An eigenfunction expansion method for the analysis of exponential decay curves

TLDR

The method addresses finite‑time data by exactly accounting for the time range, avoiding extrapolation or iteration, unlike the Fourier transform solution. The authors develop a method to analyze data consisting of random noise, an unknown baseline, and a sum (or integral) of exponential decay functions. The approach expands the solution of a Fredholm integral equation of the first kind in the kernel’s eigenfunctions, automatically handling irregular time intervals and requiring no initial estimates, with parameters determined by a linear hypothesis test corrected for nonlinearity. A computer program implementing the method is available, and simulations of two‑, three‑, and four‑component data with pseudorandom errors demonstrate its broad applicability.

Abstract

A method is developed for the analysis of data composed of random noise, plus an unknown constant ’’baseline,’’ plus a sum (or an integral over a continuous distribution) of exponential decay functions. It is based on the expansion of the solution of a Fredholm integral equation of the first kind in the eigenfunctions of the kernel. In contrast to the Fourier transform solution [Gardner et al., J. Chem. Phys. 31, 978 (1959)], the finite time range of the data is exactly accounted for, and no extrapolation or iteration is necessary. A computer program is available for the analysis of sums of exponentials. It is completely automatic in that the only input are the data (not necessarily in equal intervals of time); no potentially biased initial estimates of either the number or values of the amplitudes and decay constants are needed. These parameters and their standard deviations are decided with a linear hypothesis test corrected approximately for nonlinearity. Tests with simulated two-, three-, and four-component data containing pseudorandom errors indicate that the method has a wide range of applicability.