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A review of the parameter estimation problem of fitting positive exponential sums to empirical data. (English) Zbl 1023.65009

Summary: Exponential sum models are used frequently: in heat diffusion, diffusion of chemical compounds, time series in medicine, economics, physical sciences and technology. Thus it is important to find good methods for the estimation of parameters in exponential sums.
In this paper we review and discuss results from the last forty years of research. There are many different ways of estimating parameters in exponential sums and model of fit criterion, which gives a valid result from the fit. We find that a good choice is a weighted two-norm objective function, with weights based on the maximum likelihood (ML) criterion. If the number of exponential terms is unknown, statistical methods based on an information criterion or cross-validation can be used to determine the optimal number.
It is suitable to use hybrid Gauss-Newton and quasi-Newton algorithm to find the unknown parameters in the constrained weighted nonlinear least-squares problem formulated using an maximal likelihood objective function. The problem is highly ill conditioned and it is crucial to find good starting values for the parameters. To find the initial parameter values, a modified Prony method or a method based upon rewriting partial sums as geometrical sums is proposed.

MSC:

65D10 Numerical smoothing, curve fitting
65C60 Computational problems in statistics (MSC2010)
62F10 Point estimation
11L03 Trigonometric and exponential sums (general theory)
11Y60 Evaluation of number-theoretic constants
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