Statistical resolution probability

Multiple Pass Analysis. Pike and coworkers (13) have provided a method to increase the resolution of the ordinary least squares algorithm somewhat. It was noted that any reasonable set of assumed particle sizes constitutes a basis set for the inversion (within experimental error). Thus, the data can be analyzed a number of times with a different basis set each time, and the results combined. A statistically more-probable solution results from an average of the several equally-likely solutions. Although this "multiple pass analysis".helps locate the peaks of the distribution with better resolution and provides a smoother presentation of the result, it can still only provide limited resolution without the use of a non-negatively constrained least squares technique. We have shown, however, that the combination of both the non-negatively constrained calculation and the multiple pass analysis gives the advantages of both. 

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

Both resolution and statistical analyses of EM radioautographs have been discussed in depth by Evans and Callow (3) and Morel (7). Table 2 reveals that reducing section thickness as well as diameters of halide crystals and developed grains improve resolution of EM radioautographs. With regard to statistical analysis, Morel (7) considers the probability-circle method and cross-fire method for the quantitative analysis of EM autoradiographs. 

Let us calculate the maximum number N of compounds acceptable if we require p = 90 %, i.e. 9 chances out of 10 to have only one compound per peak, in the case of a resolution of 1000. We find N = 14. Of these 14 peaks, statistically 1 out of 10 will contain two unseparated compounds. This example shows that the statistical probability of having two compounds per peak is very high. Figure 5.1 shows these results graphically, from a more elaborate statistical analysis [7]. 

While the frequency of conglomerate-forming systems is fairly small for organic compounds, a statistical analysis of more than 500 salts has demonstrated that the probability of observing spontaneous resolution is... 

Collins, D. M. Extrapolative filtering. I. Maximization of resolution for onedimensional positive density functions. Acta Cryst. A34, 533-541 (1978). Bricogne, G. A Bayesian statistical theory of the phase problem. I. A multichannel maximum-entropy formalism for constructing generalized joint probability distributions of structure factors. Acta Cryst. A44, 517-545 (1988). 

In order to establish a clear correlation between the two parameters studied in Fig. 8, namely linewidth and anisotropy, we plotted in Fig. 9a the emission anisotropy and ZPL for 80 single (3-phase chains (circles). A clear correlation is observed with a statistical coefficient of -0.5, which translates into a probability of >99% to have these two variables correlated. The diamonds are obtained by averaging the linewidth of molecules within the same anisotropy interval of 0.1. Therefore, straight chromophores show linewidths which reach minimum values of 400 ieV, of the order of the resolution limit of the spectrometer used. 

Indeed, as the resolution of the nuclear Schrodinger equation can only be performed for very simple systems, one can often use a classical approach where nuclei are no longer considered as quantum particles but as classical ones moving on the potential energy hypersurface. Then, we may calculate reaction probabilities which are related to the reaction rate constant by an equation deduced in statistical mechanics. This can be formally written ... 

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