Skip to main content
SpringerLink
Log in

Determining the Optimal Number of Clusters

  • Reference work entry
Encyclopedia of Optimization

Article Outline

Introduction

Methods

  Dunn’s Validity Index

  Davies–Bouldin Validity Index

  Measure of Krzanowski and Lai

  Measure of Calinski and Harabasz

Applications

  A Novel Clustering Approach with Optimal Cluster Determination

  Extension for Biological Coherence Refinement

References

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 2,500.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 2,499.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Azuaje F (2002) A Cluster Validity Framework for Genome Expression Data. Bioformatics 18:319–320

    Article  Google Scholar 

  2. Bezdek JC, Pal NR (1998) Some New Indexed of Cluster Validity. IEEE Trans Syst Man Cybern 28:301–315

    Article  Google Scholar 

  3. Calinski RB, Harabasz J (1974) A Dendrite Method for Cluster Analysis. Commun Stat 3:1–27

    MathSciNet  Google Scholar 

  4. Davis DL, Bouldin DW (1979) A Cluster Separation Measure. IEEE Trans Pattern Anal Machine Intell 1(4):224–227

    Google Scholar 

  5. Dunn JC (1974) Well Separated Clusters and Optimal Fuzzy Partitions. J Cyber 4:95–104

    Article  MathSciNet  Google Scholar 

  6. Floudas CA (1995) Nonlinear and Mixed-Integer Optimization: Fundamentals and Applications. Oxford University Press, New York

    MATH  Google Scholar 

  7. Floudas CA, Aggarwal A, Ciric AR (1989) Global Optimum Search for Non Convex NLP and MINLP Problems. Comp Chem Eng 13(10):1117–1132

    Article  Google Scholar 

  8. Goodman L, Kruskal W (1954) Measures of Associations for Cross-Validations. J Am Stat Assoc 49:732–764

    Article  MATH  Google Scholar 

  9. Halkidi M, Batistakis Y, Vazirgiannis M (2002) Cluster Validity Methods: Part 1. SIGMOD Record 31(2):40–45

    Article  Google Scholar 

  10. Hubert L, Schultz J (1976) Quadratic Assignment as a General Data-Analysis Strategy. Brit J Math Stat Psych 29:190–241

    Google Scholar 

  11. Jaccard P (1912) The Distribution of Flora in the Alpine Zone. New Phytol 11:37–50

    Article  Google Scholar 

  12. Jain AK, Dubes RC (1988) Algorithms for Clustering Data. Prentice-Hall Advanced Reference Series, Prentice-Hall, New Jersey

    Google Scholar 

  13. Jung Y, Park H, Du D, Drake BL (2003) A Decision Criterion for the Optimal Number of Clusters in Hierarchical Clustering. J Global Optim 25:91–111

    Article  MathSciNet  Google Scholar 

  14. Kirkpatrick S, Gelatt CD, Vecchi MP (1983) Optimization by Simulated Annealing. Science 220(4598):671–680

    Article  MathSciNet  Google Scholar 

  15. Krzanowski WJ, Lai YT (1985) A Criterion for Determining the Number of Groups in a Data Set using Sum of Squares Clustering. Biometrics 44:23–44

    Article  MathSciNet  Google Scholar 

  16. Lukashin AV, Fuchs R (2001) Analysis of Temporal Gene Expression Profiles: Clustering by Simulated Annealing and Determining the Optimal Number of Clusters. Bioinformatics 17(5):405–414

    Article  Google Scholar 

  17. Metropolis N, Rosenbluth A, Rosenbluth M, Teller A, Teller EJ (1953) Equations of State Calculations by Fast Computing Machines. J Chem Phys 21:1087

    Article  Google Scholar 

  18. Milligan GW, Cooper MC (1985) An Examination of Procedures for Determining the Number of Clusters in a Data Set. Psychometrika 50:159–179

    Article  Google Scholar 

  19. Pauwels EJ, Fregerix G (1999) Finding Salient Regions in Images: Non-parametric Clustering for Image Segmentation and Grouping. Comput Vis Image Underst 75:73–85

    Article  Google Scholar 

  20. Rand WM (1971) Objective Criteria for the Evaluation of Clustering Methods. J Am Stat Assoc 66(336):l846–850

    Article  Google Scholar 

  21. Ray S, Turi R (1999) Determination of Number of Clusters in K-Means Clustering and Application in Color Image Segmentation. In: Proceed 4th Int Conf Advances in Pattern Recognition and Digital Techniques, 137–143

    Google Scholar 

  22. Rousseeuw PJ (1987) Silhouettes: A Graphical Aid to the Interpretation and Validation of Cluster Analysis. J Comp Appl Math 20:53–65

    Article  MATH  Google Scholar 

  23. Tan MP, Broach JR, Floudas CA (2007) A Novel Clustering Approach and Prediction of Optimal Number of Clusters: Global Optimum Search with Enhanced Positioning.J Global Optim 39:323–346

    Google Scholar 

  24. Tan MP, Broach JR, Floudas CA (2008) Evaluation of Normalization and Pre-Clustering Issues in a Novel Clustering Approach: Global Optimum Search with Enhanced Positioning. J Bioinf Comput Biol 5(4):895–913

    Article  Google Scholar 

  25. Tan MP, Broach JR, Floudas CA (2008) Microarray Data Mining: A Novel Optimization-Based Iterative Clustering Approach to Uncover Biologically Coherent Structures (submitted for publication)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Chemical Engineering, Princeton University, Princeton, USA

    Meng Piao Tan & Christodoulos A. Floudas

Authors
  1. Meng Piao Tan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Christodoulos A. Floudas
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Chemical Engineering, Princeton University, Princeton, NJ, 08544-5263, USA

    Christodoulos A. Floudas

  2. Center for Applied Optimization, Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL, 32611-6595, USA

    Panos M. Pardalos

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer-Verlag

About this entry

Cite this entry

Piao Tan, M., A. Floudas, C. (2008). Determining the Optimal Number of Clusters . In: Floudas, C., Pardalos, P. (eds) Encyclopedia of Optimization. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-74759-0_123

Download citation

Publish with us

Policies and ethics

Search

Navigation