Statistics and Computing

, Volume 25, Issue 2, pp 273–288 | Cite as

COPICA—independent component analysis via copula techniques

  • Ray-Bing Chen
  • Meihui Guo
  • Wolfgang K. Härdle
  • Shih-Feng Huang


Independent component analysis (ICA) is a modern computational method developed in the last two decades. The main goal of ICA is to recover the original independent variables by linear transformations of the observations. In this study, a copula-based method, called COPICA, is proposed to solve the ICA problem. The proposed COPICA method is a semiparametric approach, the marginals are estimated by nonparametric empirical distributions and the joint distributions are modeled by parametric copula functions. The COPICA method utilizes the estimated copula parameter as a dependence measure to search the optimal rotation matrix that achieves the ICA goal. Both simulation and empirical studies are performed to compare the COPICA method with the state-of-art methods of ICA. The results indicate that the COPICA attains higher signal-to-noise ratio (SNR) than several other ICA methods in recovering signals. In particular, the COPICA usually leads to higher SNRs than FastICA for near-Gaussian-tailed sources and is competitive with a nonparametric ICA method for two dimensional sources. For higher dimensional ICA problem, the advantage of using the COPICA is its less storage and less computational effort.


Blind source separation Canonical maximum likelihood method Givens rotation matrix Signal/noise ratio Simulated annealing algorithm 



The authors gratefully acknowledge the National Science Council in Taiwan, National Center for Theoretical Sciences (South), Tainan, Taiwan and the Deutsche Forschungsgemeinschaft through the SFB 649 “Economic Risk”, Humboldt-Universitat zu Berlin. This work was supported in part by National Science Council under grants NSC 96-2118-M-390-002- (Chen), NSC 100-2118-M-110-001-003- (Guo) and NSC 101-2118-M-390-002- (Huang).


  1. Abayomi, K., Lall, U., de la Pena, V.: Copula Based Independent Component Analysis. Working paper (2008) Google Scholar
  2. Abayomi, K., de la Pena, V., Lall, U., Levy, M.: Quantifying sustainability: methodology for and determinants of an environmental sustainability index. In: Luo, Z.W. (ed.) Green Finance and Sustainability: Environmentally-Aware Business Models and Technologies, pp. 74–89 (2011) CrossRefGoogle Scholar
  3. Bach, F.R., Jordan, M.I.: Kernel independent component analysis. J. Mach. Learn. Res. 3, 1–38 (2002) MathSciNetGoogle Scholar
  4. Bell, A.J., Sejnowski, T.J.: An information maximization approach to blind source separation and blind deconvolution. Neural Comput. 7, 1129–1159 (1995) CrossRefGoogle Scholar
  5. Blaschke, T., Wiskott, L.: CuBICA: independent component analysis by simultaneous third- and fourth-order cumulant diagonalization. IEEE Trans. Signal Process. 52, 1250–1256 (2004) CrossRefMathSciNetGoogle Scholar
  6. Chen, R.-B., Wu, Y.N.: A null space method for over-complete blind source separation. Comput. Stat. Data Anal. 51, 5519–5536 (2007) CrossRefzbMATHGoogle Scholar
  7. Comon, P.: Independent component analysis—a new concept? Signal Process. 36(3), 287–314 (1994) CrossRefzbMATHGoogle Scholar
  8. Genest, C., Nešlehová, J., Ziegel, J.: Inference in multivariate Archimedean copula models. Test 20, 223–256 (2011) CrossRefzbMATHMathSciNetGoogle Scholar
  9. Ghosh, S., Henderson, S.: Behaviour of the NORTA method for correlated random vector generation as the dimension increases. ACM Trans. Model. Comput. Simul. 13, 276V294 (2003) CrossRefGoogle Scholar
  10. Gretton, A., Bousquet, O., Smola, A.J., Schölkopf, B.: Measuring statistical dependence with Hilbert-Schmidt norms. In: ALT, pp. 63–78. Springer, Heidelberg (2005) Google Scholar
  11. Grønneberg, S., Hjort, N.L.: The Copula Information Criterion. Technical report, Department of Math., University of Oslo, Norway (2008) Google Scholar
  12. Hyvärinen, A.: Fast and robust fixed-point algorithms for independent component analysis. IEEE Trans. Neural Netw. 10, 626–634 (1999a) CrossRefGoogle Scholar
  13. Hyvärinnen, A.: Sparse code shrinkage: denoising of nongaussian data by maximum likelihood estimation. Neural Comput. 11, 1739–1768 (1999b) CrossRefGoogle Scholar
  14. Hyvärinen, A., Oja, E.: A fast fixed-point algorithm for independent component analysis. Neural Comput. 9, 1483–1492 (1997) CrossRefGoogle Scholar
  15. Hyvärinen, A., Oja, E.: Independent component analysis: algorithms and application. Neural Netw. 13, 411–430 (2000) CrossRefGoogle Scholar
  16. Kidmose, P.: Blind Separation of Heavy Tail Signals. Ph.D. Thesis, Technical University of Denmark, Lyngby (2001) Google Scholar
  17. Kirkpatrick, S., Gelatt, C.D. Jr., Vechhi, M.P.: Optimization by simulated annealing. Science 220, 671–680 (1983) CrossRefzbMATHMathSciNetGoogle Scholar
  18. Kirshner, S., Póczos, B.: ICA and ISA using Schweizer-Wolff measure of dependence. In: International Conference on Machine Learning (ICML-2008) (2008). Google Scholar
  19. Kotz, S., Nadarajah, S.: Extreme Value Distributions. Theory and Applications. Imperial College Press, London (2000) CrossRefzbMATHGoogle Scholar
  20. Learned-Miller, E.G., Fisher, J.W.: ICA using spacings estimates of entropy. J. Mach. Learn. Res. 4, 1271–1295 (2003) MathSciNetGoogle Scholar
  21. Lee, T.-W.: Independent Component Analysis. Theory and Applications. Kluwer Academic, Dordrecht (1998) zbMATHGoogle Scholar
  22. Liu, J.S.: Monte Carlo Strategies in Scientific Computing. Springer, New York (2001) zbMATHGoogle Scholar
  23. Ma, J., Sun, Z.: Copula component analysis. In: Proceedings of the 7th International Conference on Independent Component Analysis and Signal Separation, pp. 73–80. ACM, London (2007) CrossRefGoogle Scholar
  24. Mardia, K.V.: A translation family of bivariate distributions and Fréchet’s bounds. Sankhya A 32, 119–122 (1970) zbMATHMathSciNetGoogle Scholar
  25. McNeil, A.J., Nešlehová, J.: From Archimedean to Liouville copulas. J. Multivar. Anal. 101, 1772–1790 (2010) CrossRefzbMATHGoogle Scholar
  26. Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H.: Equation of state calculation by fast computing machines. J. Chem. Phys. 21, 1087–1092 (1953) CrossRefGoogle Scholar
  27. Nelsen, R.B.: An Introduction to Copulas. Springer, Berlin (2006) zbMATHGoogle Scholar
  28. Olshausen, B.A., Field, D.J.: Natural image statistics and efficient coding. Network 7, 333–339 (1996) CrossRefGoogle Scholar
  29. Schmid, F., Schmidt, R.: Nonparametric inference on multivariate versions of Blomqvist’s beta and related measures of tail dependence. Metrika 66, 323–354 (2007) CrossRefMathSciNetGoogle Scholar
  30. Schweizer, B., Wolff, E.F.: On nonparametric measures of dependence for random variables. Ann. Stat. 9, 879–885 (1981) CrossRefzbMATHMathSciNetGoogle Scholar
  31. Shen, H., Jegelka, S., Gretton, A.: Fast kernel-based independent component analysis. IEEE Trans. Signal Process. 57, 3498–3511 (2009) CrossRefMathSciNetGoogle Scholar
  32. Sklar, A.: Fonctions de Repartition à n Dimensions et Leurs Marges. Publ. Inst. Stat. Univ. Paris 8, 229–231 (1959) MathSciNetGoogle Scholar
  33. Sklar, A.: Random variables, distribution functions, and copulas—a personal look backward and forward. In: Ruschendorf, L., Schweizer, B., Taylor, M.D. (eds.) Distributions with Fixed Marginals and Related Topics, pp. 1–14. Institute of Mathematical Statistics, Hayward (1996) CrossRefGoogle Scholar
  34. Sodoyer, D., Girin, L., Jutten, C., Schwartz, J.-L.: Speech extraction based on ICA and audio-visual coherence. In: Proceedings of the 7th International Symposium on Signal Processing and Its Applications (ISSPA’03), vol. 2, pp. 65–68 (2003) CrossRefGoogle Scholar
  35. Tsai, A.C., Liou, M., Jung, T.-P., Onton, J.A., Cheng, P.E., Huang, C.-C., Duann, J.-R., Makeig, S.: Mapping single-trail EEG records on the cortical surface through a spatiotemporal modality. NeuroImage 32, 195–207 (2006) CrossRefGoogle Scholar
  36. Yu, L., Verducci, J.S., Blower, P.E.: The tau-path test for monotone association in an unspecified subpopulation. In: Application to Chemogenomic Data Mining Statistical Methodology, vol. 8, pp. 97–111 (2011) Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Ray-Bing Chen
    • 1
  • Meihui Guo
    • 2
  • Wolfgang K. Härdle
    • 3
    • 4
  • Shih-Feng Huang
    • 5
  1. 1.Dept. of StatisticsNational Cheng Kung UniversityTainanTaiwan
  2. 2.Dept. of Applied Math.National Sun Yat-sen UniversityKaohsiungTaiwan
  3. 3.Center for Applied Statistics and EconomicsHumboldt-Universität zu BerlinBerlinGermany
  4. 4.Lee Kong Chian School of BusinessSingapore Management UniversitySingaporeSingapore
  5. 5.Dept. of Applied Math.National University of KaohsiungKaohsiungTaiwan

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