Statistical model of overlap

Davis, J.M. (1997a). Justification of probability density function for resolution in statistical models of overlap. Chromatographia 44, 81. 

Rowe, K., Davis, J.M. (1995). Relaxation of randomness in two-dimensional statistical model of overlap theory and verification. Anal. Chem. 67, 2981. 

Fig. 1. Statistical model of overlap. The rectangular boxes of the gridwork correspond to the peak capacity of a 2D system with Gaussian distribution, in this case n equals approximately 160 (n x n,).

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. 

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. 

The statistical prediction error is in concentration units and represents the uncertainty in the predicted concentrations due to deviations from the model assumptions, measurement noise, and degree of overlap of the pure spectra. As the system deviates from the underlying assumptions of CLS, the residual... 

It is noteworthy that comparisons of existing assessment schemes reveal dissimilarities in the use of extrapolation methods and their input data between different jurisdictions and between prospective and retrospective assessment schemes. This is clearly apparent from, for example, a set of scientific comparisons of 5% hazardous concentration (HC5) values for different substances. Absolute HC5 values and their lower confidence values were different among the different statistical models that can be used to describe a species sensitivity distribution (SSD Wheeler et al. 2002a). As different countries have made different choices in the prescribed modeling by SSDs (regarding data quality, preferred model, etc.), it is clear that different jurisdictions may have different environmental quality criteria for the same substance. Considering the science, the absolute values could be the same in view of the fact that the assessment problem, the available extrapolation methods, and the possible set of input data are (scientifically) similar across jurisdictions. When it is possible, however, to look at the confidence intervals, the numerical differences resulting from different details in method choice become smaller because confidence intervals show overlap. 

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. 

For situations of overlapping chains, where lateral fluctuations in the segment concentration become rather small, mean-field descriptions become appropriate. The most successful of this type of theoiy is the lattice model of Scheutjens and Fleer (SF-theoiy). In chapter II.5 some aspects of this model were discussed. This theory predicts how the adsorbed amount and the concentration profile 0(z) depend on the interaction parameters and x and on the chain length N. From the statistical-thermodynamic treatment the Helmholtz energy and, hence, the surface pressure ti can also be obtcdned. When n is expressed as a function of the profile 0(z), the result may be written as ... 

In Table 1, we list the irreducible ring statistics for two crystal structures, FC-2 and BC-8, and for four models of amorphous silicon. When a bond-pair switch is introduced into the otherwise perfect FC-2 structure, the number of irreducible rings is conserved. Four 5-folds and eight 7-folds are created, but twelve 6-folds are eliminated. This conservation rule holds until the regions of bond-switching overlap, and is not grossly violated even then in the randomization process. Note that the total number of irreducible rings per atom is exactly two for the FC-2 structure and it is almost two for all the amorphous structures. 

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