Covariance-processing methods

Figures 16 and 17. Covariance processing methods can thus reduce data acquisition times for lower sensitivity experiments such as 1,1- or INADEQUATE when these data are going to be interrogated following covariance processing.50 Alternatively, when the increased s /n of...

Figure 18A shows the overlaid multiplicity-edited GHSQC and 60 Hz 1,1-ADEQUATE spectra of posaconazole (47). As will be noted from an inspection of the overlaid spectra, there is an overlap of the C46 and C47 resonances of the aliphatic side chain attached to the triazolone ring that can be seen more clearly in the expansion shown in Figure 18B. In contrast, when the data are subjected to GIC processing with power = 0.5, the overlap between the C46 and C47 resonances is clearly resolved (Figure 18C). In addition, the weak correlation between the C3 and C4 resonances of the tetrahydrofuryl moiety in the structure is also observed despite the fact that this correlation was not visible in the overlaid spectrum shown in A. This feature of the spectrum can be attributed to the sensitivity enhancement inherent to the covariance processing method.50...

The information content resulting from both processing methods is identical insofar as correlation information is concerned. The matrix-square-root transformation can minimize artefacts due to relay effects and chemical shift near degeneracy (pseudo-relay effects80-82 98). The application of covariance methods to compute HSQC-1,1-ADEQUATE spectra is described in the following section. 

The projection-reconstruction approach is a technique unrelated to covariance processing which can provide data typically inaccessible to the natural product chemist. For example, 13C-15N correlation spectra were obtained for vitamin B12 at natural abundance.104 Compared with a conventional three-dimensional 13C-15N correlation experiment, the projection-reconstruction method provides a sensitivity enhancement of two orders of magnitude. The final 13C-15N spectrum was reconstructed from data obtained from ll l5N and H- C correlations acquired using a time-shared evolution pulse sequence that allowed all the information to be obtained in one experiment.104 Martin and co-workers also demonstrated the ability to generate 13C-15N correlation spectra using unsymmetrical indirect covariance NMR with vinblastine as an example.105 In the latter case, 13C-15N correlation spectra were obtained from - C HSQC data and H-1sN HMBC data that were acquired separately. Both methods provide access to correlations that would be inaccessible for most natural products at natural abundance. 

A covariance study by Li et al. [76] dealt with the comparison of various sampling schemes and accumulation profiles with respect to their usability with covariance processing. The authors divided sparse sampling schemes into the non-uniform and the ti cut-off (CUO) schemes. While common NUS schemes employ—exponentially—increasing A(i spacing as a function of (i, a cut-off design implies the sole acquisition of signals at short <1 times up to a maximum value. The 2D NMR spectrum is then reconstructed with appropriate methods. Various standard, NUS and... 

In a subsequent study, the DemixC method was appHed to a mixture of D-glucose, L-histidine, L-lysine, serotonin hydrochloride, D-sorbitol as well as to the venom of the walking stick insect Anisomorpha buprestoides, which consists of at least six compounds [84]. The H—H TOCSY spectra of 2048 X 512 points were recorded in a screening setup at 600 MHz with a 1 mm probehead. With the help of databases, the mixture was deconvolved and the venom identified. Thus, automated mixture deconvolution in screening mode using covariance processed H—H TOCSY experiments was found feasible. 

Only a few publications in the literature have dealt with this problem. Almasy and Mah (1984) presented a method for estimating the covariance matrix of measured errors by using the constraint residuals calculated from available process data. Darouach et al. (1989) and Keller et al. (1992) have extended this approach to deal with correlated measurements. Chen et al. (1997) extended the procedure further, developing a robust strategy for covariance estimation, which is insensitive to the presence of outliers in the data set. 

As pointed out by Keller et al. (1992), if the process is truly at steady state, then estimation by the so-called direct method using the sample variance and covariance is adequate and simple to use. Let y(- be the ith element in a vector of measured variables, then the sample variance of the r repeated measurements of y, is given by... 

This procedure (based on sample variance and covariance) is referred to as the direct method of estimation of the covariance matrix of the measurement errors. As it stands, it makes no use of the inherent information content of the constraint equations, which has proved to be very useful in process data reconciliation. One shortcoming of this approach is that these r samples should be under steady-state operation, in order to meet the independent sampling condition otherwise, the direct method could give incorrect estimates. 

Nonexistent samples or reference values Samples may not exist because the plant does not exist yet (it is under construction and has not started up yet), or because the plant does not do any sampling at the process point of interest. Reference values may not exist because the plant lab is not set up to perform that particular analysis, or (worse) there is no established reference method for the analyte of interest. In any of these cases, one is left with no data to use for calibration. There are a number of ways to approach this challenge. If there are samples but no reference values, the plant samples can be sent off-site to be analyzed. The analyte concentration of interest can sometimes be estimated based on process conditions and/or the concentrations of other analytes. (This is one place where fixed covariance can come in handy.) If there is no plant yet, it may be possible to calibrate the analyzer elsewhere (different plant, semi-works, etc.). It may also be possible (or even necessary) to attempt lab value-less calibration, in which one assumes that the concentration of the analyte varies linearly with the height... 

With PLS, the covariance of the measurements with the concentrations is used in addition to the variance in R to generate U. This distinction between PLS and PCR is depicted in Figure. 89 where the only difference between the methods is in the first step (estimation of U). As with PCR, PLS is not commonly implemented using a two-step process, but is presented this way for clarity. For more details, sec Martens and Nass (1989). 

This iteration process can be repeated until a satisfactory solution is obtained. In general is not easy to determine when a solution is satisfactory. The simplest method is to investigate the value of the covariance matrix of the error estimate and break off the iteration process if this covariance matrix falls below a given value or decreases less than a given fraction from one step to the next. 

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