Skip to main content
SpringerLink
Log in

COPICA—independent component analysis via copula techniques

  • Published:
Statistics and Computing Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  • Abayomi, K., Lall, U., de la Pena, V.: Copula Based Independent Component Analysis. Working paper (2008)

  • 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)

    Chapter  Google Scholar 

  • Bach, F.R., Jordan, M.I.: Kernel independent component analysis. J. Mach. Learn. Res. 3, 1–38 (2002)

    MathSciNet  Google Scholar 

  • Bell, A.J., Sejnowski, T.J.: An information maximization approach to blind source separation and blind deconvolution. Neural Comput. 7, 1129–1159 (1995)

    Article  Google Scholar 

  • Blaschke, T., Wiskott, L.: CuBICA: independent component analysis by simultaneous third- and fourth-order cumulant diagonalization. IEEE Trans. Signal Process. 52, 1250–1256 (2004)

    Article  MathSciNet  Google Scholar 

  • Chen, R.-B., Wu, Y.N.: A null space method for over-complete blind source separation. Comput. Stat. Data Anal. 51, 5519–5536 (2007)

    Article  MATH  Google Scholar 

  • Comon, P.: Independent component analysis—a new concept? Signal Process. 36(3), 287–314 (1994)

    Article  MATH  Google Scholar 

  • Genest, C., Nešlehová, J., Ziegel, J.: Inference in multivariate Archimedean copula models. Test 20, 223–256 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  • 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)

    Article  Google Scholar 

  • 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 

  • Grønneberg, S., Hjort, N.L.: The Copula Information Criterion. Technical report, Department of Math., University of Oslo, Norway (2008)

  • Hyvärinen, A.: Fast and robust fixed-point algorithms for independent component analysis. IEEE Trans. Neural Netw. 10, 626–634 (1999a)

    Article  Google Scholar 

  • Hyvärinnen, A.: Sparse code shrinkage: denoising of nongaussian data by maximum likelihood estimation. Neural Comput. 11, 1739–1768 (1999b)

    Article  Google Scholar 

  • Hyvärinen, A., Oja, E.: A fast fixed-point algorithm for independent component analysis. Neural Comput. 9, 1483–1492 (1997)

    Article  Google Scholar 

  • Hyvärinen, A., Oja, E.: Independent component analysis: algorithms and application. Neural Netw. 13, 411–430 (2000)

    Article  Google Scholar 

  • Kidmose, P.: Blind Separation of Heavy Tail Signals. Ph.D. Thesis, Technical University of Denmark, Lyngby (2001)

  • Kirkpatrick, S., Gelatt, C.D. Jr., Vechhi, M.P.: Optimization by simulated annealing. Science 220, 671–680 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  • Kirshner, S., Póczos, B.: ICA and ISA using Schweizer-Wolff measure of dependence. In: International Conference on Machine Learning (ICML-2008) (2008). http://icml2008.cs.helsinki.fi/papers/551.pdf

    Google Scholar 

  • Kotz, S., Nadarajah, S.: Extreme Value Distributions. Theory and Applications. Imperial College Press, London (2000)

    Book  MATH  Google Scholar 

  • Learned-Miller, E.G., Fisher, J.W.: ICA using spacings estimates of entropy. J. Mach. Learn. Res. 4, 1271–1295 (2003)

    MathSciNet  Google Scholar 

  • Lee, T.-W.: Independent Component Analysis. Theory and Applications. Kluwer Academic, Dordrecht (1998)

    MATH  Google Scholar 

  • Liu, J.S.: Monte Carlo Strategies in Scientific Computing. Springer, New York (2001)

    MATH  Google Scholar 

  • 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)

    Chapter  Google Scholar 

  • Mardia, K.V.: A translation family of bivariate distributions and Fréchet’s bounds. Sankhya A 32, 119–122 (1970)

    MATH  MathSciNet  Google Scholar 

  • McNeil, A.J., Nešlehová, J.: From Archimedean to Liouville copulas. J. Multivar. Anal. 101, 1772–1790 (2010)

    Article  MATH  Google Scholar 

  • 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)

    Article  Google Scholar 

  • Nelsen, R.B.: An Introduction to Copulas. Springer, Berlin (2006)

    MATH  Google Scholar 

  • Olshausen, B.A., Field, D.J.: Natural image statistics and efficient coding. Network 7, 333–339 (1996)

    Article  Google Scholar 

  • Schmid, F., Schmidt, R.: Nonparametric inference on multivariate versions of Blomqvist’s beta and related measures of tail dependence. Metrika 66, 323–354 (2007)

    Article  MathSciNet  Google Scholar 

  • Schweizer, B., Wolff, E.F.: On nonparametric measures of dependence for random variables. Ann. Stat. 9, 879–885 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  • Shen, H., Jegelka, S., Gretton, A.: Fast kernel-based independent component analysis. IEEE Trans. Signal Process. 57, 3498–3511 (2009)

    Article  MathSciNet  Google Scholar 

  • Sklar, A.: Fonctions de Repartition à n Dimensions et Leurs Marges. Publ. Inst. Stat. Univ. Paris 8, 229–231 (1959)

    MathSciNet  Google Scholar 

  • 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)

    Chapter  Google Scholar 

  • 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)

    Chapter  Google Scholar 

  • 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)

    Article  Google Scholar 

  • 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 

Download references

Acknowledgements

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).

Author information

Authors and Affiliations

  1. Dept. of Statistics, National Cheng Kung University, Tainan, 701, Taiwan

    Ray-Bing Chen

  2. Dept. of Applied Math., National Sun Yat-sen University, Kaohsiung, 804, Taiwan

    Meihui Guo

  3. Center for Applied Statistics and Economics, Humboldt-Universität zu Berlin, Berlin, Germany

    Wolfgang K. Härdle

  4. Lee Kong Chian School of Business, Singapore Management University, Singapore, Singapore

    Wolfgang K. Härdle

  5. Dept. of Applied Math., National University of Kaohsiung, Kaohsiung, 811, Taiwan

    Shih-Feng Huang

Authors
  1. Ray-Bing Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Meihui Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Wolfgang K. Härdle
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Shih-Feng Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Meihui Guo.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chen, RB., Guo, M., Härdle, W.K. et al. COPICA—independent component analysis via copula techniques. Stat Comput 25, 273–288 (2015). https://doi.org/10.1007/s11222-013-9431-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11222-013-9431-3

Keywords

Search

Navigation