Nonlinear variable

Fig. 4.4. Comparison of the computing effort, expressed in thousands of floating point operations (Aflop), required to factor the Jacobian matrix for a 20-component system (Nc = 20) during a Newton-Raphson iteration. For a technique that carries a nonlinear variable for each chemical component and each mineral in the system (top line), the computing effort increases as the number of minerals increases. For the reduced basis method (bottom line), however, less computing effort is required as the number of minerals increases.

At this point we can see the advantage of working with the reduced problem. Most published algorithms carry a nonlinear variable for each chemical component plus one for each mineral in the system. The number of nonlinear variables in the method presented here, on the other hand, is the number of components minus the number of minerals. Depending on the size of the problem, the savings in computing effort in evaluating Equation 4.31 can be dramatic (Fig. 4.4). 

Genetic programming [137] is an evolutionary technique which uses the concepts of Darwinian selection to generate and optimise a desired computational function or mathematical expression. It has been comprehensively studied theoretically over the past few years, but applications to real laboratory data as a practical modelling tool are still rather rare. Unlike many simpler modelling methods, GP model variations that require the interaction of several measured nonlinear variables, rather than requiring that these variables be orthogonal. 

These considerations provide an impetus for the development of fast, nonlinear, variable selection QSAR methods that can avoid the aforementioned problems of linear QSAR. Several nonlinear QSAR methods have been proposed in recent years. Most of these methods are based on either artificial neural network (ANN) (50, 61, 137-142) or machine learning techniques (65,143-145). Given that optimization of many parameters is involved in these techniques, the speed of the analysis is relatively slow. More recently. Hirst reported a simple and fast nonlinear QSAR method (146), in which the activity surface was generated from the activities of training set compounds based on some predefined mathematical function. 

For illustration, we shall consider here one of the nonlinear variable selection methods that adopts a k-Nearest Neighbor (kNN) principle to QSAR [kNN-QSAR (49)]. Formally, this method implements the active analog principle that lies in the foundation of the modern medicinal chemistry. The kNN-QSAR method employs multiple topological (2D) or topographical (3D) descriptors of chemical structures and predicts biological activity of any compound as the average activity of k most similar molecules. This method can be used to analyze the structure-activity relationships (SAR) of a large number of compounds where a nonlinear SAR may predominate. 

Fig. 10-12. Simulated K breakthrough curves for P = 50, and wj oo using two-site, nonlinear variable and constant models and a two-site, linear model for (A) W2 = 1.0 and (B) 2 — . ( i and U2 are dimensionless forms of a for type-1 and type-2 sites, respectively) [from Parker and Jardine (1986), with permission].

Notice the model structure is usually in error also, e.g., the true relationship between y and x may be nonlinear, variables other than x may be required to predict y, and so on. The procedure outlined here lumps structural error into e as well, but structural error is not accounted for correctly in this way. If the structure is in serious doubt, one may pose instead model discrimination tests to choose between competing models with different stnictures [30, 31 j. 

Nonlinear variables 4, 5, 8 // Others call with only non linear variables... 

It is now desirable to deal with the nonclassical behavior of the kernel in the linear laws in a precise, formal way. Of course, one could simply try to improve the crude method just discussed such an approach is perfectly valid. However, we feel that an alternate procedure, which has almost always been used in the literature, is preferable. Mori s method allows the writing of equations with well-behaved kernels if the proper set of variables is chosen. The kernel in the linear laws is badly behaved due to the influence of the nonlinear variable. If we include the linear and nonlinear variables in the set of variables to which Mori s method is applied, the random forces and the dissipative fluxes (/ will be defined precisely in this section) will be projected orthogonal to all of these variables. The kernels in the resulting equations, the nonlinear Langevin equations, should behave classically. Thus, convolutions involving K will be converted into scalar multiplication by the classical relation. 

Mori formalism, x > Uo, and K (t), will all be of infinite rank. The manipulation of infinite-rank matrices is much harder than the manipulation of the small, finite-rank matrices that arise when writing linear laws. Thus, it is important to simplify mode-mode coupling problems as much as possible, at the very beginning, by eliminating superfluous nonlinear variables. 

In practice, mode-mode coupling calculations are almost always performed with bilinear variables alone, and with one or at most two different bilinear products at that. We are aware of no attempt to justify such extreme simplification. Our feeling is that in many cases the use of bilinear variables alone is completely correct, but that in other cases the possible importance of other nonlinear variables deserves further study. All the problems that we discuss in detail will involve only bilinear variables. 

The equations obtained by applying Mori s method to an extended set of variables are of two types. First, the time derivative of the variable of interest, Ak(0> is expressed as a sum of the values of all the linear variables and nonlinear variables at time t, with each variable multiplied by an appropriate coefficient. Second, a similar equation results for the time derivative of each nonlinear variable such equations are of interest to us only insofar as the dynamics of the nonlinear variables are required to obtain the dynamics of the nonlinear variable. 

In Kawasaki s approach to mode-mode coupling, one does not use the complete set of Mori s equations for the complete set of variables. Rather the procedure is to keep only the equation for discard the equations for the nonlinear variables, and employ various alternative techniques to solve the equation for A. This approach is the one we shall discuss, as otherwise simplification to a very few nonlinear variables is impossible. 

Bilinear, and higher, nonlinear variables containing Sn are also negligi-ble near a critical point. In estimating critical point ( ) behavior of various contributions to we have been approximating various correlation... 

In attempting to solve the nonlinear Langevin equation [Eq. (49)], one is immediately struck that a solution is not obtainable by ordinary mathematical techniques. Since the nonlinear variables A k+k-A-k are not known as a function of Ak (AA AA) Eq. (49) is simply not a closed equation for the variable of interest, A To circumvent this difficulty, it is necessary to start with the fluctuating form of the nonlinear Langevin equation. The fluctuating form of Eq. (49) is... 

Tuy, H., S. Ghannadan, A. Migdalas, and P. Varbrand, A Strongly Polynomial Algorithm for a Concave Production-Transportation Problem with a Fixed Number of Nonlinear Variables , Mathematical Programming, 72 (1996), 229-258. 

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