Independent variables selection

Reactor design is often discussed in terms of independent and dependent variables. Independent variables are choices such as reactor type and internals, catalyst type, inlet temperature, pressure, and fresH feed composition. Dependent variables result from independent variable selection. They may be constrained or unconstrained. Con-... 

The main aspects for metamodel generation are (i) Generation of training data, through an experimental design strategy (ii) Independent variable selection (iii) Parameter estimation and (iv) Metamodel validation. 

For kriging models, the structure is defined by the set of independent variables selected - including quadratic terms - and the selection of the correlation model. The parameter estimation is performed by a maximum likelihood procedure. For neural nets, the activation function to be used is defined a priori. The structure is completed by the selection of the number of neurons in the hidden layer. A backpropagation procedure has been used for training. 

Jezierska, A., Vracko, M. and Basak, S.C. (2004) Counter-propagation artificial neural network as a tool for the independent variable selection structure-mutagenicity study of aromatic amines. Mol. Div., 8, 371-377. 

In many applications satisfying the inequality, m > n, can represent a serious problem as ILS requires more standard calibration mixtures than the number of wavelengths (independent variables) selected. This can serve to limit the spectral reange or resolution of the data used in the analysis. Wavelength, variable, selection is an important issue. 

Clearly, it is possible to achieve a reasonable solution of an underdimensioned system using the Gauss factorization, only if the independent variable selection is performed adequately. 

In this chapter we have deseribed a canonieal ensemble. In fact, an ensemble is chosen according to the set of independent variables selected, i.e. volume, temperature and amount of matter of the canonical ensemble. For this ensemble, we defined a canonical partition function by relation [5.8]. 

Table 1. Independent variables selected and their solution for each forage group and dependent variable. 

Ratio and Multiplicative Feedforward Control. In many physical and chemical processes and portions thereof, it is important to maintain a desired ratio between certain input (independent) variables in order to control certain output (dependent) variables (1,3,6). For example, it is important to maintain the ratio of reactants in certain chemical reactors to control conversion and selectivity the ratio of energy input to material input in a distillation column to control separation the ratio of energy input to material flow in a process heater to control the outlet temperature the fuel—air ratio to ensure proper combustion in a furnace and the ratio of blending components in a blending process. Indeed, the value of maintaining the ratio of independent variables in order more easily to control an output variable occurs in virtually every class of unit operation. 

What is the most meaningful way to express the controllable or independent variables For example, should current density and time be taken as the experimental variables, or are time and the product of current density and time the real variables affecting response Judicious selection of the independent variables often reduces or eliminates interactions between variables, thereby leading to a simpler experiment and analysis. Also inter-relationships among variables need be recognized. For example, in an atomic absorption analysis, there are four possible variables air-flow rate, fuel-flow rate, gas-flow rate, and air/fuel ratio, but there are really only two independent variables. 

In terms of the derived general relationships (3-1) and (3-2), x, y, and h are independent variables—cost and volume, dependent variables. That is, the cost and volume become fixed with the specification of dimensions. However, corresponding to the given restriedion of the problem, relative to volume, the function g(x, y, z) =xyh becomes a constraint funedion. In place of three independent and two dependent variables the problem reduces to two independent (volume has been constrained) and two dependent as in functions (3-3) and (3-4). Further, the requirement of minimum cost reduces the problem to three dependent variables x, y, h) and no degrees of freedom, that is, freedom of independent selection. 

If an evaporation temperature (Pc) is pre-selected as a parametric independent variable, then the temperatures and enthalpies at c and e are found from (b) above the temperature T(, is also determined. If there is no heat loss, the heat balance in the HRSG between gas states 4 and 6 is... 

Interpolation of this type may be extremely unreliable toward the center of the region where the independent variable is widely spaced. If it is possible to select the values of x for which values of f(x) will be obtained, the maximum error can be minimized by the proper choices. In this particular case Chebyshev polynomials can be computed and interpolated [11]. 

Note the array R(/, J) contains the values is the number, that is the j-th measurement is the item or dimension number. For a program based on linear regression (LINREG. VALID, SHELFLIFE), since the array R(,) must have M >2 columns, it is up to the user to decide which column will be identified with abscissa X (index K), and which with ordinate Y (index L) / (/, k ) is the independent variable X, R I, L) is the dependent variable Y. K (and L, if necessary) are established by clicking on the column(s) after the file has been selected. When any program is started, the available data in the chosen file will be shown for review. 

The variable selection methods have been also adopted for region selection in the area of 3D QSAR. For example, GOLPE [31] was developed with chemometric principles and q2-GRS [32] was developed based on independent CoMFA analyses of small areas of near-molecular space to address the issue of optimal region selection in CoMFA analysis. Both of these methods have been shown to improve the QSAR models compared to original CoMFA technique. 

R r Multiple correlation coefficient. R indicates the percentage of the variability of the relative biological response that can be accounted for by the selected independent variables. 

Model formulation. After the objective of modelling has been defined, a preliminary model is derived. At first, independent variables influencing the process performance (temperature, pressure, catalyst physical properties and activity, concentrations, impurities, type of solvent, etc.) must be identified based on the chemists knowledge about reactions involved and theories concerning organic and physical chemistry, mainly kinetics. Dependent variables (yields, selectivities, product properties) are defined. Although statistical models might be better from a physical point of view, in practice, deterministic models describe the vast majority of chemical processes sufficiently well. In principle model equations are derived based on the conservation law ... 

The expression x (J)P(j - l)x(j) in eq. (41.4) represents the variance of the predictions, y(j), at the value x(j) of the independent variable, given the uncertainty in the regression parameters P(/). This expression is equivalent to eq. (10.9) for ordinary least squares regression. The term r(j) is the variance of the experimental error in the response y(J). How to select the value of r(j) and its influence on the final result are discussed later. The expression between parentheses is a scalar. Therefore, the recursive least squares method does not require the inversion of a matrix. When inspecting eqs. (41.3) and (41.4), we can see that the variance-covariance matrix only depends on the design of the experiments given by x and on the variance of the experimental error given by r, which is in accordance with the ordinary least-squares procedure. 

If the structure of the models is more complex and we have more than one independent variable or we have more than two rival models, selection of the best experimental conditions may not be as obvious as in the above example. A straightforward design to obtain the best experimental conditions is based on the divergence criterion. 

Our selection of the initial state, x0, and the value of the manipulated variables vector, u(t) determine a particular experiment. Here we shall assume that the input variables u(t) are kept constant throughout an experimental run. Therefore, the operability region is defined as a closed region in the [xoj.xo, , Xo,n, U u2,...,u,]T -space. Due to physical constraints these independent variables are limited to a very narrow range, and hence, the operability region can usually be described with a small number of grid points. 

Only two of the four state variables measured in a binary VLE experiment are independent. Hence, one can arbitrarily select two as the independent variables and use the EoS and the phase equilibrium criteria to calculate values for the other two (dependent variables). Let Q, (i=l,2,...,N and j=l,2) be the independent variables. Then the dependent ones, g-, can be obtained from the phase equilibrium relationships (Modell and Reid, 1983) using the EoS. The relationship between the independent and dependent variables is nonlinear and is written as follows... 

Partition-based modeling methods are also called subset selection methods because they select a smaller subset of the most relevant inputs. The resulting model is often physically interpretable because the model is developed by explicitly selecting the input variable that is most relevant to approximating the output. This approach works best when the variables are independent (De Veaux et al., 1993). The variables selected by these methods can be used as the analyzed inputs for the interpretation step. 

The two independent variables (the axes) show the pump speeds for the two reagents required in the analysis reaction. The initial simplex is represented by the lowest triangle the vertices represent the spectro-photometrie response. The strategy is to move toward a better response by moving away from the worst response. Since the worst response is 0.25, conditions are selected at the vortex, 0.6, and, indeed, improvement is obtained. One can follow the experimental path to the optimum, 0.721. 

This optimization method, which represents the mathematical techniques, is an extension of the classic method and was the first, to our knowledge, to be applied to a pharmaceutical formulation and processing problem. Fonner et al. [15] chose to apply this method to a tablet formulation and to consider two independent variables. The active ingredient, phenylpropanolamine HC1, was kept at a constant level, and the levels of disintegrant (corn starch) and lubricant (stearic acid) were selected as the independent variables, X and Xj. The dependent variables include tablet hardness, friability, volume, in vitro release rate, and urinary excretion rate in human subjects. 

The system selected here was also a tablet formulation. The five independent variables or formulation factors selected for this study are shown in Table 2. The dependent variables are listed in Table 3. Since each dependent variable is considered separately, any number could have been included. 

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