The masking method

Finding only the most important scales with respect to the prediction error does not indicate where in the wavenumber domain these features are located. Often it is observed that going from one resolution level to another has dramatic effect on the quality of the multivariate model. In such a case it is possible to perform a systematic search for where in the wavenumber region for the particular scale the important features are approximately localized. The method described here is only applied to the cluster analysis example, however it could have been used for other multivariate modelling techniques also. In cluster analysis, instead of prediction error, the interest is focused on properties such as overlap between clusters, cluster separation, cluster variance etc. In general, some interesting property H is observed to change with respect to the resolution level. 

One way to find the important wavenumber regions is to look through all the possible combinations of wavelet coefficients being multiplied with zero or one for the scale in question. In total there are 1 binary mask vectors that can be multiplied with a wavelet coefficient vector w at scale k. The structure of the new wavelet coefficient vector w is 

This masking operation is applied to all the wavelet coefficient vectors of the spectra in the data set and then reconstructed. Again, it should be stressed that reconstruction is not necessary it is sufficient to only use all wavelet coefficients up to the current scale. 

All the wavelet coefficients above resolution level k are set to zero and do not contribute. For each combination i the chosen multivariate method is applied to the masked data. Fig. 5 illustrates the basic idea behind the method. 

It is now necessary to measure the effect the various combinations have on E. For instance, let H be the degree of cluster overlap. In this case it is possible to observe how different masks affect E. Of course, it is vital to have decided on which objects belong to a cluster. In other words, the analysis is temporarily turned into a supervised rather than an unsupervised problem which for complex cases can be solved by methods like CART [45] and discriminant PLS [46-48] and Quinlan s C4.5 algorithm [49]. 

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Cluster analysis. In this section, the application of the simple multiscale approach to cluster analysis is demonstrated. The masking method will also be used to localise important features. There are several possible cluster analysis algorithms, however only discriminant function analysis (DFA) will be used here. Before discussing the results from the simple multiscale analysis, this section will first present DFA and how it can applied to both unsupervised and supervised classification, followed by how the cluster properties S are measured at each resolution level. 

Several 1,4-dicarbonyl compounds are prepared based on this oxidation. Typically, the 1,4-diketone 10 or the 1,4-keto aldehyde 12 can be prepared by the allylation of a ketone[24] or aldehyde[61,62], followed by oxidation. The reaction is a good annulation method for cyclopentenones (11 and 13). Syntheses of pentalenene[78], laurenene[67], descarboxyquadrone[79], muscone (14 R = Me)[80]) and the coriolin intermediate 15[71] have been carried out by using allyl group as the masked methyl ketone (facing page). 

When the analytical method s selectivity is insufficient, it may be necessary to separate the analyte from potential interferents. Such separations can take advantage of physical properties, such as size, mass or density, or chemical properties. Important examples of chemical separations include masking, distillation, and extractions. 

The etching method is one where the work is masked with a resist such as wax and successively etehed as the masking is extending from the regions that are to remain the highest to the lowest. The contour steps initially might... 

Bioorganic molecules sometimes have the same or similar behavior as the analytes of interest in the analytical methods used to make chemical measurements of soil. Thus, they can falsely indicate the presence of more molecules than is actually the case, or even that an analyte or contaminant is present when it is not or has not been added from an external source. These compounds may also mask analytes of interest by reacting or simply associating with them (see Chapters 13 and 14 for a further discussion and specifics of this topic). 

In another elegant application of the masked disilene method, the anionic polysilanyl chain ends were chemically bound to substrate surfaces to form end-graft polysilanes. These systems are discussed in Section 3.11.4.2.3. 

Still-gauging methods are adequate for only the largest leaks. The accuracy of most tank-installed liquid level gauges is usually /s in. at best. A product loss reflected by a 0.062-in. level drop for a 100-ft-diameter tank translates to a 306 gal/day leak. If this Vi6-in. drop in product level is not discernible from the masking effects of fluid expansion, losses in excess of 2650 barrels annually will go undetected. At, for example, 20 per barrel, this loss amounts to over 53,000 for a single tank. Most importantly, the associated liability risks of groundwater contamination could involve much greater potential costs. 

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