Skip to main content
SpringerLink
Log in

Construction Algorithm of Principal Curves in the Sense of Limit

  • Conference paper
AI 2005: Advances in Artificial Intelligence (AI 2005)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 3809))

Included in the following conference series:

  • 1752 Accesses

Abstract

Principal curves have been defined as self-consistent, smooth, one-dimensional curves which pass through the middle of a multidimensional data set. They are nonlinear generalization of the first Principal Component. In this paper, we take a new approach by defining principal curves as continuous curves based on the local tangent space in the sense of limit. It is proved that this new principal curves not only satisfy the self-consistent property, but also are the unique existence for any given open covering. Based on the new definition, a new practical algorithm for constructing principal curves is given. And the convergence properties of this algorithm are analyzed. The new construction algorithm of principal curves is illustrated on some simulated data sets.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 189.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Hastie, T., Stuetzle, W.: Principal Curves. Journal of the American Statistical Association 84, 502–516 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  2. Tarpey, T., Flury, B.: Self-consistency: A fundamental concept in statistics. Statistical Science 11(3), 229–243 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  3. Tibshirani, R.: Principal Curves Revisited. Statistics and Computation 2, 183–190 (1992)

    Article  Google Scholar 

  4. Kégl, B., Krzyzak, A., Linder, T., Zeger, K.: Learning and design of principal curves. IEEE Trans. on Pattern Analysis and Machine Intelligence 22(3), 281–297 (2000)

    Article  Google Scholar 

  5. Delicado, P.: Another Look at Principal Curves and Surfaces. Journal of Multivariate Analysis 77, 84–116 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  6. Kui-yu, C., Ghosh, J.: A Unified Model for Probabilistic Principal Surfaces. IEEE Trans. on Pattern Analysis and Machine Intelligence 23(1), 22–41 (2001)

    Article  Google Scholar 

  7. Duchamp, T., Stuezle, W.: Geometric Properties of Principal Curves in the Plane. Robust Statistics. Data Analysis, and Computer Intensive Methods, 135–152 (1995)

    Google Scholar 

  8. Duchamp, T., Stuetzle, W.: Extremal Properties of Principal Curves in the Plane. Annals of Statistics 24(4), 1511–1520 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  9. Martínez-Morales, J.L.: Extremal Properties of Principal Embeddings. J. Math. Pures Appl. 78, 913–923 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  10. Junping, Z., Jue, W.: An Overview of Principal Curves. Chinese Journal of Computers 26(2), 129–146 (2003)

    MathSciNet  Google Scholar 

  11. Wanli, M., Ke, L., Xiaofei, H.: Locality Pursuit Embedding. Pattern Recognition 37, 781–788 (2004)

    Article  MATH  Google Scholar 

  12. Koetsier, J., Han, Y., Fyfe, C.: Twinned principal curves. Neural Networks 17, 399–409 (2004)

    Article  MATH  Google Scholar 

  13. Kambhatla, N., Leen, T.K.: Dimension reduction by local principal component analysis. Neural Computation 9, 1493–1516 (1997)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Computer and Information Technology, Beijing Jiaotong University, Beijing, 100044, China

    Lianwei Zhao, Siwei Luo & Chao Shao

  2. Faculty of Information Technology, University of Technology, Sydney, Australia

    Yanchang Zhao

Authors
  1. Lianwei Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yanchang Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Siwei Luo
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Chao Shao
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Guangxi Normal University, College of CS and IT, Guilin, China, and University of Technology, Faculty of Engineering and Information Technology, Sydney, Australia

    Shichao Zhang

  2. Department of Electrical and Computer Systems Engineering, Monash University, 3800, Melbourne, Victoria, Australia

    Ray Jarvis

Rights and permissions

Reprints and permissions

Copyright information

© 2005 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Zhao, L., Zhao, Y., Luo, S., Shao, C. (2005). Construction Algorithm of Principal Curves in the Sense of Limit. In: Zhang, S., Jarvis, R. (eds) AI 2005: Advances in Artificial Intelligence. AI 2005. Lecture Notes in Computer Science(), vol 3809. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11589990_29

Download citation

  • DOI: https://doi.org/10.1007/11589990_29

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-30462-3

  • Online ISBN: 978-3-540-31652-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation