Least-squares approach

Let us consider that the niunber of echoes M and the incident wavelet (/) (e.g., a normalized comer echo) are known. Least Squares approach for estimating parameter vectors x and requires the solution to the nonlinear least squares problem ... 

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... 

The response produced by Eq. (8-26), c t), can be found by inverting the transfer function, and it is also shown in Fig. 8-21 for a set of model parameters, K, T, and 0, fitted to the data. These parameters are calculated using optimization to minimize the squarea difference between the model predictions and the data, i.e., a least squares approach. Let each measured data point be represented by Cj (measured response), tj (time of measured response),j = 1 to n. Then the least squares problem can be formulated as ... 

The aim of this work is the determination of several nutritional parameters, such as Energetic Value, Protein, Fat, and Carbohydrates content, in commercially available yoghurt samples by using Attenuated Total Reflectance Fourier Transform Infrared (ATR-FT-IR) spectrometry and a partial least square approach. 

If the UCKRON expression is simplified to the form recommended for reactions controlled by adsorption of reactant, and if the original true coefficients are used, it results in about a 40% error. If the coefficients are selected by a least squares approach the approximation improves significantly, and the numerical values lose their theoretical significance. In conclusion, formalities of classical kinetics are useful to retain the basic character of kinetics, but the best fitting coefficients have no theoretical significance. 

B. Waiczack and D.L. Massart, The radial basis functions — partial least squares approach as a flexible non-linear regression technique. Anal. Chim. Acta, 331 (1996) 177-185. 

We will begin our discussion by demonstrating that, for a non-overdetermined system of equations, the algebraic approach and the least-square approach provide the same solution. We will then extend the discussion to the case of an overdetermined system of equations. 

Wold, S Ruhe, A., Wold, H Dunn, W. J. I. SIAM J. Sci. Stat. Comput. 5, 1984, 735-743. The collinearity problem in linear regression. The partial least squares approach to generalized inverses. 

The least-squares approach can become very unreliable if outliers are present in the data (see Section 4.4). In this case, it is more advisable to minimize another function of the errors which results in more robust regression estimates. Although with the OLS approach the Equations 4.22 and 4.23 can always be applied, it is advisable to use the following assumptions for obtaining reliable estimates ... 

Wold, H. Soft Modeling by Latent Variables the Nonlinear Iterative Partial Least Squares Approach," Ed. J. Gani, in Perspective in Probability and Statistics - Papers in Honor of M. S. Bartlett, Academic Press, London 1975, pp. 117-142... 

Let us fit the probabilistic model, = Po + r, to the same data (see Figure 5.10). If the least squares approach to the fitting of this model is employed, the appropriate matrices and results are exactly those given in Section 5.2 where the same model was fit to the different factor levels j , = 3, yj, = 3, = 6, y, = 5. This identical... 

In case, the network is overdetermined a least square approach is possible. The information obtained by MS from tracer studies can be combined with metabolite balancing for the estimation of flux parameters [25, 27]. Measurement errors can be included to predict uncertainties for the obtained flux parameters [23]. 

Adopting Eu=ql and Ey=0, then Equation l6 reduces to Equation 5 With Eu=ql and Ey=rl, Equation l6 has a format which is identical to the solution derived in (2T) through a deterministic minimum least squares approach for time-invariant systems. This is to be expected, because the Wiener filtering technique may be in fact Included as part of the general theory of least squares. 

The procedure described here clearly gives somewhat better results at each assumed magnitude of errors than the least squares approach. 

H. Wold, Soft modelling by latent variables the non-linear iterative partial least squares approach, in J. Gani (ed.), Papers in Honour of M. S. Bartlett Perspectives in Probability and Statistics, Academic Press, London, 1975. 

A useful empirical approach to the design of heterogeneous chemical reactors often consists of selecting a suitable equation, such as one in Table 3.3 which, with numerical values substituted for the kinetic and equilibrium constants, represents the chemical reaction in the absence of mass transfer effects. Graphical methods are often employed to aid the selection of an appropriate equation140 and the constants determined by a least squares approach<40). It is important to stress, however, that while the equation selected may well represent the experimental data, it does not... 

Figure 5.1 shows the unconstrained straight line model. Vn — A)+ Pixi, passing exactly through the two experimental points. The matrix least squares approach can... 

It was noted above that reliable pitch estimation is necessary for two-voice separation using the solution of Equation (9.75). Under a harmonic assumption, since the function in Equation (9.73) is nonlinear in the fundamental frequencies, a simple closed-form solution for pitch based on a least squares approach does not exist. Under certain... 

The phase problem and the problem of arbitration. Fibrous structures are usually made up of linear polymers with helical conformations. Direct or experimental solution of the X-ray phase problem is not usually possible. However, the extensive symmetry of helical molecules means that the molecular asymmetric unit is commonly a relatively small chemical unit such as one nucleotide. It is therefore not difficult to fabricate a preliminary model (which incidently provides an approximate solution to the phase problem) and then to refine this model to provide a "best" solution. This process, however, provides no assurance that the solution is unique. Other stereochemically plausible models may have to be considered. Fortunately, the linked-atom least-squares approach provides a very good framework for objective arbitration independent refinements of competing models can provide the best models of each kind the final values of n or its components (eqn. xxiv) provide measures of the acceptability of various models these measures of relative acceptability can be compared using standard statistical tests (4) and the decision made whether or not a particular model is significantly superior to any other. 

Numerical simulations of the data were conducted with the algorithms discussed above, with the added twist of optimizing the model to fit the data collected in the laboratory by adjusting the collision efficiency and the fractal dimension (no independent estimate of fractal dimension was made). Thus, a numerical solution was produced, then compared with the experimental data via a least squares approach. The best fit was achieved by minimizing the least squared difference between model solution and experimental data, and estimating the collision efficiency and fractal dimension in the process. The best model fit achieved for the data in Fig. 10a is plotted in Fig. 10b, and that for Fig. 11a is shown in Fig. lib. The collision efficiencies estimated were 1 x 10-4 and 2 x 10-4, and the fractal dimensions were 1.5 and 1.4, respectively. As expected, collision efficiency and fractal dimension were inversely correlated. However, the values of the estimates are, in both cases, lower than might be expected. The lower values were attributed to the following ... 

The matrix of the regression parameters is estimated by using the least squares approach A = (X7 X) (XrY), where the index T means the transposed... 

Y,(calc) is the value obtained by using the correlation equation with the set of coefficients (to be determined) that will minimize s this is the least-squares approach. Rather than try all possible coefficient sets, expressions in terms of the Y and X values for the coefficients can be obtained. Differentiation with respect to each of the a in turn, leads to m + simultaneous equations from which the m coefficient values and intercept are obtained. The resulting equations become more numerous as the number of parameters increase consequently, calculations must be done by computer. Chemistry is greatly aided through computing. 

In addition to the studies of various crystalline forms of domestic hen egg-white lysozyme, the structures of human and tortoise egg-white lysozymes have been determined (for crystal data see Table III). Artymiuk and Blake (1981) refined the structure of the human enzyme to 1.5 A resolution. The main objectives of this study were to determine the extent of differences in structure from that of the hen egg-white protein, to discover the location of water molecules, and to test the validity of the method of restrained refinement. The particular restrained least-squares approach to refinement described in their paper appears to have been validated. The two proteins were found to be closely homologous, but there were small differences (e.g., in a helices), details of which can be obtained from consulting their paper. 

The coefficients A, B, and C are fitted using a least-squares approach. Results from this approach are given in Table 6. Each data point for the occluded moisture at a given time is a mean value of at least six assays from samples sterilized and dried in different locations within the autoclave. The model predicts mean moisture values within the 95% confidence interval of the measured mean moisture values. Using the data in Table 6 we selected 12 h as the time required to adequately dry the stoppers, since 16 h did not offer a significant decrease in moisture over the 12-h results. 

D. Garvin, V. B. Parker, D. D. Wagman, and W. H. Evans, A Combined Least Sums and Least Squares Approach to the Solution of Thermodynamic Data Networks, The Proceedings of the Fifth Biennial International CODATA Conference (Pergamon, New York, 1977). 

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