X-variable

Quadralically Convergent or Second-Order SCF. As mentioned in Section 3.6, the variational procedure can be formulated in terms of an exponential transformation of the MOs, with the (independent) variational parameters contained in an X matrix. Note that the X variables are preferred over the MO coefficients in eq. (3.48) for optimization, since the latter are not independent (the MOs must be orthonormal). The exponential may be written as a series expansion, and the energy expanded in terms of the X variables describing the occupied-virtual mixing of the orbitals. 

The first and second derivatives of the energy with respect to the X variables ( 0) and "(O)) can be written in term of Fock matrix elements and two-electron integrals in the MO basis. For an RHF type wave function these are given as... 

We will consider two other variables obtained by a non-linear transformation y = e and z = e. The minimum energy is at x = 0, corresponding to y = z = 1. Consider now an NR optimization starting at x = -0.5, corresponding to y = 1.6487 and z = 0.6065. Table 14.1 shows that the NR procedure in the x-variable requires four iterations before X is less than 10"". In the y-variable the optimization only requires one step to reach the y = 1 minimum exactly The optimization in the z-variable takes six iterations before the value is within 10"" of the minimum. 

Consider now the same system starting from. x = l(y = 0.3679 and z = 2.7183). The x-variable optimization now diverges toward the x = cxd limit, the y-variable optimization again converges (exactly) in one step, and the z-variable optimization also diverges toward the z = co limit. The reason for this behaviour is seen when plotting the three functional forms as shown in Figures 14.2-14.4. The horizontal axis... 

Example 58 Fig. 4.33 explains the theoretical basis and Fig. 4.34 depicts increasingly sophisticated ways of reducing the risk of having a single measurement beyond the specification limit. Between two and 10 measurements are performed and the mean is calculated in order to decide whether a product that is subject to the specifieation limits ( 95. .. 105% of nominal) can or cannot be released. A normal distribution ND(/x = variable, = 1.00) is assumed the case x ,ean = 105 is explained the same arguments apply for Xmean = 95, of course. Tables 4.30-4.32 give the key figures. 

If we are given n-things where we choose x variables, taken r at a time, the individual probability of choosing (1 + x) will be ... 

This is already a considerable improvement. The natural question then is Which linear combination of K-variables yields the highest R when regressed on the X-variables in a multiple regression Canonical correlation analysis answers this question. 

The particular linear combinations of the X- euid F-variables achieving the maximum correlation are the so-called first canonical variables, say tj = Xw, and u.-Yq,. The vectors of coefficients Wj and q, in these linear combinations are the canonical weights for the X-variables and T-variables, respectively. For the data of Table 35.5 they are found to be Wj = [0.583, -0.561] and qj = [0.737,0.731]. The correlation between these first canonical variables is called the first canonical correlation, p,. This maximum correlation turns out to be quite high p, = 0.95 R = 0.90), indicating a strong relation between the first canonical dimensions of X and Y. 

Equation (35.15) represents the projection of each K-variable onto the space spanned by the X-variables, i.e. each 7-variable is replaced by its fit from multiple regression on X. 

Here, the loading vector p, contains the coefficients of the separate univariate regressions of the individual X-variables on t,. The / element of p, p j, represents the regression coefficient of it regressed on t, py = tj j The full vector of... 

Let us now consider a new set of values measured for the various X-variables, collected in a supplementary row vector x. From this we want to derive a row vector y of expected T-values using the predictive PLS model. To do this, the same sequence of operations is followed transforming x into a set of factor scores r, t 2, t A pertaining to this new observation. From these t -scores y can be... 

Fig. 35.11. Loadings of the two dominant PLS factors on the X-variables (lowercase) and F-variables (upper case).

The PLOT statement usually appears in the DYNAMIC part of the model and controls the output to the screen during a run after a START command. Only two variables may be specified. The format of this command is PLOT X variable, y variable, xmin, xmax, ymin, ymax... 

Figure 10-4. The double- and single-site titration models for His and Asp groups [42]. (A) In the double site model, only one X is used for describing the equilibrium between the protonated and deprotonated forms, while the tautomer interversion process is represented by the variable x. (B) In the single-site model, protonation at different sites is represented by different X variables. HSP refers to the doubly protonated form of histidine. HSD and HSE refer to the singly protonated histidine with a proton on the h and e nitrogens, respectively. ASP1 and ASP2 refer to the protonated carboxylic acid with a proton on either of the carboxlate oxygens...

In those days, Statistics was more highly regarded than it is now, and the analytical chemists then knew the fundamental requirements of doing calibration work. There are several we need not go into all of them now, but the one that is pertinent to our current discussion is the one that states that, while the y-variable may contain error, the X-variable must be known without error. Now, in the real world this is never true, since all quantities are the result of some measurement, which will therefore have error... 

Finally, and of most interest, is the data in Table 25-1C. Here we have taken the same errors as in Table 25-IB and applied them to the X variable rather than the Y variable. By symmetry arguments, we might expect that we should find the same results as in Table 25-1B. In fact, however, the results are different, in several notable ways. In the first place, we arrive at the wrong model. We know that this model is not correct because we know what the right model is, since we predetermined it. This is the first place that what the statisticians have told us about the results are seen. In statistical parlance, the presence of error in the X variable biases the coefficient toward zero , and so we find the slope is decreased (always decreased) from the correct value (of unity, with this data) to 0.96+. So the first problem is that we obtain the wrong model. 

Now we come to the Standard Error of Estimate and the PRESS statistic, which show interesting behavior indeed. Compare the values of these statistics in Tables 25-IB and 25-1C. Note that the value in Table 25-1C is lower than the value in Table 25-1B. Thus, using either of these as a guide, an analyst would prefer the model of Table 25-1C to that of Table 25-1B. But we know a priori that the model in Table 25-1C is the wrong model. Therefore we come to the inescapable conclusion that in the presence of error in the X variable, the use of SEE, or even cross-validation as an indicator, is worse than useless, since it is actively misleading us as to the correct model to use to describe the data. 

---