Numerical estimation of adsorption energy distributions from adsorption isotherm data with the expectation-maximization method

Abstract

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.

Characterization of binding sites in molecularly imprinted polymers

“…The other analytical method, proposed by Guiochon and co-workers [1,10,[52][53][54][55][56], is called the expectation maximation (EM) method and has been mainly used in the characterization of MIPs developed for enantiomers separations [10][11][12][13]16,50]. The method is very simple as in this case the energy affinity distribution is obtained exclusively from the adsorption isotherm experimental values, without any previous assumption on the binding model.…”

“…These models make use of either analytically simple approximations previously employed in solving similar algebraic problems in other fields [48] or the well known numerical approaches for solving integral equations. The affinity distribution analysis using the simple approximations were explored by the group of Shimizu which developed the affinity spectrum (AS) approach to characterize the MIP's distribution of binding sites [33,34,38,40,49] while the groups of Guiochon, Spivak and Stanley [50][51][52][53][54][55][56][57][58] have proposed the expectation maximization (EM) model making use of numerical approaches. Both models are known formerly under the name affinity distribution (AD) or affinity energy distribution (AED) methods as they provide a mathematical picture of the surface binding site distribution.…”

Characterization of the adsorption site energies and heterogeneous surfaces of porous materials

“…Stanley and Guiochon suggested the application of an Expectation-Maximization (EM) algorithm as a robust method to obtain the site energy distribution function from the inverse equation of the fundamental equation of the adsorption of heterogeneous surfaces. 72 This method does not require any prior knowledge of the adsorption isotherm model describing the overall adsorption data, nor any smoothing of the experimental data. The EM method is applied directly to the raw experimental data and therefore involves a low amount of artifactual information from a numerical standpoint, and so provides a relatively unbiased solution.…”

“…The amount q(p j ) at concentration p j is iteratively determined. At the k th step of the iteration process, the amount adsorbed is given by: 72 ; j E [1, M]…”

“…The use of the above equation requires a set of initial estimation of F(E i ) as the method maximizes the influence of data over the final distribution. The initial guess used to determine the site energy distribution function is based on the 'total ignorance' guess: [72][73][74] (57)…”