Constrained least-squares

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.

Let us consider constrained least squares estimation of unknown parameters in algebraic equation models first. The problem can be formulated as follows ... 

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,... 

Modified Gauss-Newton Algorithm for Constrained Least Squares... 

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. 

If incorrect phase behavior is predicted by the EOS then constrained least squares (CLS) estimation should be performed and new parameter estimates be obtained. Subsequently, the phase behavior should be computed again and if the fit is found to be acceptable for the intended applications, then the CLS estimates should suffice. This was found to be the case for the carbon dioxide-n-hexane system presented later in this chapter. 

Figure 14.3 Vapor-liquid equilibrium data and calculated values for the carbon dioxide-n-hexane system. Calculations were done using interaction parameters from implicit and constrained least squares (LS) estimation, x and y are the mote fractions in the liquid and vapor phase respectively [reprinted from the Canadian Journal of Chemical Engineering with permission]...

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. 

Englezos, P. and N. Kalogerakis, "Constrained Least Squares Estimation of Binary Interaction Parameters in Equations of State", Computers Chem. Eng.. 17. 117-121 (1993). 

Gill, P.E. and W. Murray, "Nonlinear Least Squares and Nonlinearly Constrained Optimization", Lecture Notes in Mathematics No 506, G.A. Watson, Ed, Springer-Verlag, Berlin and Heidelberg, pp 134-147 (1975). 

In practice, the choice of parameters to be refined in the structural models requires a delicate balance between the risk of overfitting and the imposition of unnecessary bias from a rigidly constrained model. When the amount of experimental data is limited, and the model too flexible, high correlations between parameters arise during the least-squares fit, as is often the case with monopole populations and atomic displacement parameters [6], or with exponents for the various radial deformation functions [7]. 

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. 

The constrained least squares problem for the overall plant can now be replaced by the equivalent two-problem formulation. 

The data reconciliation problem can be generally stated as the following constrained weighted least-squares estimation problem ... 

Using the Q-R orthogonal factorization method described in Chapter 4, the constrained weighted least-squares estimation problem (5.4) is transformed into an unconstrained one. The following steps are required ... 

As shown before, the general data reconciliation procedure for the overall system must solve the following constrained least squares problem ... 

In Chapters 3 and 4 we have shown that the vector of process variables can be partitioned into four different subsets (1) overmeasured, (2) just-measured, (3) determinable, and (4) indeterminable. It is clear from the previous developments that only the overmeasured (or overdetermined) process variables provide a spatial redundancy that can be exploited for the correction of their values. It was also shown that the general data reconciliation problem for the whole plant can be replaced by an equivalent two-problem formulation. This partitioning allows a significant reduction in the size of the constrained least squares problem. Accordingly, in order to identify the presence of gross (bias) errors in the measurements and to locate their sources, we need only to concentrate on the largely reduced set of balances... 

The constrained least-square method is developed in Section 5.3 and a numerical example treated in detail. Efficient specific algorithms taking errors into account have been developed by Provost and Allegre (1979). Literature abounds in alternative methods. Wright and Doherty (1970) use linear programming methods that are fast and offer an easy implementation of linear constraints but the structure of the data is not easily perceived and error assessment inefficiently handled. Principal component analysis (Section 4.4) is more efficient when the end-members are unknown. 

Although satisfactory criteria for deciding whether data are better analyzed by distributions or multiexponential sums have yet to established, several methods for determining distributions have been developed. For pulse fluorometry, James and Ware(n) have introduced an exponential series method. Here, data are first analyzed as a sum of up to four exponential terms with variable lifetimes and preexponential weights. This analysis serves to establish estimates for the range of the preexponential and lifetime parameters used in the next step. Next, a probe function is developed with fixed lifetime values and equal preexponential factors. An iterative Marquardt(18) least-squares analysis is undertaken with the lifetimes remaining fixed and the preexponential constrained to remain positive. When the preexponential... 

Here there are four measured frequencies with which to constrain three independent force constants, so the best-fitting force constants can be determined through an iterative least squares fit, minimizing S(v eas-Vcaic) - Assuming average atomic masses of 51.996 and 15.9994 for chromium and oxygen, respectively, the best-fit force constants are Ai = 495.2 Newtons/m,iT = 21.3 Newtons/m, andFjv = 44.7 Newtons/m. These force constants show the typical relationship K H,Fj j,. Calculated frequencies are ... 

Butene exists as an equilibrium mixture of two conformations, Me-skew and Me-syn (21). The most reliable composition to date is 83 17% according to combined ED, microwave (MW), and ab initio MO analysis (133). This study includes the MM (CFF)-ED-MW analysis of this molecule for comparison, which gave a final skew/syn ratio of 80 20. The molecular orbital constrained electron diffraction (MOCED) results appear to agree better with the observed data than does the MM constrained analysis, the R value of the least-squares analysis of the latter being 20% higher than that of MOCED. However, one may ask whether such a small difference in R values justifies the enormous difference in computer time between the ab initio (about 200 hr on an IBM 370/155) and MM (less than a minute) methods used in this work. 

Equations (4.40) and (4.41) are easily implemented in an existing least-squares program and give both the constrained and the unconstrained results in a single refinement cycle. However, the method fails if the unconstrained refinement corresponds to a singular matrix, as would be the case, for example, if all population parameters, including those of the core functions, were to be refined in addition to the scale factor k. 

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