Skip to main content
SpringerLink
Log in

Functional Sufficient Dimension Reduction for Functional Data Classification

  • Published:
Journal of Classification Aims and scope Submit manuscript

Abstract

We consider two novel functional classification methods for binary response and functional predictor. We extend the most popular functional sufficient dimension reduction methods such as functional sliced inverse regression (FSIR) and functional sliced average variance estimation (FSAVE) by introducing a regularized estimation procedure and incorporating the localized information of the functional predictor in the analysis. Compared to the existing FSIR and FSAVE, the proposed methods are appealing because they are capable of estimating more than one effective dimension reduction direction, whereas FSIR detects only one such direction and FSAVE produces inefficient estimation in the case of binary response. Moreover, our methods make use of the localized information of the functional predictor, thereby more efficiently capturing the nonlinear relation between the binary response and the functional predictor. Furthermore, the proposed methods can be extended to incorporate the ancillary unlabeled data in semi-supervised learning. The empirical performance and the applications of the proposed methods are demonstrated by simulation studies and real applications.

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.

Institutional subscriptions

Similar content being viewed by others

References

  • AMATO, U., ANTONIADIS, A., and FEIS, I.D. (2006), “Dimension Reduction in Functional Regression with Application”, Computational Statistics and Data Analysis, 50, 2422–2446.

    Article  MathSciNet  MATH  Google Scholar 

  • ALONSO, A., CASADO, D., and ROMO, J. (2012), “Supervised Classification for Functional Data: A Weighted Distance Approach”, Computational Statistics and Data Analysis, 56, 2334–2346.

    Article  MathSciNet  MATH  Google Scholar 

  • ALONSO, A., CASADO, D., LÓPEZ-PINTADO, S., and ROMO, J. (2014), “Robust Functional Supervised Classification for Time Series”, Journal of Classification, 31, 325–350.

    Article  MathSciNet  MATH  Google Scholar 

  • ARAKI, Y., KONISHI, S., KAWANO, S., and MATSUI, H. (2009), “Functional Logistic Discrimination via Regularized Basis Expansions”, Communications in Statistics-Theory and Methods, 38, 2944–2957.

    Article  MathSciNet  MATH  Google Scholar 

  • BERLINET, A., BIAU, G., and ROUVIRE, L. (2008), “Functional Classification with Wavelets”, Annals of the Institute of Statistics of the University of Paris, 52, 61–80.

    MathSciNet  Google Scholar 

  • BENNETT, K., and DEMIRIZ, A. (1998), “Semi-Supervised Support Vector Machines”, in Advances in Neural Information Processing Systems, 11, Cambridge, MA: MIT Press, pp. 368–374.

    Google Scholar 

  • BONGIORNO, E.G., SALINELLI, E., GOIA, A., and VIEU, P. (Eds.) (2014), Contributions in Infinite-Dimensional Statistics and Related Topics, Bologna: Societa Editrice Esculapio.

    Google Scholar 

  • BIAU, G., BUNEA, F., and WEGKAMP, M.H. (2005), “Functional Classification in Hilbert Spaces”, IEEE Transactions on Information Theory, 51, 2163–2172.

    Article  MathSciNet  MATH  Google Scholar 

  • CARDOT, H., FERRATY, F., and SARDA, P. (2003), “Spline Estimators for the Functional Linear Model”, Statistica Sinica, 13, 571–591.

    MathSciNet  MATH  Google Scholar 

  • CHAMROUKHI, F., SAME, A., GOVAERT, G., and AKNIN, P. (2010),“A Hidden Process Regression Model for Functional Data Description: Application to Curve Discrimination”, Neurocomputing, 73, 1210–1221.

  • CHEN, K.H., and LEI, J. (2015), “Localized Functional Principal Component Analysis”, Journal of the Americal Statistical Association, 110(511), 1266–1275.

    Article  MathSciNet  MATH  Google Scholar 

  • CUEVAS, A., FEBRERO, M., and FRAIMAN, R. (2007), “Robust Estimation and Classification for Functional Data via Projection-Based Depth Notions”, Computational Statistics, 22, 481–497.

    Article  MathSciNet  MATH  Google Scholar 

  • COOK, R.D., LI, B., and CHIAROMONTE, F. (2007), “Dimension Reduction in Regression Without Matrix Inversion”, Biometrika, 94, 569–584.

    Article  MathSciNet  MATH  Google Scholar 

  • DELAIGLE, A., and HALL, P. (2012), “Achieving Near-Perfect Classification for Functional Data”, Journal of the Royal Statistical Society Series B, 74, 267–286.

    Article  MathSciNet  Google Scholar 

  • DONOHO, D., and GRIMES, C. (2003), “Hessian Eigenmaps: New Locally Linear Embedding Techniques for High Dimension Data”, Proceedings of the National Academy of Science, 100, 5591–5596.

    Article  MATH  Google Scholar 

  • EPIFANIO, I. (2008), “Shape Descriptors for Classification of Functional Data”, Technometrics, 50, 284–294.

    Article  MathSciNet  Google Scholar 

  • FERRATY, F., and VIEU, P. (2003), “Curves Discrimination: A Nonparametric Functional Approach”, Computation Statistical Data Analysis, 4, 161–173.

    Article  MathSciNet  MATH  Google Scholar 

  • FERRATY, F., and VIEU, P. (2006), Nonparametric Functional Data Analysis: Theory and Practice, New York: Springer.

    MATH  Google Scholar 

  • FERRÉ, L., and YAO, A.F. (2003), “Functional Sliced Inverse Regression Analysis”, Statistics, 37, 475–488.

    Article  MathSciNet  MATH  Google Scholar 

  • FERRÉ, L., and YAO, A.F. (2005), “Smoothed Functional Inverse Regression”, Statistics Sinica, 15, 665–683.

    MathSciNet  MATH  Google Scholar 

  • FROMONT, M., and TULEAU, C. (2006), “Functional Classification with Margin Conditions”, in Learning Theory: Proceedings of the 19th Annual Conference on Learning Theory, Pittsburgh, June 22nd-25th, eds. J.G. Carbonell and J. Siekmann, New York: Springer.

  • GOIA, A., and VIEU, P. (2015),“A Partitioned Single Functional Index Model”, Journal of Computational Statistics, 30(3), 673–692.

    Article  MathSciNet  MATH  Google Scholar 

  • HALL, P., POSKITT, D., and PRESNELL, B. (2001), “A Functional Data-Analytic Approach to Signal Discrimination”, Technometrics, 43, 1–9.

    Article  MathSciNet  MATH  Google Scholar 

  • HORVÁTH, L., and KOKOSZKA, P. (2012), Inference for Functional Data with Applications, New York: Springer.

    Book  MATH  Google Scholar 

  • HUANG, D.S., and ZHENG, C.H. (2006), “Independent Component Analysis-Based Penalized Discriminant Method for Tumor Classification Using Gene Expression Data”, Bioinformatics, 22, 1855–1862.

    Article  Google Scholar 

  • JAMES, G., and HASTIE, T.J. (2001), “Functional Linear Discriminant Analysis for Irregularly Sampled Curves”, Journal of the Royal Statistical Society Series B, 63, 533–550.

    Article  MathSciNet  MATH  Google Scholar 

  • LENG, X.Y., and MÜLLER, H.G. (2006), “Classification Using Functional Data Analysis for Temporal Gene Expression Data”, Bioinformatics, 22, 68–76.

    Article  Google Scholar 

  • LÓPEZ-PINTADO, S., and ROMO, J. (2006), “Depth-Based Classification for Functional Data”, in DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Vol. 72, Providence: American Mathematical Society, pp. 103–120.

  • LI, K.C. (1991), “Sliced Inverse Regression for Dimension Reduction (With Discussion)”, Journal of American Statistical Association, 86, 316–342.

    Article  MathSciNet  MATH  Google Scholar 

  • LIAN, H., and LI, G.R. (2014), “Series Expansion for Functional Sufficient Dimension Reduction”, Journal of Multivariate Analysis, 124, 150–165.

    Article  MathSciNet  MATH  Google Scholar 

  • LIAN, H. (2015), “Functional Sufficient Dimension Reduction: Convergence Rates and Multiple Functional Case”, Journal of Statistical Planning and Inference, 167, 58–68.

    Article  MathSciNet  MATH  Google Scholar 

  • LIANG, F., MUKHERJEE, S., and WEST, M. (2007), “Understanding the Use of Unlabelled Data in Predictive Modeling”, Statistical Science, 22, 189–205.

    Article  MathSciNet  MATH  Google Scholar 

  • PREDA, C., SAPORTA, G., and LEVEDER, C. (2007), “PLS Classification of Functional Data”, Computational Statistics 22, 223–235.

    Article  MathSciNet  MATH  Google Scholar 

  • RAMSAY, J.O., and SILVERMAN, B.W. (2002), Applied Functional Data Analysis: Methods and Case Studies, New York: Springer.

    Book  MATH  Google Scholar 

  • RAMSAY, J.O., and SILVERMAN, B.W. (2005), Functional Data Analysis (2nd ed.), New York: Springer.

    MATH  Google Scholar 

  • ROSSI, F., and VILLA, N. (2006), “Support Vector Machine for Functional Data Classification”, Neurocomputing, 69, 730–742.

    Article  Google Scholar 

  • ROWEIS, S., and SAUL, L. (2000), “Nonlinear Dimensionality Reduction by Locally Linear Embedding”, Science 290, 2323–2326.

    Article  Google Scholar 

  • SHIN, H. (2008), “An Extension of Fishers Discriminant Analysis for Stochastic Processes”, Journal of Multivariate Analysis, 99, 1191–1216.

    Article  MathSciNet  MATH  Google Scholar 

  • SILVERMAN, B.W. (1996), “Smoothed Functional Principal Components Analysis by Choice of Norm”, The Annals of Statistics, 24, 1–24.

    Article  MathSciNet  MATH  Google Scholar 

  • SONG, J.J., DENG, W., LEE, H.J., and KWON, D. (2008), “Optimal Classification for Time-Course Gene Expression Data Using Functional Data Analysis”, Computational Biology and Chemistry, 32, 426–432.

    Article  MathSciNet  MATH  Google Scholar 

  • TIAN, S.T., and JAMES, G. (2010), “Interpretable Dimensionality Reduction for Classification with Functional Data”, Computational Statistics and Data Analysis, 57, 282–296.

    Article  Google Scholar 

  • VILAR, J.A., and PERTEGA, S. (2004), “Discriminant and Cluster Analysis for Gaussian Stationary Processes: Local Linear Fitting Approach”, Journal Nonparametric Statistics, 16, 443–462.

    Article  MathSciNet  MATH  Google Scholar 

  • WANG, G.C., FENG, X., and CHEN, M. (2016), “Functional Partial Linear Single-Index Model”, Scandinavian Journal of Statistics, 43(1), 261-274.

    Article  MathSciNet  MATH  Google Scholar 

  • WANG, G.C., LIN, N., and ZHANG, B.X. (2013a), “Functional Contour Regression”, Journal of Multivariate Analysis, 116, 1–13.

    Article  MathSciNet  MATH  Google Scholar 

  • WANG, G.C., LIN, N., and ZHANG, B.X. (2013b), “Dimension Reduction in Functional Regression Using Mixed Data Canonical Correlation Analysis”, Statistics and Its Interface, 6, 187–196.

    Article  MathSciNet  MATH  Google Scholar 

  • WANG, G.C., LIN, N., and ZHANG, B.X. (2014), “Functional K-mean Inverse Regression”, Computational Statistics and Data Analysis, 70, 172–182.

    Article  MathSciNet  Google Scholar 

  • WANG, G.C., ZHOU, J.J., WU, W.Q., and CHEN, M. (2017), “Robust Functional Sliced Inverse Regression”, Statistical Papers, 58, 227–245.

    Article  MathSciNet  MATH  Google Scholar 

  • WANG, G.C., ZHOU, Y., FENG, X.N., and ZHANG, B.X. (2015), “The Hybrid Method of FSIR and FSAVE for Functional Effective Dimension Reduction”, Computational Statistics and Data Analysis, 91, 64–77.

    Article  MathSciNet  MATH  Google Scholar 

  • WANG, X.H., RAY, S., and MALLICK, B.K. (2007), “Bayesian Curve Classification Using Wavelets”, Journal of American Statistical Association, 102, 962–973.

    Article  MathSciNet  MATH  Google Scholar 

  • WU, Y., and LIU, Y. (2013), “Adaptively Weighted Large Margin Classification”, Journal of Computational and Graphical Statistics, 22, 416–432.

    Article  MathSciNet  Google Scholar 

  • WU, Q., FENG, L., and SAYAN, M. (2010), “Localized Sliced Inverse Regression”, Journal of Computational and Graphical Statistics, 19, 843–860.

    Article  MathSciNet  Google Scholar 

  • YAO, F., LEI, E., and WU, Y. (2015), “Effective Dimension Reduction for Sparse Functional Data”, Biometrika, 102(2), 421–437.

    Article  MathSciNet  MATH  Google Scholar 

  • ZIPUNNIKOV, V., CAFFO, B., YOUSEM, D.M., DAVATZIKOS, C., SCHWARTZ, B.S., and CRAINICEANU, C. (2011), “Multilevel Functional Principal Component Analysis for High-Dimensional Data”, Journal of Computational and Graphical Statistics, 20, 852–873.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Jinan University, Guangzhou, China

    Guochang Wang

  2. The Chinese University of Hong Kong, Shatin, Hong Kong, China

    Xinyuan Song

Authors
  1. Guochang Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xinyuan Song
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xinyuan Song.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, G., Song, X. Functional Sufficient Dimension Reduction for Functional Data Classification. J Classif 35, 250–272 (2018). https://doi.org/10.1007/s00357-018-9256-z

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00357-018-9256-z

Keywords

Search

Navigation