78
M. Congedo
learning [5] has recently been shown to be very promising in other fields of research.
Studies testing its performance on ERP data are not conclusive so far. An approach
that features at the same time good accuracy, good generalization and good adaptation capabilities in the case of ERP data has been recently borrowed from the field of
differential geometry. This approach makes use of the Riemannian geometry on the
manifold of symmetric positive definite (SPD) matrices. Covariance matrices are of
this kind. A very simple classifier can be obtained based on the minimum distance
to mean (MDM) method [2]: every sweep is represented as a covariance matrix,
i.e., as a point on the multidimensional space of SPD matrices. The training set is
used to estimate the center of mass of training points for each class, i.e., a point best
representing the class. An unlabeled sweep is then simply assigned to the class the
center of mass of which is the closest to the unlabeled sweep. This approach as well
as other classifiers based on Riemannian geometry have been shown to possess good
accuracy, generalization and robustness properties [19, 66, 103, 105]
References
1. S. Arndt, T. Cizadlo, N.C. Andreasen et al., Tests for comparing images based on randomization and permutation methods. J. Cereb. Blood Flow Metab. 16, 1271–1279 (1996)
2. A. Barachant, S. Bonnet, M. Congedo et al., Multi-class brain computer interface classification
by riemannian geometry. IEEE Trans. Biomed. Eng. 59, 920–928 (2012)
3. Y. Benjamini, Y. Hochberg, Controlling the false discovery rate: a practical and powerful
approach to multiple testing. J. Roy. Stat. Soc. Ser. B Stat. Methodol. 57, 289–300 (1995)
4. Y. Benjamini, D. Yukutieli, The control of the false discovery rate in multiple testing under
dependency. Ann. Stat. 29, 1165–1188 (2001)
5. Y. Bengio, Learning deep architectures for AI. Found. Trends Mach. Learn. 2, 1–12 (2009)
6. R.C. Blair, J.F. Troendle, R.W. Beck, Control of familywise errors in multiple assessments
via stepwise permutation tests. Stat. Med. 15, 1107–1121 (1996)
7. P. Bloomfield, Fourier Analysis of Time Series. An Introduction, 2nd edn. (Wiley, Hoboken,
New Jersey, 2000), p. 261
8. A. Burns, Fourier-, Hilbert- and wavelet-based signal analysis: are they really independent
approaches? J. Neurosci. Methods 137, 321–332 (2004)
9. G. Buszáki, C.A. Anastassiou, C. Koch, The origin of extracellular fields and currents—EEG,
ECoG, LFP and spikes. Nat. Rev. Neurosci. 13, 407–420 (2012)
10. R.T. Canolty, E. Edwards, S.S. Dalal et al., High gamma power is phase-locked to theta
oscillations in human neocortex. Science 313, 1626–1628 (2006)
11. J.-F. Cardoso, Blind signal separation: statistical principles. Proc. IEEE 86, 2009–2025 (1998)
12. R.M. Chapman, J.W. McCrary, EP component identification and measurement by principal
component analysis. Brain Cogn. 27, 288–310 (1995)
13. M. Chavez, M. Besserve, C. Adam et al., Towards a proper estimation of phase synchronization
from time series. J. Neurosci. Methods 154, 149–160 (2006)
14. M.X. Cohen, Analyzing Neural Time Series Data: Theory and Practice (The MIT Press,
Cambridge, Massachusetts, 2014), p. 600
15. M.X. Cohen, J.F. Cavanagh, Single-trial regression elucidates the role of prefrontal theta
oscillations in response conflict. Front. Psychol. 2, 30 (2011)
16. L.L. Colgin, Theta-gamma coupling in the entorhinal-hippocampal system. Curr. Opin. Neurobiol. 31, 45–50 (2015)
17. P. Comon, C. Jutten (eds.), Handbook of Blind Source Separation, Independent Component
Analysis and Applications (Academic Press, Cambridge, MA, 2010)
M. Congedo
learning [5] has recently been shown to be very promising in other fields of research.
Studies testing its performance on ERP data are not conclusive so far. An approach
that features at the same time good accuracy, good generalization and good adaptation capabilities in the case of ERP data has been recently borrowed from the field of
differential geometry. This approach makes use of the Riemannian geometry on the
manifold of symmetric positive definite (SPD) matrices. Covariance matrices are of
this kind. A very simple classifier can be obtained based on the minimum distance
to mean (MDM) method [2]: every sweep is represented as a covariance matrix,
i.e., as a point on the multidimensional space of SPD matrices. The training set is
used to estimate the center of mass of training points for each class, i.e., a point best
representing the class. An unlabeled sweep is then simply assigned to the class the
center of mass of which is the closest to the unlabeled sweep. This approach as well
as other classifiers based on Riemannian geometry have been shown to possess good
accuracy, generalization and robustness properties [19, 66, 103, 105]
References
1. S. Arndt, T. Cizadlo, N.C. Andreasen et al., Tests for comparing images based on randomization and permutation methods. J. Cereb. Blood Flow Metab. 16, 1271–1279 (1996)
2. A. Barachant, S. Bonnet, M. Congedo et al., Multi-class brain computer interface classification
by riemannian geometry. IEEE Trans. Biomed. Eng. 59, 920–928 (2012)
3. Y. Benjamini, Y. Hochberg, Controlling the false discovery rate: a practical and powerful
approach to multiple testing. J. Roy. Stat. Soc. Ser. B Stat. Methodol. 57, 289–300 (1995)
4. Y. Benjamini, D. Yukutieli, The control of the false discovery rate in multiple testing under
dependency. Ann. Stat. 29, 1165–1188 (2001)
5. Y. Bengio, Learning deep architectures for AI. Found. Trends Mach. Learn. 2, 1–12 (2009)
6. R.C. Blair, J.F. Troendle, R.W. Beck, Control of familywise errors in multiple assessments
via stepwise permutation tests. Stat. Med. 15, 1107–1121 (1996)
7. P. Bloomfield, Fourier Analysis of Time Series. An Introduction, 2nd edn. (Wiley, Hoboken,
New Jersey, 2000), p. 261
8. A. Burns, Fourier-, Hilbert- and wavelet-based signal analysis: are they really independent
approaches? J. Neurosci. Methods 137, 321–332 (2004)
9. G. Buszáki, C.A. Anastassiou, C. Koch, The origin of extracellular fields and currents—EEG,
ECoG, LFP and spikes. Nat. Rev. Neurosci. 13, 407–420 (2012)
10. R.T. Canolty, E. Edwards, S.S. Dalal et al., High gamma power is phase-locked to theta
oscillations in human neocortex. Science 313, 1626–1628 (2006)
11. J.-F. Cardoso, Blind signal separation: statistical principles. Proc. IEEE 86, 2009–2025 (1998)
12. R.M. Chapman, J.W. McCrary, EP component identification and measurement by principal
component analysis. Brain Cogn. 27, 288–310 (1995)
13. M. Chavez, M. Besserve, C. Adam et al., Towards a proper estimation of phase synchronization
from time series. J. Neurosci. Methods 154, 149–160 (2006)
14. M.X. Cohen, Analyzing Neural Time Series Data: Theory and Practice (The MIT Press,
Cambridge, Massachusetts, 2014), p. 600
15. M.X. Cohen, J.F. Cavanagh, Single-trial regression elucidates the role of prefrontal theta
oscillations in response conflict. Front. Psychol. 2, 30 (2011)
16. L.L. Colgin, Theta-gamma coupling in the entorhinal-hippocampal system. Curr. Opin. Neurobiol. 31, 45–50 (2015)
17. P. Comon, C. Jutten (eds.), Handbook of Blind Source Separation, Independent Component
Analysis and Applications (Academic Press, Cambridge, MA, 2010)