Linear least-square systems

We assume that a random variable vector Y of (here upper-case is used to indicate not a matrix but an ordered set of m random variables) distributed as a multivariate normal distribution has been measured through an adequate analytical protocol (e.g., CaO concentration, the 87Sr/86Sr ratio.). The outcome of this measurement is the data vector jm. Here ym is the mean of a large number of measurements with expected 

We see that for Sy=Im, the cumbersome equations (5.4.23) and (5.4.24) reduce to the standard least-square solutions. The covariance matrix S on x can be obtained through equation (4.3.4) 

The nxm covariance matrix cov(i, j ) between the model x and the observations y can be obtained easily. By linearity 

In order to get the correlation coefficients, this matrix must be pre-multiplied by the inverse of the diagonal matrix having the standard deviations of x on the diagonal line, and post-multiplied by the inverse of the diagonal matrix having the standard deviations of y on the diagonal line. 

We will now investigate the sampling properties of the statistic representing the weighted sum of squared residuals i1 given by equation (5.4.13). We first observe that the slightly different expression (y — l)rSy i(y —is zero since 

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In engineering we often encounter conditionally linear systems. These were defined in Chapter 2 and it was indicated that special algorithms can be used which exploit their conditional linearity (see Bates and Watts, 1988). In general, we need to provide initial guesses only for the nonlinear parameters since the conditionally linear parameters can be obtained through linear least squares estimation. 

Crisponi, G., Nurchi, V., and Ganadu, M.L. (1990), An Approach to Obtaining an Optimal Design in the Non-Linear Least Squares Determination of Binding Parameters in a Complex Biochemical System, J. Chemom., 4, 123-133. 

The process of research in chemical systems is one of developing and testing different models for process behavior. Whether empirical or mechanistic models are involved, the discipline of statistics provides data-based tools for discrimination between competing possible models, parameter estimation, and model verification for use in this enterprise. In the case where empirical models are used, techniques associated with linear regression (linear least squares) are used, whereas in mechanistic modeling contexts nonlinear regression (nonlinear least squares) techniques most often are needed. In either case, the statistical tools are applied most fruitfully in iterative strategies. 

The rapid development of computer technology has yielded powerful tools that make it possible for modem EIS analysis software not only to optimize an equivalent circuit, but also to produce much more reliable system parameters. For most EIS data analysis software, a non-linear least squares fitting method, developed by Marquardt and Levenberg, is commonly used. The NLLS Levenberg-Marquardt algorithm has become the basic engine of several data analysis programs. 

The thorough treatment of the experimental data does allow one to obtain reliable values of the reactivity ratios. The results of such a treatment are presented in Table 6.3 for some concrete system let us form a notion about an accuracy of the reactivity ratios estimations. The detailed analysis of such a significant problem in the case of the well-studied copolymerization of styrene with methyl methacrylate is reported in Ref. [227]. Important results on the comparison of the precision of rj, r2 estimates by means of different methods are presented by O Driscoll et al. [228]. Such a comparison of six well-known linear least-squares procedures [215-218,222,223] with the statistically correct non-linear least-squares method leads to the conclusion that some of them [216, 217, 222] can provide rather precise rls r2 estimates when the experiment is properly planned. 

The values of k reported in Tables VII and VIII for the systems studied were determined by applying a linear least-squares calculation to the experimental data. The average absolute deviation (A.A.D.) of logio(tts/ ) was determined at fixed intervals of z from the individual... 

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