Sigmoidal transformation

Figure 11.15 The difference between local space average color and the color of the current pixel was used to compute the output color. The linear transformation was used for the images shown in (a) and (b). The sigmoidal transformation was used for the images shown in (c) and (d).

The desirability function (g) that scaled the total analysis time (T) was also a sigmoidal transformation that gave values close to zero for analysis times greater than 45 min and values approaching one for total analysis times close to 6 min. Preliminary experiments, mostly performed by a univariate approach, were used to set these limits. 

This equation relates the fraction transformed to the nucleation rate, the growth rate, and the time elapsed since the start of the transformation (at constant temperature) and predicts sigmoidal transformation behavior, meaning that the fraction transformed first increases exponentially in time before slowing down and asymptotically approaching complete transformation. 

In order to describe the nonlinear stress-strain relations of the marine soft soil, a single hidden layer BP model was setup with the use of neural network technology. For the model, the input values are bias stress, confining pressure and time, the output value is the strain. Therefore, nodes of the input layer is 3, the number of nodes of the output layer is 1. The number of hidden layer units ranging from 5 to 25, and it need to be determined based on the training and fitting results. The neurons in the hidden layer is a sigmoid transform function, the neurons of the... 

A sigmoid (s-shaped) is a continuous function that has a derivative at all points and is a monotonically increasing function. Here 5,p is the transformed output asymptotic to 0 < 5/,p I and w,.p is the summed total of the inputs (- 00 < Ui p < -I- 00) for pattern p. Hence, when the neural network is presented with a set of input data, each neuron sums up all the inputs modified by the corresponding connection weights and applies the transfer function to the summed total. This process is repeated until the network outputs are obtained. 

The concentration of the unknown is then read off the standard curve opposite its B/Bq value. This sigmoid shaped standard curve, because of its linear portion, simplifies data handling. A mathematical transform of the B/Bq vs log dose is shown in Figure 2. This logit of B/Bq vs log dose is a widely used method of standard curve presentation (5,6,7). Logit B/B is defined as follows ... 

This latter transform is used to linearize the usual sigmoid curve produced in plotting B/Bq vs log dose. This transform may be accomplished by using a computer, logit paper or a table of logit values. 

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). 

A transformation of the peak voltammogram to the sigmoidal shape shown in the preceding section, Fig. 5.13, is achieved by the convolution analysis method proposed by K. Oldham. The experimental function j(t) = j[T(E — Ej)/v] is transformed by convolution integration... 

The probit relationship of Equation 2-4 transforms the sigmoid shape of the normal response versus dose curve into a straight line when plotted using a linear probit scale, as shown in Figure 2-10. Standard curve-fitting techniques are used to determine the best-fitting straight line. 

Figure 2-10 The probit transformation converts the sigmoidal response vs. log dose curve into a straight line when plotted on a linear probit scale. Source D. J. Finney, Probit Analysis, 3d ed. (Cambridge Cambridge University Press, 1971), p. 24. Reprinted by permission.

Probit Equation The probit equation has been used in an attempt to quantitatively correlate hazardous material concentration, duration of exposure, and probability of effect/injury, for several types of exposures. The objective of such use is to transform the typical sigmoidal (S-shaped) relationship between cause and effect to a straight-line relationship (Mannan, Lees Loss Prevention in the Process Industries, 3d ed., p. 9/68, 2005). 

Very often, the dose-effect curve is redrawn using a logarithmic scale for the dose. This gives rise to a sigmoid curve, as shown in Fig. 5.2. It is a mathematical transformation, which shows an approximate linear portion for the 20-80% maximal effect scale, which is usually the dose level for a therapeutic drug. Doses above 80% provide very little increase in therapeutic effects but with a concomitant rise in the risk of adverse reactions. 

The variation of the peak current with the electrode kinetic parameter k and chemical kinetic parameter e is shown in Fig. 2.31. When the quasireversible electrode reaction is fast (curves 1 and 2 in Fig. 2.31) the dependence is similar as for the reversible case and characterized by a pronounced minimum If the electrode reaction is rather slow (curves 3-5), the dependence A fJ, vs. log( ) transforms into a sigmoidal curve. Although the backward chemical reaction is sufficiently fast to re-supply the electroactive material on the time scale of the reverse (reduction) potential pulses, the reuse of the electroactive form is prevented due to the very low kinetics of the electrode reaction. This situation corresponds to the lower plateau of curves 3-5 in Fig. 2.31. 

Under hydrothermal conditions (150-180 °C) maghemite transforms to hematite via solution probably by a dissolution/reprecipitation mechanism (Swaddle Olt-mann, 1980 Blesa Matijevic, 1989). In water, the small, cubic crystals of maghemite were replaced by much larger hematite rhombohedra (up to 0.3 Lim across). Large hematite plates up to 5 Lim across were produced in KOH. The reaction conditions influenced both the extent of nucleation and crystal morphology. The transformation curve was sigmoidal and the kinetic data in water and in KOH fitted a first order, random nucleation model (Avrami-Erofejev), i.e. 

The first layer transmits the value of the predictors to the second— hidden—layer. All the neurons of the input layer are connected to the / neurons of the second layer by means of weight coefficients, meaning that the / elements of the hidden layer receive, as information, a weighted sum S of the values from the input layer. They transform the information received (S) by means of a suitable transfer function, frequently a sigmoid. 

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. 

Equations (6.15) and (6.17) phenomenologically describe the overall growth kinetics after the initial nucleation took place and further nucleation is still occurring. Indeed, the sigmoidal form of the X(t) curve represents a wide variety of transformation reactions. Equation (6.13) is named after Johnson, Mehl, and Avrami [W. A. Johnson, R. E Mehl (1939) M. Avrami (1939)]. Let us finally mention two points. 1) Plotting Vin (1 -X) vs. t should give a straight line with slope km. 2) The time ty of the inflection point (d2X/dt2 = 0) on X(t) is suitable to derive either m or km, namely... 

The function ((t) in Eq. 21.12 has a characteristic sigmoidal shape with a maximum rate of transformation at intermediate times. Examples are shown in Fig. 21.2. The d = 3 form of Eq. 21.12 is commonly known as the Johnson-Mehl-Avrami equation. 

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

Figure 11.3 The dose-response relationship, (a) Five segments of the sigmoidal dose-response curve as described in the text, (b) Linearized dose-response relationship through log (dose)-probit (effect) transformations. Locations of the LD50 and LD05 are depicted.

Sigmoidal curves of the concentration-response data are presented in Figure 1, and their probit transformations in Figure 2, where a log-normal model is assumed. Here, the log-concentration transformation (pT-scale) is paired with the probit parameter, which is indicative of the proportion of percentage inhibition . 

Figure 2. Probit transformation of the sigmoidal concentration-effect curves of Figure 1 for calculation of effective concentrations as IC50, IC20, or IC19.

The controlled-extraction procedure which has been studied by Lee (6, 7) allows the transformation of monolithically loaded beads into beads with drug concentration gradients, which are either parabolic or sigmoidal in nature. Dependent on the gradient, the release is predictably slowed down from very hydrophilic hydrogel beads sigmoidally distributed OX is released at an almost constant rate for up to 3 hours (7). The extraction can be carried out with water or... 

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