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If the individual profiles are not known, a deconvolution of overlapping profiles is still possible. However, assumptions of a model for the peak shape must be made. A simple model is the gaussian one, which is defined by a standard deviation (or width) and a central moment, in our case the retention time. Because the gaussian model itself is a nonlinear function, fitting a combination of unknown gaussian profiles to an observed chromatographic profile is usually performed by means of a nonlinear least-squares procedure, for instance, the Marquardt al-gorithm . Such an algorithm searches for the optimal combination of the parameters (peak widths and retention times for all components) by evaluating the change in a least-squares criterion [see Eq. (21)] as a function of changes in the parameter values. The as-sumption of a symmetrical gaussian peak shape, however, can introduce significant errors in the estimated values for plate count, and resolution. More complex, but more realistic, peak shapes have also been used, such as the exponentially modified gaussian profile (EMG) [133,134]. Obviously, the introduction of additional degrees of freedom in the shapes of the profile (i.e., the asymmetry introduced by the tailing factor in the EMG) further complicates the curve-fitting procedures.
Many of the techniques applied to chromatograms are used similarly in other domains, most notably on UV spectra obtained by scanning or photodiode-array UV detectors. Both smoothing and filtering have been applied, as well as spectral deconvolution using the spectra of the pure compounds (see also the next section). Because the distinctive features of UV spectra are somewhat limited, various methods have been applied to enhance minor features in the spectra. Examples include difference and second-derivative spectroscopy [135,1361. Again, higher-order derivatives can easily be calculated by means of convolution functions.
The use of UV spectra to identify LC peaks has created a need for library searching, for the retrieval of identical or similar spectra from a collection. Again, because of the limited number of distinctive features in UV spectra, the libraries will generally be smaller than the extensive (commercial) libraries available for infrared (IR) and nuclear magnetic resonance
Figure 10 Two gaussian profiles (dashed lines) fitted to an observed chromatographic profile (solid line, top panel) to estimate the contributions of the individual components. The bottom panel shows the minimized difference dA between the observed profile and the sum of the two gaussian profiles.
(NMR) spectroscopy and for mass spectrometry (MS). The UV spectral libraries are usually dedicated to a specific sub-set of chemical substances. Thus, the emphasis on search algorithms is shifted from the efficient screening of large numbers of spectra, to a thorough comparison of complete spectra. The often minute differences between similar compounds need to be enhanced. The retrieval of UV spectra from a library for the identification LC peaks is further complicated because the experimental conditions can significantly impinge on the spectral characteristics. For instance, the type of organic modifier or the pH of the mobile phase can affect the location and intensities of the bands in the spectra. Library spectra have to be
Computers and Liquid Chromatography
recorded under similar circumstances to permit a correct identification. An excellent overview of the different ways to compare UV spectra and of the factors affecting the results is available .
A comparison of UV spectra usually involves some kind of point-to-point comparison of normalized spectra. The normalization has to take place to remove concentration effects. The results of the comparison are then summarized in one number to express the similarity (or dissimilarity) of the spectra. Often the similarity between spectra is expressed by means of a correlation coefficient [see e.g., Ref. 138]. Its value is calculated from the line resulting from a linear regression through the points representing the absorbances at the same wavelengths in the two spectra x and ó values. Thus, the correlation coefficient r is given by:
Si=l («I «avg) (^1 ^avg)
r = -p......— ^ (23)
(«,¦ - <w2 27-1 ¹, ~ *..g)2
where at and bt represent the absorbance observed in the two spectra at wavelength i, and aavg and ba„s are the average absorbance values for each spectrum.
Another way to visualize this comparison is by representing the UV spectra as vectors in an /ã-dimensional space, where n equals the number of observations or wavelengths in the spectra. Each coordinate corresponds to the absorbance at a given wavelength. The comparison of two spectra is now equivalent to determining the angle between the two vectors. If the spectra are identical, this angle will be zero. The larger the difference, the farther the vectors will be apart and the larger the angle will be. The angle can be calculated from the dot product: