Squared terms

Vcaic,i is obtained by feeding the appropriate r,- value into the regression equation. Anothe common squared term is the residual sum of squares (RSS), which is the sum of square of the differences between the observed and calculated y values. TSS is equal to the sur of RSS and ESS. The is then given by ... 

There is an obvious similarity between equation 5.15 and the standard deviation introduced in Chapter 4, except that the sum of squares term for Sr is determined relative toy instead of y, and the denominator is - 2 instead of - 1 - 2 indicates that the linear regression analysis has only - 2 degrees of freedom since two parameters, the slope and the intercept, are used to calculate the values ofy . 

To calculate the standard deviation for the analyte s concentration, we must determine the values for y and E(x - x). The former is just the average signal for the standards used to construct the calibration curve. From the data in Table 5.1, we easily calculate that y is 30.385. Calculating E(x - x) looks formidable, but we can simplify the calculation by recognizing that this sum of squares term is simply the numerator in a standard deviation equation thus,... 

For a more complete discussion of critical phenomena, we consider the scattering intensity S(q) from concentration fluctuations (q is the wavenumber of the scattering) which can be derived when one supplements Eq. (1) by a gradient-square term " ... 

It is the gradient-squared term that caught Frieden s attention. Frieden noticed that it almost always appears in the Langrangians for various physical phenomena. The Langrangian for classical mechanics, for example, is dq/dt) — V. The La-grangian for the Schroedinger equation contains the term Vtp. The reader can perhaps recall at least a half-dozen other simple examples from basic physics (see Table 1.1 in [frieden98]). 

In recent years there has been much activity to devise methods for multivariate calibration that take non-linearities into account. Artificial neural networks (Chapter 44) are well suited for modelling non-linear behaviour and they have been applied with success in the field of multivariate calibration [47,48]. A drawback of neural net models is that interpretation and visualization of the model is difficult. Several non-linear variants of PCR and PLS regression have been proposed. Conceptually, the simplest approach towards introducing non-linearity in the regression model is to augment the set of predictor variables (jt, X2, ) with their respective squared terms (xf,. ..) and, optionally, their possible cross-product... 

Canonical analysis, or canonical reduction, is a technique used to reduce a second-order regression equation, such as Eq. (9), to an equation consisting of a constant and squared terms, as follows ... 

Screening designs are mainly used in the intial exploratory phase to identify the most important variables governing the system performance. Once all the important parameters have been identified and it is anticipated that the linear model in Eqn (2) is inadequate to model the experimental data, then second-order polynomials are commonly used to extend the linear model. These models take the form of Eqn (3), where (3j are the coefficients for the squared terms in the model and 3-way and higher-order interactions are excluded. 

They have the property of allowing both the potential and the kinetic energies to be written as sums of square terms that is. all cross terms vanish. This result can be demonstrated by substituting for [ and 2 in Eq- (95) into Eqs. (75) and (76). The results are... 

The tortuosity factor appears as a squared term because it decreases the concentration gradient and increases the diffusive path length. Using a cubic lattice model and inquiring how many steps a diffusing molecule needs to take to get around an obstacle, 0 was derived to be... 

Hinshelwood (51) used reasoning based on statistical mechanics to show that the energy probability factor in the kinetic theory expressions (e E,RT) is strictly applicable only to processes for which the energy may be represented in two square terms. Each translational and rotational degree of freedom of a molecule corresponds to one squared term, and each vibrational degree of freedom corresponds to two squared terms. If one takes into account the energy that may be stored in 5 squared terms, the correct probability factor is... 

N Pe Peclet number S number of squared terms contributing... 

The improvement in the fit from the quadratic polynomial applied to the nonlinear data indicated that the square term was indeed an important factor in fitting that data. In fact, including the quadratic term gives well-nigh a perfect fit to that data set, limited only by the computer truncation precision. The coefficient obtained for the quadratic term is comparable in magnitude to the one for linear term, as we might expect from the amount of curvature of the line we see in Anscombe s plot [7], The coefficient of the quadratic term for the normal data, on the other hand, is much smaller than for the linear term. 

Parameter Coefficient when using only linear term t-value when using only linear term Coefficient using square term t-value using square term... 

The performance statistics, the SEE and the correlation coefficient show that including the square term in the fitting function for Anscombe s nonlinear data set gives, as we noted above, essentially a perfect fit. It is clear that the values of the coefficients obtained are the ones he used to generate the data in the first place. The very large /-values of the coefficients are indicative of the fact that we are near to having only computer round-off error as operative in the difference between the data he provided and the values calculated from the polynomial that included the second-degree term. 

We now need to solve this expression for Z. We begin by expanding the square term ... 

The gradient-squared term in the above equation represents the energy of the interface separating the phases the constant C can be interpreted as a measure of the interaction range. The bulk potential /(< )) has the Landau-Ginzburg (GL) 4>4 (double-well) structure... 

The internal energy is, as indicated above, connected to the number of degrees of freedom of the molecule that is the number of squared terms in the Hamiltonian function or the number of independent coordinates needed to describe the motion of the system. Each degree of freedom contributes jRT to the molar internal energy in the classical limit, e.g. at sufficiently high temperatures. A monoatomic gas has three translational degrees of freedom and hence, as shown above, Um =3/2RT andCy m =3/2R. 

The reason why the rate equation includes the square term [MV+ ]2 rather than just [MV+ ] should not surprise us. Notice that we could have written the equation for the chemical reaction in a slightly different way, as... 

The maximum-squared term ensures that a positive penalty is incurred only when the gj < 0 constraint is violated. 

This is explicitly demonstrated in Appendix 3.A2. The mathematical process involved is appropriately called a diagonalization because both the kinetic and the potential energy terms in Q, only involve diagonal (squared) terms. Thus, in terms of the new coordinates which are usually called the normal coordinates... 

In Equation 3.A1.14b it has been indicated that the sum is independent of the order of summation. In Equation 3.A1.14c, the order of the terms has been interchanged for convenience and the transpose of the B matrix has been introduced, bit = by. In Equation 3.A1.14d, it has been recognized that the sum over i and j carried out in Equation 3.A1.14c produces the ks th element of the triple matrix product B,FmB. In Equation 3.A1.14e, we see that B in Equation 3.A1.14d is identical to B-1 and therefore the triple matrix product is just the diagonal matrix from the right hand side of Equation 3.A1.7. Thus, the matrix diagonalization has led to a coordinate transformation, which makes the potential a sum of square terms. 

We need to appreciate from the above calculation that a huge difference in activity is represented by a very small difference between E and in part because of the square term in the Nemst-equation denominator. This explains why it is essential to have a good quality voltmeter (i.e. one having a nearinfinite resistance R and with the ability to display the emf to several significant figures) and to take readings only when true equilibrium has been reached. 

A treatment similar to that above can be applied to other single equilibria. If the stoichiometry condition akin to (1.146), the zero net rate condition at final equilibrium as in (1.148), and the neglect of squared terms in the deviation concentrations are applied to the rate equation similar to (1.147), it is found that there is always a linear relation of the form... 

The close agreement between the experimental and calculated (Equation 9) ratios of 18 2/18 3 support exclusion of the 4-hydroxylphenyl analogue from the calculations. Examination of Equation 9 shows an interdependence between the biological activity and the hydrophobic properties of the chemical used, commonly found with many QSAR equations. This interdependent relationship is determined by the and terms, respectively. These terms control phenomena of hydrophobic interactions with receptors and phenomena of transport and distribution within the total biological systems. The occurrence of squared terms of the hydrophobic parameter in structure-activity correlations has been explained on the assumption that the compound has to penetrate several lipophilic-hydrophilic barriers or compartments on its way to the site of action (16, 17). This is consistent with the uptake of pyridazinones by roots and sbsequent translocation to the shoots (chloroplast) as the site of action (13). 

The steady-state expression for v for this scheme is rather complex, containing squared terms in [S] and in [A] in... 

Fromm and Rudolph have discussed the practical limitations on interpreting product inhibition experiments. The table below illustrates the distinctive kinetic patterns observed with bisubstrate enzymes in the absence or presence of abortive complex formation. It should also be noted that the random mechanisms in this table (and in similar tables in other texts) are usually for rapid equilibrium random mechanism schemes. Steady-state random mechanisms will contain squared terms in the product concentrations in the overall rate expression. The presence of these terms would predict nonhnearity in product inhibition studies. This nonlin-earity might not be obvious under standard initial rate protocols, but products that would be competitive in rapid equilibrium systems might appear to be noncompetitive in steady-state random schemes , depending on the relative magnitude of those squared terms. See Abortive Complex... 

It becomes painfully and quickly obvious that for an analysis requiring many data points, the number of correction terms becomes cumbersome. However, Brunauer was able to show that the squared terms do not contribute significantly and the above equation can be rewritten as... 

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