Data lateral, diagonal

Symmetry relationship of cross peak locations has been used for improving the quality of correlation spectra for quite a long time in NMR spectroscopy both for diagonally [21] and laterally symmetric spectra [22]. Such data processing procedures have their advantage but may introduce artifacts and remove real information and therefore should be used with caution [10]. 2Q-HoMQC spectra can be symmetrized directly using the appropriate symmetry function [33], but most commercial software do not provide such option. Also, fine structure of the direct correlation peaks in 2Q-H0MQC spectra is antisymmetric in the SQ dimension which requires extra attention. 

The expansions in even powers of normal frequencies are of special interest, because they provide means for obtaining explicit relations between the equations of motion and the thermodynamic quantities, through the use of the method of moments The sum of over all the normal vibrations can be expressed as the trace, or the sum of all the diagonal elements, of a matrix H" obtained by multiplying the Hamiltonian matrix H of the system by itself (n — 1) times. Such expansions thus enable us to estimate the thermodynamic functions and their isotope effects from known force fields and structures without solving the secular equations, or alternatively, to estimate the force fields from experimental data on the thermodynamic quantities and their isotope effects. The expansions explicitly correlate the motions of particles with the thermodynamic quantities. They can also be used to evaluate analytically a characteristic temperature associated with the system, such as the cross-over temperature of an isotope exchange equilibrium. Such possible applications, however, are useful only if the expansion yields a sufficiently close approximation. The precision of results obtainable with orthogonal polynomial expansions will be explored later. 

Another approach then is to first identify the fixed effects and then build the random effects matrix. However, there is an interaction between the random effects matrix and fixed effects such that the exclusion of a random effect may fail to identify a significant fixed effect. So what is an analyst to do A common strategy is to first treat all structural parameters in the model as independent random effects, i.e., to use a diagonal covariance matrix. Random effects with near zero variance are treated as fixed effects. Second, whenever possible, use an unstructured covariance matrix between those random effects identified in the first step as being important. If the unstructured covariance model does not minimize successfully, then treat the covariance as a simple matrix (no covariances between diagonal elements). Lastly, once the final model is selected, obtain the EBE (which are discussed later in the chapter) for the random effects and generate a scatter plot correlation matrix. EBEs that appear correlated should then have a covariance term included in the covariance, otherwise the covariance is set equal to zero. Keep in mind, however, that this approach is sequential in that A — B — C, but model building is not necessarily sequential. The process may be iterative such that the process may need to be modified based on the data and model. 

Here is where the matrix S comes into play. In the matrix S, all the elements are zero, except for the diagonal elements, which are called singular values. The singular values are a measure of the contribution the associated columns in the U and V matrices make to the experimental data set in matrix A, that is, they can be considered as a kind of weighting. These values are relatively large in the first columns but tend rapidly to zero in the later columns. In an ideal situation, the rank of the matrix A would be just the number of non-zero values in the diagonal of S. In practice, however, this is not the case because of the noise in the system and it can be difficult to locate the cut-off between values associated with signal and those with noise. 

---