In many applications, it is important to recognise the distribution of empirical data in almost real time. One of the specific applications is the identification of statistical models for fading in wireless systems of the base station receivers. This is one of the most important problems in spatial diversity. In this paper, we describe the methodology and the results of a nonlinear regression approach for recognising the distribution of the input signal with the values of its parameters. Furthermore, the proposed approach could be used for the real-time recognition of the probability distributions without any prior knowledge about the input signal. To prove its performance, the Levenberg–Marquardt nonlinear least-squares algorithm is tested on a large set of randomly generated signals with the Gamma, Rayleigh, Rician, Nakagami-m, and Weibull distributions. The experimental results demonstrate that this approach is accurate in recognizing statistical distributions from the signal.
1. Motulsky, H. J. and Ransnas, L. A. Fitting curves to data using nonlinear regression: a practical and nonmathematical review. FASEB J., 1987, 1(5), 365–374.
https://doi.org/10.1096/fasebj.1.5.3315805
2. Draper, N. R. and Smith, H. Applied Regression Analysis. John Wiley & Sons, 1998.
https://doi.org/10.1002/9781118625590
3. Tsherniak, A., Vazquez, F., Montgomery, P. G., Weir, B. A., Kryukov, G., Cowley, G. S. et al. Defining a cancer dependency map. Cell, 2017, 170(3), 564-576.e16.
https://doi.org/10.1016/j.cell.2017.06.010
4. Rashid, U., Kumari, N., Signal, N., Taylor, D. and Vandal, A. C. On nonlinear regression for trends in split-belt treadmill training. Brain Sci., 2020, 10(10), 737.
https://doi.org/10.3390/brainsci10100737
5. Hahne, J. M., Bießmann, F., Jiang, N., Rehbaum, H., Farina, D., Meinecke, F. C. et al. Linear and nonlinear regression techniques for simultaneous and proportional myoelectric control. IEEE Trans. Neural Syst. Rehabil. Eng., 2014, 22(2), 269–279.
https://doi.org/10.1109/TNSRE.2014.2305520
6. Bennink, E., Riordan, A. J., Horsch, A. D., Dankbaar, J. W., Velthuis, B. K. and de Jong, H. W. A fast nonlinear regression method for estimating permeability in CT perfusion imaging. J. Cereb. Blood Flow Metab., 2013, 33(11), 1743–1751.
https://doi.org/10.1038/jcbfm.2013.122
7. Tufaner, F. and Demirci, Y. Prediction of biogas production rate from anaerobic hybrid reactor by artificial neural network and nonlinear regressions models. Clean Technol. Environ. Policy, 2020, 22(3), 713–724.
https://doi.org/10.1007/s10098-020-01816-z
8. Li, W., Shao, L., Wang, W., Li, H., Wang, X., Li, Y. et al. Air quality improvement in response to intensified control strategies in Beijing during 2013–2019. Sci. Total Environ., 2020, 744, 140776.
https://doi.org/10.1016/j.scitotenv.2020.140776
9. Tümer, A. E. and Edebalı, S. Prediction of wastewater treatment plant performance using multilinear regression and artificial neural networks. In Proceedings of the 2015 International Symposium on Innovations in Intelligent SysTems and Applications (INISTA), Madrid, Spain, 2–4 September 2015. IEEE, 1–5.
https://doi.org/10.1109/INISTA.2015.7276742
10. Wang, J., Li, W., Yang, B., Cheng, X., Tian, Z. and Guo, H. Impact of hydrological factors on the dynamic of COVID-19 epidemic: A multi-region study in China. Environ. Res., 2021, 198, 110474.
https://doi.org/10.1016/j.envres.2020.110474
11. Luo, Y., Yan, J. and McClure, S. Distribution of the environmental and socioeconomic risk factors on COVID-19 death rate across continental USA: a spatial nonlinear analysis. Environ. Sci. Pollut. Res., 2021, 28(6), 6587–6599.
https://doi.org/10.1007/s11356-020-10962-2
12. Namasudra, S., Dhamodharavadhani, S. and Rathipriya, R. Nonlinear neural network based forecasting model for predicting COVID-19 cases. Neural Process. Lett., 2021.
https://doi.org/10.1007/s11063-021-10495-w
13. Zhang, L., Shi, Z., Cheng, M.-M., Liu, Y., Bian, J.-W., Zhou, J. T. et al. Nonlinear regression via deep negative correlation learning. IEEE Trans. Pattern Anal. Mach. Intell., 2021, 43(3), 982–998.
https://doi.org/10.1109/TPAMI.2019.2943860
14. Panic, S., Stefanovic, M., Anastasov, J. and Spalevic, P. Fading and Interference Mitigation in Wireless Communications. CRC Press, Boca Raton, 2015.
https://doi.org/10.1201/b16275
15. Simon, M. K. and Alouini, M.-S. Digital Communication Over Fading Channels. Wiley, 2005.
https://doi.org/10.1002/0471715220
16. Rice, J. A. Estimation of parameters and fitting of probability distributions. In Mathematical Statistics and Data Analysis. Cengage Learning, 2006.
17. Wang, N., Song, X. and Cheng, J. Generalized method of moments estimation of the Nakagami-m fading parameter. IEEE Trans. Wirel. Commun., 2012, 11(9), 3316–3325.
https://doi.org/10.1109/TWC.2012.071612.111838
18. Bouhlel, N. and Dziri, A. Maximum likelihood parameter estimation of Nakagami–Gamma shadowed fading channels. IEEE Commun. Lett., 2015, 19(4), 685–688.
https://doi.org/10.1109/LCOMM.2015.2394293
19. Alymani, M. Fading Channel Parameter Estimation Using Deep Learning. Stevens Institute of Technology, Hoboken, NJ, 2021.
20. Stüber, G. L. Principles of Mobile Communication. Springer International Publishing, Cham, 2017.
https://doi.org/10.1007/978-3-319-55615-4
21. Suraweera, H. A., Louie, R. H. Y., Li, Y., Karagiannidis, G. K. and Vucetic, B. Two hop amplify-and-forward transmission in mixed rayleigh and rician fading channels. IEEE Commun. Lett., 2009, 13(4), 227–229.
https://doi.org/10.1109/LCOMM.2009.081943
22. Bernadó, L., Zemen, T., Karedal, J., Paier, A., Thiel, A., Klemp, O. et al. Multi-dimensional K-factor analysis for V2V radio channels in open sub-urban street crossings. In Proceedings of the 21st Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications, Istanbul, Turkey, 26–30 September 2010. IEEE 58–63.
https://doi.org/10.1109/PIMRC.2010.5671774
23. Stefanović, D., Panic, S. and Spalevic, P. Second-order statistics of SC macrodiversity system operating over Gamma shadowed Nakagami-m fading channels. AEU – Int. J. Electron. Commun., 2011, 65, 413–418.
https://doi.org/10.1016/j.aeue.2010.05.001
24. Reig, J. Multivariate Nakagami-m distribution with constant correlation model. AEU – Int. J. Electron. Commun., 2009, 63(1), 46–51.
https://doi.org/10.1016/j.aeue.2007.10.009
25. Panić, S. R., Stefanović, M., Anastasov, J. and Spalević, P. Modeling of fading channels. In Fading and Interference Mitigation in Wireless Communications. CRC Press, 2014.
26. Chen, Y. and Beaulieu, N. C. Estimation of Ricean and Nakagami distribution parameters using noisy samples. In Proceedings of 2004 IEEE International Conference on Communications, Vol. 1, Paris, France, 20–24 June 2004. IEEE, 562–566.
https://doi.org/10.1109/ICC.2004.1312552
27. Elzhov, T. V., Mullen, K. M., Spiess, A.-N. and Bolker, B. minpack.lm: R interface to the Levenberg–Marquardt nonlinear least-squares algorithm found in MINPACK, plus support for bounds, 2022.
28. Gavin, H. P. The Levenberg–Marquardt algorithm for nonlinear least squares curve-fitting problems. Dep. Civ. Environ. Eng. Duke Univ., 2019, 19.
29. Burnham, K. P. and Anderson, D. R. Multimodel inference: understanding AIC and BIC in model selection. Sociol. Methods Res., 2004, 33(2), 261–304.
https://doi.org/10.1177/0049124104268644
30. Kuha, J. AIC and BIC: Comparisons of assumptions and performance. Sociol. Methods Res., 2004, 33(2), 188–229.
https://doi.org/10.1177/0049124103262065
