Direct covariance

Figure 7 Results from a field comparison of derived from direct covariance (O), deliberate tracers ( ), and the atmospheric profile technique for CO2 (A) and DMS ( ) plotted against wind speed. Also shown are the parametrizations of LM86 (dashed line), W92 (dotted line), and a cubic fit to the data (solid line) (reproduced by permission of Elsevier from Afar. Chem., 2001b, 75, 267-280).

Wu and Wu (2002) compared three different covariate screening methods nonlinear least-squares based method (NL-based), EBE-based method, and direct covariate screening by inclusion in the model and LRT. In the NL-based method, the same model is fit to each individual using nonlinear regression and the parameter estimates for that subject are obtained. Correlation tests or regression-based models between the individual parameter estimates and individual covariates may then be used to determine whether a significant relationship exists between the variables. This method is difficult to implement in practice because it requires rich data for each subject. For Phase 3 studies where sparse pharmacokinetic data are often collected, this method is impractical since many subjects will have insufficient data to support even simple pharmacokinetic... 

The potential to extend 2-dimensional covariance NMR to higher dimensionality has its foundations in Eq. (5.13). Thus, Snyder et al. [16] laid the basis for the computation of 4D NOESY spectra. In their strategy, the critical entry point consisted of considering a 4D data set an array of 2D data or a plane-of-planes. In order to Ulustrate the calculations, the terms donor and acceptor planes in combination with donor and acceptor pairs were coined. It should be noted that an acceptor plane is associated with each donor pair 01,0)2) at frequencies < i and < 2- Mapping of either the acceptor planes onto the donor planes or vice versa describes the projection of a dimension onto another, which leads to an increase for direct covariance or to a decrease for indirect covariance in dimensionality of resolution. 

The direct covariance matrix multiplication of a homonuclear NMR spectrum yields a homonuclear covariance map whose square root is equivalent to the NMR spectrum obtained from Fourier transformation. As can be readily seen from Fig. 5.1 A, the higher resolution of the direct dimension is mapped onto the indirect dimension. Hence, a symmetric spectrum... 

Indirect covariance processing as depicted in Fig. 5.IB is another type of single spectmm manipulation. Here, a heteronuclear correlation spectmm is transformed into a correlation spectmm of the indirect dimension of the original data. In the current example, a H—C HSQC—TOCSY spectmm is converted into a C—C correlation spectmm, containing only correlations of carbon atoms that share the connectivity information of their attached protons. Quaternary carbons thus remain unobserved due to the principle of the H-C HSQC—TOCSY. While the direct covariance processing leads to an increase in resolution, the indirect covariance processing results in a formal decrease. Yet, the reduction of signals under retention of the information content in the case of the latter transformation compensates for the decrease in dimensionafity. It is also possible to use the indirect covariance procedure on homonuclear spectra to obtain homo-decoupled... 

The apphcation of direct covariance processing to either NMR spectra or mixed time—frequency data matrices was preferentially used for spectral enhancement with respect to resolution. As the resolution of the indirect or incremented dimension is endowed with that of the direct or acquisition dimension, homonuclear spectra were treated almost exclusively. 

A study was conducted which did not employ a frequency symmetric experiment prior to direct covariance processing. Zhang et al. [80] subjected a 2D C—C INADEQUATE data set of 2048 x 1024 data points to covariance NMR. They thus transformed the common double-quantum representation of an INADEQUATE spectrum into a symmetric singlequantum map. The resulting symmetric spectrum was considered easier to interpret. Menthol 2 was used as reference compound. 

Covariance spectra with 512 and 256 increments in the indirect dimension from C-labeUed microcrystaUine proteins were visually inspected with respect to their signal-to-noise and resolution enhancement [86]. The C-C correlation spectra with PARIS recoupling were recorded on the C- and N-labeUed protein Crh which is an 85 amino acid domain-swapped homodimer of catabolite repression phosphocarrier protein of approximately 21 kDa. The spectra were judged to display no difference in resolution or sensitivity. Thus direct covariance processing was... 

With respect to covariance NMR, STOCSY themes can be interpreted as a form of direct covariance, where the direct dimension consists of the proton chemical shift and the indirect dimension is represented by sample variations. 

In comparison to other spectroscopic techniques, where generahzed covariance was performed to obtain synchronous and asynchronous correlation maps and where both correlation maps were interpreted, the asynchronous map was rarely exploited if considered at aU for NMR purposes. Yet, the synchronous map as an equivalent to the direct covariance spectrum served to correlate species in different samples for a few studies. In contrast, analysis of the sample variation by statistical total correlation NMR, STOCSY, has become a comer stone of metabolomics investigations this field was considered beyond the scope of this chapter, hence only the current variants of STOCSY and their purposes were briefly presented. 

In liquid solutions, nuclear spins are correlated through J-couplings, whereas in solids dipolar interactions come into play as well. The first report on the application of covariance to soHd-state NMR was published by Hu et al. in which an indirect covariance homonuclear correlation (HOMCOR) spectrum was obtained from a 2D-FT heteronuclear correlation (HETCOR) spectrum. The indirect covariance scheme, which will be discussed in Section 5.1, contrasts with the original direct covariance spectroscopy, where the covariance processing is performed over an array of one-dimensional spectra. 

Hu et al. also reported on the first appHcation of direct covariance spectroscopy to sohd-state NMR to reveal correlations in amino acids. 

Indirect covariance spectroscopy was proposed by Zhang and Briischweiler [85] soon after publication of their pioneering work on covariance spectroscopy [58]. Its apphcation to soHd-state NMR was reported a few years later by Hu et al. [86]. The procedure of the indirect covariance spectroscopy starts with a 2D Fourier-transformed FIETCOR spectrum. This is in contrast to the original direct covariance spectroscopy, in which covariance processing is appHed to an array of one-dimensional spectra so that Fourier transformation is performed only with respect to the direct dimension. From a single I—S 2D FiETCOR spectrum, a pair of HOMCOR spectra are... 

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