Variable orthogonalization

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... 

Statistical manipulations on the USDA database (cluster analysis, principal component analysis with varimax rotation e.g., Everitt, 1980) revealed subsets of represent ve species, as idealized in Fig 2b, but with dif ent variables (orthogonal principal components) than traditional fractions as measured by USDA. A set cf species from each orthogonal subset appears in Table 1. The Latin names, and where available, the common names of the biomass species are given. The extractives ranges are ash content, 4 to 17% protein content, 5 to 14% polyphenol, 3 to 11% and oil content, 1 to 4%. However no species contains extremes of all 4 variables. Nor can species be found, retaining native compositions, at extremes of just one extractive composition, while the other fractions are present at constant levels. Thus we use orthogonal but non-intuitive compositions in this work, then rank pyrolysis effects in terms of traditional extractives content to get an understanding of their impact on biomass pyrolysis. 

In the 1990s several research efforts on bend modelling were reported. In the results published by Lutters et al. (1995), the described Equihb-rium Model assumes a plane strain situation. The influence of shear stress and variable orthogonal reaction forces at the contact points with the die, as well as friction forces are incorporated in the model. The reported deviations between calculated and measured punch displacements, are of order of magnitude 0.1-0.2 mm, once the... 

Figure 5 Optimization of a conical intersection in the subspace containing X and one variable orthogonal to the plane containing X, and Xj...

Deuflhard P and Wulkow M 1989 Computational treatment of polyreaction kinetics by orthogonal polynomials of a discrete variable Impact of Computing in Science and Engineering vol 1... 

By applying this method, they demonstrated that the removal of insignificant variables increases the quality and reliability of the models despite the fact that the correlation coefficient, r, always decreases, although only shghtly. For example, the characteristics of a model with six orthogonalized descriptors were r = 0.99288, s = 0.9062, F = 127.4 and the quality of this model was sufficiently improved after removal of the two least significant descriptors, to r = 0.9925, s = 0.8553,... 

Packages exist that use various discretizations in the spatial direction and an integration routine in the time variable. PDECOL uses B-sphnes for the spatial direction and various GEAR methods in time (Ref. 247). PDEPACK and DSS (Ref. 247) use finite differences in the spatial direction and GEARB in time (Ref. 66). REACOL (Ref. 106) uses orthogonal collocation in the radial direction and LSODE in the axial direction, while REACFD uses finite difference in the radial direction both codes are restricted to modeling chemical reactors. 

Here and below, T , 1, , and e, i, j = 1,. . . , 5, denote atomic position vectors, atom-atom distances, and the corresponding unit vectors, respectively. In order to construct a correctly closed conformation, variables qi,. . . , q4 are considered independent, and the last valence angle q is computed from Eq. (7) as follows. Variables qi,.. ., q4 determine the orientation of the plane of q specified by vector 634 and an in-plane unit vector 6345 orthogonal to it. In the basis of these two vectors, condition (7) results in... 

If there is more than one constraint, one additional multiplier term is added for each constraint. The optimization is then performed on the Lagrange function by requiring that the gradient with respect to the x- and A-variable(s) is equal to zero. In many cases the multipliers A can be given a physical interpretation at the end. In the variational treatment of an HF wave function (Section 3.3), the MO orthogonality constraints turn out to be MO energies, and the multiplier associated with normalization of the total Cl wave function (Section 4.2) becomes the total energy. 

In pi actice, loads are not necessarily uniformly distributed nor uniaxial, and cross-sectional areas are often variable. Thus it becomes necessary to define the stress at a point as the limiting value of the load per unit area as the area approaches zero. Furthermore, there may be tensile or compressive stresses (O,, O, O ) in each of three orthogonal directions and as many as six shear stresses (t, , T ). The... 

Power Series Expansions and Formal Solutions (a) Helium Atom. If the method of superposition of configurations is based on the use of expansions in orthogonal sets, the method of correlated wave functions has so far been founded on power series expansions. The classical example is, of course, Hyl-leraas expansion (Eq. III.4) for the ground state of the He atom, which is a power series in the three variables... 

The governing variables are accurately known, are independent of each other (orthogonal) and span wide value ranges. 

Methods based on the partitioning of a reaction system into fast and slow components have been proposed by several authors [158-160], A key assumption made in this context is the separation of the space of concentration variables into two orthogonal subspaces and Qf spanned by the slow and fast reactions. With this assumption the time variation of the species concentrations is given as... 

The analytical determination of the derivative dEtotldrir of the total energy Etot with respect to population n, of the r-th molecular orbital is a very complicated task in the case of methods like the BMV one for three reasons (a), those methods assume that the atomic orbital (AO) basis is non-orthogonal (b), they involve nonlinear expressions in the AO populations (c) the latter may have to be determined as Mulliken or Lbwdin population, if they must have a physical significance [6]. The rest of this paper is devoted to the presentation of that derivation on a scheme having the essential features of the BMV scheme, but simplified to keep control of the relation between the symbols introduced and their physical significance. Before devoting ourselves to that derivation, however, we with to mention the reason why the MO occupation should be treated in certain problems as a continuous variable. 

The columns of V are the abstract factors of X which should be rotated into real factors. The matrix V is rotated by means of an orthogonal rotation matrix R, so that the resulting matrix F = V R fulfils a given criterion. The criterion in Varimax rotation is that the rows of F obtain maximal simplicity, which is usually denoted as the requirement that F has a maximum row simplicity. The idea behind this criterion is that real factors should be easily interpretable which is the case when the loadings of the factor are grouped over only a few variables. For instance the vector f, =[000 0.5 0.8 0.33] may be easier to interpret than the vector = [0.1 0.3 0.1 0.4 0.4 0.75]. It is more likely that the simple vector is a pure factor than the less simple one. Returning to the air pollution example, the simple vector fi may represent the concentration profile of one of the pollution sources which mainly contains the three last constituents. 

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