How accurately can parameters from exponential models be estimated? A Bayesian view

Estimating the amplitudes and decay rate constants of exponentially decaying signals is an important problem in NMR. Understanding how the uncertainty in the parameter estimates depends on the data acquisition parameters and on the “true” but unknown values of the exponential signal parameters is an important step in designing experiments and determining the amount and quality of the data that must be gathered to make good parameter estimates. In this article, Bayesian probability theory is applied to this problem. Explicit relationships between the data acquisition parameters and the “true” but unknown exponential signal parameters are derived for the cases of data containing one and two exponential signal components. Because uniform prior probabilities are purposely employed, the results are broadly applicable to experimental parameter estimation. © 2005 Wiley Periodicals, Inc. Concepts Magn Reson Part A 27A: 73–83, 2005

[1]  R. T. Cox,et al.  The Algebra of Probable Inference , 1962 .

[2]  P. Laplace.  A Philosophical Essay On Probabilities , 1902 .

[3]  R. Baierlein.  Probability Theory: The Logic of Science , 2004 .

[4]  G. L. Bretthorst,et al.  Bayesian analysis. V : Amplitude estimation for multiple well-separated sinusoids , 1992 .

[5]  H. Jeffreys,et al.  Theory of probability , 1896 .

[6]  A. Istratov,et al.  Exponential analysis in physical phenomena , 1999 .

[7]  T. Bayes,et al.  Studies in the History of Probability and Statistics: IX. Thomas Bayes's Essay Towards Solving a Problem in the Doctrine of Chances , 1958 .

[8]  G. L. Bretthorst.  Bayesian analysis. I. Parameter estimation using quadrature NMR models , 1990 .

[9]  T. Bayes.  An essay towards solving a problem in the doctrine of chances , 2003 .

[10]  G. Larry Bretthorst.  Generalizing the Lomb-Scargle periodogram , 2001 .

[11]  G. L. Bretthorst.  Bayesian ANALYSIS. IV. Noise and computing time considerations , 1991 .

[12]  G. Larry Bretthorst,et al.  Bayesian analysis. II. Signal detection and model selection , 1990 .

[13]  G. L. Bretthorst,et al.  Site-specific thermodynamics and kinetics of a coiled-coil transition by spin inversion transfer NMR. , 1998, Biophysical journal.

[14]  H. Cramér.  Mathematical methods of statistics , 1947 .

[15]  John Skilling,et al.  Maximum Entropy and Bayesian Methods , 1989 .

[16]  C. R. Rao,et al.  Information and the Accuracy Attainable in the Estimation of Statistical Parameters , 1992 .

[17]  Kenneth Holmström,et al.  A review of the parameter estimation problem of fitting positive exponential sums to empirical data , 2002, Appl. Math. Comput..

[18]  G. L. Bretthorst,et al.  Estimating the Ratio of Two Amplitudes in Nuclear Magnetic Resonance Data , 1992 .

[19]  C. Ray Smith,et al.  Maximum-entropy and Bayesian methods in science and engineering , 1988 .

[20]  G. L. Bretthorst.  Nonuniform sampling: Bandwidth and aliasing , 2001 .

[21]  G. L. Bretthorst.  Generalizing the Lomb-Scargle periodogram—the nonsinusoidal case , 2001 .

[22]  G. Larry Bretthorst,et al.  Bayesian analysis. III. Applications to NMR signal detection, model selection, and parameter estimation , 1990 .

[23]  G. Bromage.  A QUANTIFICATION OF THE HAZARDS OF FITTING SUMS OF EXPONENTIALS TO NOISY DATA , 1983 .

[24]  R. Shrager,et al.  Some pitfalls in curve-fitting and how to avoid them: a case in point. , 1998, Journal of Biochemical and Biophysical Methods.

[25]  Probability Theory and Multiexponential Signals, How Accurately Can the Parameters be Determined? , 1996 .

[26]  G. Barnard.  STUDIES IN THE HISTORY OF PROBABILITY AND STATISTICS: IX. THOMAS BAYES'S ESSAY TOWARDS SOLVING A PROBLEM IN THE DOCTRINE OF CHANCESReproduced with the permission of the Council of the Royal Society from The Philosophical Transactions (1763), 53, 370-418 , 1958 .

[27]  G. L. Bretthorst.  AN INTRODUCTION TO MODEL SELECTION USING PROBABILITY THEORY AS LOGIC , 1996 .

[28]  G. L. Bretthorst,et al.  Thermodynamics and kinetics of a folded-folded' transition at valine-9 of a GCN4-like leucine zipper. , 1999, Biophysical journal.

[29]  L. M. M.-T..  Theory of Probability , 1929, Nature.

[30]  G. Larry Bretthorst,et al.  Physics and Probability: On the Difference in Means , 1993 .

[31]  G. L. Bretthorst,et al.  AN INTRODUCTION TO PARAMETER ESTIMATION USING BAYESIAN PROBABILITY THEORY , 1990 .

[32]  Marvin H. J. Guber.  Bayesian Spectrum Analysis and Parameter Estimation , 1988 .

[33]  G. L. Bretthorst,et al.  Temperature dependence of the folding and unfolding kinetics of the GCN4 leucine zipper via 13C(alpha)-NMR. , 2001, Biophysical journal.

[34]  A. Vecchio,et al.  Bayesian bounds on parameter extraction accuracy for compact coalescing binary gravitational wave signals , 1997, gr-qc/9705064.