ANALYSIS OF 1ST-ORDER RATE-CONSTANT SPECTRA WITH REGULARIZED LEAST-SQUARES AND EXPECTATION MAXIMIZATION .1. THEORY AND NUMERICAL CHARACTERIZATION

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 regularizor 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. AN three methods are able to yield high-resolution 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 cass'' limiting data. For discrete distributions comprising data of high signal-to-noise 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.