Non-linear least square procedures

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. 

A quarter of a century ago Behnken [224] as well as Tidwell and Mortimer [225] pointed out that the linearization transforms the error structure in the observed copolymer composition with the result that such errors after transformation have no longer zero mean and constant variances. It means that such transformed variables do not meet the requirements for the least-squares procedure. The only statistically accurate means of estimation of the reactivity ratios from the experimental data is based on the non-linear least-squares procedure. An effective computing program for this purpose has been published by Tidwell and Mortimer (TM) [225]. Their method is considered to be such a modification of the curve-fitting procedure where the sum of the squares of the difference between the observed and computed polymer compositions is minimized. 

Numerous reports are available [19,229-248] on the development and analysis of the different procedures of estimating the reactivity ratio from the experimental data obtained over a wide range of conversions. These procedures employ different modifications of the integrated form of the copolymerization equation. For example, intersection [19,229,231,235], (KT) [236,240], (YBR) [235], and other [242] linear least-squares procedures have been developed for the treatment of initial polymer composition data. Naturally, the application of the non-linear procedures allows one to obtain more accurate estimates of the reactivity ratios. However, majority of the calculation procedures suffers from the fact that the measurement errors of the independent variable (the monomer feed composition) are not considered. This simplification can lead in certain cases to significant errors in the estimated kinetic parameters [239]. Special methods [238, 239, 241, 247] were developed to avoid these difficulties. One of them called error-in-variables method (EVM) [239, 241, 247] seems to be the best. EVM implies a statistical approach to the general problem of estimating parameters in mathematical models when the errors in all measured variables are taken into account. Though this method requires more information than do ordinary non-linear least-squares procedures, it provides more reliable estimates of rt and r2 as well as their confidence limits. 

Avdeef has recently reported the refinement of partition coefficients and ionization constants of multi-protic substances based on a generalized, weighted, non-linear least-squares procedure and pH titration curve. This method allows for the determination of pKa and logP values of multiprotic substances with fairly close ionization constants. 

Ten ml of 1 mM or 5 mM aqueous solutions of the samples were pre-acidified to pH 1.8-2.0 with 0.5 M HCl, and were then titrated alkalimetrically to some appropriate high pH (maximum 12.5). The titrations were carried out at 25.0 0.1 °C, at I = 0.1 M ionic strength using NaCl, and under N2 atmosphere. The initial estimates of pJC values were obtained by difference plots (nH vs. pH, where nn is the average number of bound protons) and were then refined by a weighted non linear least-squares procedure (Avdeef, 1992,1993). For each molecule a minimum of three and occasionally five or more separate titrations were performed and the average pK values along with the standard deviations were calculated."... 

Figure D.2 provides a contrasting picture of the variation of ysv and ysL with Y2. The Langmuir isotherm fits the j sv data very well throughout the range of concentrations studied. Moreover, ygv [23,55.4) mN/m is substantially larger than the range ysL e [0,6.12)mN/m. The fit to the S/V data is excellent, but the fit to the S/L data is modest up to Y2 = 0.4 the data between (0.4, 1] did not follow the fit. The non-linear least squares procedure could not be made to converge when the ysr data for Y2 > 0.4 was included in the array. Apparently, the structure of the solution/SS304 interface does not satisfy the conditions required for the Langmuir isotherm to be valid.

Again, an effluent profile may be fitted to this analytic form by a non-linear least-squares procedure on these four quantities (and the pulse size). The actual form of this analytic result is rather lengthy and is given elsewhere (De Smedt and Wierenga, 1979 Van Geneuchten, 1981). 

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