Citation :

Manoj Kumar Mukul, Mukesh Kumar Ojha, Raj Kishore Prasad, "Nonlinear Complexity Features Add Nothing beyond the Power Spectrum in Motor-Imagery BCI: A Surrogate-Controlled Reassessment," International Journal of Electrical and Electronics Engineering, vol. 13, no. 10, pp. 62-82, 2026. Crossref, https://doi.org/10.14445/23488379/IJEEE-V13I10P106

Abstract

A significant amount of research in the field of Motor-Imagery (MI) brain–computer interfaces have shown that nonlinear and higher-order features-such as fractal dimensions, entropies, detrended fluctuation analysis, variational-mode complexity, Phase–Amplitude Coupling (PAC), and bispectrum/bicoherence-enhance the decoding of sensorimotor rhythms. However, the past studies have drawn their conclusions based on within-session cross-validation, per-subject tuning, fusion over a spatial–spectral baseline, and the absence of a surrogate control. The present study poses a more focused and testable question: do these features contain information that goes beyond that provided by the power spectrum when assessed fairly? Using the BCI Competition IV datasets 2a and 2b (18 subjects) under the official independent-session protocol, a single matched shrinkage-LDA classifier is applied to decode fifteen families of features, each family is subjected to an Iterative Amplitude-Adjusted Fourier Transform (IAAFT) surrogate test, and all the families of tests are corrected for multiple comparisons. The results are clear. Nonlinear-7 achieves 0.576 compared to 0.691 for plain band power on dataset 2a and 0.668 compared to 0.735 on dataset 2b; when compared to a filter-bank CSP and Riemannian ensemble, the additional value of the surrogate-corrected nonlinear and PAC candidates is only +0.0000 Cohen's κ, and mutual-information selection includes them in 0.0% of the feature slots even though they make up 33.3% of the candidate pool. The higher-order spectra make the mechanism evident: across all 18 subjects, the magnitude of the bispectrum is statistically no different from that of band power (0.715 versus 0.721, p=0.59) and the accuracy on a per-subject basis correlates with band-power accuracy at r=0.958; in contrast, the normalized bicoherence, which removes that power scaling, achieves a decoding accuracy of 0.546-significantly lower than band power (p=3.5×10−6)-and falls below the level expected from its spectrally matched null (z=−0.16). A montage factorial that separates electrode count from spatial derivation shows that the apparent competitiveness of the complexity features on low-channel montages is not due to spatial focusing. This indicates the robustness of the results: when the number of channels, the number of electrodes, and the spatial derivation are kept the same across datasets, the ordering between nonlinear and band power remains unchanged, and the accuracy of the nonlinear features correlates with that of band power across the entire montage ladder (r=0.77 on 2a, r=0.84 on 2b). The results do not mean that all useful features disappear after controlling for the spectral information. In 3 of the 4 dataset-and-descriptor combinations, a small amount of information remains above chance even after correcting for false discoveries. However, this leftover effect is very small-only 4 to 6 percentage points above chance. By comparison, band power is 19 to 24 points above chance. And when these remaining features are added to the best-performing pipeline, they provide no additional benefit. This paper reports that nonlinear complexity features do not carry any usable information beyond the power spectrum for MI decoding, and the authors make available a reusable surrogate redundancy test that any candidate feature should pass before a nonlinear gain is claimed.

Keywords

Motor imagery, Brain-Computer Interface, Nonlinear features, Power spectrum, Surrogate data, IAAFT.

References

1. G. Pfurtscheller, and F.H. Lopes da Silva, “Event-Related EEG/MEG Synchronization and Desynchronization: Basic Principles,” Clinical Neurophysiology, vol. 110, no. 11, pp. 1842-1857, 1999.
   [CrossRef] [Google Scholar] [Publisher Link]
2. H. Ramoser, J. Muller-Gerking, and G. Pfurtscheller, “Optimal Spatial Filtering of Single Trial EEG during Imagined Hand Movement,” IEEE Transactions on Rehabilitation Engineering, vol. 8, no. 4, pp. 441-446, 2000.
   [CrossRef] [Google Scholar] [Publisher Link]
3. A. Babloyantz, J.M. Salazar, and C. Nicolis, “Evidence of Chaotic Dynamics of Brain Activity during the Sleep Cycle,” Physics Letters A, vol. 111, no. 3, pp. 152-156, 1985.
   [CrossRef] [Google Scholar] [Publisher Link]
4. Edward N. Lorenz, “Deterministic Nonperiodic Flow,” Journal of the Atmospheric Sciences, vol. 20, no. 2, pp. 130-141, 1963.
   [CrossRef] [Google Scholar] [Publisher Link]
5. B.B. Mandelbrot, The Fractal Geometry of Nature/Revised and Enlarged Edition/, W.H. Freeman and Co, New York, 1983.
   [Google Scholar] [Publisher Link]
6. James Theiler et al., “Testing for Nonlinearity in Time Series: The Method of Surrogate Data,” Physica D: Nonlinear Phenomena, vol. 58, no. 1-4, pp. 77-94, 1992.
   [CrossRef] [Google Scholar] [Publisher Link]
7. Thomas Schreiber, and Andreas Schmitz, “Improved Surrogate Data for Nonlinearity Tests,” Physical Review Letters, vol. 77, no. 4, pp. 635-638, 1996.
   [CrossRef] [Google Scholar] [Publisher Link]
8. Klaus Lehnertz, and Christian E. Elger, “Can Epileptic Seizures be Predicted? Evidence from Nonlinear Time Series Analysis of Brain Electrical Activity,” Physical Review Letters, vol. 80, no. 22, pp. 5019-5022, 1998.
   [CrossRef] [Google Scholar] [Publisher Link]
9. Klaus Lehnertz, and Christian E. Elger, “Spatio-Temporal Dynamics of the Primary Epileptogenic Area in Temporal Lobe Epilepsy,” Electroencephalography and Clinical Neurophysiology, vol. 95, no. 2, pp. 108-117, 1995.
   [CrossRef] [Google Scholar] [Publisher Link]
10. Ralph G. Andrzejak et al., “Indications of Nonlinear Deterministic and Finite-Dimensional Structures in Time Series of Brain Electrical Activity,” Physical Review E, vol. 64, no. 6, pp. 1-8, 2001.
    [CrossRef] [Google Scholar] [Publisher Link]
11. Denis Jordan et al., “Electroencephalographic Order Pattern Analysis for the Separation of Consciousness and Unconsciousness,” Anesthesiology, vol. 109, no. 6, pp. 1014-1022, 2008.
    [CrossRef] [Google Scholar] [Publisher Link]
12. Amr F. Mohamed, and Vacius Jusas, “Advancing Fractal Dimension Techniques to Enhance Motor Imagery Tasks using EEG for Brain–Computer Interface Applications,” Applied Sciences, vol. 15, no. 11, pp. 1-27, 2025.
    [CrossRef] [Google Scholar] [Publisher Link]
13. Venkatesh Kanagaluru, and M. Sasikala, “Two Class Motor Imagery EEG Signal Classification for BCI using LDA and SVM,” Signal Processing, vol. 41, no. 5, pp. 2743-2749, 2024.
    [CrossRef] [Google Scholar] [Publisher Link]
14. Chengzhen Wu et al., “The Application of Entropy in Motor Imagery Paradigms of Brain–Computer Interfaces,” Brain Sciences, vol. 15, no. 2, pp. 1-22, 2025.
    [CrossRef] [Google Scholar] [Publisher Link]
15. Alois Schlögl et al., “A Fully Automated Correction Method of EOG Artifacts in EEG Recordings,” Clinical Neurophysiology, vol. 118, no. 1, pp. 98-104, 2007.
    [CrossRef] [Google Scholar] [Publisher Link]
16. Kai Keng Ang et al., “Filter Bank Common Spatial Pattern Algorithm on BCI Competition IV Datasets 2a and 2b,” Frontiers in Neuroscience, vol. 6, pp. 1-9, 2012.
    [CrossRef] [Google Scholar] [Publisher Link]
17. Peter Grassberger, and Itamar Procaccia, “Measuring the Strangeness of Strange Attractors,” Physica D: Nonlinear Phenomena, vol. 9, no. 1-2, pp. 189-208, 1983.
    [CrossRef] [Google Scholar] [Publisher Link]
18. Michael T. Rosenstein, James J. Collins, and Carlo J. De Luca, “A Practical Method for Calculating Largest Lyapunov Exponents from Small Data Sets,” Physica D: Nonlinear Phenomena, vol. 65, no. 1-2, pp. 117-134, 1993.
    [CrossRef] [Google Scholar] [Publisher Link]
19. Floris Takens, “Detecting Strange Attractors in Turbulence,” Dynamical Systems and Turbulence Warwick 1980, Proceedings of a Symposium Held at the University of Warwick 1979/80, Heidelberg, Berlin, Springer, vol. 898, pp. 366-381, 1981.
    [CrossRef] [Google Scholar] [Publisher Link]
20. Tomoyuki Higuchi, “Approach to an Irregular Time Series on the Basis of the Fractal Theory,” Physica D: Nonlinear Phenomena, vol. 31, no. 2, pp. 277-283, 1988.
    [CrossRef] [Google Scholar] [Publisher Link]
21. Joshua S. Richman, and J. Randall Moorman, “Physiological Time-Series Analysis using Approximate Entropy and Sample Entropy,” American Journal of Physiology-Heart and Circulatory Physiology, vol. 278, no. 6, pp. H2039-H2049, 2000.
    [CrossRef] [Google Scholar] [Publisher Link]
22. Christoph Bandt, and Bernd Pompe, “Permutation Entropy: A Natural Complexity Measure for Time Series,” Physical Review Letters, vol. 88, no. 17, pp. 1-5, 2002.
    [CrossRef] [Google Scholar] [Publisher Link]
23. C.‐K. Peng et al., “Quantification of Scaling Exponents and Crossover Phenomena in Nonstationary Heartbeat Time Series,” Chaos, vol. 5, no. 1, pp. 82-87, 1995.
    [CrossRef] [Google Scholar] [Publisher Link]
24. Adriano B.L. Tort et al., “Measuring Phase-Amplitude Coupling between Neuronal Oscillations of Different Frequencies,” Journal of Neurophysiology, vol. 104, no. 2, pp. 1195-1210, 2010.
    [CrossRef] [Google Scholar] [Publisher Link]
25. C.L. Nikias, and M.R. Raghuveer, “Bispectrum Estimation: A Digital Signal Processing Framework,” Proceedings of the IEEE, vol. 75, no. 7, pp. 869-891, 1987.
    [CrossRef] [Google Scholar] [Publisher Link]
26. Alexandre Barachant et al., “Multiclass Brain-Computer Interface Classification by Riemannian Geometry,” IEEE Transactions on Biomedical Engineering, vol. 59, no. 4, pp. 920-928, 2012.
    [CrossRef] [Google Scholar] [Publisher Link]
27. Konstantin Dragomiretskiy, and Dominique Zosso, “Variational Mode Decomposition,” IEEE Transactions on Signal Processing, vol. 62, no. 3, pp. 531-544, 2014.
    [CrossRef] [Google Scholar] [Publisher Link]
28. J. Richard Landis, and Gary G. Koch, “The Measurement of Observer Agreement for Categorical Data,” Biometrics, vol. 33, no. 1, pp. 159-174, 1977.
    [CrossRef] [Google Scholar] [Publisher Link]
29. Benjamin Blankertz et al., “Single-Trial Analysis and Classification of ERP Components—A Tutorial,” NeuroImage, vol. 56, no. 2, pp. 814-825, 2011.
    [CrossRef] [Google Scholar] [Publisher Link]
30. He He, and Dongrui Wu, “Transfer Learning for Brain-Computer Interfaces: A Euclidean Space Data Alignment Approach,” IEEE Transactions on Biomedical Engineering, vol. 67, no. 2, pp. 399-410, 2020.
    [CrossRef] [Google Scholar] [Publisher Link]