Abstract
Cluster analysis of genome-wide expression data from DNA microarray hybridization studies is a useful tool for identifying biologically relevant gene groupings (DeRisi et al. 1997; Weiler et al. 1997). It is hence important to apply a rigorous yet intuitive clustering algorithm to uncover these genomic relationships. In this study, we describe a novel clustering algorithm framework based on a variant of the Generalized Benders Decomposition, denoted as the Global Optimum Search (Floudas et al. 1989; Floudas 1995), which includes a procedure to determine the optimal number of clusters to be used. The approach involves a pre-clustering of data points to define an initial number of clusters and the iterative solution of a Linear Programming problem (the primal problem) and a Mixed-Integer Linear Programming problem (the master problem), that are derived from a Mixed Integer Nonlinear Programming problem formulation. Badly placed data points are removed to form new clusters, thus ensuring tight groupings amongst the data points and incrementing the number of clusters until the optimum number is reached. We apply the proposed clustering algorithm to experimental DNA microarray data centered on the Ras signaling pathway in the yeast Saccharomyces cerevisiae and compare the results to that obtained with some commonly used clustering algorithms. Our algorithm compares favorably against these algorithms in the aspects of intra-cluster similarity and inter-cluster dissimilarity, often considered two key tenets of clustering. Furthermore, our algorithm can predict the optimal number of clusters, and the biological coherence of the predicted clusters is analyzed through gene ontology.
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References
Adams W.P. and Sherali H.D. (1990). Linearization strategies for a class of zero-one mixed integer programming problems. Operat. Res. 38(2): 217–226
Aggarwal A. and Floudas C.A. (1990). Synthesis of general separation sequences - nonsharp separations. Comput. Chem. Eng 14: 631–653
Beer M. and Tavazoie S. (2004). Predicting gene expression from sequence. Cell 117: 185–198
Bezdek J.C. (1981). Pattern Recognition with Fuzzy Objective Function Algorithms. Plenum Press, New York
Brooke A., Kendrick D. and Meeraus A. (1988). GAMS: A User’s Guide. The Scientific Press, San Francisco, CA
Carpenter G. and Grossberg S. (1990). ART3: hierarchical search using chemical transmitters in self-organizing patterns recognition architectures. Neural Networks 3: 129–152
Ciric A.R. and Floudas C.A. (1989). A retrofit approach of heat exchanger networks. Comput. Chem. Eng 13: 703–715
Claverie J. (1999). Computational methods for the identification of differential and coordinated gene expression. Human Mol. Genet. 8: 1821–1832
Davis D.L. and Bouldin D.W. (1979). A cluster separation measure. IEEE Trans. Pattern Anal. Machine Intell. 1(4): 224–227
Dempster A.P., Laird N.M. and Rudin D.B. (1977). Maximum likelihood from incomplete data via the EM algorithm. J. Roy. Stat. Soc. B. 39(1): 1–38
DeRisi J.L., Iyer V.R. and Brown P.O. (1997). Exploring the metabolic and genetic control of gene expression on a genomic scale. Science 278: 680–686
Dhillon, I.S., Guan, Y.: Information theoretic clustering of sparse co-occurrence data. Proceedings of the Third IEEE International Conference on Data Mining (ICDM) (2003)
Dunn J.C. (1973). A fuzzy relative of the ISODATA process and its use in detecting compact well-separated clusters. J. Cybernet. 3: 32–57
Dunn J.C. (1974). Well separated clusters and optimal fuzzy partitions. J. Cybernet. 4: 95–104
Duran M.A. and Odell P.L. (1974). Cluster Analysis: A Survey. Springer Verlag, New York
Eisen M.B., Spellman P.T., Brown P.O. and Botstein D. (1998). Cluster analysis and display of genome-wide expression patterns. Proc. Nat. Acad. Sci. U.S.A. 95(25): 14863–14868
Floudas C.A., Akrotirianakis I.G., Caratzoulas S., Meyer C.A. and Kallrath J. (2005). Global optimization in the 21st Century: advances and challenges. Comput. Chem. Eng. 29: 1185–2002
Floudas, C.A. Deterministic Global Optimization: Theory, Algorithms, and Applications. Kluwer Academic Publishers (2000)
Floudas, C.A.: Nonlinear and Mixed-Integer Optimization: Fundamentals and Applications. Oxford University Press (1995)
Floudas C.A., Aggarwal A. and Ciric A.R. (1989). Global optimum search for non convex NLP and MINLP problems. Comp. Chem. Eng. 13(10): 1117–1132
Floudas C.A. and Anastasiadis S.H. (1988). Synthesis of general distillation sequences with several multicomponent feeds and products. Chem. Eng. Sci. 43: 2407–2419
Floudas C.A. and Grossmann I.E. (1987). Synthesis of flexible heat exchanger networks with uncertain flow rates and temperatures. Comput. Chem. Eng 11: 319–336
Geoffrion A.M. (1973). Generalized benders decomposition. J. Optim. Theory Appl. 10(4): 237
Goodman L. and Kruskal W. (1954). Measures of associations for cross-validations. J. Am. Stat. Assoc. 49: 732–764
Gower J.C. and Ross G.J.S. (1969). Minimum spanning trees and single-linkage cluster analysis. Appl. Stat. 18: 54–64
Halkidi M., Batistakis Y. and Vazirgiannis M. (2002). Cluster validity methods: Part 1. SIGMOD record 31(2): 40–45
Hansen P. and Jaumard B. (1997). Cluster analysis and mathematical programming. Math. Program. 79: 191–215
Hartigan J.A. (1975). Clustering Algorithms. John Wiley & Sons, New York
Hartigan J.A. and Wong M.A. (1979). Algorithm AS 136: a K-means clustering algorithm. Appl. Stat. J. Roy. St. C. 28: 100–108
Herrero J., Valencia A. and Dopazo J. (2001). A hierarchical unsupervised growing neural network for clustering gene expression patterns. Bioinformatics 17(2): 126–136
Heyer L.J., Kruglyak S. and Yooseph S. (1999). Exploring expression data: identification and analysis of co-expressed genes. Genome Res. 9: 1106–1115
Hubert L. and Schultz J. (1976). Quadratic assignment as a general data-analysis strategy. Br. J. Math. Stat. Psychol. 29: 190–241
Jaccard P. (1912). The distribution of flora in the alpine zone. New Phytol. 11: 37–50
Jain A.K., Murty M.N. and Flynn P.J. (1999). Data clustering: a review. ACM Comput. Surv. 31(3): 264–323
Jain A.K. and Dubes R.C. (1988). Algorithms for Clustering Data. Prentice-Hall Advanced Reference Series, Prentice-Hall, Inc., Englewood Cliffs, New Jersey
Johnson, R.E.: The role of cluster analysis in assessing comparability under the US transfer pricing regulations. Business Economics (April 2001)
Jung Y., Park H., Du D. and Drake B.L. (2003). A decision criterion for the optimal number of clusters in hierarchical clustering. J. Global Optimiz. 25: 91–111
Kirkpatrick S., Gelatt C.D. and Vecchi M.P. (1983). Optimization by simulated annealing. Science 220(4598): 671–680
Kohonen T. (1984). Self Organization and Associative Memory. Springer Information Science Series, Springer Verlag, Berlin, Heidelberg, New York
Kohonen T. (1997). Self-Organizing Maps. Springer Verlag, Berlin
Kokossis A.C. and Floudas C.A. (1994). Optimization of complex reactor networks - II. Nonisothermal operation.. Chem. Eng. Sci 49: 1037–1051
Leisch, F., Weingessel, A., Dimitriadou, E.: Competitive learning for binary valued data. In: Niklasson L., Bod’en M., Ziemke T. (eds.) Proceedings of the 8th International Conference on Artificial Neural Networks (ICANN 98), vol. 2, pp. 779–784. Sk"ovde, Sweden, Springer (1998)
Likas A., Vlassis N. and Vebeek J.L. (2003). The global K-means clustering algorithm. Pattern Recogn. 36: 451–461
Lin X., Floudas C., Wang Y. and Broach J.R. (2003). Theoretical and computational studies of the glucose signaling pathways in yeast using global gene expression data. Biotechnol. Bioeng. 84(7): 864–886
Lukashin A.V. and 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
McQueen, J.: Some methods for classification and analysis of multivariate observations. In: Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, pp. 281–297 (1967)
Metropolis N., Rosenbluth A., Rosenbluth M., Teller A. and Teller E.J. (1953). Equations of State calculations by fast computing machines. J. Chem. Phys. 21: 1087–1091
Paules G.E. IV. and Floudas C.A. (1989). APROS: Algorithmic development methodology for discrete-continuous optimization problems. Oper. Res. J. 37: 902–915
Pauwels E.J. and Frederix G. (1999). Finding salient regions in images: non-parametric clustering for image segmentation and grouping. Comput. Vision Image Understand. 75: 73–85
Pipenbacher P., Schliep A., Schneckener S., Schonhuth A., Schomburg D. and Schrader R. (2002). ProClust: improved clustering of protein sequences with an extended graph-based approach. Bioinformatics 18(Suppl 2): S182–S191
Rand W.M. (1971). Objective criteria for the evaluation of clustering methods. J. Am. Stat. Assoc. 66(336): 846–850
Rousseeuw P.J. (1987). Silhouettes: A graphical aid to the interpretation and validation of cluster analysis. J. Comp. App. Math 20: 53–65
Ruspini E.H. (1969). A new approach to clustering. Inf. Control 15: 22–32
Schneper L., Düvel K. and Broach J.R. (2004). Sense and sensibility: nutritional response and signal integration in yeast. Curr. Opin. Microbiol. 7(6): 624–630
Sherali H.D. and Desai J. (2005a). A global optimization RLT-based approach for solving the hard clustering problem. J. Global Optimiz. 32(2): 281–306
Sherali H.D. and Desai J. (2005b). A global optimization RLT-based approach for solving the fuzzy clustering approach. J. Global Optimiz. 33(4): 597–615
Slonim N., Atwal G.S., Tkačik G. and Bialek W. (2005). Information based clustering. Proc. Nat. Acad. Sci. U.S.A. 102(51): 18297–18302
Sokal R.R. and Michener C.D. (1958). A statistical method for evaluating systematic relationships. Univ. Kans. Sci. Bull. 38: 1409–1438
Sorlie T., Tibshirani R., Parker J., Hastie T., Marron J.S., Nobel A., Deng S., Johnsen H., Pesich R., Geisler S., Demeter J., Perou C.M., Lonning P.E., Brown P.O., Borresen-Dala A.L. and Botstein D. (2003). Repeated observations of breast tumor subtypes in independent gene expression data sets. Proc. Nat. Acad. Sci. U.S.A. 100: 8418–8423
Tishby, N., Pereira, F., Bialek, W.: The information bottleneck method; proceedings of the 37th annual allerton conference on communication. Control Comput. 368–377 (1999)
Troyanskaya O.G., Dolinski K., Owen A.B., Altman R.B. and Botstein D. (2003). A Bayesian framework for combining heterogeneous data sources for gene function prediction (in Saccharomyces cerevisiae). Proc. Nat. Acad. Sci. U.S.A. 100: 8348–8353
Wang Y., Pierce M., Schneper L., Guldal C.G., Zhang X., Tavazoie S. and Broach J.R. (2004). Ras and Gpa2 mediate one branch of a redundant glucose signaling pathway in yeast. Plos Biol. 2(5): 610–622
Weiler J., Gausepohl H., Hauser N., Jensen O.N. and Hoheisel J.D. (1997). Hybridization-based DNA screening on peptide nucleic acid (PNA) oligomer arrays. Nuclei Acids Res. 25: 2792–2799
Wu Z. and Leahy R. (1993). An optimal graph theoretic approach to data clustering: theory and its application to image segmentation. IEEE Trans. Pattern Recogn. Mach. Intell. 15(11): 1101–1113
Xu R. and Wunsch D. (2005). Survey of clustering algorithms. IEEE Trans. Neural Networks 16(3): 645–678
Zahn C.T. (1971). Graph theoretical methods for detecting and describing gestalt systems. IEEE Trans. Comput. C- 20: 68–86
Zhang, B., Hsu, M., Dayal, U.: K-Harmonic Means – A Data Clustering Algorithm. Hewlett-Packard Research Laboratory Technical Report (June 1999)
Zhang, B.: Generalized K-Harmonic Means: Boosting in Unsupervised Learning. Hewlett-Packard Research Laboratory Technical Report (October 2000)
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Tan, M.P., Broach, J.R. & Floudas, C.A. A novel clustering approach and prediction of optimal number of clusters: global optimum search with enhanced positioning. J Glob Optim 39, 323–346 (2007). https://doi.org/10.1007/s10898-007-9140-6
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DOI: https://doi.org/10.1007/s10898-007-9140-6