Fitness maximization

FIGURE 11.9 Outliers, (a) Dose-response curve fit to all of the data points. The potential outlier value raises the fit maximal asymptote, (b) Iterative least squares algorithm weighting of the data points (Equation 11.25) rejects the outlier and a refit without this point shows a lower-fit maximal asymptote. 

The Kohonen Self-Organizing Maps can be used in a. similar manner. Suppose Xj., k = 1,. Nis the set of input (characteristic) vectors, Wy, 1 = 1,. l,j = 1,. J is that of the trained network, for each (i,j) cell of the map N is the number of objects in the training set, and 1 and j are the dimensionalities of the map. Now, we can compare each with the Wy of the particular cell to which the object was allocated. This procedure will enable us to detect the maximal (e max) minimal ( min) errors of fitting. Hence, if the error calculated in the way just mentioned above is beyond the range between e and the object probably does not belong to the training population. 

Using a GA with a population of 10 members, find the values of the controller gain K and the tachogenerator constant that maximizes the fitness function... 

Validate, or otherwise, using a gene tie algorithm that this value of K maximizes the fitness funetion... 

Table 11.3 One pass (read left to right) through the step.s of a basic genetic algorithm scheme to maximize the fitness function f x) = using a population of six 6-bit chromosomes. The crossover notation aina2) means that chromosomes Ca, and Ca2 exchange bits beyond the bit. The underlined bits in the Mutation Operation column are the only ones that have undergone random mutation. See text for other details.

There are statistical procedures available to determine whether the data can be fit to a model of dose-response curves that are parallel with respect to slope and all share a common maximal response (see Chapter 11). In general, dose-response data can be fit to a three-parameter logistic equation of the form... 

If the antagonism is insurmountable, then there are a number of molecular mechanisms possible. The next question to ask is if the maximal response to the agonist can be completely depressed to basal levels. If this is not the case, then there could be partial allosteric alteration of the signaling properties of the receptor. Alternatively, this could be due to a hemi-equilibrium condition that produces a partial shortfall to true competitive equilibrium, leading to incomplete depression of the maximal response but also antagonist concentration-related dextral displacement of the concentration response curve to the agonist (see Figure 10.19a). The model (see Section 10.6.5) used to fit these data is discussed in Section 6.5 and shown in... 

FIGURE 11.12 Overextrapolation of data, (a) Nonlinear curve fitting techniques estimate an ordinate maximal asymptote that is nearly 100% beyond the last available data point, (b) The curve fitting procedure estimates a maximal asymptote much closer to the highest available data point. A useful rule is to reject fits that cause an estimated maximal asymptote that is >25% the value of the highest available data point. 

The mean maximal response for the five curves is 96.1 and the mean slope is 1.27. The five curves are then refit to three-parameter logistic functions utilizing the mean maximal response and mean slope. The ECso for the curves fit in this manner are shown in Table 12.6b (ECS0 values in column labeled Mean). 

A statistical test is performed to determine whether or not the data may be fit to a set of curves of common maximal response and slope or if they must be fit to individual equations. For this example, Aikake s information criteria are calculated (see... 

General Procedure Dose-response curves are obtained for an agonist in the absence and presence of a range of concentrations of the antagonist. The dextral displacement of these curves (ECSo values) are fit to a hyperbolic equation to yield the potency of the antagonist and the maximal value for the cooperativity constant (a) for the antagonist. 

The inhibition curves are fit to an appropriate function to allow estimation of the half maximal value for blockade (IC50). For example, the data... 

On a rank-by-rank (i.e. factor-by-factor) basis, we rotate, or perturb, each pair of factors, (1 spectral factor and its corresponding concentration factor) towards each other to maximize the fit of the linear regression between the projections of the spectra onto the spectral factor with the projections of the concentrations onto the concentration factor. 

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