Peculiarities of the Rice Statistical Distribution: Mathematical Substantiation

Tatiana Yakovleva

doi:doi:10.11648/j.acm.20180704.12

| Peer-Reviewed

Peculiarities of the Rice Statistical Distribution: Mathematical Substantiation

Tatiana Yakovleva

Published in Applied and Computational Mathematics (Volume 7, Issue 4)

Received: 16 September 2018 Published: 18 September 2018

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

Abstract

The Rice statistical distribution has recently become a subject of increasing scientific interest due to its wide applicability in various fields of science and technology, such as the magnetic-resonance visualization and ultrasound diagnostics in medicine, the radio and radar signals’ analysis and processing, the optical measurements, etc. The common feature of the tasks that are adequately described by the Rician statistical model consists in the fact that the value to be measured and analyzed is an amplitude, or an envelope of the output signal which is composed as a sum of the initially determined informative component and a random noise component being formed by many independent normally-distributed summands. The efficient reconstruction of the Rician signal’s informative component against the noise background is shown to be achieved only on the basis of joint determination of both a priori unknown Rician parameters. The Rice statistical distribution possesses some peculiarities that allow solving rather complicated tasks connected with the stochastic data processing. The paper considers the issues of the strict mathematical substantiations of the Rice distribution properties that are meaningful for its efficient application, namely: the Rician likelihood function features and the stable character of the Rice distribution. There are provided the rigorous proofs of the mentioned properties.

Published in	Applied and Computational Mathematics (Volume 7, Issue 4)
DOI	10.11648/j.acm.20180704.12
Page(s)	188-196
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2018. Published by Science Publishing Group

Keywords

Rice Distribution, Probability Density, Likelihood Function, Data Processing

References

[1]	S. O. Rice, “Mathematical Analysis of Random Noise”, Bell Syst. Tech. Journal, 1944. Vol. 23. p. 282-322.
[2]	T. R. Benedict, T. T. Soong, “The joint estimation of signal and noise from the sum envelope”, IEEE Trans. Inf. Theory, 1967. Vol. IT-13. No. 3. p. 447-454.
[3]	K. K. Talukdar, W. D. Lawing, “Estimation of the parameters of Rice distribution”, J. Acoust. Soc. Amer., 1991. Vol. 89. No. 3. p. 1193-1197.
[4]	J. Sijbers, A. J. den Dekker, P. Scheunders, D. VanDyck, “Maximum-Likelihood Estimation of Rician Distribution Parameters”, IEEE Transactions on Medical Imaging, 1998. Vol. 17. No 3. p. 357-361.
[5]	T. V. Yakovleva, “Conditions of Rice statistical model applicability and estimation of the Rician signal’s parameters by maximum likelihood technique”, Computer Research and Modeling, 2014, vol. 6, no. 1, pp. 13-25.
[6]	T. M. Cover, J. A. Thomas Elements of Information Theory. — John Wiley and Sons, 2006. 776p.
[7]	A. J. den Dekker, J. Sijbers “Data distributions in magnetic resonance images: A review”, Physica Medica: European Journal of Medical Physics, 2014. Vol. 30. Issue 7. p. 723-741.
[8]	T. V. Yakovleva, “A Theory of Signal Processing at the Rice Distribution”, Dorodnicyn Computing Centre, RAS, Moscow, 2015. – 268p.
[9]	C. F. M. Carobbi, M. Cati, L. M. Millanta, “A New Procedure for Evaluating the Performance of the Site for Radiation Test or Antenna Calibration”, EMC Europe 2004, International Symposium on Electromagnetic Compatibility, Symposium Record, Eindhoven, The Netherlands, 2004. Vol. 2. p. 702-706.
[10]	T. V. Yakovleva, A. V. Kniazkov, “Speckle-noise computing by two-parameter analysis of the reflected light’s periodic variation”, Optical Memory & Neural Networks (Information Optics), 2014. Vol. 23. №4. p. 233-241.
[11]	T. V. Yakovleva, A. V. Kniazkov, “Comparison of Two Techniques of Electro-Optical Coefficient Evaluation”, Physics Procedia, 2015. Vol. 73, p. 189 – 192.
[12]	T. V. Yakovleva, N. S. Kulberg, “Special features of the Likelihood Function of the Rice Statistical Distribution”, Doklady Mathematics, 2014. Vol. 90. No. 1. p. 472–475.
[13]	T. V. Yakovleva, N. S. Kulberg, “Mathematical Statistics Methods as a Tool of Two-Parameter Magnetic-Resonance Image Analysis”, Informatics and its application, 2014. v. 8. is. 3. p. 79-89.
[14]	T. V. Yakovleva, N. S. Kulberg, “Methods of Mathematical Statistics in Two-Parameter Analysis of Rician Signals”, Doklady Mathematics. 2014. Vol. 90, No. 3. P. 1–5.
[15]	Tatiana V. Yakovleva, Nicolas S. Kulberg, “Noise and Signal Estimation in MRI: Two-Parametric Analysis of Rice-Distributed Data by Means of the Maximum Likelihood Approach”, American Journal of Theoretical and Applied Statistics, 2013. Vol. 2. No. 3. p. 67-79.
[16]	M. Abramovits, I. Stegun, Handbook of Mathematical Functions, М.:Nauka, 1979. 832 p.
[17]	J. H. Park, Jr., “Moments of generalized Rayleigh distribution”, Q. Appl. Math., 1961. Vol. 19. No. 1. p. 45-49.

Cite This Article

Plain Text BibTeX RIS

APA Style

Tatiana Yakovleva. (2018). Peculiarities of the Rice Statistical Distribution: Mathematical Substantiation. Applied and Computational Mathematics, 7(4), 188-196. https://doi.org/10.11648/j.acm.20180704.12

Copy | Download

ACS Style

Tatiana Yakovleva. Peculiarities of the Rice Statistical Distribution: Mathematical Substantiation. Appl. Comput. Math. 2018, 7(4), 188-196. doi: 10.11648/j.acm.20180704.12

Copy | Download

AMA Style

Tatiana Yakovleva. Peculiarities of the Rice Statistical Distribution: Mathematical Substantiation. Appl Comput Math. 2018;7(4):188-196. doi: 10.11648/j.acm.20180704.12

Copy | Download

@article{10.11648/j.acm.20180704.12,
  author = {Tatiana Yakovleva},
  title = {Peculiarities of the Rice Statistical Distribution: Mathematical Substantiation},
  journal = {Applied and Computational Mathematics},
  volume = {7},
  number = {4},
  pages = {188-196},
  doi = {10.11648/j.acm.20180704.12},
  url = {https://doi.org/10.11648/j.acm.20180704.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20180704.12},
  abstract = {The Rice statistical distribution has recently become a subject of increasing scientific interest due to its wide applicability in various fields of science and technology, such as the magnetic-resonance visualization and ultrasound diagnostics in medicine, the radio and radar signals’ analysis and processing, the optical measurements, etc. The common feature of the tasks that are adequately described by the Rician statistical model consists in the fact that the value to be measured and analyzed is an amplitude, or an envelope of the output signal which is composed as a sum of the initially determined informative component and a random noise component being formed by many independent normally-distributed summands. The efficient reconstruction of the Rician signal’s informative component against the noise background is shown to be achieved only on the basis of joint determination of both a priori unknown Rician parameters. The Rice statistical distribution possesses some peculiarities that allow solving rather complicated tasks connected with the stochastic data processing. The paper considers the issues of the strict mathematical substantiations of the Rice distribution properties that are meaningful for its efficient application, namely: the Rician likelihood function features and the stable character of the Rice distribution. There are provided the rigorous proofs of the mentioned properties.},
 year = {2018}
}

Copy | Download

TY  - JOUR
T1  - Peculiarities of the Rice Statistical Distribution: Mathematical Substantiation
AU  - Tatiana Yakovleva
Y1  - 2018/09/18
PY  - 2018
N1  - https://doi.org/10.11648/j.acm.20180704.12
DO  - 10.11648/j.acm.20180704.12
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 188
EP  - 196
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20180704.12
AB  - The Rice statistical distribution has recently become a subject of increasing scientific interest due to its wide applicability in various fields of science and technology, such as the magnetic-resonance visualization and ultrasound diagnostics in medicine, the radio and radar signals’ analysis and processing, the optical measurements, etc. The common feature of the tasks that are adequately described by the Rician statistical model consists in the fact that the value to be measured and analyzed is an amplitude, or an envelope of the output signal which is composed as a sum of the initially determined informative component and a random noise component being formed by many independent normally-distributed summands. The efficient reconstruction of the Rician signal’s informative component against the noise background is shown to be achieved only on the basis of joint determination of both a priori unknown Rician parameters. The Rice statistical distribution possesses some peculiarities that allow solving rather complicated tasks connected with the stochastic data processing. The paper considers the issues of the strict mathematical substantiations of the Rice distribution properties that are meaningful for its efficient application, namely: the Rician likelihood function features and the stable character of the Rice distribution. There are provided the rigorous proofs of the mentioned properties.
VL  - 7
IS  - 4
ER  -

Copy | Download

Author Information

Tatiana Yakovleva

Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences, Moscow, Russia

Download PDF

Sections

Plain Text BibTeX RIS

APA Style

Tatiana Yakovleva. (2018). Peculiarities of the Rice Statistical Distribution: Mathematical Substantiation. Applied and Computational Mathematics, 7(4), 188-196. https://doi.org/10.11648/j.acm.20180704.12

Copy | Download

ACS Style

Tatiana Yakovleva. Peculiarities of the Rice Statistical Distribution: Mathematical Substantiation. Appl. Comput. Math. 2018, 7(4), 188-196. doi: 10.11648/j.acm.20180704.12

Copy | Download

AMA Style

Tatiana Yakovleva. Peculiarities of the Rice Statistical Distribution: Mathematical Substantiation. Appl Comput Math. 2018;7(4):188-196. doi: 10.11648/j.acm.20180704.12

Copy | Download

@article{10.11648/j.acm.20180704.12,
  author = {Tatiana Yakovleva},
  title = {Peculiarities of the Rice Statistical Distribution: Mathematical Substantiation},
  journal = {Applied and Computational Mathematics},
  volume = {7},
  number = {4},
  pages = {188-196},
  doi = {10.11648/j.acm.20180704.12},
  url = {https://doi.org/10.11648/j.acm.20180704.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20180704.12},
  abstract = {The Rice statistical distribution has recently become a subject of increasing scientific interest due to its wide applicability in various fields of science and technology, such as the magnetic-resonance visualization and ultrasound diagnostics in medicine, the radio and radar signals’ analysis and processing, the optical measurements, etc. The common feature of the tasks that are adequately described by the Rician statistical model consists in the fact that the value to be measured and analyzed is an amplitude, or an envelope of the output signal which is composed as a sum of the initially determined informative component and a random noise component being formed by many independent normally-distributed summands. The efficient reconstruction of the Rician signal’s informative component against the noise background is shown to be achieved only on the basis of joint determination of both a priori unknown Rician parameters. The Rice statistical distribution possesses some peculiarities that allow solving rather complicated tasks connected with the stochastic data processing. The paper considers the issues of the strict mathematical substantiations of the Rice distribution properties that are meaningful for its efficient application, namely: the Rician likelihood function features and the stable character of the Rice distribution. There are provided the rigorous proofs of the mentioned properties.},
 year = {2018}
}

Copy | Download

TY  - JOUR
T1  - Peculiarities of the Rice Statistical Distribution: Mathematical Substantiation
AU  - Tatiana Yakovleva
Y1  - 2018/09/18
PY  - 2018
N1  - https://doi.org/10.11648/j.acm.20180704.12
DO  - 10.11648/j.acm.20180704.12
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 188
EP  - 196
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20180704.12
AB  - The Rice statistical distribution has recently become a subject of increasing scientific interest due to its wide applicability in various fields of science and technology, such as the magnetic-resonance visualization and ultrasound diagnostics in medicine, the radio and radar signals’ analysis and processing, the optical measurements, etc. The common feature of the tasks that are adequately described by the Rician statistical model consists in the fact that the value to be measured and analyzed is an amplitude, or an envelope of the output signal which is composed as a sum of the initially determined informative component and a random noise component being formed by many independent normally-distributed summands. The efficient reconstruction of the Rician signal’s informative component against the noise background is shown to be achieved only on the basis of joint determination of both a priori unknown Rician parameters. The Rice statistical distribution possesses some peculiarities that allow solving rather complicated tasks connected with the stochastic data processing. The paper considers the issues of the strict mathematical substantiations of the Rice distribution properties that are meaningful for its efficient application, namely: the Rician likelihood function features and the stable character of the Rice distribution. There are provided the rigorous proofs of the mentioned properties.
VL  - 7
IS  - 4
ER  -

Copy | Download