Analysis of first-order rate constant spectra with regularized least-squares and expectation maximization. 1. Theory and numerical characterization

Abstract

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.

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.…”

Potential of adsorption isotherm measurements for closer elucidating of binding in chiral liquid chromatographic phase systems

“…We will see that calculation of the AED is an important step in the adsorption isotherm model discrimination process. The AED could be solved by many different methods [38,39].…”

Kinetics and Equilibrium of Cadmium Complexation with Heterogeneous Complexants: A Stopped-Flow Study at Na-Illite

“…In the last few years, different numeri-THEORY cal techniques have been applied on such problems like the Equilibrium ion exchange of Cd 2/ at montmorillonite (8) or the dissociation of Cu 2/ from humic acids (9). Additionally these algoAt minerals with a heterogeneous surface, many different rithms have been analyzed theoretically (10). If kinetic types of binding sites Z i exist which may bind metal ions.…”

“…[10], which is the difference between the molarity of Cd 2/ in equilibrium at time t and with the total molarity of …”