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 JAMP  Vol.7 No.4 , April 2019
Analytical and Numerical Investigations of Probabilistic Monochromatic Problem
Abstract: A probabilistic formalism, relying on Bayes’ theorem and linear Gaussian inversion, is adapted, so that a monochromatic problem can be investigated. The formalism enables an objective test in probabilistic terms of the quantities and model concepts involved in the problem at hand. With this formalism, an amplitude (linear parameter), a frequency (non-linear parameter) and a hyperparameter of the Gaussian amplitude prior are inferred jointly given simulated data sets with Gaussian noise contributions. For the amplitude, an analytical normal posterior follows which is conditional on the frequency and the hyperparameter. The remaining posterior estimates the frequency with an uncertainty of MHz, while the convolution of a standard approach would achieve an uncertainty of some GHz. This improvement in the estimation is investigated analytically and numerically, revealing for instance the positive effect of a high signal-to-noise ratio and/or a large number of data points. As a fixed choice of the hyperparameter imposes certain results on the amplitude and frequency, this parameter is estimated and, thus, tested for plausibility as well. From abstract point of view, the model posterior is investigated as well.
Cite this paper: Schmuck, S. and Svensson, J. (2019) Analytical and Numerical Investigations of Probabilistic Monochromatic Problem. Journal of Applied Mathematics and Physics, 7, 793-808. doi: 10.4236/jamp.2019.74054.
References

[1]   Sivia, D. and Skilling, J. (2006) Data Analysis: A Bayesian Tutorial. Oxford University Press, Oxford.

[2]   Jaynes, E.T. (1987) Maximum Entropy and Bayesian Spectral Analysis and Estimation Problems. Fundamental Theories of Physics, 21, 1-37.

[3]   Bretthorst, G.L. (1988). Bayesian Spectrum Analysis and Parameter Estimation. Springer-Verlag, Berlin, Heidelberg.
https://doi.org/10.1007/978-1-4684-9399-3

[4]   Schmuck, S. and Svensson, J. (2017) Fourier Spectroscopy: A Bayesian Way. International Journal of Spectroscopy, 2017, Article ID: 9265084.
https://doi.org/10.1155/2017/9265084

[5]   Schmuck, S., Fessey, J., Boom, J.E., Meneses, L., Abreu, P., Belonohy, E. and Lupelli, I. (2016) Electron Cyclotron Emission Spectra in X- and O-Mode Polarisation at JET: Martin-Puplett Interferometer, Absolute Calibration, Revised Uncertainties, Inboard/Outboard Temperature Profile, and Wall Properties. Review of Scientific Instruments, 87, Article ID: 093506.
https://doi.org/10.1063/1.4962809

 
 
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