Skip to main content
Springer Nature Link
Log in

Fuzzy relational clustering based on comparing two proximity matrices with utilization of particle swarm optimization

  • Original Paper
  • Published:
Soft Computing Aims and scope Submit manuscript

Abstract

The first stage of knowledge acquisition and reduction of complexity concerning a group of entities is to partition or divide the entities into groups or clusters based on their attributes or characteristics. Clustering algorithms normally require both a method of measuring proximity between patterns and prototypes and a method for aggregating patterns. However sometimes feature vectors or patterns may not be available for objects and only the proximities between the objects are known. Even if feature vectors are available some of the features may not be numeric and it may not be possible to find a satisfactory method of aggregating patterns for the purpose of determining prototypes. Clustering of objects however can be performed on the basis of data describing the objects in terms of feature vectors or on the basis of relational data. The relational data is in terms of proximities between objects. Clustering of objects on the basis of relational data rather than individual object data is called relational clustering. The premise of this paper is that the proximities between the membership vectors, which are obtained as the objective of clustering, should be proportional to the proximities between the objects. The values of the components of the membership vector corresponding to an object are the membership degrees of the object in the various clusters. The membership vector is just a type of feature vector. Based on this premise, this paper describes another fuzzy relational clustering method for finding a fuzzy membership matrix. The method involves solving a rather challenging optimization problem, since the objective function has many local minima. This makes the use of a global optimization method such as particle swarm optimization (PSO) attractive for determining the membership matrix for the clustering. To minimize computational effort, a Bayesian stopping criterion is used in combination with a multi-start strategy for the PSO. Other relational clustering methods generally find local optimum of their objective function.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  • Babu GP and Murti MN (1994). Clustering with evolution strategies. Pattern Recognit 27: 321–329

    Article  Google Scholar 

  • Bezdek JC, Pal NR (1995) Cluster validation with generalized Dunn’s indices. In: Artificial Neural Networks and Expert Systems, 1995. Proceedings., Second New Zealand International Two-Stream Conference on, pp 190–193

  • Bezdek JC and Pal NR (1998). Some new indexes of cluster validity. Syst Man Cybern B IEEE Trans 28: 301–315

    Article  Google Scholar 

  • Bezdek JC, Hataway RJ and Windham MP (1991). Numerical comparison of the RFCM and AP Algorithms for clustering relational data. Pattern Recognit 24: 783–791

    Google Scholar 

  • Bezdek J (1981). Pattern recognition with fuzzy objective function algorithms. Plenum, NY

    MATH  Google Scholar 

  • Bhuyan JN, Raghavan VV, Elayavalli VK (1991) Genetic algorithms for clustering with an ordered representation. In: International conference on genetic algorithms, pp 408–15

  • Borg I and Groenen P (1997). Modern multidimensional scaling theory and applications. Spinger, Berlin

    MATH  Google Scholar 

  • Ching-Yi C, Fun Y (2004) Particle swarm optimization algorithm and its application to clustering analysis. In: Networking, Sensing and Control, 2004 IEEE International Conference on, pp 789–794

  • Dave RN and Sen S (2002). Robust fuzzy clustering of relational data. Fuzzy Syst IEEE Trans 10: 713–727

    Article  Google Scholar 

  • Davidson ML (1983). Multidimensional scaling. Wiley, New York

    Google Scholar 

  • Dixon LCW, Gomulka J and Szego GP (1975). Towards a global optimization technique. In: Dixon, LCW and Szego, GP (eds) Towards Global Optimization, vol 1, pp 29–45. North-Holland, Amsterdam

    Google Scholar 

  • Doring C, Borgelt C, Kruse R (2004) Fuzzy clustering of quantitative and qualitative data. In: Annual Meeting of the North American Fuzzy Information Processing Society NAFIPS ’04., Banff, Canada, pp 84–89

  • Dunn JC (1973). A Fuzzy relative of the ISODATA process and its use in detecting compact well-separated clusters. J Cybern 3: 32–57

    Article  MATH  MathSciNet  Google Scholar 

  • Dunn JC (1976). Indices of partition fuzziness and detection of clusters in large data sets. In: Gupta, MM (eds) Fuzzy automata and decision processes, pp. Elsevier, New York

    Google Scholar 

  • Eberhart R, Kennedy J (1995) A new optimizer using particle swarm theory. In: Proceedings of the Sixth International Symposium on Micro Machine and Human Science, pp 39–43

  • Feng H-M, Chen C-Y and Ye F (2006). Adaptive hyper-fuzzy partition particle swarm optimization clustering algorithm. Cybern Syst 37: 463–479

    Article  MATH  Google Scholar 

  • Fourie PC and Groenwold AA (2002). The particle swarm optimization algorithm in size and shape optimization. Struct Multidiscip Optim 23: 259–267

    Article  Google Scholar 

  • Hall LO, Bezdek JC, Boggavarpu S, Bensaid A (1994) Genetic fuzzy clustering. In: NAFIPS/IFIS/NASA ’94. Proceedings of the First International Joint Conference of the North American Fuzzy Information Processing Society Biannual Conference. The Industrial Fuzzy Control and Intelligent Systems Conference, and the NASA Joint Technolo, pp 411–415

  • Hall LO, Ozyurt IB and Bezdek JC (1999). Clustering with a genetically optimized approach. Evol Comput IEEE Trans 3: 103–112

    Article  Google Scholar 

  • Hathaway RJ and Bezdek JC (1994). Nerf c-means: non-Euclidean relational fuzzy clustering. Pattern Recognit 27: 429–437

    Article  Google Scholar 

  • Hathaway RJ, Davenport JW and Bezdek JW (1989). Relational duals of the c-means clustering algorithms. Pattern Recognit 22: 205–212

    Article  MATH  MathSciNet  Google Scholar 

  • Hoppner F, Klawonn F, Kruse R and Runkler T (1999). Fuzzy cluster analysis. Wiley, New York

    Google Scholar 

  • Hubert L and Arabie P (1985). Comparing partitions. J Classifi 2: 193–198

    Article  Google Scholar 

  • Jain AK and Dubes RC (1988). Algorithms for clustering data. Prentice Hall, Upper Saddle River, NJ

    MATH  Google Scholar 

  • Kaufman L and Rousseeuw PJ (1990). Finding groups in data: an introduction to cluster analysis. Wiley, New York

    Google Scholar 

  • Kennedy J, Eberhart R (1995) Particle swarm optimization. In: Proceedings of the IEEE International Conference on Neural Networks, pp 1942–1948

  • Kennedy J, Eberhart RC (1997) A discrete binary version of the particle swarm algorithm. In: IEEE International Conference on ‘Computational Cybernetics and Simulation’, pp 4104–4108

  • Kok S, Wood DW, Groenwold AA (2007) A particle swarm minimization algorithm with enhanced hill climbing capability. S Afr J Sci Technol

  • Krishnapuram R, Joshi A, Nasraoui O and Yi L (2001). Low-complexity fuzzy relational clustering algorithms for we mining. IEEE Trans Fuzzy Syst 9: 595–607

    Article  Google Scholar 

  • Kruskal JB and Wish M (1989). Multidimensional scaling. Sage Publications, Beverly Hills, CA

    Google Scholar 

  • Maulik U and Bandyopadhyay S (2000). Genetic algorithm-based clustering technique. Pattern Recognit 33: 1455–1465

    Article  Google Scholar 

  • Maulik U and Bandyopadhyay S (2002). Performance evaluation of some clustering algorithms and validity indices. Trans Pattern Anal Mach Intell 24: 1650–1654

    Article  Google Scholar 

  • Nasraoui O, Krishnapuram R, Joshi A (1999) Relational clustering based on a new robust estimator with application to web mining. NAFIPS

  • Pal NK and Bezdek JC (1995). On clustering validity for the fuzzy c-means model. IEEE Trans Fuzzy Syst 3: 370–379

    Article  Google Scholar 

  • Roubens M (1978). Pattern classification problems and fuzzy sets. Fuzzy Sets Syst 1: 239–253

    Article  MATH  MathSciNet  Google Scholar 

  • Snyman JA and Fatti LP (1987). A multi-start global minimization algorithm with dynamic search trajectories. J Optim Theory Appl 54: 121–141

    Article  MATH  MathSciNet  Google Scholar 

  • Stein B, Eissen SMz, Wissbrock F (2003) On cluster validity and the information need of users. In: 3rd IASTED International Conference on ArtificialIntelligence and Applications, Benalmadena, Spain, pp 216–221

  • Stevens SS (1946) On the theory of scales of measurement. Science 103(2684):677–680 (Friday, June 7, 1946)

    Google Scholar 

  • Thompson NJ (2003). J—the natural language for analytic computing. Research Studies Press, Hertfordshire, UK

    Google Scholar 

  • Vinod H (1969). Integer programming and theory of grouping. J Am Stat Assoc 64: 506–517

    Article  MATH  Google Scholar 

  • Wilke DN, Kok S, Groenwold AA (2005) Comparison of linear and classical velocity update rules in particle swarm optimization: Notes on Diversity. International Journal of Numerical Methods in Engineering

  • Windham MP (1985). Numerical classification of proximity data with assignment measures. J Classifi 2: 157–172

    Article  Google Scholar 

  • Xie XL and Beni G (1991). A validity measure for fuzzy clustering. Trans Pattern Anal Mach Intell 13: 841–847

    Article  Google Scholar 

  • Zhang D, Liu X, Guan Z (2006) A dynamic clustering algorithm based on PSO and its application in Fuzzy identification. In: IIH-MSP’06 International Conference on Intelligent Information Hiding and Multimedia Signal Processing, pp 232–235

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical and Mechatronics Engineering, University of Stellenbosch, Private Bag X1, Matieland, 7602, Stellenbosch, South Africa

    Roelof K. Brouwer & Albert Groenwold

  2. Department of Computing Science, Thompson Rivers University, 900 McGill Road, Kamloops, BC, V2C 5N3, Canada

    Roelof K. Brouwer

Authors
  1. Roelof K. Brouwer
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Albert Groenwold
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Roelof K. Brouwer.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Brouwer, R.K., Groenwold, A. Fuzzy relational clustering based on comparing two proximity matrices with utilization of particle swarm optimization. Soft Comput 13, 577–589 (2009). https://doi.org/10.1007/s00500-008-0334-8

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00500-008-0334-8

Keywords