A Modified Prony Algorithm for Exponential Function Fitting

A modification of the classical technique of Prony for fitting sums of exponential functions to data is considered. The method maximizes the likelihood for the problem (unlike the usual implementation of Prony’s method, which is not even consistent for transient signals), proves to be remarkably effective in practice, and is supported by an asymptotic stability result. Novel features include a discussion of the problem parametrization and its implications for consistency. The asymptotic convergence proofs are made possible by an expression for the algorithm in terms of circulant divided difference operators.

[1]  A. Krall.  Applied Analysis , 1986 .

[2]  Linda Kaufman,et al.  A Variable Projection Method for Solving Separable Nonlinear Least Squares Problems , 1974 .

[3]  Petre Stoica,et al.  Study of the statistical performance of the Pisarenko harmonic decomposition method , 1988 .

[4]  Robert J. Mulholland,et al.  State-variable canonical forms for Prony's method , 1986 .

[5]  Gordon K. Smyth,et al.  Coupled and separable iterations in nonlinear estimation , 1985 .

[6]  M. R. Osborne,et al.  On the consistency of Prony's method and related algorithms , 1992 .

[7]  W. Wiscombe,et al.  Exponential-sum fitting of radiative transmission functions , 1977 .

[8]  R. D. Murphy,et al.  Iterative solution of nonlinear equations , 1994 .

[9]  Gordon K. Smyth,et al.  A Modified Prony Algorithm for Fitting Functions Defined by Difference Equations , 1991, SIAM J. Sci. Comput..

[10]  Yoram Bresler,et al.  Exact maximum likelihood parameter estimation of superimposed exponential signals in noise , 1986, IEEE Trans. Acoust. Speech Signal Process..

[11]  Hideaki Sakai,et al.  Statistical analysis of Pisarenko's method for sinusoidal frequency estimation , 1984 .

[12]  Hamza Ouibrahim,et al.  Prony, Pisarenko, and the matrix pencil: a unified presentation , 1989, IEEE Trans. Acoust. Speech Signal Process..

[13]  M. R. Osborne,et al.  A modified prony algorithm for estimating sinusoidal frequencies , 1994 .

[14]  C. K. Yuen,et al.  Digital spectral analysis , 1979 .

[15]  Randall J. LeVeque,et al.  On least squares exponential sum approximation with positive coefficients , 1980 .

[16]  M. R. Smith,et al.  Decomposition of Multicomponent Exponential Decays by Spectral Analytic Techniques , 1976 .

[17]  D. Kammler.  Least Squares Approximation of Completely Monotonic Functions by Sums of Exponentials , 1979 .

[18]  Douglas M. Bates,et al.  Nonlinear Regression Analysis and Its Applications , 1988 .

[19]  M. R. Osborne.  Some Special Nonlinear Least Squares Problems , 1975 .

[20]  Gordon K. Smyth,et al.  Partitioned algorithms for maximum likelihood and other non-linear estimation , 1996, Stat. Comput..

[21]  R. Barham,et al.  An Algorithm for Least Squares Estimation of Nonlinear Parameters When Some of the Parameters Are Linear , 1972 .

[22]  Richard Bellman,et al.  Introduction to Matrix Analysis , 1972 .

[23]  V. Kroupa.  Digital spectral analysis , 1983, Proceedings of the IEEE.

[24]  M. R. Osborne.  Nonlinear least squares — the Levenberg algorithm revisited , 1976, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.

[25]  Steven Kay,et al.  Modern Spectral Estimation: Theory and Application , 1988 .

[26]  J. Varah.  A Spline Least Squares Method for Numerical Parameter Estimation in Differential Equations , 1982 .

[27]  L. Kaufman.  A variable projection method for solving separable nonlinear least squares problems , 1974 .

[28]  M. Benson,et al.  Parameter fitting in dynamic models , 1979 .

[29]  R. Jennrich.  Asymptotic Properties of Non-Linear Least Squares Estimators , 1969 .

[30]  Liara Aparecida dos Santos Leal,et al.  Diagonalization and spectral decomposition of factor block circulant matrices , 1988 .

[31]  Roger D. Nussbaum,et al.  Circulant matrices and differential-delay equations , 1985 .

[32]  Emil Pelikán.  Spectral analysis of ARMA processes by Prony's method , 1984, Kybernetika.

[33]  James M. Ortega,et al.  Iterative solution of nonlinear equations in several variables , 2014, Computer science and applied mathematics.

[34]  Hans D. Mittelmann,et al.  Multi-Grid Solution of Bifurcation Problems , 1985 .

[35]  Alan C. Wilde.  Differential equations involving circulant matrices , 1983 .

[36]  A. E. Gilmour.  Circulant matrix methods for the numerical solution of partial differential equations by FFT convolutions , 1988 .

[37]  C. Lanczos.  Applied Analysis , 1961 .

[38]  Axel Ruhe.  Fitting Empirical Data by Positive Sums of Exponentials , 1980 .

[39]  Haluk Derin,et al.  Estimating Components of Univariate Gaussian Mixtures Using Prony's Method , 1987, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[40]  E. A. Sylvestre,et al.  Elimination of Linear Parameters in Nonlinear Regression , 1971 .

[41]  David G. Cantor,et al.  On Approximation by Positive Sums of Powers , 1970 .

[42]  Debasis Kundu,et al.  Estimating the Parameters of Undamped Exponential Signals , 1993 .

[43]  J. Varah.  On Fitting Exponentials by Nonlinear Least Squares , 1982 .