Constrained least-squares analysis

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-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... 

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. 

Multiple Pass Analysis. Pike and coworkers (13) have provided a method to increase the resolution of the ordinary least squares algorithm somewhat. It was noted that any reasonable set of assumed particle sizes constitutes a basis set for the inversion (within experimental error). Thus, the data can be analyzed a number of times with a different basis set each time, and the results combined. A statistically more-probable solution results from an average of the several equally-likely solutions. Although this "multiple pass analysis".helps locate the peaks of the distribution with better resolution and provides a smoother presentation of the result, it can still only provide limited resolution without the use of a non-negatively constrained least squares technique. We have shown, however, that the combination of both the non-negatively constrained calculation and the multiple pass analysis gives the advantages of both. 

A modification of this procedure was proposed in the literature [389] and applied to determine the time constant distribution function [379]. This method is based on the predistribution of time constants uniformly on the logarithmic scale, and to improve the quality of the analysis, a Mmite Carlo technique was used to increase the number of analyzed time constants. Approximation was carried out using a constrained least-squares method and led to a continuous distribution function. This procedure converted the nonlinear problem to a linear one from which versus r , were obtained and produced positive values of the distribution function. The procedure was also applied to the distribution of the dielectric constants [379,389]. 

Angeles, J and Liu, Z. "A Constrained Least-square Method for the Optimization of Spherical Four-bar Path Generators", in Mechanism Synthesis and Analysis, 1990, Ed. s McCarthy, M., Derby, S. and Pisano, A... 

Although Foster et al. (2000) conservatively fit only 1 or 2 different model As spectra to each sample spectrum, least-squares fitting does not impose limitations on the number of models used in the fit, nor is their any parameter besides the fit residual to guide selection of the models used in the fits. Well-constrained fits can be obtained using principal component analysis (PCA) to guide selection of the type and number of components used in linear least-squares fits (Ressler et al., 2000). However, use of PCA... 

In the modern waxd crystallographic analysis, waxd experimental results are always combined with conformational and packing energy calcidations to obtain more precise crystal unit cell dimensions and atomic positions in a crystal lattice. Under constrained conditions, the least-square calculation reduces the error to a degree that the diffraction positions and intensities can be semiquan-titatively fitted to the experimental observations. However, force fields between chain molecules in crystals are difficult to establishe and defects existing in real crystals limit the accuracy of the calculations. 

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