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Univariate normalization of bispectrum using Hölder's inequality.
Forooz Shahbazi1, Arne Ewald2, Guido Nolte3
1Fraunhofer Institute FOKUS, Kaiserin Augusta Allee. 31, 10589 Berlin, Germany; Technische Universität Berlin, Machine Learning Group, Marchstr.23, 10587 Berlin, Germany.
Journal of Neuroscience Methods
|July 1, 2014
Summary
A new univariate normalization method for cross-bispectrum analysis was developed. This method ensures bounded bicoherence values, crucial for studying complex biological systems like the brain.
Area of Science:
- Neuroscience
- Complex Systems Analysis
- Signal Processing
Background:
- Biological systems, including the brain, are complex non-linear systems.
- Studying their dynamics requires methods that detect non-linearities.
- Cross-bispectrum (third-order cumulant) measures interfrequency interactions between signals.
Purpose of the Study:
- To propose a novel univariate normalization factor for cross-bispectra.
- To ensure the normalized measure (bicoherence) is bounded between zero and one.
- To compare the statistical significance of this new normalization against existing methods.
Main Methods:
- Development of a univariate normalization factor for cross-bispectra.
- Mathematical proof using a generalization of Hölder's inequality to establish bounds.
- Comparison with three existing normalizations using resampling tests on real EEG data.
Main Results:
- The proposed univariate normalization ensures bicoherence values are bounded between 0 and 1.
- Statistical significance of bicoherence values showed slight improvements with univariate normalization.
- Differences in significance were minimal or negligible across subjects.
Conclusions:
- The normalization factor plays a minor role in the statistical power of bicoherence values.
- Univariate normalization is the only method satisfying all criteria for proper normalization.
- This method is suitable for analyzing non-linear dynamics in biological systems.
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