Least Squares Procedure

Extension of the streamline Petrov -Galerkin method to transient heat transport problems by a space-time least-squares procedure is reported by Nguen and Reynen (1984). The close relationship between SUPG and the least-squares finite element discretizations is discussed in Chapter 4. An analogous transient upwinding scheme, based on the previously described 0 time-stepping technique, can also be developed (Zienkiewicz and Taylor, 1994). 

Procedure. Compute the slope of the function by a linear least squares procedure and obtain a value of Boltzmann s constant. How many particles do you expect to find 125 pm above the reference point Take the uncertainty you have calculated for the slope, as the uncertainty in k. Is the modem value of = 1.381 x 10 within these enor limits ... 

The least-squares procedure can be appHed to the transformed variables of any of the equations in Table 2, where a simple transformation of one or both of the variables results in a linearized expression. The sums for equations 83 and 84 must be formed from the transformed variables rather than from the original data. 

Processing all peaks for the model by nonlinear least square procedure ... 

Thermodynamic Functions of the Gases. To apply Eqs. (1-10), the free energies of formation, Ag , for all gaseous species as a function of temperature are required. Tabulated data were fit by a least-squares procedure to derive an analytical equation for AG° of each vapor species. For the plutonium oxide vapor species, the data calculated from spectroscopic data (3 ) were used for 0(g) and 02(g), the JANAF data (.5) were used and for Pu(g), data from the compilation of Oetting et al. (6) were used. The coefficients of the equations for AG° of the gaseous species are included in Table I. 

The temperature-factor parameter B and the scale factor k were determined by a least-squares procedure/ with observational equations set up in logarithmic form and with weights obtained from those in equation (9) by multiplying by (G (obs.))2. Since a semi-logarithmic plot of G2 (obs.)/Gf (calc.) against B showed a pronounced deviation from linearity for the last five lines, these lines were omitted from the subsequent treatments. They were much broader than the others, and apparently their intensities were underestimated. The temperature-factor parameter B was found by this treatment to have the value 1-47 A2. 

The parameters were then further refined by four successive least-squares procedures, as described by Hughes (1941). Only hk() data were used. The form factor for zinc was taken to be 2-4 times the average of the form factors for magnesium and aluminum. The values of the form factor for zinc used in making the average was corrected for the anomalous dispersion expected for copper Kot radiation. The customary Lorentz, polarization, temperature, and absorption factors were used. A preliminary combined scale, temperature, and absorption factor was evaluated graph-... 

When comparisons are to be drawn among scales derived with different criteria of physical validity, we believe this point to be especially appropriate. The SD is the explicit variable in the least-squares procedure, after all, while the correlation coefficient is a derivative providing at best a non linear acceptability scale, with good and bad correlations often crowded in the range. 9-1.0. The present work further provides strong confirmation of this conclusion. 

Steinier, J., Termonia, Y., and Deltour, J., Comments on Smoothing and Differentiation of Data by Simplified Least Square Procedure, Anal. Chem. 44, 1972, 1906-1909. 

Savitzky, A. and Golay, M. J. E., Smoothing and Differentiation of Data by Simplified Least-Squares Procedures, Anal. Chem. 36, July 1964, 1627-... 

A. Savitzky and M.J.E. Golay, Smoothing and differentiating of data by simplified least-squares procedures. Anal. Chem., 36 (1964) 1627-1639. 

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. 

Ej is determined by the weights (through Oj which is a function of NET, see eq. (44.8)). Note that this error is in fact the same as the error term used in a usual least squares procedure. 

The graphically deduced constants are subsequently refined by a weighted nonlinear least squares procedure [472]. Although the potentiometric method can be used in discovery settings to calibrate high-throughput solubility methods and computational procedures, it is too slow for HTS applications. It is more at home in a preformulation lab. 

Thus, we may obtain a best value fork from a modification of the normal least squares procedure. As k , F hkl will only significantly differ from Fm when Fm is small and Fumi is large, i.e. the reflections carrying the most information about k are those very reflections that are poorly observed. Hence, this is not a reliable method. 

If we are interested only in the determination of a molecular structure, as most chemists have been, it suffices to approximate the true molecular electron density by the sum of the spherically averaged densities of the atoms, as discussed in Section 6.4. A least-squares procedure fits the model reference density preKr)t0 the observed density pobs(r) by minimizing the residual density Ap(r), defined as follows ... 

One may also use the methods of the statistician to determine average rate constants (e.g., the standard unweighted least squares procedure). 

For frequency calculations one usually starts out with a set of approximate existent force constants (e.g. taken over from similar, already treated molecules under the preliminary tentative assumption of transferability), and subsequently varies the force constants in a systematic way by means of a least-squares procedure until the calculated frequencies (square roots of the eigenvalues of Eq. (10)) agree satisfactorily with the experimental values. Clearly, if necessary, the analytical form of the force field is also to be modified in the course of this fitting process. 

Most of the force fields described in the literature and of interest for us involve potential constants derived more or less by trial-and-error techniques. Starting values for the constants were taken from various sources vibrational spectra, structural data of strain-free compounds (for reference parameters), microwave spectra (32) (rotational barriers), thermodynamic measurements (rotational barriers (33), nonbonded interactions (1)). As a consequence of the incomplete adjustment of force field parameters by trial-and-error methods, a multitude of force fields has emerged whose virtues and shortcomings are difficult to assess, and which depend on the demands of the various authors. In view of this, we shall not discuss numerical values of potential constants derived by trial-and-error methods but rather describe in some detail a least-squares procedure for the systematic optimisation of potential constants which has been developed by Lifson and Warshel some time ago (7 7). Other authors (34, 35) have used least-squares techniques for the optimisation of the parameters of nonbonded interactions from crystal data. Overend and Scherer had previously applied procedures of this kind for determining optimal force constants from vibrational spectroscopic data (36). 

The p.c.s. measurements were carried out using a Malvern multibit correlator and spectrometer together with a mode stabilized Coherent Krypton-ion laser. The resulting time correlation functions were analysed using a non-linear least squares procedure on a PDP11 computer. The latex dispersions were first diluted to approximately 0.02% solids after which polymer solution of the required concentration was added. 

Another problem with real data is that due to random indeterminate errors (Chapter 1), the analyst cannot expect the measured points to fit a straight-line graph exactly. Thus it is often true that we draw the best straight line that can be drawn through a set of data points and the unknown is determined from this line. A linear regression, or least squares, procedure is then done to obtain the correct position of the line and therefore the correct slope, etc. 

The DMC method uses the same statistical mathematics that are used in a standard least-squares procedure for determining the best values of parameters of an equation to fit a number of data points. In the DMC approach, we would like to have NP future output responses match some optimum trajectory by finding the best values of NC future changes in the manipulated variables. This is exactly the concept of a least-squares problem of fitting NP data points with an equation with NC coefficients. This is a valid least-squares problem as long as NP is greater than NC. 

Use of Multiple-Curve and Weighted Least-Squares Procedures with Confidence Band Statistics... 

Two procedures for improving precision in calibration curve-based-analysis are described. A multiple curve procedure is used to compensate for poor mathematical models. A weighted least squares procedure is used to compensate for non-constant variance. Confidence band statistics are used to choose between alternative calibration strategies and to measure precision and dynamic range. 

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