Simple Linearization Linear Normalization Illustration

Under this method, the scaled values are computed using Equations 6.6 and 6.7 and are given in Table 6.21. 

Scaled Criteria Values by Simple Linearization (Example 6.5) 

Note that the best and worst values of each criterion are 1 and 0, respectively and all the criteria values are now to be maximized. The revised weighted sums are as follows  

The rankings are unchanged with Supplier A as the best, followed by 

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Figure 6.5 Simulated time-effect data where the intercept was normally distributed with a mean of 100 and a standard deviation of 60. The effect was linear over time within an individual with a slope of 1.0. No random error was added to the model—the only predictor in the model is time and it is an exact predictor. The top plot shows the data pooled across 50 subjects. Solid line is the predicted values under the simple linear model pooled across observations. Dashed line is the 95% confidence interval. The coefficient of determination for this model was 0.02. The 95% confidence interval for the slope was —1.0, 3.0 with a point estimate of 1.00. The bottom plot shows how the data in the top plot extended to an individual. The bottom plot shows perfect correspondence between effect and time within an individual The mixed effects coefficient of determination for this data set was 1.0, as it should be. This example was meant to illustrate how the coefficient of determination using the usual linear regression formula is invalid in the mixed effects model case because it fails to account for between-subject variability and use of such a measure results in a significant underestimation of the predictive power of a covariate.

In Section 4.3.f it was shown that there are 3N — 5 normal vibrations in a linear molecule and 3N — 6 in a non-linear molecule, where N is the number of atoms in the molecule. There is a set of fairly simple rules for determining the number of vibrations belonging to each of the symmetry species of the point group to which the molecule belongs. These rules involve the concept of sets of equivalent nuclei. Nuclei form a set if they can be transformed into one another by any of the symmetry operations of the point group. For example, in the C2 point group there can be, as illustrated in Figure 6.18, four kinds of set ... 

From the viewpoint of developing quantitative correlations it is desirable to seek a linear relationship between descriptor and property, but a nonlinear or curvilinear relationship is adequate for illustrating relationships and interpolating purposes. In this handbook we have elected to use the simple descriptor of molar volume at the normal boiling point as estimated by the Le Bas method (Reid et al. 1987). This parameter is very easily calculated and proves to be adequate for the present purposes of plotting property versus relationship without seeking linearity. 

Whenever the solute and solvent exhibit significant degrees of mutual attraction, deviations from the simple relationships will be observed. The properties of these nonideal solutions must be determined by the balance of attractive and disruptive forces. When a definite attraction can exist between the solute and solvent, the vapor pressure of each component is normally decreased. The overall vapor pressure of the system will then exhibit significant deviations from linearity in its concentration dependence, as is illustrated in Fig. 10B. 

Consider a simplified illustration of the foregoing QSAR examples. Consider a list of normal alkanes together with their water solubility and boiling points. A plot shows that solubility (in logarithmic form) is linear with number of carbon atoms and that boiling point is nonlinear. Such a relation is a QSAR based on the simple structure feature, number of carbon atoms. A linear equation captures all the structure information available in this data set. (The structure information could, of course, be represented in other ways, such as number of methylene groups, number of hydrogen atoms, number of carbon-carbon bonds, etc.) It is important to note here that no assumption has been made about the relation between water solubility and number of carbon atoms. This is an example of what Adamson has called a mechanism-free model. 

Furthermore, the square of a variable is an F so that we can represent one-degree-of-freedom f-tests involving the linear and quadratic contrasts and the two-degree-of-freedom F-tests in a simple diagram. This has been done in Figure 25.3, where, however the asymptotic case has been considered. This means that instead of f-statistics we can have standard Normal statistics and that we can replace the F-statistic by twice its value which, asymptotically, has a chi-square distribution with two degrees of freedom. The tests illustrated are at the 5% level. 

The three fundamental modes of vibration are shown in Figure 4.12. In the case of a triatomic molecule, it is simple to deduce that the three modes of vibration are composed of two stretching modes (symmetric and asymmetric) and a bending mode. However, for larger molecules it is not so easy to visualize the modes of vibration. We return to this problem in the next section. The three normal modes of vibration of SO2 all give rise to a change in molecular dipole moment and are therefore IR active. A comparison of these results for CO2 and SO2 illustrates that vibrational spectroscopy can be used to determine whether an X3 or XY2 species is linear or bent. 

To illustrate the essential features of NNMI, we first point out that the characteristics of global atmospheric motions can be examined from the solutions of a linearized system of the basic hydrostatic prediction equations discussed in Section III.B. A simple perturbation system is the one linearized around the atmosphere at rest with a basic temperature that depends only on height. Solutions of such a linear system with appropriate boundary conditions are called normal modes. 

Once the specimen turns to a superconducting state, the obtained superconductor-insulator-normal metal (SIN) spectrum probes the quasiparticle excitation in the superconductor, which directly reflects the symmetry of the order parameter A(k). If A(k) has simple s-wave symmetry, as is realized in conventional low-temperature superconductors, one expects a finite gap of A with overshooting peaks just outside the gap in N(E), as illustrated in fig. 6. Even if A(k) possesses anisotropic s-wave symmetry, a finite gap, corresponding to the minimum gap, appears. In dx2-yi superconductors with A(k) = coslkx - cos 2, in contrast, N(E) is gapless with linear N(E) for E A. It is noted that the extended-s wave A(x) = cos 2kx + cos 2ky is also characterized to possess a gapless feature with two singularities bX E = A and A2. 

The very nearly linear behavior of isochores or isometrics, which are lines of constant molal volume or density upon P vs. T coordinates, provides motivation for seeking a simple, albeit approximate equation of state. Figure 1 illustrates isochores for normal hydrogen. Those identified by V are from Ref. 2 by p are from Ref. 1. Experimental data usually are in the form of isotherms, from which it is not possible to make an isochore plot directly. Tedious graphical or mathematical interpolation procedures are required. The compendium of Woolley, Scott, and Brickwedde [1] and the linear isochores of Stewart and Johnson [2] provide the necessary interpolations. Such data, therefore, are not directly experimental. A linear isochore may be expressed in terms of its intercept and slope by (1). This may be compared with Van der Waals (2), wherein v is molal volume,... 

Description of System. With the elementary methods described in this chapter, even as simple a molecule as water is rather too cumbersome to Ik, used as an illustration. In later chapters much more powerful inclhods (which are, however, based on those in this chapter) will be developed. Until then, an artificial example may prove helpful in illustrating the idea of normal vibrations and normal coordinates. Such. an example is provided by a linear system of two point masses and three weightless springs as shown in Fig. 2-4. The springs 1 and 2 are fastened lo fixed points so that no question of rotation or translation enters. I urthermore, only linear motions will be considered. Therefore, only two... 

As a simple illustrative example, let us consider a linear molecule of XMX type. If we disregard the bending motion, there can be two normal vibrations, one being even and the other odd. The normal coordinates are... 

This section considers the behavior of polymeric liquids in steady, simple shear flows - the shear-rate dependence of viscosity and the development of differences in normal stress. Also considered in this section is an elastic-recoil phenomenon, called die swell, that is important in melt processing. These properties belong to the realm of nonlinear viscoelastic behavior. In contrast to linear viscoelasticity, neither strain nor strain rate is always small, Boltzmann superposition no longer applies, and, as illustrated in Fig. 3.16, the chains are displaced significantly from their equilibrium conformations. The large-scale organization of the chains (i.e. the physical structure of the liquid, so to speak) is altered by the flow. The effects of finite strain appear, much as they do when a polymer network is deformed appreciably. 

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