Least-squares analysis technique

The stability constant for the formation of ZrF from ZrF" (g = 2 in Eq.(V.23)), was determined using a weighted (least squares) analysis technique (see Figure V-16). The selected log Q AT° determined from the regression is ... 

Thus, the solid surface tension can be determined from experimental contact angles and liquid surface tensions when is known. The latter was determined experimentally for a given set of yiv and 6 data measured on one and the same type of solid surface by a least-squares analysis technique. A weighted mean was calculated as 0.000 1247 (m /mJ) (6). It was found that calculations of y y values with slightly different values have very little effect on the outcome (25). However, it is still an open question as to whether or not p in equation (7.27) is a universal constant, i.e. independent of the solid surface. Such a question can be addressed only after an even larger body of accurate contact angle data on various solids has been generated. 

This book, and many standard texts, emphasizes graphical techniques for htting data. These methods give valuable qualitative insights that may be missed with too much reliance on least-squares analysis. 

While principal components models are used mostly in an unsupervised or exploratory mode, models based on canonical variates are often applied in a supervisory way for the prediction of biological activities from chemical, physicochemical or other biological parameters. In this section we discuss briefly the methods of linear discriminant analysis (LDA) and canonical correlation analysis (CCA). Although there has been an early awareness of these methods in QSAR [7,50], they have not been widely accepted. More recently they have been superseded by the successful introduction of partial least squares analysis (PLS) in QSAR. Nevertheless, the early pattern recognition techniques have prepared the minds for the introduction of modem chemometric approaches. 

The theta (0) conditions for the homopolymers and the random copolymers were determined in binary mixtures of CCl and CyHw at 25°. The cloud-point titration technique of Elias (5) as moaified by Cornet and van Ballegooijen (6) was employed. The volume fraction of non-solvent at the cloud-point was plotted against the polymer concentration on a semilogarithmic basis and extrapolation to C2 = 1 made by least squares analysis of the straight line plot. Use of concentration rather than polymer volume fraction, as is required theoretically (6, 7 ), produces little error of the extrapolated value since the polymers have densities close to unity. 

The success of the gas phase electron diffraction analysis of cis-and /ra 5-decalin (123) is another example of the use of MM calculations as an auxiliary technique. Minimum energy conformations and vibrational ampUtudes were calculated by both the Lifson and Boyd force fields (30,31) and were used as the starting values for refinement of the geometrical and vibrational parameters for the least-squares analysis. The results revealed no appreciable strain in cj5-decalin (123) other than that expected from gauche interactions. 

The next section of this paper describes the use of classical least-squares analysis of FTIR data to determine coal mineralogy. This is followed by promising preliminary results obtained using factor analysis techniques. 

Infrared data in the 1575-400 cm region (1218 points/spec-trum) from LTAs from 50 coals (large data set) were used as input data to both PLS and PCR routines. This is the same spe- tral region used in the classical least-squares analysis of the small data set. Calibrations were developed for the eight ASTM ash fusion temperatures and the four major ash elements as oxides (determined by ICP-AES). The program uses PLSl models, in which only one variable at a time is modeled. Cross-validation was used to select the optimum number of factors in the model. In this technique, a subset of the data (in this case five spectra) is omitted from the calibration, but predictions are made for it. The sum-of-squares residuals are computed from those samples left out. A new subset is then omitted, the first set is included in the new calibration, and additional residual errors are tallied. This process is repeated until predictions have been made and the errors summed for all 50 samples (in this case, 10 calibrations are made). This entire set of... 

Experience in this laboratory has shown that even with careful attention to detail, determination of coal mineralogy by classical least-squares analysis of FTIR data may have several limitations. Factor analysis and related techniques have the potential to remove or lessen some of these limitations. Calibration models based on partial least-squares or principal component regression may allow prediction of useful properties or empirical behavior directly from FTIR spectra of low-temperature ashes. Wider application of these techniques to coal mineralogical studies is recommended. 

Tavare and Garside ( ) developed a method to employ the time evolution of the CSD in a seeded isothermal batch crystallizer to estimate both growth and nucleation kinetics. In this method, a distinction is made between the seed (S) crystals and those which have nucleated (N crystals). The moment transformation of the population balance model is used to represent the N crystals. A supersaturation balance is written in terms of both the N and S crystals. Experimental size distribution data is used along with a parameter estimation technique to obtain the kinetic constants. The parameter estimation involves a Laplace transform of the experimentally determined size distribution data followed a linear least square analysis. Depending on the form of the nucleation equation employed four, six or eight parameters will be estimated. A nonlinear method of parameter estimation employing desupersaturation curve data has been developed by Witkowki et al (S5). 

Significance of the data was evaluated by analysis of variance with appropriate contrasts, and least square difference techniques. A probability value of less than 0.05 was judged to be statistically significant. 

In nonlinear least squares analysis we search for those parameter values I that mirumize the sum of squares of the differences between the measured values and the calculated values for all the data points. Many software programs are available to find these parameter values and all one has to do is to enter the data. The PO LYMATH software will be used to illustrate this technique. In order to carry out the search efficiently, in some cases one has to enter initial estimates of the parameter values close to the actual values. These estimates can be obtained using the linear-least-squares technique just discussed. 

To determine the oil, water, and solids contents simultaneously, sophisticated statistical techniques must usually be applied, such as partial least-squares analysis (PLS) and multivariate analysis (MVA). This approach requires a great deal of preparation and analysis of standards for calibration. Near-infrared peaks can generally be quantified by using Beer s law consequently, NIRA is an excellent analytical tool. In addition, NIRA has a fast spectral acquisition time and can be adapted to fiber optics this adaptability allows the instrument to be placed in a control room somewhat isolated from the plant environment. 

The enthalpy of formation was obtained from the equilibrium data of Brewer and Lofgren (1 ). They studied the reaction x Cu(cr) + X HCl(g) Cu2Cl2(g) + x/2 H2(g) by measuring the amount of CuCl formed when various ratios of HC1 H2 were passed over heated copper. Brewer and Lofgren analyzed the data by a least squares fitting technique and deduced partial pressures of monomer and triraer. The monomer pressures were subjected to 2nd and 3rd law analysis and gave A H (298.15 K) = 46.85 3.2 and 43.83... 

Equations (25) are linear with respect to x and this classification technique is referred to as /inear discriminant analysis, with the discriminant function obtained by least squares analysis, analogous to multiple regression analysis. 

In this chapter the experimental ECD and NIMS procedures for studying the reactions of thermal electrons with molecules and negative ions are described. Gas phase electron affinities and rate constants for thermal electron attachment, electron detachment, anion dissociation, and bond dissociation energies are obtained from ECD and NIMS data. Techniques to test the validity of specific equipment and to identify problems are included. Examples of the data reduction procedure and a method to include other estimates of quantities and their uncertainties in a nonlinear least-squares analysis will be given. The nonlinear least-squares procedure for a simple two-parameter two-variable case is presented in the appendix. 

A different approach to mathematical analysis of the solid-state C NMR spectra of celluloses was introduced by the group at the Swedish Forest Products Laboratory (STFI). They took advantage of statistical multivariate data analysis techniques that had been adapted for use with spectroscopic methods. Principal component analyses (PCA) were used to derive a suitable set of subspectra from the CP/MAS spectra of a set of well-characterized cellulosic samples. The relative amounts of the I and I/3 forms and the crystallinity index for these well-characterized samples were defined in terms of the integrals of specific features in the spectra. These were then used to derive the subspectra of the principal components, which in turn were used as the basis for a partial least squares analysis of the experimental spectra. Once the subspectra of the principal components are validated by relating their features to the known measures of variability, they become the basis for analysis of the spectra of other cellulosic samples that were not included in the initial analysis. Here again the interested reader can refer to the original publications or the overview presented earlier. ... 

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