Tests for Gaussianity of a Stationary Time Series

Authors: Adnan Al-Smadi

Abstract:

One of the primary uses of higher order statistics in signal processing has been for detecting and estimation of non- Gaussian signals in Gaussian noise of unknown covariance. This is motivated by the ability of higher order statistics to suppress additive Gaussian noise. In this paper, several methods to test for non- Gaussianity of a given process are presented. These methods include histogram plot, kurtosis test, and hypothesis testing using cumulants and bispectrum of the available sequence. The hypothesis testing is performed by constructing a statistic to test whether the bispectrum of the given signal is non-zero. A zero bispectrum is not a proof of Gaussianity. Hence, other tests such as the kurtosis test should be employed. Examples are given to demonstrate the performance of the presented methods.

Keywords: Non-Gaussian, bispectrum, kurtosis, hypothesistesting, histogram.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1058019

References:

[1] A. Al-Smadi and M. Smadi," Study of the reliability of a binary symmetric channel under non-Gaussian disturbances," International Journal of Communication Systems, vol. 16, no. 10, pp. 865-973, December 2003.
[2] S. Zabin and D. Furbeck, "Efficient identification of non-Gaussian mixtures," IEEE Trans. Comm., vol. 48, pp. 106-117, January 2000.
[3] J. M Mendel," Tutorial on higher order-statistics (spectra) in signal processing and system theory: Theoretical results and some applications," Proceedings of the IEEE, vol. 79, pp. 278-305, March 1991.
[4] C.L. Nikias and M.R. Raghveer, "Bispectrum estimation: A digital signal processing framework," Proceedings of the IEEE, vol.75, no. 7, pp. 869-891, July, 1987.
[5] A. Al-Smadi & D. M. Wilkes, "Robust and Accurate ARX and ARMA Model Order Estimation of Non-Gaussian Processes", IEEE Trans. On Signal Processing, vol. 50, no. 3, pp. 759 -763, March 2002.
[6] A. Al-Smadi, "Cumulant based approach to FIR system identification", International Journal of Circuit Theory and Applications, vol. 31, no. 6, pp. 625-636, November, 2003.
[7] D.R. Brillinger, "An Introduction to Polyspectra," Ann. Math. Statist., vol. 36, pp. 1351-1374, 1965.
[8] M.Choudhury, S. Shah, and N. Thornhill, "Diagnosis of poor controlloop performance using higher order statistics," Automatica, vol. 40, pp. 1719-1728, 2004.
[9] D. Montgomery, Introduction to Statistical Quality Control, John Wiley & Sons, New York, 2000.
[10] D.C. Montgomery and G. C. Runger, Applied Statistics and probability for Engineers, John Wiley, New York, 1999.
[11] R. Pond, Fundamentals of Statistical Quality Control, Macmillan College Publishing, New York, 1994.
[12] J. A. Cadzow, "Blind deconvolution via cumulant extrema," IEEE Signal Processing Mag., vol. 13, no. 5, pp. 24-42, 1996.
[13] Y.Nishiguchi, N. Toda, and S. Usui, "Parametric estimation of Higherorder spectra by tensor product expansion with coordinate transformation," Electronics and Communications in Japan, vol. 87, no. 1, pp. 75-83, 2004.
[14] S.A. Kassam, Signal detection in non-Gaussian noise, Springer-Verlag, New York, 1988.
[15] J. Caillec and R. Garello, "Comparison of statistical indices using third order statistics for nonlinearity detection," Signal Processing, vol. 84, pp. 499- 525, 2004.