BiSmoothed Functional Independent Component Analysis for EEG Artifact Removal
Abstract
:1. Introduction
2. Smoothed Functional Independent Component Analysis
2.1. Preliminaries
2.2. Functional ICA of a Smoothed Principal Component Expansion
 The FPCA of ${S}^{2}\left({x}_{i}\right)$ with respect to ${\langle \xb7,\xb7\rangle}_{\lambda}$, ${S}^{2}\left({x}_{i}\right)={\sum}_{j}{z}_{ij}{\gamma}_{\lambda ,j}.$
 The FPCA of $S\left({x}_{i}\right)$ with respect to $\langle \xb7,\xb7\rangle $, $S\left({x}_{i}\right)={\sum}_{j}{z}_{ij}{S}^{1}\left({\gamma}_{\lambda ,j}\right).$
 The FPCA of X with respect to ${\langle \xb7,\xb7\rangle}_{S}$, ${x}_{i}={\sum}_{j}{z}_{ij}{S}^{2}\left({\gamma}_{\lambda ,j}\right),$with ${\langle f,g\rangle}_{S}=\langle S\left(f\right),S\left(g\right)\rangle ={\langle {S}^{2}\left(f\right),{S}^{2}\left(g\right)\rangle}_{\lambda}.$
3. Basis Expansion Estimation Using a PSpline Penalty
4. Parameter Tuning
Penalty Parameter Selection
Algorithm 1.baseline crossvalidation 
Input: $A,{\varphi}_{j}\phantom{\rule{4pt}{0ex}}(j=1,\dots ,p),\mathcal{G},{P}_{2},{\lambda}_{k}={({\lambda}_{1},\dots ,{\lambda}_{m})}^{{\scriptstyle \mathrm{T}}}$ Output: ${\lambda}^{\u2022}.$ for each$\lambda $ in ${\lambda}_{k}$:
end for ${\lambda}^{\u2022}\leftarrow $ argmin${}_{\lambda}$bcv. 
5. Simulation Study
Algorithm 2.functional artifact subtraction 
Input: $A,{\varphi}_{j}\phantom{\rule{4pt}{0ex}}(j=1,\dots ,p),\mathcal{G},{P}_{2},\lambda ,q$ Output: ${d}_{j}.$

6. Estimating Brain Signals from Contaminated EventRelated Potentials
7. Discussion
Author Contributions
Funding
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations and Abbreviations
Abbreviations
Abbreviations
EEG  Electroencephalography 
FICA  Functional Independent Component Analysis 
FPCA  Functional Principal Component Analysis 
ICA  Independent Component Analysis 
KL  Karhunen–Loève 
PCA  Principal Component Analysis 
References
 Wang, X. Neurophysiological and Computational Principles of Cortical Rhythms in Cognition. Physiol. Rev. 2010, 90, 1195–1268. [Google Scholar] [CrossRef] [PubMed]
 Buzsáki, G. Rhythms of the Brain; Oxford University Press: Oxford, UK, 2006. [Google Scholar]
 Castellanos, N.P.; Makarov, V.A. Recovering EEG Brain Signals: Artifact Suppression with Wavelet Enhanced Independent Component Analysis. J. Neurosci. Methods 2006, 158, 300–312. [Google Scholar] [CrossRef]
 Nordhausen, K.; Oja, H. Independent Component Analysis: A Statistical Perspective. WIREs Comput. Stat. 2018, 10, 1–23. [Google Scholar] [CrossRef]
 Hyvärinen, A.; Karhunen, J.; Oja, E. Independent Component Analysis; John Wiley & Sons, Ltd: New York, NY, USA, 2001. [Google Scholar]
 Ramsay, J.; Silverman, B.W. Functional Data Analysis; Springer: New York, NY, USA, 2005. [Google Scholar]
 Hsing, T.; Eubank, R. Theoretical Foundations of Functional Data Analysis, with an Introduction to Linear Operators; Willey: Chichester, UK, 2015. [Google Scholar]
 Wang, J.L.; Chiou, J.M.; Müller, H.G. Functional Data Analysis. Annu. Rev. Stat. Its Appl. 2016, 3, 257–295. [Google Scholar] [CrossRef][Green Version]
 Mehta, N.; Gray, A. FuncICA for Time Series Pattern Discovery. In Proceedings of the 2009 SIAM International Conference on Data Mining, Sparks, NV, USA, 30 April–2 May 2009; pp. 73–84. [Google Scholar]
 Peña, C.; Prieto, J.; Rendón, C. Independent Components Techniques Based on Kurtosis for Functional Data Analysis; Working Paper 14–10 Statistics and Econometric Series (06); Universidad Carlos III de Madrid: Madrid, Spain, 2014. [Google Scholar]
 Li, B.; Bever, G.V.; Oja, H.; Sabolová, R.; Critchley, F. Functional Independent Component Analysis: An Extension of the FourthOrder Blind Identification; Technical Report; Université de Namur: Namur, Belgium, 2019. [Google Scholar]
 Virta, J.; Li, B.; Nordhausen, K.; Oja, H. Independent Component Analysis for Multivariate Functional Data. J. Multivar. Anal. 2020, 176, 1–19. [Google Scholar] [CrossRef][Green Version]
 Ash, R.B.; Gardner, M.F. Topics in Stochastic Processes; Academic Press: New York, NY, USA, 1975. [Google Scholar]
 Xiao, L.; Zipunnikov, V.; Ruppert, D.; Crainiceanu, C. Fast Covariance Estimation for HighDimensional Functional Data. Stat. Comput. 2016, 26, 409–421. [Google Scholar] [CrossRef][Green Version]
 Hasenstab, K.; Scheffler, A.; Telesca, D.; Sugar, C.A.; Jeste, S.; DiStefano, C.; Sentürk, D. A MultiDimensional Functional Principal Components Analysis of EEG Data. Biometrics 2017, 3, 999–1009. [Google Scholar] [CrossRef][Green Version]
 Nie, Y.; Wang, L.; Liu, B.; Cao, J. Supervised Functional Principal Component Analysis. Stat. Comput. 2018, 28, 713–723. [Google Scholar] [CrossRef]
 Pokora, O.; Kolacek, J.; Chiu, T.; Qiu, W. Functional Data Analysis of SingleTrial Auditory Evoked Potentials Recorded in the Awake Rat. Biosystems 2018, 161, 67–75. [Google Scholar] [CrossRef] [PubMed]
 Scheffler, A.; Telesca, D.; Li, Q.; Sugar, C.A.; Distefano, C.; Jeste, S.; Şentürk, D. Hybrid Principal Components Analysis for RegionReferenced Longitudinal Functional EEG Data. Biostatistics 2018, 21, 139–157. [Google Scholar] [CrossRef] [PubMed]
 Urigüen, J.A.; GarciaZapirain, B. EEG Artifact Removal—StateoftheArt and Guidelines. J. Neural Eng. 2015, 12, 1–23. [Google Scholar] [CrossRef] [PubMed]
 Akhtar, M.T.; Mitsuhashi, W.; James, C.J. Employing Spatially Constrained ICA and Wavelet Denoising, for Automatic Removal of Artifacts from Multichannel EEG Data. Signal Process. 2012, 92, 401–416. [Google Scholar] [CrossRef]
 Mammone, N.; Morabito, F.C. Enhanced Automatic Wavelet Independent Component Analysis for Electroencephalographic Artifact Removal. Entropy 2014, 16, 6553–6572. [Google Scholar] [CrossRef][Green Version]
 Bajaj, N.; Requena Carrión, J.; Bellotti, F.; Berta, R.; De Gloria, A. Automatic and Tunable Algorithm for EEG Artifact Removal Using Wavelet Decomposition with Applications in Predictive Modeling During Auditory Tasks. Biomed. Signal Process. Control. 2020, 55, 101624. [Google Scholar] [CrossRef]
 Eilers, P.H.C.; Marx, B.D. Flexible Smoothing with BSplines and Penalties (with Discussion). Stat. Sci. 1996, 11, 89–121. [Google Scholar] [CrossRef]
 Currie, I.D.; Durban, M. Flexible Smoothing with PSplines: A Unified Approach. Stat. Model. 2002, 2, 333–349. [Google Scholar] [CrossRef]
 Aguilera, A.M.; AguileraMorillo, M.C. Penalized PCA Approaches for BSpline Expansions of Smooth Functional Data. Appl. Math. Comput. 2013, 219, 7805–7819. [Google Scholar] [CrossRef]
 AguileraMorillo, M.; Aguilera, A.; Escabias, M. Penalized Spline Approaches for Functional Logit Regression. TEST 2013, 22, 251–277. [Google Scholar] [CrossRef]
 Aguilera, A.; AguileraMorillo, M.; Preda, C. Penalized Versions of Functional PLS Regression. Chemom. Intell. Lab. Syst. 2016, 154, 80–92. [Google Scholar] [CrossRef]
 AguileraMorillo, M.; Aguilera, A.; Durbán, M. Prediction of Functional Data with Spatial Dependence: A Penalized Approach. Stoch. Environ. Res. Risk Assess. 2017, 31, 7–22. [Google Scholar] [CrossRef]
 AguileraMorillo, M.; Aguilera, A. MultiClass Classification of Biomechanical Data: A Functional LDA Approach Based on MultiClass Penalized Functional PLS. Stat. Model. 2020, 20, 592–616. [Google Scholar] [CrossRef]
 Vidal, M.; Aguilera, A.M. pfica: Independent Component Analysis for Univariate Functional Data; R Package Version 0.1.2. 2021. Available online: https://CRAN.Rproject.org/package=pfica (accessed on 25 May 2021).
 Ghanem, R.; Spanos, P. Stochastic Finite Elements: A Spectral Approach; Springer: New York, NY, USA, 1991. [Google Scholar]
 Acal, C.; Aguilera, A.M.; Escabias, M. New Modeling Approaches Based on Varimax Rotation of Functional Principal Components. Mathematics 2020, 8, 2085. [Google Scholar] [CrossRef]
 Delaigle, A.; Hall, P. Defining Probability Density for a Distribution of Random Functions. Ann. Stat. 2010, 38, 1171–1193. [Google Scholar] [CrossRef][Green Version]
 Gutch, H.W.; Theis, F.J. To Infinity and Beyond: On ICA over Hilbert spaces. In Latent Variable Analysis and Signal Separation; Theis, F.J., Cichocki, A., Yeredor, A., Zibulevsky, M., Eds.; Springer: Berlin/Heidelberg, Germany, 2012; pp. 180–187. [Google Scholar]
 Silverman, B.W. Smoothed Functional Principal Components Analysis by Choice of Norm. Ann. Stat. 1996, 24, 1–24. [Google Scholar] [CrossRef]
 Qi, X.; Zhao, H. Some Theoretical Properties of Silverman’s Method for Smoothed Functional Principal Component Analysis. J. Multivar. Anal. 2011, 102, 742–767. [Google Scholar] [CrossRef][Green Version]
 Lakraj, G.P.; Ruymgaart, F. Some Asymptotic Theory for Silverman’s Smoothed Functional Principal Components in an Abstract Hilbert Space. J. Multivar. Anal. 2017, 155, 122–132. [Google Scholar] [CrossRef]
 Ocaña, F.A.; Aguilera, A.M.; Valderrama, M.J. Functional Principal Component Analysis by Choice of Norm. J. Multivar. Anal. 1999, 71, 262–276. [Google Scholar] [CrossRef][Green Version]
 Aguilera, A.M.; AguileraMorillo, M.C. Comparative Study of Different BSpline Approaches for Functional Data. Math. Comput. Model. 2013, 58, 1568–1579. [Google Scholar] [CrossRef]
 Ocaña, F.A.; Aguilera, A.M.; Escabias, M. Computational Considerations in Functional Principal Component Analysis. Comput. Stat. 2007, 22, 449–465. [Google Scholar] [CrossRef]
 Kollo, T. Multivariate Skewness and Kurtosis Measures with an Application in ICA. J. Multivar. Anal. 2008, 99, 2328–2338. [Google Scholar] [CrossRef][Green Version]
 Loperfido, N. A New Kurtosis Matrix, with Statistical Applications. Linear Algebra Its Appl. 2017, 512, 1–17. [Google Scholar] [CrossRef]
 Rice, J.A.; Silverman, B.W. Estimating the Mean and Covariance Structure Nonparametrically when the Data are Curves. J. R. Stat. Soc. Ser. B 1991, 53, 233–243. [Google Scholar] [CrossRef]
 Schäfer, J.; Strimmer, K. A Shrinkage Approach to LargeScale Covariance Matrix Estimation and Implications for Functional Genomics. Stat. Appl. Genet. Mol. Biol. 2005, 4, 1–29. [Google Scholar] [CrossRef] [PubMed][Green Version]
 Noury, N.; Hipp, J.F.; Siegel, M. Physiological Processes Nonlinearly Affect Electrophysiological Recordings During Transcranial Electric Stimulation. Neuroimage 2016, 140, 99–109. [Google Scholar] [CrossRef] [PubMed]
 Novi, S.L.; Roberts, E.; Spagnuolo, D.; Spilsbury, B.M.; Price, D.C.; Imbalzano, C.A.; Forero, E.; Yodh, A.G.; Tellis, G.M.; Tellis, C.M.; et al. Functional NearInfrared Spectroscopy for Speech Protocols: Characterization of Motion Artifacts and Guidelines for Improving Data Analysis. Neurophotonics 2020, 7, 1–15. [Google Scholar] [CrossRef] [PubMed]
 Artoni, F.; Delorme, A.; Makeig, S. Applying Dimension Reduction to EEG data by Principal Component Analysis Reduces the Quality of its Subsequent Independent Component Decomposition. Neuroimage 2018, 175, 176–187. [Google Scholar] [CrossRef]
 Tong, S.; Thakor, N.V. Quantitative EEG Analysis Methods and Clinical Applications; Artech House: Boston, MA, USA, 2009. [Google Scholar]
 Ieva, F.; Paganoni, A.M.; Tarabelloni, N. Covariancebased Clustering in Multivariate and Functional Data Analysis. J. Mach. Learn. Res. 2016, 17, 1–21. [Google Scholar]
 Zhang, X.; Wang, J.L. From Sparse to Dense Functional Data and Beyond. Ann. Stat. 2016, 44, 2281–2312. [Google Scholar] [CrossRef]
 Xiao, L. Asymptotic Theory of Penalized Splines. Electron. J. Stat. 2019, 13, 747–794. [Google Scholar] [CrossRef]
Trial  ${\mathit{j}}_{0}$  q  $\mathit{\lambda}$  logbcv$\left(\mathit{\lambda}\right)$  var (%)  var (%) 

$\mathbf{\lambda}$  $\mathbf{\lambda}=\mathbf{0}$  
Nodding  6  5  ${10}^{8}$  10.66  99.40  94.43 
Arm mov.  4  2  4000  13.91  75.85  62.42 
Blinks  4  3  400.0  13.76  97.50  93.56 
Chewing  5  4  0.300  13.01  68.23  68.03 
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Vidal, M.; Rosso, M.; Aguilera , A.M. BiSmoothed Functional Independent Component Analysis for EEG Artifact Removal. Mathematics 2021, 9, 1243. https://doi.org/10.3390/math9111243
Vidal M, Rosso M, Aguilera AM. BiSmoothed Functional Independent Component Analysis for EEG Artifact Removal. Mathematics. 2021; 9(11):1243. https://doi.org/10.3390/math9111243
Chicago/Turabian StyleVidal, Marc, Mattia Rosso, and Ana M. Aguilera . 2021. "BiSmoothed Functional Independent Component Analysis for EEG Artifact Removal" Mathematics 9, no. 11: 1243. https://doi.org/10.3390/math9111243