Spectral space

When we calculate the eigenvectors for the two different data spaces (concentration and spectral spaces) we find the corresponding spectral and concentration vectors are shifted by different amounts in different directions. This is a consequence of the independence of the noises in the concentration and spectral spaces. So, just as the noise destroyed the perfect congruence between the noise-free spectral and concentration data points, it also destroyed... 

Although gross outliers should be deleted in advance (see the initial paragraphs in Section 4.4), it is likely to occur that some sample with some particular characteristic (spectrum, noise, etc.) shows a suspicious or different behaviour. Two plots inspecting the space of the spectra and the relationship of the X-space with the Y-space will be useful here. The X-spectral space is searched for by the z-z scores plot". Here the scores of the second, third, fourth, etc.. 

Figure 4.19 Diagnostic plots t va t scores to inspect for anomalous samples on the spectral space (X-block).

An important feature that affects the numerical solution strategy is that these equations are written in the spectral space, either in the three dimensional space of wave-vectors (/-propagated UPPE) or in a two-dimensional space of transverse wave-vectors plus a one dimensional angular-frequency space (z-propagated UPPE). At the same time, the nonlinear material response must be calculated in the real-space representation. Consequently, a good implementation of Fast Fourier Transform is essential for a UPPE solver. 

A distinct feature of NMR spectroscopy, the possibility to simultaneously observe hundreds of atoms in complex macromolecules, finds its foundation in the invention of multidimensional experiments almost 40 years ago [1,2]. The approach, however, has an important caveat the ultimate resolution obtained in multidimensional experiments comes at a very high price, the long data collection times needed to systematically sample the large multidimensional spectral space. The number of measured data points increases polynomially with the spectrometer field and the desired spectral resolution, and exponentially with the number of dimensions. The problem of lengthy sampling compromises or even prohibits many applications of multidimensional spectroscopy in chemistry and molecular biology. Fortunately, the advent of fast NMR spectroscopy offers a number of solutions. 

The linewidths ABm of the two mini-exciton ESR lines M+ and M" depend in a very characteristic manner on the spectral spacing of the two associated lines - B+ and A - B, i.e. on the difference of the fine structures of the two inequivalendy-oriented molecules A and B ABm has a finite value when the A- B spacing goes to zero and then increases quadratically with increasing A-B spacing (Eig. 7.16). In frequency units, (Acom = giiBABulh), the empirical result is then... 

Spectral space charge limit The onset of space... 

This clustering algorithm operates in a two-pass mode. In the first pass, the program reads through the data set and sequentially builds clusters (group of points in spectral space). There is a mean vector associated with each cluster. In the second pass, a minimum distance classification to mean vector algorithm is applied, pixel wise, where each pixel is assigned to one mean vector created in pass 1. 

R, di radius in spectral space used to determine when a new cluster should be formed. 

C, a spectral space distance parameter used when merging clusters. 

The locally weighted regression (LWR) philosophy assumes that the data can be efficiently modeled over a short span with linear methods. The first step in LWR is to determine the N samples that are most similar with the unknown sample to be analyzed. Similarity can be defined by distance between samples in the spectral space [25] by projections into the principal component space [26] and by employing estimates of the property of interest [27]. Once the N nearest standards are determined, either PLS or PCR is employed to calculate the calibration model. 

Figure 5.20 Diagnostic plots t vi. u scores to inspect for anomalous samples and strong non-linearities in the relationship between the spectral space (X-block) and the property of interest (Y-block), CS2 example.

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