Overlap, component, statistical

Davis, J.M., Giddings, J.C. (1983). Statistical theory of component overlap in multicomponent chromatograms. Anal. Chem. 55, 418-424. 

Statistical Theory of Component Overlap in Multicomponent Chromatograms, J. M. Davis and J. C. Giddings, Anal. Chem., 55, 418 (1983). 

Test of the Statistical Model of Component Overlap by Computer-Generated Chromatograms, J. C. Giddings, J. M. Davis, and M. R. Schure, in S. Ahuja, Ed., Ultrahigh Resolution Chromatography, ACS Symposium Series No. 250, American Chemical Society, Washington, DC, 1984, pp. 9-26. 

Test of the Statistical Model of Component Overlap by Computer-Generated Chromatograms... 

An earlier statistical model of component overlap Is reviewed. It Is argued that a statistical treatment Is necessary to reasonably Interpret and optimize high resolution chromatograms having substantial overlap. 

As samples become more complex, the ability of a particular separation method to resolve all components decreases. A statistical study of component overlap has shown that a chromatogram must be approximately 95% vacant to provide a 90% probability that a given component of interest will appear as an isolated peak (9). This is shown graphically in Fig. 10, where the probability of separation is plotted as a function of the system peak capacity for cases where the number... 

For many applications, quantitative band shape analysis is difficult to apply. Bands may be numerous or may overlap, the optical transmission properties of the film or host matrix may distort features, and features may be indistinct. If one can prepare samples of known properties and collect the FTIR spectra, then it is possible to produce a calibration matrix that can be used to assist in predicting these properties in unknown samples. Statistical, chemometric techniques, such as PLS (partial least-squares) and PCR (principle components of regression), may be applied to this matrix. Chemometric methods permit much larger segments of the spectra to be comprehended in developing an analysis model than is usually the case for simple band shape analyses. 

The limitations of one-dimensional (ID) chromatography in the analysis of complex mixtures are even more evident if a statistical method of overlap (SMO) is applied. The work of Davis and Giddings (26), and of Guiochon and co-workers (27), recently summarized by Jorgenson and co-workers (28) and Bertsch (29), showed how peak capacity is only the maximum number of mixture constituents which a chromatographic system may resolve. Because the peaks will be randomly rather than evenly distributed, it is inevitable that some will overlap. In fact, an SMO approach can be used to show how the number of resolved simple peaks (5) is related to n and the actual number of components in the mixture (m) by the following ... 

Certainly two-dimensional techniques have far greater peak capacity than onedimensional techniques. However, the two-dimensional techniques don t utilize the separation space as efficiently as one-dimensional techniques do. These theories and simulations utilized circles as the basis function for a two-dimensional zone. This was later relaxed to an elliptical zone shape for a more realistic zone shape (Davis, 2005) with better understanding of the surrounding boundary effects. In addition, Oros and Davis (1992) showed how to use the two-dimensional statistical theory of spot overlap to estimate the number of component zones in a complex two-dimensional chromatogram. 

The statistical model of peak overlap clearly explains that the number of observed peaks is much smaller than the number of components present in the sample. The Fourier analysis of multicomponent chromatograms can not only identify the ordered or disordered retention pattern but also estimate the average spot size, the number of detectable components present in the sample, the spot capacity, and the saturation factor (Felinger et al., 1990). Fourier analysis has been applied to estimate the number of detectable components in several complex mixtures. 

Figure 8. Figure (a) after Clayton et al. (1976, 1977). The scales are as in Figure 1. The O isotopic compositions of the different meteorite classes are represented ordinary chondrites (H, L, LL), enstatite chondrites (EFl, EL), differentiated meteorites (eucrites, lAB irons, SNCs) and some components of the carbonaceous chondrites. As the different areas do not overlap, a classification of the meteorites can be drawn based on O isotopes. Cr (b) and Mo (c) isotope compositions obtained by stepwise dissolution of the Cl carbonaceous chondrite Orgueil (Rotaru et al. 1992 Dauphas et al. 2002), are plotted as deviations relative to the terrestrial composition in 8 units. Isotopes are labeled according to their primary nucleosynthetic sources. ExpSi is for explosive Si burning and n-eq is for neutron-rich nuclear statistical equilibrium. The open squares represent a HNOj 4 N leachate at room temperature. The filled square correspond to the dissolution of the main silicate phase in a HCl-EIF mix. The M pattern for Mo in the silicates is similar to the s-process component found in micron-size SiC presolar grains as shown in Figure 7.

There are apparently many multivariate statistical methods partly overlapping in scope [11]. For most problems occurring in practice, we have found the use of two methods sufficient, as discussed below. The first method is called principal component analysis (PCA) and the second is the partial least-squares projection to latent structures (PLS). A detailed description of the methods is given in Appendix A. In the following, a brief description is presented. 

With sufficiently complex samples, particularly biological and environmental samples, the frequency of overlap can be estimated by statistical means. In a statistical model developed by Davis and this author [33], far-reaching conclusions follow from a simple basic assumption the probability that any small interval dx along the separation path x is occupied by a component peak center is A dx, where A is a constant. This assumption defines a Poisson process and leads to well-known statistical conclusions. 

We note that m is a statistical number related to basic constant A it may differ slightly from the true component number m. However, because of peak overlap, the true m cannot be obtained by counting the number of peaks appearing in the chromatogram. Thus m, if obtainable, becomes our best approximation for m. 

With V— 102-5 x 104 and z = 1-10, as above, we see that from 10 to 800 distinct peaks can theoretically be resolved in electrophoresis. (In practice the number of resolvable components is much less than nc due to statistical peak overlap as explained in Section 6.7). This enormous resolving power is consistent with observation (see Figures 8.2 and 8.4). Theory thus provides an explanation of the unusual power of electrophoresis and an insight into the variables (z and V) that must be manipulated for increased performance. 

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