Least-square constraints

The first and second moment conditions can be very easily introduced into the r5-fit method as least-squares constraints [7,54] if the number of isotopomers is sufficient for a complete restructure. The effect on the coordinates is not expected to be particularly unbalanced unless the moment conditions are required for the sole purpose of locating atoms that could not be substituted (e.g., fluorine or phosphorus) or that have a near-zero coordinate. While all coordinates may change, the small coordinates will, of course, change more. In the cases tested, the coordinate values of the rs-fit with constraints and those of the corresponding r/e-fit (not of the r0-fit), including errors and correlations, differed by only a small fraction of the respective errors, i.e., much less than reported above. This was true under the provision that all atoms could be substituted and that the planar moments that were excluded from the r -fit because of substitution on a principal plane or axis, were also omitted from the r/E-fit. With these modifications, the basic physical considerations and the input data are the same in both cases, and the results should be identical in the limit where the number of observations equals that of the variables. 

Multiple sets of Burnett data were obtained for each isotherm—three sets for ethylene and two sets for helium. Each set consisted of data from a series of four consecutive expansions from the highest to the lowest pressure compatible with our optimum accuracy and precision. The initial pressure for each set was selected so as to intersperse the data from all of the sets over the entire pressure range of interest, 0.3 MPa to 3.7 MPa. Consistent with the extent of the nonideal behavior of the gas, the density-series generalized equation was applied to the ethylene data and the pressure-series generalized equation was applied to the helium data. The parameters in the resulting overdetermined sets of equations then were evaluated using the least-squares constraint. 

The selection to minimize absolute error [Eq. (6)] calls for optimization algorithms different from those of the standard least-squares problem. Both problems have simple and extensively documented solutions. A slight advantage of the LP solution is that it does not need to be solved for the points for which the approximation error is less than the selected error threshold. In contrast, the least squares problem has to be solved with every newly acquired piece of data. The LP problem can effectively be solved with the dual simplex algorithm, which allows the solution to proceed recursively with the gradual introduction of constraints corresponding to the new data points. 

Mathematical Models. As noted previously, a mathematical model must be fitted to the predicted results shown In each factorial table generated by each scientist. Ideally, each scientist selects and fits an appropriate model based upon theoretical constraints and physical principles. In some cases, however, appropriate models are unknown to the scientists. This Is likely to occur for experiments Involving multifactor, multidisciplinary systems. When this occurs, various standard models have been used to describe the predicted results shown In the factorial tables. For example, for effects associated with lognormal distributions a multiplicative model has been found useful. As a default model, the team statistician can fit a polynomial model using standard least square techniques. Although of limited use for Interpolation or extrapolation, a polynomial model can serve to Identify certain problems Involving the relationships among the factors as Implied by the values shown In the factorial tables. 

The Rietveld Fit of the Global Diffraction Pattern. The philosophy of the Rietveld method is to obtain the information relative to the crystalline phases by fitting the whole diffraction powder pattern with constraints imposed by crystallographic symmetry and cell composition. Differently from the non-structural least squared fitting methods, the Rietveld analysis uses the structural information and constraints to evaluate the diffraction pattern of the different phases constituting the diffraction experimental data. 

Fig. 8.2 A least-squares superimposition of the unmodified X-ray structure of the protein-bound ligand 21 (dark grey) and the corresponding constrained optimized stmcture (grey) using flat-bottomed Cartesian constraints with a half-width of 0.8A. The RMS value is 0.43 A. Hydrogens are removed for clarity.

Given a set of data points (x y,), i=l,...,N and a mathematical model of the form, y = f(x,k), the objective is to determine the unknown parameter vector k by minimizing the least squares objective function subject to the equality constraint, namely... 

The point where the constraint is satisfied, (x0,yo), may or may not belong to the data set (xj,yj) i=l,...,N. The above constrained minimization problem can be transformed into an unconstrained one by introducing the Lagrange multiplier, to and augmenting the least squares objective function to form the La-grangian,... 

Solution of the above constrained least squares problem requires the repeated computation of the equilibrium surface at each iteration of the parameter search. This can be avoided by using the equilibrium surface defined by the experimental VLE data points rather than the EoS computed ones in the calculation of the stability function. The above minimization problem can be further simplified by satisfying the constraint only at the given experimental data points (Englezos et al. 1989). In this case, the constraint (Equation 14.25) is replaced by... 

In Equation 14.27, cT, oP and ax are the standard deviations of the measurements of T, P and x respectively. All the derivatives are evaluated at the point where the stability function cp has its lowest value. We call the minimization of Equation 14.24 subject to the above constraint simplified Constrained Least Squares (simplified CLS) estimation. 

Copp and Everet (1953) have presented 33 experimental VLE data points at three temperatures. The diethylamine-water system demonstrates the problem that may arise when using the simplified constrained least squares estimation due to inadequate number of data. In such case there is a need to interpolate the data points and to perform the minimization subject to constraint of Equation 14.28 instead of Equation 14.26 (Englezos and Kalogerakis, 1993). First, unconstrained LS estimation was performed by using the objective function defined by Equation 14.23. The parameter values together with their standard deviations that were obtained are shown in Table 14.5. The covariances are also given in the table. The other parameter values are zero. 

In order to answer these questions, the kinetic and network structure models were used in conjunction with a nonlinear least squares optimization program (SIMPLEX) to determine cure response in "optimized ovens ". Ovens were optimized in two different ways. In the first the bake time was fixed and oven air temperatures were adjusted so that the crosslink densities were as close as possible to the optimum value. In the second, oven air temperatures were varied to minimize the bake time subject to the constraint that all parts of the car be acceptably cured. Air temperatures were optimized for each of the different paints as a function of different sets of minimum and maximum heating rate constants. 

The adjustment of measurements to compensate for random errors involves the resolution of a constrained minimization problem, usually one of constrained least squares. Balance equations are included in the constraints these may be linear but are generally nonlinear. The objective function is usually quadratic with respect to the adjustment of measurements, and it has the covariance matrix of measurements errors as weights. Thus, this matrix is essential in the obtaining of reliable process knowledge. Some efforts have been made to estimate it from measurements (Almasy and Mah, 1984 Darouach et al., 1989 Keller et al., 1992 Chen et al., 1997). The difficulty in the estimation of this matrix is associated with the analysis of the serial and cross correlation of the data. 

Parameter estimation is also an important activity in process design, evaluation, and control. Because data taken from chemical processes do not satisfy process constraints, error-in-variable methods provide both parameter estimates and reconciled data estimates that are consistent with respect to the model. These problems represent a special class of optimization problem because the structure of least squares can be exploited in the development of optimization methods. A review of this subject can be found in the work of Biegler et al. (1986). 

When the system is nonestimable, the estimated value of x (x) is not a unique solution to the least squares problem. In this case a solution is only possible if additional information is incorporated. This must be introduced via the process model equations (constraint equations). They occur in practice when some or all of the system variables must conform to some relationships arising from the physical constraints of the process. 

To incorporate the constraints in our least squares problem, we consider the Lagrangian equation... 

That is, the least squares estimate can be finally expressed as the contribution of three terms. The first one arises from the solution of the original problem, without constraints (for data reconciliation xo = y) the next is a correction term due to the presence of constraints and the last one takes into account failures in the model (systematic errors). 

If one or more leaks are considered, the constraint model for the process must be modified to take them into account. Now, the least squares formulation of the problem, when measurement bias are absent, can be stated as... 

If combinations of leaks and measurement biases are considered, both the measurement model and the process constraints equations need to be modified. The formulation for the least squares problem is now... 

Recall that, in the absence of gross errors, the measurement and linear constraint models are given by Eqs. (7.1) and (7.4), respectively. Furthermore, the solution of the least square estimation problem of x variables is... 

Let us suppose that an initial data reconciliation problem has been resolved using a set of process constraints and the covariance matrix for the estimated variables (Eold) is available. If a set of constraints B is incorporated into the least square estimation problem, the covariance matrix E for the new case (Enew) can be estimated using the previous one by means of the formula... 

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