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Optimization and testing in linear non-Gaussian component analysis. (English) Zbl 07260626

Summary: Independent component analysis (ICA) decomposes multivariate data into mutually independent components (ICs). The ICA model is subject to a constraint that at most one of these components is Gaussian, which is required for model identifiability. Linear non-Gaussian component analysis (LNGCA) generalizes the ICA model to a linear latent factor model with any number of both non-Gaussian components (signals) and Gaussian components (noise), where observations are linear combinations of independent components. Although the individual Gaussian components are not identifiable, the Gaussian subspace is identifiable. We introduce an estimator along with its optimization approach in which non-Gaussian and Gaussian components are estimated simultaneously, maximizing the discrepancy of each non-Gaussian component from Gaussianity while minimizing the discrepancy of each Gaussian component from Gaussianity. When the number of non-Gaussian components is unknown, we develop a statistical test to determine it based on resampling and the discrepancy of estimated components. Through a variety of simulation studies, we demonstrate the improvements of our estimator over competing estimators, and we illustrate the effectiveness of our test to determine the number of non-Gaussian components. Further, we apply our method to real data examples and show its practical value.

MSC:

62-XX Statistics
68-XX Computer science

Software:

JADE; LNGCA; BSSasymp; ProDenICA; R
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Full Text: DOI arXiv

References:

[1] D. M. Bean, Non-gaussian component analysis, Ph.D. Thesis, Univ. of California, Berkeley, CA, 2014.
[2] Anthony J. Bell and Terrence J. Sejnowski,An information-maximization approach to blind separation and blind deconvolution, Neural Comput. 7(6) (1995), 1129-1159.
[3] Gilles Blanchard et al.,Non-gaussian component analysis: a semi-parametric framework for linear dimension reduction, inAdvances in neural information processing systems, MIT Press, Cambridge, MA, 2006, 131-138.
[4] J.-F. Cardoso,Source separation using higher order moments, in1989 International Conference on Acoustics, Speech, and Signal Processing, 1989. ICASSP-89, IEEE, Glasgow, UK 1989, 2109-2112.
[5] Jean-François Cardoso and Antoine Souloumiac,Blind beamforming for non-gaussian signals, IEE Proc. F Radar Signal Process. 140 (1993), 362-370.
[6] Trevor Hastie and Rob Tibshirani,Independent components analysis through product density estimation, inAdvances in neural information processing systems, Suzanna Becker, Sebastian Thrun, Klaus Obermayer, MIT Press, Cambridge, MA, 2002, 665-672 https://dl.acm.org/citation.cfm?id=2968 701.
[7] T. Hastie and R. Tibshirani. Prodenica: Product density estimation for ICA using tilted gaussian density estimates.Rpackage version, 1, 2010.
[8] Aapo Hyvärinen and Erkki Oja,A fast fixed-point algorithm for independent component analysis, Neural Comput. 9(7) (1997), 1483-1492. · Zbl 0893.94034
[9] Aapo Hyvärinen, Juha Karhunen, and Erkki Oja,Independent component analysis, Vol 46, John Wiley & Sons, New York, USA 2004. · Zbl 0893.94034
[10] Pauliina Ilmonen et al.,A new performance index for ICA: properties,computationandasymptoticanalysis,LatentVar.Anal. Signal Sep. LNCS 6365 (2010), 229-236 https://link.springer.com/ chapter/10.1007/978-3-642-15995-4_29
[11] Carlos M. Jarque and Anil K. Bera,A test for normality of observations and regression residuals, Int. Stat. Rev. 55 (1987), 163-172. · Zbl 0616.62092
[12] Z. Jin and D. S. Matteson, Independent component analysis via energy-based and kernel-based mutual dependence measures, arXiv preprint arXiv:1805.06639, 2018a.
[13] Ze Jin and David S. Matteson,Generalizing distance covariance to measure and test multivariate mutual dependence via complete and incomplete v-statistics, J. Multivar. Anal. 168 (2018b), 304-322. · Zbl 1401.62073
[14] Z. Jin, Benjamin B Risk, and David S Matteson. LNGCA: Linear non-gaussian component analysis,Rpackage version 1.0, 2019. · Zbl 1478.62155
[15] Motoaki Kawanabe et al.,A new algorithm of non-gaussian component analysis with radial kernel functions, Ann. Inst. Stat. Math. 59(1) (2007), 57-75. · Zbl 1147.62349
[16] Ruoyu Li et al.,Fast preconditioning for accelerated multi-contrast mri reconstruction, inInternational Conference on Medical Image Computing and Computer-Assisted Intervention, N. Navab et al., Eds., Springer, Munich, Germany 2015, 700-707.
[17] David S. Matteson and Ruey S. Tsay,Independent component analysis via distance covariance, J. Amer. Stat. Assoc. 112(518) (2017), 623-637.
[18] J. Miettinen, K. Nordhausen, and S. Taskinen,Blind source separation based on joint diagonalization in R: The packages jade and bssasymp, J. Stat. Softw. 76 (2017) https://www.jstatsoft.org/article/view/v076i02.
[19] Klaus Nordhausen et al.,Asymptotic and bootstrap tests for the dimension of the non-gaussian subspace, IEEE Signal Process. Lett. 24(6) (2017), 887-891.
[20] Christos H. Papadimitriou and Kenneth Steiglitz,Combinatorial optimization: algorithms and complexity, Prentice-Hall, Inc, Upper Saddle River, NJ, USA 1982. · Zbl 0503.90060
[21] Benjamin B. Risk et al.,An evaluation of independent component analyses with an application to resting-state fmri, Biometrics 70(1) (2014), 224-236. · Zbl 1419.62430
[22] Benjamin B. Risk, David S. Matteson, and David Ruppert,Linear non-gaussian component analysis via maximum likelihood, J. Amer. Stat. Assoc. (2017) To appear https://www.tandfonline.com/doi/ref/10.1080/ 01621459.2017.1407772. · Zbl 1478.62155
[23] Hiroaki Sasaki, Gang Niu, and Masashi Sugiyama,Non-gaussian component analysis with log-density gradient estimation, inProceedings of the 19th International Conference Artificial intelligence and statistics, JMLR: W&CP, Cadiz, Spain, 2016, 1177-1185.
[24] H. Shiino, H. Sasaki, G. Niu, and M. Sugiyama. Whitening-free least-squares non-gaussian component analysis, arXiv preprint arXiv:1603.01029, 2016.
[25] Fabian J. Theis, Motoaki Kawanabe, and Klaus-Robert Muller,Uniqueness of non-gaussianity-based dimension reduction, IEEE Trans. Signal Process. 59(9) (2011), 4478-4482. · Zbl 1391.62109
[26] J. Virta, K. Nordhausen, and H. Oja, Joint use of third and fourth cumulants in independent component analysis, arXiv preprint arXiv:1505.02613, 2015.
[27] J.
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