Parsimonious basis selection in exponential spectral analysis

Abstract

Sums of decaying real exponentials (SDREs) are frequently used in models of time-varying processes. First-order compartmental models are widely employed to describe mass transit in chemical and biological systems. In these models the evolution of compartment concentration versus time is represented as the convolution of an input function with an SDRE. In exponential spectral analysis (ESA) the nonlinear problem of estimating the SDRE rate constants is replaced by the linear estimation of the coefficients of a preselected set of exponential basis functions (EBFs). This work addresses the problem of selecting the number of EBFs and the rate constant of each basis element. Basis dimension is established via model selection, in which approximation error and parameter redundancy are the criteria. The latter is estimated via simulation of the fitted model over multiple noise realizations. A constrained Cramér–Rao lower bound is derived for ESA parameters. The resulting parsimonious ESA algorithm (PESA) ameliorates the inherent problem of non-uniqueness in ESA parameters. Consequently, sets of time series may be compared in a statistically meaningfully way in terms of physically or physiologically significant parameters. PESA is applied to compare the retention of two radiotracers in the artificially perfused rabbit heart.