variety of different forms for p x) satisfying this desired limiting property, any of which could also have been chosen, this sigmoid function is convenient because it allows us to analyze the system with tools borrowed from statistical mechanics. 

NH O and NH O). Because there are significant differences between covalent and van der Waals radii of oxygen and nitrogen atoms, the relationships between optimum energy and optimum distances were sought separately for the three above-mentioned subsets [48]. Those relationships were not expected to be linear because energy values approach zero as distance increases. In that study a sigmoid function was used ... 

The net input, NET, is then passed to the transfer function that transforms it into the output signal of the unit. Different transfer functions may be used, the most common non-linear one being the sigmoidal function (Fig. 44.5b). 

The transfer function of the hidden units in MLF networks is always a sigmoid or related function. As can be seen in Fig. 44.5b, 0, represents the offset, and has the same function as in the simple perceptron-like networks. P determines the slope of the transfer function. It is often omitted in the transfer function since it can implicitly be adjusted by the weights. The main function of the transfer function is modelling the non-linearities in the data. In Fig. 44.11 it can be seen that there are five different response regions in the sigmoidal function ... 

Fig. 44.11. Different response regions in a sigmoid function. For explanation, see text.

When the MLF is used for classification its non-linear properties are also important. In Fig. 44.12c the contour map of the output of a neural network with two hidden units is shown. It shows clearly that non-linear boundaries are obtained. Totally different boundaries are obtained by varying the weights, as shown in Fig. 44.12d. For modelling as well as for classification tasks, the appropriate number of transfer functions (i.e. the number of hidden units) thus depends essentially on the complexity of the relationship to be modelled and must be determined empirically for each problem. Other functions, such as the tangens hyperbolicus function (Fig. 44.13a) are also sometimes used. In Ref. [19] the authors came to the conclusion that in most cases a sigmoidal function describes non-linearities sufficiently well. Only in the presence of periodicities in the data... 

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

A commonly used sigmoidal function is the logistic function (Figure 2.22). 

The sigmoidal function generates a different output signal for each input signal, so the neuron can pass on information about the size of the input in a fashion that is not possible with a step function, which can transmit only an on/off signal. A network composed of neurons with sigmoidal functions can learn complicated behavior. Most importantly, it can learn to model nonlinear functions, and because nonlinear behavior is ubiquitous in science, this ability is crucial in producing a scientific tool of wide applicability. 

The limiting cases are limvo 0 a = 1 and limy. x a = 0. To evaluate the saturation matrix we restrict each element to a well-defined interval, specified in the following way As for most biochemical rate laws na nt 1, the saturation parameter of substrates usually takes a value between zero and unity that determines the degree of saturation of the respective reaction. In the case of cooperative behavior with a Hill coefficient = = ,> 1, the saturation parameter is restricted to the interval [0, n] and, analogously, to the interval [0, n] for inhibitory interaction with na = 0 and n = , > 1. Note that the sigmoidality of the rate equation is not specifically taken into account, rather the intervals for hyperbolic and sigmoidal functions overlap. 

IC50 values were determined at 48 h by nonlinear least-square fit to sigmoidal functions. ... 

This is a sigmoidal function of T as plotted in Figure 6-1. Note that at low temperature, X = rk(T) so X increases rapidly with T as k(T), while at high temperature X —> 1 as the reaction goes to completion. Note also that X = when kx = 1. 

These neurons transmit information to the third—output—layer, as a weighted combination (Z) of values. The neurons in the output layer correspond to the response variables which, in the case of classification, are the coded class indices. The output neurons transform the information Z, from the hidden layer, by means of a further sigmoid function or a semilinear function. 

While the direction of the axial velocity does not change in many of the stagnation flows, in some it does. Certainly the opposed flows Section 6.10) have both positive and negative velocities. So the convective difference formulas must change depending on the velocity direction. A sigmoid function can be used to switch the difference formula in a smoothly varying way as... 

Figure 8.17 shows a very specific case of a feed-forward network with four inputs, three hidden nodes, and one output. However, such networks can vary widely in their design. First of all, one can choose any number of inputs, hidden nodes, and number of outputs in the network. In addition, one can even choose to have more than one hidden layer in the network. Furthermore, it is common to perform scaling operations on both the inputs and the outputs, as this can enable more efficient training of the network. Finally, the transfer function used in the hidden layer (f) can vary widely as well. Many feed-forward networks use a non-linear function called the sigmoid function, defined as ... 

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