Bicoherence

The bispectrum is defined as the triple product

${\displaystyle B(f_{1},f_{2})=F(f_{1})F(f_{2})F^{*}(f_{1}+f_{2})}$

where ${\displaystyle B}$ is the bispectrum evaluated at frequencies ${\displaystyle f_{1}}$ and ${\displaystyle f_{2}}$, ${\displaystyle F}$ is the Fourier transform of the signal, and ${\displaystyle ^{*}}$ denotes the complex conjugate. The Fourier transform is a complex quantity, and so is the bispectrum. From complex multiplication, the magnitude of the bispectrum is equal to the product of the magnitudes of each of the frequency components, and the phase of the bispectrum is the sum of the phases of each of the frequency components.

Suppose that the three Fourier components ${\displaystyle F(f_{1})}$, ${\displaystyle F(f_{2})}$ and ${\displaystyle F(f_{1}+f_{2})}$ were perfectly phase locked. Then if the Fourier transform was calculated several times from different parts of the time series, the bispectrum will always have the same value. If we add together all of the bispectra, they will sum without cancelling. On the other hand, suppose that the phases of each of these frequencies was random. Then, the bispectrum will have the same magnitude (assuming that the magnitude of the frequency components is the same) but the phase will be randomly oriented. Adding together all of the bispectra will result in cancellation, because of the random phase orientation, and so the sum of the bispectra will have a small magnitude. Detecting phase coupling requires summation over a number of independent samples- this is the first motivation for defining the bicoherence. Secondly, the bispectrum is not normalized, because it still depends on the magnitudes of each of the frequency components. The bicoherence includes a normalization factor that removes the magnitude dependence.

There is some inconsistency with the definition of the bicoherence normalization constant. Some of the definitions that have been used are

${\displaystyle b(f_{1},f_{2})={\frac {\left|\sum \limits _{n}F_{n}(f_{1})F_{n}(f_{2})F_{n}^{*}(f_{1}+f_{2})\right|}{\sqrt {\sum \limits _{n}|F_{n}(f_{1})|^{2}|F_{n}(f_{2})|^{2}|F_{n}^{*}(f_{1}+f_{2})|^{2}}}}}$

which was provided in Sigl and Chamoun 1994, but does not appear to be correctly normalized. Alternatively, plasma physics typically uses

${\displaystyle b^{2}(f_{1},f_{2})={\frac {|\langle F_{n}(f_{1})F_{n}(f_{2})F_{n}^{*}(f_{1}+f_{2})\rangle |^{2}}{\langle |F_{n}(f_{1})F_{n}(f_{2})|^{2}\rangle \langle |F_{n}^{*}(f_{1}+f_{2})|^{2}\rangle }}}$

where the angle brackets denote averaging. Note that this is the same as using a sum, because ${\displaystyle n}$ is the same in the numerator and the denominator. This definition is directly from Nagashima 2006, and is also referred to in He 2009 and Maccarone 2005.

Finally, one of the most intuitive definitions comes from Hagihira 2001 and Hayashi 2007, which is

${\displaystyle b(f_{1},f_{2})={\frac {\left|\sum \limits _{n}F_{n}(f_{1})F_{n}(f_{2})F_{n}^{*}(f_{1}+f_{2})\right|}{\sum \limits _{n}|F_{n}(f_{1})F_{n}(f_{2})F_{n}^{*}(f_{1}+f_{2})|}}}$

The numerator contains the magnitude of the bispectrum summed over all of the time series segments. This quantity is large if there is phase coupling, and approaches 0 in the limit of random phases. The denominator, which normalizes the bispectrum, is given by calculating the bispectrum after setting all of the phases to 0. This corresponds to the case where there is perfect phase coupling, because all of the samples have zero phase. Therefore, the bicoherence has a value between 0 (random phases) and 1 (total phase coupling).

Among the three normalizations above, the second one can be interpreted as a correlation coefficient defined between energy-supplying and energy-receiving parties in an second order nonlinear interaction, whereas the bispectrum has been proven to be the corresponding covariance.[2] Therefore, just as correlation cannot sufficiently demonstrate the presence of causality, a significant bicoherence peak also cannot sufficiently substantiate the existence of an nonlinear interaction.

• Coherence (signal processing)

• Hagihira, S., Takashina, M., Mori, T., Mashimo, T., & Yoshiya, I. (2001). Practical Issues in Bispectral Analysis of Electroencephalographic Signals. Anesthesia & Analgesia, 93(4), 966-970. Retrieved from http://www.anesthesia-analgesia.org/content/93/4/966.abstract
• Hayashi, K., Tsuda, N., Sawa, T., & Hagihira, S. (2007). Ketamine increases the frequency of electroencephalographic bicoherence peak on the alpha spindle area induced with propofol. British Journal of Anaesthesia, 99(3), 389-95. doi:10.1093/bja/aem175
• Nagashima, Y., Itoh, K., Itoh, S.-I., Hoshino, K., Fujisawa, A., Ejiri, A., Takase, Y., et al. (2006). Observation of coherent bicoherence and biphase in potential fluctuations around geodesic acoustic mode frequency on JFT-2M. Plasma Physics and Controlled Fusion, 48(5A), A377-A386. doi:10.1088/0741-3335/48/5A/S38
• He, H. (2009). The Canonical Bicoherence -€” Part I : Definition, Multitaper Estimation, and Statistics. Signal Processing, IEEE Transactions, 57(4), 1273-1284. Retrieved from http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=4749274
• Maccarone, T. J., & Schnittman, J. D. (2004). The bicoherence as a diagnostic for models of high-frequency quasi-periodic oscillations. Monthly Notices of the Royal Astronomical Society, 357(1), 12-16. doi:10.1111/j.1365-2966.2004.08615.x
• Mendel JM. "Tutorial on higher-order statistics (spectra) in signal processing and system theory: theoretical results and some applications." Proceedings of the IEEE, 79, 3, 278-305
• M J Hinich, "Testing for Gaussianity and linearity of a stationary time series", Journal of Time Series Analysis 3(3), 1982 pp 169–176.
• HOSA - Higher Order Spectral Analysis Toolbox. (shareware for Microsoft Windows-type personal computers.)
• Sigl, J.C. and N.G. Chamoun. 1994. An introduction to bispectral analysis for the electroencephalogram. Journal of Clinical Monitoring 10:392-404.
• T.H. Bullock, J.Z. Achimowicz et al., "Bicoherence of intracranial EEG in awake, sleep and seizures", Journal of Clinical Neurophysiology and EEG, 1997,vol.231,pp. 130–142.
• J.L. Shils, M. Litt, B.E. Skolnick, M.M. Stecker, "Bispectral analysis of visual interactions in humans", Electroencephalography and Clinical Neurophysiology, 1996;98:113-125.