Abstract.
Estimating the amplitudes and decay rate constants of exponentially decaying signals is an important problem in science. Understanding how the uncertainty in the parameter estimates depends on the experimental parameters is important as an aid in understanding how to improve the reliability of the parameter estimates. In this paper, probability theory has been applied to this problem with the intent of understanding the relevant experimental parameters. In the case of a single exponential, the uncertainty in the estimated decay rate depends directly on the three halves power of the true decay rate constant, inversely on the signal-to-noise ratio, and inversely on the square root of the number of data values. The uncertainty in the amplitude estimate depends directly on the square root of the true decay rate constant, directly on the noise level, and inversely on the square root of the number of data values. The case of two exponentials has also been analyzed with similar results. However, here the presence of the second signal introduces interference effects which make the estimate more uncertain.
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References
Bretthorst, G. Larry, “Bayesian Analysis. III. Applications to NMR Signal Detection, Model Selection and Parameter Estimation,” J. Magn. Reson. 88, pp. 571–595 (1990).
Jaynes, E. T., Bayesian Spectrum and Chirp Analysis, in “Maximum-Entropy and Bayesian Spectral Analysis and Estimation Problems” (C. R. Smith and G. J. Erickson Eds.), p. 1, Reidel, Dordrecht, The Netherlands, 1987.
Bretthorst, G. Larry, Smith, C. Ray, “Bayesian Analysis of Signals from Closely-Spaced Objects,” SPIE Vol.1050 Infrared System and Components 111 (1989).
Kotyk, J. John, Hoffman, G. Norman, Hutton, C. William, Bretthorst, G. Larry, and Ackerman, J.H. Joseph “Comparison of Fourier and Bayesian Analysis of NMR Signals. I. Well-Separated Resonances (The Single-Frequency Case), ” J. Magn. Reson. 98, pp. 483–500 (1992).
Neil, J. Jeffrey, Bretthorst, G. Larry, On the Use of Bayesian Probability Theory for Analysis of Exponential Decay Data: An Example Taken from Intravoxel Incoherent Motion Experiments, Magnetic Resonance in Medicine, 29, pp. 642–647 (1993).
Bretthorst, G. Larry, An Introduction to Parameter Estimation Using Bayesian Probability Theory, in “Maximum Entropy and Bayesian Methods” (P. Fougère Ed.), p. 53, Kluwer Academic, Dordrecht, The Netherlands, 1989.
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© 1996 Springer Science+Business Media Dordrecht
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Ramaswami, A., Bretthorst, G.L. (1996). Probability Theory and Multiexponential Signals, How Accurately Can the Parameters be Determined?. In: Heidbreder, G.R. (eds) Maximum Entropy and Bayesian Methods. Fundamental Theories of Physics, vol 62. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8729-7_21
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DOI: https://doi.org/10.1007/978-94-015-8729-7_21
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