Canonical variates

It may be iisefiil to mention here one currently widely applied approximation for barrierless reactions, which is now frequently called microcanonical and canonical variational transition state theory (equivalent to the minimum density of states and maximum free energy transition state theory in figure A3,4,7. This type of theory can be understood by considering the partition fiinctions Q r ) as fiinctions of r similar to equation (A3,4.108) but with F (r ) instead of V Obviously 2(r J > Q so that the best possible choice for a... 

Poliak E 1990 Variational transition state theory for activated rate processes J. Chem. Phys. 93 1116 Poliak E 1991 Variational transition state theory for reactions in condensed phases J. Phys. Chem. 95 533 Frishman A and Poliak E 1992 Canonical variational transition state theory for dissipative systems application to generalized Langevin equations J. Chem. Phys. 96 8877... 

Hu X and Hase W L 1989 Properties of canonical variational transition state theory for association reactions without potential energy barriers J. Rhys. Chem. 93 6029-38... 

Several VTST techniques exist. Canonical variational theory (CVT), improved canonical variational theory (ICVT), and microcanonical variational theory (pVT) are the most frequently used. The microcanonical theory tends to be the most accurate, and canonical theory the least accurate. All these techniques tend to lose accuracy at higher temperatures. At higher temperatures, excited states, which are more difficult to compute accurately, play an increasingly important role, as do trajectories far from the transition structure. For very small molecules, errors at room temperature are often less than 10%. At high temperatures, computed reaction rates could be in error by an order of magnitude. 

CVT (canonical variational theory) a variational transition state theory technique... 

Fig. 33.1. Canonical variate plot for three classes with different thyroid status. The boundaries are obtained by linear discriminant analysis [2].

We can go one step further, however. Each of the above multiple regression relations is between a single variable (response) of one data set and a linear combination of the variables (predictors) from the other set. Instead, one may consider the multiple-multiple correlation, i.e. the correlation of a linear combination from one set with a linear combination of the other set. Such linear combinations of the original variables are variously called factors, components, latent variables, canonical variables or canonical variates (also see Chapters 9,17, 29, and 31). 

The next pair of canonical variates, t2 and U2 also has maximum correlation P2, subject, however, to the condition that this second pair should be uncorrelated to the first pair, i.e. t t2 = u U2 = 0. For the example at hand, this second canonical correlation is much lower p2 = 0.55 R = 0.31). For larger data sets, the analysis goes on with extracting additional pairs of canonical variables, orthogonal to the previous ones, until the data table with the smaller number of variables has been... 

Canonical structure correlations between the original variables (x, y) and their canonical variates (r, u). 

We have seen that PCR and RRR form two extremes, with CCA somewhere in between. RRR emphasizes the fit of Y (criterion ii). Thus, in RRR the X-components t, preferably should correlate highly with the original T-variables. Whether X itself can be reconstructed ( back-fitted ) from such components t, is of no concern in RRR. With standard PCR, i.e. top-down PCR, the emphasis is initially more on the X-side (criterion i) than on the T-side. CCA emphasizes the importance of correlation whether the canonical variates t and u account for much variance in each respective data set is immaterial. Ideally, of course, one would like to have the best of all three worlds, i.e. when the major principal components of X (as in PCR) and the major principal components of Y (as in RRR) happen to be very similar to the major canonical variables (as in CCA). Is there a way to combine these three desiderata — summary of X, summary of Y and a strong link between the two — into a single criterion and to use this as a basis for a compromise method The PLS method attempts to do just that. 

While principal components models are used mostly in an unsupervised or exploratory mode, models based on canonical variates are often applied in a supervisory way for the prediction of biological activities from chemical, physicochemical or other biological parameters. In this section we discuss briefly the methods of linear discriminant analysis (LDA) and canonical correlation analysis (CCA). Although there has been an early awareness of these methods in QSAR [7,50], they have not been widely accepted. More recently they have been superseded by the successful introduction of partial least squares analysis (PLS) in QSAR. Nevertheless, the early pattern recognition techniques have prepared the minds for the introduction of modem chemometric approaches. 

The rate of hydrogen transfer can be calculated using the direct dynamics approach of Truhlar and co-workers which combines canonical variational transition state theory (CVT) [82, 83] with semi-classical multidimensional tunnelling corrections [84], The rate constant is calculated using [83] ... 

Canonical variational transition state theory, with transitional modes treated as harmonic oscillators refs. S... 

Of course, one is not really interested in classical mechanical calculations. Thus in normal practice the partition functions used in TST, as discussed in Chapter 4, are evaluated using quantum partition functions for harmonic frequencies (extension to anharmonicity is straightforward). On the other hand rotations and translations are handled classically both in TST and in VTST, which is a standard approximation except at very low temperatures. Later, by introducing canonical partition functions one can direct the discussion towards canonical variational transition state theory (CVTST) where the statistical mechanics involves ensembles defined in terms of temperature and volume. There is also a form of variational transition state theory based on microcanonical ensembles referred to by the symbol p,. Discussion of VTST based on microcanonical ensembles pVTST is beyond the scope of the discussion here. It is only mentioned that in pVTST the dividing surface is... 

TST = conventional Transition State Theory, ICVT = Improved Canonical Variational Transition state theory, ICVT/SCT = ICVT/Small Curvature Tunneling, ICVT/p,OMT = ICVT/Microcanonical Optimized Multidimensional Tunneling. 

Cluster analysis is far from an automatic technique each stage of the process requires many decisions and therefore close supervision by the analyst. It is imperative that the procedure be as interactive as possible. Therefore, for this study, a menu-driven interactive statistical package was written for PDP-11 and VAX (VMS and UNIX) series computers, which includes adequate computer graphics capabilities. The graphical output includes a variety of histograms and scatter plots based on the raw data or on the results of principal-components analysis or canonical-variates analysis (14). Hierarchical cluster trees are also available. All of the methods mentioned in this study were included as an integral part of the package. 

Lipopolysaccharide extracts from different pathogenic and nonpatho-genic . coli strains were also analyzed by FT-IR with principle component analysis and canonical variate analysis (Kim et al, 2006b). The data showed that E. coli strains can be discriminated with >95% accuracy. Listeria species were also reliably classified by FT-IR coupled with an artificial neural network technology with a success rate of 96% (Rebuffo et al, 2006), while the identification rate for L. monocytogenes alone was 99.2%. 

Canonical variate itt correlations, the value of an observation after transforming the direction and scale of the coordinate axes. Canonical transformation is the procedure by which this is done. 

A.D. Isaacson, D.G. Truhlar, Polyatomic canonical variational theory for chemical reaction rates. Separable-mode formalism with application to OH+I-p H2O+H, J. Chem. Phys. 76 (1982) 1380. 

The mechanism of cyclopropenations of alkynes with ethyl diazoacetate, catalysed by (AcO)4Rh2 and (DPTI)3Rh2(OAc), has been studied by a combination of kinetic isotope effects and theoretical calculations. With each catalyst, a significant normal 13C KIE was observed for the terminal acetylenic carbon, while a very small 13C KIE was detected at the internal acetylenic carbon. These isotope effects are consistent with the canonical variational transition structures for cyclopropenations with intact tetrabridged rhodium carbenoids but not with a 2 + 2-cycloaddition on a tribridged rhodium carbenoid structure.99... 

The use of a direct combined (or polyphasic) approach can create highly specific soil fingerprints from normal constituents. This, in addition to the application of appropriate statistical analysis, would make soil analysis a more effective tool for routine forensic work, thus considerably extending its applicability. Indeed, combinations of different data each with its own discriminatory potential may result in probabilities of association or disassociation that even surpass those of techniques such as human DNA. Initial work using a canonical variate analysis has shown discrimination between soil types can be improved by including more analytical data. Figure 11.11 illustrates... 

Figure 11.11 Canonical variate plots showing increases in discrimination among three visually similar topsoils as soil attributes (analyses) are combined. A is a cultivated podzol/improved pasture B is a brown earth/improved pasture and C is a brown earth/seminatural grassland. Four replicate values sampled 4 m apart plus 90% confidence ellipses.

Figure 20.4. Spectral patterns corresponding to the canonical variates 1 (—) and 2 (— - —) for FTIR data (Dufour et al., 2000 see text for details).

---