Axis-parallel projection methods

Non-axis-parallel projection methods use a (linear/nonUnear) combination of two or more dimensions for an axis of the projection plane. Principal-component analysis is a well-established technique in this category. 

Axis-parallel projection methods use existing dimensions as axes of the projection plane. One of the existing dimensions is selected as the horizontal axis, and another as the vertical axis, to make a familiar and comprehensible presentation. Sometimes, other dimensions can be mapped as color, size, length, angle, and so on. 

Projection methods in category 2, axis-parallel projection methods, have been used by researchers in machine learning, data mining, and informalion visualization. In this category, 2D projections (visualized in scatterplots ) arc made by choosing one dimension for the xr-axis and another dimension for the y-axis. 

These automatic projection pursuit methods using a series of low-dimensional projections have made impressive gains in the problem of multidimensional data analysis, but they have limitations. One of the most important problems is the difficulty in interpreting the solutions from the automatic projection pursuit. Since the axes are the linear/nonlinear combination of the variables (or dimensions) of the original data, it is hard to determine what the projection actually means to users. Conversely, this is one of the reasons that axis-parallel projections (projection methods in category 2) are used in many multidimensional analysis tools (Guo, 2003 Ward, 1994). 

Similarly, a twofold axis parallel to 6, since it fixes pairs of atoms with the same y coordinate, should cause the OkO reflections to be, on the average, twice as strong as the general hkl reflections. The number of reflections along a central i qw of the reciprocal lattice may not, however, be large enough to make the statistical method applicable. A twofold axis also makes a pro jection along this axis a centrosymmetric projection. 

Figure 3.2. Equilibrium linear susceptibility in reduced units X = x Hi[/m) versus temperature for three different ellipsoidal systems with equation x ja +y lb + jc < I, resulting in a system of N dipoles arranged on a simple cubic lattice. The points shown are the projection of the spins to the xz plane. The probing field is applied along the anisotropy axes, which are parallel to the z axis. The thick lines indicate the equilibrium susceptibility of the corresponding noninteracting system (which does not depend on the shape of the system and is the same in the three panels) thin lines show the susceptibility including the corrections due to the dipolar interaction obtained by thermodynamic perturbation theory [Eq. (3.22)] the symbols represent the susceptibility obtained with a Monte Carlo method. The dipolar interaction strength is itj = d/ 2o = 0.02.

Figure 12. The upper panels show one-dimensional velocity distributions of NO from the photodissociation of 2-chloro-2-nitrosopropane at 650 nm using the projection TOF method. The NO ions were produced by REMPI via the A2Z+ <- X2n transition using the branches indicated in the figure. The velocity distributions were taken with sD and ePR collinear and both cPR and the probe laser propagation direction perpendicular the axis of the detector, kD. The symbols O and indicate velocity distributions taken with eD parallel and perpendicular to kD, respectively. The lower panel shows the difference spectra of the velocity distributions, which are sensitive to the yS°(20) bipolar moment. The solid line shows a fit to the difference spectra using equations discussed in [42] and [170]. [Reprinted with permission from R. Uberna, R. D. Hinchliffe, and J. I. Cline, J. Chem. Phys., 105(22), 9847 (1996). Copyright 1996 American Institute of Physics.

An example of such spectra is shown in Fig. 16 (n-hexane). The traces plotted for various values of co how that parallel to the coj-axis, the full multiplet structure is retained whereas in the coj direction the completely decoupled spectrum results. The undecoupled spectrum is to be considered as a projection of the spectra for various values of onto the coj-axis. This rather involved technique is less sensitive than ordinary FTNMR. It has been used mainly in C-NMR spectroscopy. A number of applications as well as the solution to problems connected with this method have been reported -307>... 

---