Extended least squares method

The spline surface S(x,y) consists of a set of bicubic polynomials, one in each panel, joined together with continuity up to the second derivative across the panel boundaries. Because each B-spline only extends over four adjacent knot intervals, the functions B.(x)C.(y) are each non-zero only over a rectangle composed of 16 adjacent panels in a 4 x 4 arrangement. The amount of calculation required to evaluate the coefficients y may be reduced by making use of this property. As before, least-squares methods may be used if the number of data exceeds (h+4)(jJ+4), which is usually the case. 

Two approaches have been applied to estimate the net charges of atoms from the observed electron-density distribution in crystals. The first method is a direct integration of observed density in an appropriate region around an atom (hereafter abbreviated as DI method) (64). The second is the so-called extended L-shell method (ELS method) (19, 81) in which a valence electron population of an atom is calculated by a least-squares method on the observed and calculated structure amplitudes. 

Another approach to weighting the data uses the data itself to develop the variance equation. This is the extended least squares (ELS) method. Parameters for the variance equations are included in the fitting... 

The above-mentioned nonlinear least square method for the case with two parameters (a,j8) is the basic one and easily extended to the cases with more parameters. Considering the possibility of obtaining the reliable Sh of this NMR titration experiment, data treatment should be carried out with three parameters for the better regression. The programs of spreadsheet software for this three-parameters-method are developed and shown in Appendix 2.4 [24]. 

In this chapter we will describe some of the more sophisticated uses of least squares, especially those for fitting experimental data to specific mathematical functions. First we will describe fitting data to a function of two or more independent parameters, or to a higher-order polynomial such as a quadratic. In section 3.3 we will see how to simplify least-squares analysis when the data are equidistant in the dependent variable (e.g., with data taken at fixed time intervals, or at equal wavelength increments), and how to exploit this for smoothing or differentiation of noisy data sets. In sections 3.4 and 3.5 we will use simple transformations to extend the reach of least-squares analysis to many functions other than polynomials. Finally, in section 3.6, we will encounter so-called non-linear least-squares methods, which can fit data to any computable function. 

Parameters for the simple model were determined graphically by Eadie-Hofstee plotting of initial reaction rates and substrate concentrations. Details are given elsewhere (30). As has been observed in hydrolysis of other solid substrates, a residue of non-lysed substrate was found at extended reaction times, when dY/dt tended toward zero. The extent of reaction was strongly dependent on initial substrate and enzyme concentrations (33,34). An empirical funciton for Y was fitted to the ultimate turbidity data for lysis runs at a variety of initial yeast and enzyme concentrations using a least squares method. The calculated values for Yco were used in the simulations (30). Figure 3 shows results of the simple model. 

These completely empirical methods assume that atoms, bonds or groups constituting the molecule carry their own contribution to the property into consideration. Such a contribution is calculated by a least square method over a number of compounds containing the envisaged groups more than once. This procedure has been applied to the AG functions by Hine and Mookerj ie(96), to the properties by Guthrie(99), recently Cabani et al.(lOO) have extended this method to AH, AC° Yi quantities and have repeated the cal-... 

Extended Kalman filtering has been a popular method used in the literature to solve the dynamic data reconciliation problem (Muske and Edgar, 1998). As an alternative, the nonlinear dynamic data reconciliation problem with a weighted least squares objective function can be expressed as a moving horizon problem (Liebman et al., 1992), similar to that used for model predictive control discussed earlier. 

The nonlinear iterative partial least-squares (NIPALS) algorithm, also called power method, has been popular especially in the early time of PCA applications in chemistry an extended version is used in PLS regression. The algorithm is efficient if only a few PCA components are required because the components are calculated step-by-step. 

Hence, we adopted their analytical way and extended their method to analyze the v p bands. It is necessary to resolve the overlap of those transitions by least-squares fitting in order to obtain the widths of the component bands. For the least-squares procedure, we have to determine analytical functions for fundamental transitions, v, a, v, p, and for hot band transitions the v,p band is reproduced as a sum of three Lorentzian curves of the v, p, v,Mp, and v,h2p. We also took into account the presence of the v2hla and v2h a band for the v,a band. The observed spectra in the 2320-2220 cm range were deconvoluted using... 

Similarly, Lippmaa and coworkers evaluated the relative acidities of linear and branched carboxylic acids from the variation with degree of protonation of the measured 13C NMR shifts.23 The method was then extended to secondary deuterium IEs, evaluated from the variation with degree of protonation of the measured 13C NMR shifts of a mixture of isotopologues.24 The data were fit by nonlinear least squares to Equation (17), where 6H and 6D are the observed chemical shifts of undeuterated and deuterated isotopologues, 6H and <5d are those chemical shifts in the deprotonated form, < >]) and are those chemical shifts in the protonated form, R = K /K, and n is the degree of protonation of the undeuterated material. This is the same equation as Equation (15), but adapted to deuteration, and again n is evaluated from chemical shifts as (hH — <5h)/(<5h - h)-... 

The methods of Chapter 6 are not appropriate for multiresponse investigations unless the responses have known relative precisions and independent, unbiased normal distributions of error. These restrictions come from the error model in Eq. (6.1-2). Single-response models were treated under these assumptions by Gauss (1809, 1823) and less completely by Legendre (1805), co-discoverer of the method of least squares. Aitken (1935) generalized weighted least squares to multiple responses with a specified error covariance matrix his method was extended to nonlinear parameter estimation by Bard and Lapidus (1968) and Bard (1974). However, least squares is not suitable for multiresponse problems unless information is given about the error covariance matrix we may consider such applications at another time. 

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