Spectrum residual

Figure 84 also contains a plot of the differences between the original and the regenerated spectrum. This is identical to the residual spectrum. The residuals of this spectrum look comfortably like pure random noise. 

Reactant and product analyses were obtained from the intensities of infrared absorption bands by successive subtraction of absorptions by known species. Low noise reference spectra for UDMH and several reaction products were generated for this purpose in order to minimize the increase in the noise level of the residual spectrum with each stage of subtraction. 

Figure 2, Selected residual IR spectra from the UDMH + NO experiments (a) spectrum of un-known(s) formed in approximately 3 h from UDMH + NO in the dark (b) residual spectrum at 19 min into the UDMH + NO irradiation (c) residual spectrum at 200 min into the irradiation.

Slow, efficient progreuaing required and large data storage for reference files. Identification of multiple components of mixtures can lie achieved by using residual spectrum. 

Independently exainining the different contributions to F can help determine wh) a sample is excluded from a particular class. The PCA contribution reflects structure in the residual spectrum, which is an indication of additional sources of variation present in the unknown measurement vector (e.g., increased noise level, an unmodeled interferent, or a noise spike). The distance contribution becomes significant when the magnitude of the features in the unknown are unlike the training set data. This can occur when additional sources of variation are present or wt en the concentrations of the expected components are outside the training set range. 

FIGURE 5.7. Graphical representation of Equation 5.13, estimation of the residual spectrum. 

Values Whereas the sample leverage is based on the score vector of a sample, the value is based on the residual spectrum. The values are a convenient screening diagnostic for identifj ing samples that need closer examination. 

Given the estimated concentrations, measurement residuals can be calculated for each sample and are examined for diagnostic purposes. The measurement residual is the portion of the mixture spectrum, r, that is not described using the pure spectra. To generate a residual spectrum, a reconstructed mixture spectrum, f, is first generated using the pure spectra (S) and the estimated concentrations (c) ... 

Figure 8.24a, for example, shows the FTIR spectrum before the photolysis of mixtures of DMS in air with h2o2 as the OH source and the residual spectrum after 5 min of photolysis (Barnes et al., 1996). The reactants, as well as the product S02 have been subtracted out in Fig. 8.24b. Dimethyl sulfoxide (DMSO) as well as dimethyl sulfone, CH3S02CH3 (DMS02), and small amounts of COS are observed as products. DMSO is so reactive that it is rapidly converted into DMS02 in this system and hence both are observed in Fig. 8.24b. However, Barnes and co-workers calculate that the DMSO yield corrected for secondary oxidation is about the same as the fraction of the OH-DMS reaction that proceeds by addition under these conditions, i.e., that the major fate of the adduct is reaction (47). Turnipseed et al. (1996) measured the yield of H02 from reaction (47) to be 0.50 + 0.15 at both 234 and 258 K, suggesting that there are other reaction paths than (47) as well. The mechanism of formation of COS is not clear but may involve the oxidation of thioformaldehyde (H2C=S). The implications for the global budget of COS are discussed by Barnes et al. (1994b, 1996).

Figure 11.13 shows a typical DO AS spectrum measured in air after correcting for atmospheric background light and an electronic offset (Stutz and Platt, 1997). Below the spectrum are shown reference spectra for the gases that contribute to the atmospheric spectrum, scaled by the a, factors determined using Eq. (H). In this case, O, N02, SOz, and HCHO all contribute, leaving a residual spectrum with a peak-to-peak absorbance of 6 X 10 4. 

Figure 8. Analysis of the product spectrum in Figure 6. (A) residual spectrum from Figure 6B (see text) (B) reference spectrum of C2H5OOH (C) reference spectrum of C2H5OH. Signals in (A) were truncated at 20% absorption.

Figure 13. Spectral data resulting from the photolysis of n-C4H9ONO (5.7 ppm) in 1 Torr of 02 and 700 Torr of N2. (A) before irradiation (B) after 5 min irradiation and (C) residual spectrum from (B) (see text). The absorbance scale for (Q was expanded by a factor of 5 as compared with those for (A) and (B).

Figure 15. Analysis of the product spectra in Figure 14. (A) difference spectrum of Figures 14B and 14C (B) residual spectrum obtained from (A) above and (C) reference spectrum of formic anhydride. Values in parentheses are concentrations in ppm. Values without parentheses are frequencies.

Figure 19. Effects of added S02 on the Os + C2H4 reaction. (A) initial mixture contained 03 (5 mppm), C2H4 (10 ppm) and S02 (5 ppm) in air at 700Torr and (B) residual spectrum of (A) above.

Figure 20. Residual spectrum from an ozone-propylene-S02 reaction and H2S04 aerosol spectrum from the HO-initiated oxidation of S02 (see text).

Figure 22. Spectral data in the frequency region 900-3700cm-1 from a mixture containing 03 (3 ppm) and tetramethylethylene (7 ppm) in 700Torr of air. (A) synthesized spectrum corresponding to the initial reactant mixture prior to the reaction (B) recorded after approximately 1 min of mixing and (C) residual spectrum from (B) above.

Figure 24. Product spectra from the dark reaction of N03 with TME in the presence of N02 and air. (A) mixture containing N2Os (1.4 ppm), N02 (4.0 ppm), and HN03 (0.7 ppm) (B) addition of TME (2 ppm) to (A) above and (C) residual spectrum from (B).

Fig. 12.7. Analysis of Raman spectra obtained in vitro for Bruch s membrane, showing component spectra (a) and (b) for heme and collagen, respectively, and a residual spectrum (c) assigned to Raman responses of age-related glycation end products and proteins. Adapted from [29]...

X-variables. This leads to the presence of model residuals (E in Equations 8.19 and 8.35). The residuals of the model can be used to indicate the nature of unmodeled information in the calibration data. For process analytical spectroscopy, plots of individual sample residuals versus wavelength ( residual spectra ) can be used to provide some insight regarding chemical or physical effects that are not accounted for in the model. In cases where a sample or variable outlier is suspected in the calibration data, inspection of that sample or variable s residual can be used to help determine whether the sample or variable should be removed from the calibration data. When a model is operating on-line, the X-residuals of prediction (see Equation 8.55) can be used to determine whether the sample being analyzed is appropriate for application to a quantitative model (see Section 8.4.3). In addition, however, one could also view the prediction residual vector ep as a profile (or residual spectrum ) in order to provide some insight into the nature of the prediction sample s inappropriateness. 

Fig. 24. Results from the fit of a single voxel from a. /-refocused PRESS MRSI data set acquired at 4.1 T with 40 ms TE. (a) The top spectrum is the spectral data (thin line) overlaid with the fit result (heavy line), the middle spectrum is spectral data with only the baseline fit overlaid, and the bottom spectrum is the residual spectrum, (b) An expanded plot of the metabolite signal only (with baseline fit subtracted) in the top spectrum and the summed metabolite fits (without baseline) in the middle spectrum. The bottom spectrum contains the labelled, individual metabolite fits, with the vertical scale increased by 2. Reproduced with permission from B. J. Soher, K. Young, V. Govindaraju and A. A. Maudsley, Magn. Reson. Med., 1998, 40, 822. 1998 John Wiley and Sons.

If the contamination is great enough, the total variance in an unknown spectrum explained by the eigenvectors will be significantly less than 99.9%. This is easily demonstrated by calculating and plotting an unknown sample s residual spectrum using k factors. 

In Equation 4.43, ru is the sample s residual vector or residual spectrum, a is the pretreated spectrum, and the quantity tuVT is the unknown sample s reproduced spectrum. The scores for the spectrum must be determined from Equation 4.42 using the basis vectors V determined from a data set that does not contain the contamination. 

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