Square model

If the four subunits are arranged in a square so that subunit-subunit interactions exist only between nearest neighbors, then most of the equations of this section apply with some reinterpretation. Instead of Qapys, defined in (3.6.1) for the tetrahedral model, we now have 

For = 2, we have two possible configurations either L or //alternate or two L subunits follow two //subunits. The PF is almost the same as (3.6.16) except for reinterpretation of the A = 2 term. 

and h = 0.01. Both curves were calculated using Eq. (3.6.37). The tetrahedral and the square model differ in the explicit form of the Q(i), using either (3.6.1) or (3.6.59), respectively. One can loosely say that the tetrahedral model is more cooperative in the sense that the binding curve falls below the corresponding curve of the square model. 

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The validity of least squares model fitting is dependent on four prineipal assumptions eoneerning the random error term , whieh is inherent in the use of least squares. The assumptions as illustrated by Baeon and Downie [6] are as follows ... 

As noted earlier, the x -test for goodness-of-fit gives a more balanced view of the concept of fit than does the pure least-squares model however, there is no direct comparison between x and the reproducibility of an analytical method. 

Overdetermination of the system of equations is at the heart of regression analysis, that is one determines more than the absolute minimum of two coordinate pairs (xj/yi) and xzjyz) necessary to calculate a and b by classical algebra. The unknown coefficients are then estimated by invoking a further model. Just as with the univariate data treated in Chapter 1, the least-squares model is chosen, which yields an unbiased best-fit line subject to the restriction ... 

S. Wold, Non-linear partial least squares modelling. II. Spline inner relation. Chemom. Intell. Lab. Syst., 14(1992)71-84. 

Restori, R. and Schwarzenbach, D. (1995) Maximum-entropy versus least-squares modelling of the electron-density in K2PtCl6, Acta Cryst., B51, 261-263. 

We originally proposed NNM to be present in metallic beryllium [30] based on analysis of the X-ray diffraction data measured by Larsen and Hansen [24], Based on Fourier maps and elaborate multipole least-squares modeling, indisputable evidence... 

The quantification should start with a rough estimation of the order of magnitude of each uncertainty contribution pi and pij. Insignificant one can be neglected because the uncertainty components are added according to a squared model. The significant values should be refined in subsequent stages and converted to parameters u(pf) which correspond to standard deviations. 

In the square model, the sites are identical only in a weak sense. This means that there is only one (first) intrinsic binding constant, but we have two different pair correlations, which are denoted by g and for the nearest and next-nearest... 

Compare with Eq. (6.4.2). The difference arises because here each subunit is in contact with three subunits, while in the square model it is in contact with two subunits. Compare also with Eq. (6.3.2). The direct correlations in this system are simple S, 5, and 5 . 

Judging from the shapes of the Bis, we should conclude that the square model is more cooperative than the tetrahedral model. Indeed, all the cooperativities in this system are larger for the square model. We have computed all the correlations for these two models, using the parameters in (6.6.5). These yield for the square model... 

The average correlation, plotted as f(C)- 1, shows that the square model starts initially with a small positive value and increases monotonously to the very large value of 37,348 at C -> < . On the other hand, the g(C) - 1 curve for the tetrahedral model starts from a very small value and reaches the value of about 21,058 at very high concentrations. Clearly, both of the Bis appear as positive cooperative, but with much stronger cooperativity for the square model, in apparent defiance of the density of interaction argument. 

Note that in the square model only nearest-neighbor pairs are assigned an energy parameter a, while in the tetrahedral model each pair of ligands contributes a factor a. Both of these functions gave a good fit to experimental data with k = 0.033, and a = 12 for the square model and a = 12 = 5.2 for the tetrahedral model. 

Referring to -RT In a as interaction energy, it was only natural to assume that each connected pair would contribute a factor a. This is equivalent to the assumption of pairwise additivity of both the triplet and quadruplet correlation. In our language, these assumptions are equivalent for the square model to... 

M. Sjostrom, S. Wold, W. Lindberg, J.A. Persson and H. Martens, A multivariate calibration problem in analytical chemistry solved by partial least squares models in latent variables. Anal. Chim. Acta, 150, 61-70 (1983). 

When the model used for Fcalc is that obtained by least-squares refinement of the observed structure factors, and the phases of Fca,c are assigned to the observations, the map obtained with Eq. (5.9) is referred to as a residual density map. The residual density is a much-used tool in structure analysis. Its features are a measure for the shortcomings of the least-squares minimization, and the functions which constitute the least-squares model for the scattering density. 

Factor The result of a transformation of a data matrix where the goal is to reduce the dimensionality of the data set. Estimating factors is necessary to construct principal component regression and partial least-squares models, as discussed in Section 5.3.2. (See also Principal Component.)... 

D. C. Baxter and J. Ohman, Multi-component standard additions and partial least squares modelling, a multivariate calibration approach to the resolution of spectral interferences in graphite furnace atomic absorption spectrometry, Spectrochim. Acta, Part B, 45(4 5), 1990, 481 491. 

D. C. Baxter, W. Freeh and 1. Berglund, Use of partial least squares modelling to compesate for spectral interferences in electrothermal atomic... 

Sjoestroem, M., Wold, S., Lindberg, W., Persson, J.A. and Martens, H., A Multivariate Calibration Problem in Analytical Chemistry Solved by Partial Least Squares Models in Latent Variables Anal. Chim. Acta 1983, 150, 61-70. 

Apart from optimization, a problem is often set for mathematical modeling or interpolation. The optimum does not interest us in that case but the model that adequately describes the obtained results in the experimental field. A subdomain is not chosen in that case, but the polynomial order is moved up until an adequate model is obtained. When a linear or incomplete square model (with no members with a square factors) is adequate it means that the research objective corresponds to the optimization objective. 

Factorial methods - factor analysis (FA) - principal components analysis ( PCA) - partial least squares modeling (PLS) - canonical correlation analysis Finding factors (causal complexes)... 

Unfortunately, the square model does not hold for a [PtCl2(5a) complex, where significant distortion of the oxaphos-phinane ringes were observed. Ref. [32]. 

Usually, linear models are preferable (linear ordinary, i.e., unweighted, least squares regression model is not appropriate in many cases, in which weighted least squares model should be applied), but, if necessary, nonlinear (e.g., second order) models can be used [60],... 

A more relevant conclusion arising from the computational work provides the relative stability of different dimers. There is a clear favouring of the (ZnO)2 square structure over possible O - Zn - N macrocycles and more markedly over the OZnO, NZnN isomeric macrocycle. Structures and relative enthalpies are shown in Fig. 19. Within the square model, homo- and heterochiral dimers are closely similar in energy and show no discernable structural differences that could indicate why one is catalytically active and the other not. 

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