Practical Aspects of the Maximum Entropy Inversion of the Laplace Transform for the Quantitative Analysis of Multi-Exponential Data

Abstract

The number, position, area, and width of the bands in a lifetime distribution give the number of exponentials present in time-resolved data and their time constants, amplitudes, and heterogeneities. The maximum entropy inversion of the Laplace transform (MaxEnt-iLT) provides a lifetime distribution from time-resolved data, which is very helpful in the analysis of the relaxation of complex systems. In some applications both positive and negative values for the lifetime distribution amplitudes are physical, but most studies to date have focused on positive-constrained solutions. In this work, we first discuss optimal conditions to obtain a sign-unrestricted maximum entropy lifetime distribution, i.e., the selection of the entropy function and the regularization value. For the selection of the regularization value we compared four methods: the χ² criterion and Bayesian inference (already used in sign-restricted MaxEnt-iLT), and the L-curve and the generalized cross-validation methods (not yet used in MaxEnt-iLT to our knowledge). Except for the frequently used χ² criterion, these methods recommended similar regularization values, providing close to optimum solutions. However, even when an optimal entropy function and regularization value are used, a MaxEnt lifetime distribution will contain noise-induced errors, as well as systematic distortions induced by the entropy maximization (regularization-induced errors). We introduce the concept of the apparent resolution function in MaxEnt, which allows both the noise and regularization-induced errors to be estimated. We show the capability of this newly introduced concept in both synthetic and experimental time-resolved Fourier transform infrared (FT-IR) data from the bacteriorhodopsin photocycle.