Analysis of 1st‐Order Rate Constant Spectra With Regularized Least‐Squares and Expectation Maximization: 1. Theory and Numerical Characterization

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Analytical Chemistry

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Analysis of parallel, first-order rate processes by deconvolutlon of slngle-exponential kernels from experlmentai data is performed with regularized least squares and the method of expectation maximlzatlon (EM). These methods may be used in general for the unbiased numerical analysis of ilnear Fredholm integrals of the first kind with optimal results. Regularized least squares is performed uslng a smoothing regularizor with an adaptive choice for the regularization parameter (CONTIN) and by ridge regression uslng the generalized cross-validation choice for the regularlzation parameter (GCV). The resolution and performance of the methods are studied as a function of data type (continuous or discrete dlstrlbutbns of single exponentials), data sampling, and superimposed noise. Ail three methods are able to yield hlgh-resolution estimates and are statlstically valid. However, subtle differences dependent on the data exist that suggest that the most probablistic estimate, or maximum Iikelihood estimate, is dependent on the uitlmate valldlty of the spectfic model used to describe the data. Therefore, qualitative comparison of the three methods In terms of maximum entropy is conddered for “worst case” ilmiting data. For discrete distributions comprising data of hlgh slgnal-to-noise ratio (SNR), the order EM > CONTIN > GCV is observed for the entropy of the solutions. For continuous distributions of hlgh SNR, the order EM > GCV > CONTIN is observed. For either type of underlying distribution and low SNR, the three methods converge to comparable performance whlle breaklng down In terms of the quality and accuracy of the estimations. The EM algorlthm Is suggested as the maximum Ilkellhood (or maximum entropy) method when a high response to model error is not desired. The GCV algorithm yields a maximum likelihood estimate hlghly dependent on the model valldlty. The CONTIN algorithm provides a compromlse between the two.

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