Standard least-squares method

Phase identification was done on the basis of both d-spacing and the peak height intensity of all the x-ray lines. These values were compared with values obtained for the end-member (unsubstituted) compounds and also calculated by means of the Lazy-Pulverix computer program (9). Precision lattice parameters were obtained by the Debye-Scherrer method with a 114.6 mm dia. camera and filtered Cr Ka radiation standard least-squares methods were used. 

The eight-coordinate tetrakis(diethyldithiocarbamate) lanthanide complexes are isomor-phous and the lanthanum complex has a quasi-tetrahedral configuration of the four CS2 chelate groups. The transition frequencies and dipole strengths of these complexes available over the accessible f-f manifold allowed the extraction of the Judd-Ofelt intensity parameters Qx for k = 2, 4, 6 by standard least squares methods [112,113]. The values of observed Qx and calculated Q, values are given in Table 8.10. The three components of f2() are... 

An improved method was developed by Chirlian and Francl and called CHELP (CHarges from ELectrostatic Potentials). Their method, which uses a Lagrangian multiplier method for fitting the atomic charges, is fast and noniterative and avoids the initial guess required in the standard least-squares methods. In this approach, the best least-squares fit is obtained by minimizing y ... 

The standard least squares method would minimize the following relation ... 

There are several numerical methods to determine the parameters below we describe, as examples, the standard least-squares method and the chi-square method. 

The standard least-squares approach provides an alternative to the Galerkin method in the development of finite element solution schemes for differential equations. However, it can also be shown to belong to the class of weighted residual techniques (Zienkiewicz and Morgan, 1983). In the least-squares finite element method the sum of the squares of the residuals, generated via the substitution of the unknown functions by finite element approximations, is formed and subsequently minimized to obtain the working equations of the scheme. The procedure can be illustrated by the following example, consider... 

For a specified mean and standard deviation the number of degrees of freedom for a one-dimensional distribution (see sections on the least squares method and least squares minimization) of n data is (n — 1). This is because, given p and a, for n > 1 (say a half-dozen or more points), the first datum can have any value, the second datum can have any value, and so on, up to n — 1. When we come to find the... 

This least-squares method is a standard technique. 

The parameter values found by the two methods differ slightly owing to the different criteria used which were the least squares method for ESL and the maximum-likelihood method for SIMUSOLV and because the T=10 data point was included with the ESL run. The output curve is very similar and the parameters agree within the expected standard deviation. The quality of parameter estimation can also be judged from a contour plot as given in Fig. 2.41. 

Firstly, it has been found that the estimation of all of the amplitudes of the LI spectrum cannot be made with a standard least-squares based fitting scheme for this ill-conditioned problem. One of the solutions to this problem is a numerical procedure called regularization [55]. In this method, the optimization criterion includes the misfit plus an extra term. Specifically in our implementation, the quantity to be minimized can be expressed as follows [53] ... 

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. 

Another approach is to prepare a stock solution of high concentration. Linearity is then demonstrated directly by dilution of the standard stock solution. This is more popular and the recommended approach. Linearity is best evaluated by visual inspection of a plot of the signals as a function of analyte concentration. Subsequently, the variable data are generally used to calculate a regression line by the least-squares method. At least five concentration levels should be used. Under normal circumstances, linearity is acceptable with a coefficient of determination (r2) of >0.997. The slope, residual sum of squares, and intercept should also be reported as required by ICH. 

If m and ri2 are unity, r2/cA is plotted versus ca/cr. Then ki is obtained from the intersection of the resulting straight line and the ordinate, whereas ki is its slope. Standard mathematical methods, such as linear- and multiple regression, or search techniques based on least-squares-methods to minimize the deviation of measured and calculated reaction rates, must be applied to determine the rate constants when m and m are different from unity. 

Fig. 2. Compensation plot for cracking and related (see text) reactions on nickel (Table I, A). The error in the individual points is indicated by the size of each cross the line was calculated by the least squares method (Appendix II),and standard deviations of slope (erj and intercept (nB) are indicated,...

By application of least-squares methods, most probable values and standard deviations for the parameters Em and a of this distribution have been calculated, and from the former a value for the temperature-independent rate constant 0 has been derived ... 

The crystal structure of phenanthrene has been solved by Basak (1950), and his (two-dimensional) data refined by Mason (1961) using least-squares methods. The accuracy of this analysis is not high because of the limited number of data used the standard deviation in bond length is 0-05 A. The results of a three-dimensional analysis at 95°K, which is reported to be in progress (Mason, 1961), are awaited with interest. 

The structure of the closely related molecule, 1,2-cyclopentenophen-anthrene, has been determined and refined with partial three-dimensional data by least-squares methods by Entwhistle and Iball (1961). Independent confirmation of the correctness of this structure has been provided by Basak and Basak (1959) who did not, however, carry out any refinement of the structure. Entwhistle and Iball s results show that the molecule is not planar the deviations of the carbon atoms from the mean molecular plane are shown in Fig. 9 (the standard deviations of the atomic coordinates lie between 0-009 and 0-015 A). The three aromatic rings appear to be linked in a slightly twisted arrangement. Atoms H and K, which are bonded to the overcrowded hydrogen atoms, are displaced almost the same distance on opposite sides of the mean plane. In the five-membered ring, atoms C and E are below the molecular plane by about 0-10 A while atom D lies 0-18 A... 

To solve this problem, some assumptions should be made on the relationship between the error of the regression line and the concentration. As a rule, one assumes that the error of the regression line is proportional to the concentration. The variance function Var(X) is obtained by plotting the standard error vs. the concentration. The function is consequently estimated with the least-squares method Var(X) = Sl = (c -T d cone)2. An alternative approach is described in the ISO 11483-2 standard, which uses an iterative procedure to estimate the variance function [18]. 

---