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Analysis of parallel, first-order rate processes by deconvolution of single-exponential kernels from experimental data Is performed with regularized least squares and the method of expectation maximization (EM). These methods may be used In general for the unbiased numerical analysis of linear Fredholm Integrals of the first kind with optimal results. Regularized least squares Is performed using a smoothing regularlzor with an adaptive choice for the regularization parameter (CONTIN) and by ridge regression using the generalized cross-validation choice for the regularization parameter (GCV). The resolution and performance of the methods are studied as a function of data type (continuous or discrete distributions of single exponentials), data sampling, and superimposed noise. All three methods are able to yield hlgh-resokition estimates and are statistically valid. However, subtle differences dependent on the data exist that suggest that the most probabilistic estimate, or maximum likelihood estimate, Is dependent on the ultimate validity of the specific model used to describe the data. Therefore, qualitative comparison of the three methods In terms of maximum entropy Is considered for "worst case" limiting data. For discrete distributions comprising data of high slgnal-to-nolse ratio (SNR), the order EM > CONTIN > GCV Is observed for the entropy of the solutions. For continuous distributions of high SNR, the order EM > GCV > CONTIN Is observed. For either type of underlying distribution and low SNR, the three methods converge to comparable performance while breaking down In terms of the quality and accuracy of the estimations. The EM algorithm Is suggested as the maximum likelihood (or maximum entropy) method when a high response to model error Is not desired. The GCV algorithm yields a maximum likelihood estimate highly dependent on the model validity.The CONTIN algorithm provides a compromise between the two.

The expectation-maximization (EM) method of parameter estimation is used to calculate adsorption energy distributions of molecular probes from their adsorption isotherms. EM does not require prior knowledge of the distribution function or the isotherm, requires no smoothing of the isotherm data, and converges with high stability towards the maximum-likelihood estimate. The method is therefore robust and accurate at high iteration numbers. The EM "algorithm is tested with simulated energy distributions corresponding to unimodal Gaussian, bimodal Gaussian, Poisson distributions, and the distributions resulting from Misra isotherms. Theoretical isotherms are generated from these distributions using the Langmuir model, and then chromatographic band profiles are computed using the ideal model of chromatography. Noise is then introduced in the theoretical band profiles comparable to those observed experimentally. The isotherm is then calculated using the elution-by-characteristic points method. The energy distribution given by the EM method is compared to the original one. The results are contrasted to those obtained with the House and Jaycock algorithm HILDA, and shown to be superior in terms of both robustness, accuracy, and information theory. The effect of undersampling of the high-pressure/low-energy region of the adsorption is reported and discussed for the EM algorithm, as well as the effect of signal-to-noise ratio on the degree of heterogeneity that may be estimated experimentally.