Skip to main content
SpringerLink
Log in

Guaranteed Parameter Set Estimation for Exponential Sums: The Three-Terms Case

  • Published:
Reliable Computing

Abstract

In this paper the problem of parameter estimation for exponential sums with three terms is considered. This task consists of finding the set of parameters (amplitudes as well as decay constants) such that the exponential sum attains values in specified intervals at prescribed time data points. These intervals represent uncertainties in the measurements. An interval variant of Prony’s method is given by which intervals can be found containing all the consistent values of the respective parameters. By the use of interval arithmetic these enclosures can also be guaranteed in the presence of rounding errors.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Anderson, D. H.: Compartmental Modeling and Tracer Kinetics, Lect. Notes Biomaths. 50, Springer-Verlag, Berlin, Heidelberg, New York, 1983.

  2. Bareiss E.H. (1969) Numerical Solution of Linear Equations with Toeplitz and Vector Toeplitz Matrices. Numer. Math. 13, 404–424

    Article  MATH  MathSciNet  Google Scholar 

  3. Blahut R.S. (1985) Fast Algorithms for Digital Signal Processing. Addison-Wesley Publishing Company, Reading

  4. Esposito, W. R. and Floudas, C. A.: Global Optimization in Parameter Estimation of Nonlinear Algebraic Models via the Error-in-Variables Approach, Industrial and Engineering Chemistry Research 37, (1998), pp. 1841–1858.

  5. Esposito W.R., Floudas C.A. (1998) Parameter Estimation of Nonlinear Algebraic Models via Global Optimization. Computers and Chemical Engineering 22, 213–220

    Article  Google Scholar 

  6. Garloff J. (1986) Solution of Linear Equations Having a Toeplitz Interval Matrix as Coefficient Matrix. Opuscula Mathematica 2, 33–45

    MathSciNet  Google Scholar 

  7. Garloff J. (1993) The Bernstein Algorithm. Interval Computations 2, 154–168

    MathSciNet  Google Scholar 

  8. Garloff J., Granvilliers L., Smith A.P. (2005). Accelerating Consistency Techniques and Prony’s Method for Reliable Parameter Estimation of Exponential Sums. In: Jermann C., Neumaier A., Sam D. (eds) Global Optimization and Constraint Satisfaction, Lect Notes Comp Sci 3478. Springer-Verlag, Berlin Heidelberg, pp. 31–45

    Google Scholar 

  9. Gau C.-Y., Stadtherr M.A. (1999) Nonlinear Parameter Estimation Using Interval Analysis. AIChE Symp. Ser. 94 (304): 444–450

    Google Scholar 

  10. Gibaldi M., Perrier D.(1982). Pharmacokinetics. Marcel Dekker, New York

    Google Scholar 

  11. Goodman D.K. (1970). Determination of the Number of Terms Necessary for a Class of Approximation Procedures. In: Anderssen R.S., Osborne M.R. (eds) Data Representation. University of Queensland Press, St Lucia, pp. 77–93

    Google Scholar 

  12. Goodman, D. K.: On a Class of Order Determination Problems, PhD thesis, University of New South Wales, Kensington, 1970.

  13. Goodman, D. K. and Hiller, J.: On the Fitting of Curves Specified in an Interval Sense, in: Lin, Shu (ed.), Proceedings of the Fourth Hawaii International Conference on System Sciences, Western Periodicals, North Hollywood, 1971.

  14. Granvilliers L., Cruz J., Barahona P. (2004) Parameter Estimation Using Interval Computations. SIAM Journal on Scientific Computing 26 (2):591–612

    Article  MATH  MathSciNet  Google Scholar 

  15. Henrici P. (1974). Applied and Computational Complex Analysis, Vol I. John Wiley & Sons, New York, London, Sydney

    Google Scholar 

  16. Hofer, E. P., Tibken, B., and Vlach, M.: Traditional Parameter Estimation Versus Estimation of Guaranteed Parameter Sets, in: Krämer, W. and Wolff von Guddenberg, J. (eds), Scientific Computing, Validated Numerics, Interval Methods, Kluwer Academic Publishers, Boston, Dordrecht, London, 2001, pp. 241–254.

  17. Jaulin L. (2000) Interval Constraint Propagation with Application to Bounded-Error Estimation. Automatica 36, 1547–1552

    Article  MATH  MathSciNet  Google Scholar 

  18. Jaulin L., Kieffer M., Didrit O., Walter É (2001). Applied Interval Analysis. Springer, London, Berlin, Heidelberg

    MATH  Google Scholar 

  19. Jaulin L., Walter É (1993) Set Inversion via Interval Analysis for Nonlinear Bounded-Error Estimation. Automatica 29(4): 1053-1064

    Article  MATH  MathSciNet  Google Scholar 

  20. Julius R.S. (1972) The Sensitivity of Exponentials and Other Curves to Their Parameters. Computers and Biomedical Research 5, 473–478

    Article  Google Scholar 

  21. Karlin S. (1968). Total Positivity, Vol I. Stanford University Press, Stanford

    Google Scholar 

  22. Lanczos C. (1956). Applied Analysis. Prentice Hall, Englewood Cliffs

    Google Scholar 

  23. Neumaier, A.: Interval Methods for Systems of Equations, Cambridge University Press (Encyclopedia of Mathematics and Its Applications), Cambridge, 1990.

  24. Popova E.D. (2004) Parametric Interval Linear Solver. Numerical Algorithms 37, 345–356

    Article  MATH  Google Scholar 

  25. Prony, R.: Essai experimental et analytique sur les lois de la dilatabilité edes fluides élastiques et sur celles de la force expansive de la vapeur de l’eau et de la vapeur de l’alkool, à différentes températures, Journal de l’Ecole Polytechnique 1 (2) (1795), pp. 24–76.

  26. Rokne J. (1972) Explicit Calculation of the Lagrangian Interval Interpolating Polynomial. Computing 9, 149–157

    Article  MATH  MathSciNet  Google Scholar 

  27. Sheppard C.W., Householder A.S. (1951) The Mathematical Basis of the Interpretation of Tracer Experiments in Closed Steady-State Systems. Journal of Applied Physics 22, 510–520

    Article  MATH  MathSciNet  Google Scholar 

  28. Trench W.F. (1964) An Algorithm for the Inversion of Finite Toeplitz Matrices. J. SIAM 12 (3): 515–522

    MATH  MathSciNet  Google Scholar 

  29. Vaughan D.P., Dennis M.J. (1979) Number of Exponential Terms Describing the Solution of an N-Compartmental Mammillary Model: Vanishing Exponentials. Journal of Pharmacokinetics and Biopharmaceutics 7 (5): 511–525

    Article  Google Scholar 

  30. Weiss M. (1990). Theoretische Pharmakokinetik. Verlag Gesundheit, Berlin

    Google Scholar 

  31. Weiß, C.: Einschließung der Nullstellenmengen von Intervallpolynomen kleinen Grades, Diploma thesis, University of Applied Sci./FH Konstanz, Fac. of Comp. Sci, 2004.

  32. Zettler M., Garloff J. (1998) Robustness Analysis of Polynomials with Polynomial Parameter Dependency. IEEE Trans. Automat. Control 43, 425–431

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute for Applied Research, University of Applied Sciences / HTWG Konstanz, Postfach 100543, D–78405, Konstanz, Germany

    Jürgen Garloff, Ismail Idriss & Andrew P. Smith

Authors
  1. Jürgen Garloff
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ismail Idriss
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Andrew P. Smith
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jürgen Garloff.

Additional information

Support from the Ministry of Education and Research of the Federal Republic of Germany under contract no. 1705803 and from the DAAD program PROCOPE under contract no. D/0205730 is gratefully acknowledged.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Garloff, J., Idriss, I. & Smith, A.P. Guaranteed Parameter Set Estimation for Exponential Sums: The Three-Terms Case. Reliable Comput 13, 351–359 (2007). https://doi.org/10.1007/s11155-007-9034-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11155-007-9034-9

Keywords

Search

Navigation