Variability spectral

Principal Component Analysis (PCA). Principal component analysis is an extremely important method within the area of chemometrics. By this type of mathematical treatment one finds the main variation in a multidimensional data set by creating new linear combinations of the raw data (e.g. spectral variables) [4]. The method is superior when dealing with highly collinear variables as is the case in most spectroscopic techniques two neighbor wavelengths show almost the same variation. 

Some of the challenges with manual MALDI MS data analysis include the variability in relative intensity and peak appearance of MALDI MS ions typically observed for direct microorganism analysis. This spectral variability creates a challenge for manual data analysis where visual intensity differences are more noticeable. Replicate MALDI spectra of the same sample visually show significant differences that are hard to manually deal with effectively. 

The Mahalanobis Distance statistic provides a useful indication of the first type of extrapolation. For the calibration set, one sample will have a maximum Mahalanobis Distance, Z) ax. This is the most extreme sample in the calibration set, in that, it is the farthest from the center of the space defined by the spectral variables. If the Mahalanobis Distance for an unknown sample is greater than ZTax, then the estimate for the sample clearly represents an extrapolation of the model. Provided that outliers have been eliminated during the calibration, the distribution of Mahalanobis Distances should be representative of the calibration model, and ZEax can be used as an indication of extrapolation. 

In 2000 two major petrochemical companies installed process NMR systems on the feed streams to steam crackers in their production complexes where they provided feed forward stream characterization to the Spyro reactor models used to optimize the production processes. The analysis was comprised of PLS prediction of n-paraffins, /xo-paraffins, naphthenes, and aromatics calibrated to GC analysis (PINA) with speciation of C4-C10 for each of the hydrocarbon groups. Figure 10.22 shows typical NMR spectral variability for naphtha streams. Table 10.2 shows the PLS calibration performance obtained with cross validation for... 

This filtering preprocessing method can be used whenever the variables are expressed as a continuous physical property. One example is dispersive or Fourier-Transform spectral data, where the spectral variables refer to a continuous series of wavelength or wavenumber values. In these cases, derivatives can serve a dual purpose (I) they can remove baseline offset variations between samples, and (2) they can improve the resolution of overlapped spectral features. 

This procedure can consist of different stages with one sublibrary being constructed at each stage. The first one enables the identification of the product within an ample library, a second defines a characteristic that produces an important variability in the spectra (e.g. particle size) and a third defines a smaller spectral variability (difference in the content of impurities, origin of manufacture, etc.). 

The body of samples selected is split into two subsets, namely the calibration set and the validation set. The former is used to construct the calibration model and the latter to assess its predictive capacity. A number of procedures for selecting the samples to be included in each subset have been reported. Most have been applied to situations of uncontrolled variability spanning much wider ranges than those typically encountered in the pharmaceutical field. One especially effective procedure is that involving the selection of as many samples as required to span the desired calibration range and encompassing the whole possible spectral variability (i.e. the contribution of physical properties). The choice can be made based on a plot of PCA scores obtained from all the samples. 

A proof of this relation may be found in Bracewell (1978). Note that the spectral variable used in this and the next chapter is the same as that defined in Eqs. (7) and (8). Now consider a spatial distribution /(x) and its Fourier spectrum F(w) that come close to satisfying the equality in Eq. (4). We may take Ax and Aw as measures of the width, and hence the resolution, of the respective functions. To see how this relates to more realistic data, such as infrared spectral lines, consider shifting the peak function /(x) by various amounts and then superimposing all these shifted functions. This will give a reasonable approximation to a set of infrared lines. To discuss quantitatively what is occurring in the frequency domain, note that the Fourier spectrum of each shifted function by the shift theorem is given simply by the spectrum of the unshifted function multiplied by a constant phase factor. The superimposed spectrum would then be... 

We shall end this chapter with a few practical remarks concerning the calculation of the inverse-filtered spectrum. In this research the Fourier transform of the data is divided by the Fourier transform of the impulse response function for the low frequencies. Letting 6 denote the inverse-filtered estimate and n the discrete integral spectral variable, we would have for the inverse-filtered Fourier spectrum... 

However, for illustration, only one side of the interferogram and its spectrum will be shown, usually the function of the positive spatial and spectral variable. In other operating modes of the interferometer, asymmetric interferograms are produced that have a complex Fourier transform. Asymmetric interferograms will not be treated in this work. For a more complete discussion of Fourier transform spectroscopy, the reader should consult Bell (1972), Vanasse and Strong (1958), Vanasse and Sakai (1967), Steel (1967), Mertz (1965), the Aspen International Conference on Fourier Spectroscopy (Vanasse et al., 1971), and the two volumes of Spectrometric Techniques (Vanasse, 1977, 1981). A review of early work, which includes several major contributions of his own, is given by Connes (1969). Another interesting paper on the earlier historical development of Fourier transform spectroscopy is that by Loewenstein (1966). 

Asteroids in the outer asteroid belt show considerable spectral variability, due in part to differences in the degree of aqueous alteration. However, alteration alone is not sufficient to explain all the compositional variability observed in meteorites derived from these objects. Laboratory studies of carbonaceous chondrites show significant differences in the compositions and proportions of the various primary components, demonstrating that accreted materials in the asteroid belt were not uniform. 

The number of spectral variables and the collinearity problem are the two main drawbacks of the ILS model when applied to spectroscopic data. 

Although these factors are fairly similar to those described in Chapter 3, there are some relevant differences. In effect, in PCA (and hence in PCR), the factors explain most of the variance in the X-domain regardless of whether such variance is or is not related to the analyte, whereas in PLS they are calculated so that they explain not only variance in the spectral variables but also variance which is related to the property of interest. A typical example might be a series of graphite furnace measurements where a baseline shift appears due to some... 

Hence, the decision rule can also be written as follows If p (K Cj)p(Cj) > p(K Ci)p(Ci), the sample whose spectral variables are given by K should be assigned to class Cj. 

In a third method of evaluating the data, a chi-square analysis was used to asses near-IR spectral variability. For... 

A considerable amount of (unpublished) work has been performed by Ciurczak on counterfeit tablets. Using the same algorithms that have been applied to discriminate between placebos and active products, counterfeit products may be easily identified. The spectral variability stems from different raw materials and manufacturing processes, even though the active may be present at the correct level. 

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