Abstract
This Chapter is about dividing a dataset or its subset in two parts. If both parts are to be clusters, this is referred to as divisive clustering. If just one part is to be a cluster, this will be referred to as separative clustering. Iterative application of divisive clustering builds a binary hierarchy of which we will be interested at a partition of the dataset. Iterative application of separative clustering builds a set of clusters, possibly overlapping. The first three sections introduce three different approaches in divisive clustering: Ward clustering, Spectral clustering and Single link clustering. Ward clustering is an extension of K-means clustering dominated by the so-called Ward distance between clusters; also, this is a natural niche for conceptual clustering in which every division is made over a single feature to attain immediate interpretability of the hierarchy branches and clusters. Spectral clustering gained popularity with the so-called Normalized Cut approach to divisive clustering. A relaxation of this combinatorial problem appears to be equivalent to optimizing the Rayleigh quotient for a Laplacian transformation of the similarity matrix under consideration. In fact, other approaches under consideration, such as uniform clustering and semi-average clustering, also may be treated within the spectral approach. Single link clustering formalizes the nearest neighbor approach and is much related to graph-theoretic concepts: components and maximum spanning trees. One may think of divisive clustering as a process for building a binary hierarchy, which goes “top-down” in contrast to agglomerative clustering (in Sect. 4.6), which builds a binary hierarchy “bottom-up”. Two remaining sections describe two separative clustering approaches as extensions of popular approaches to the case. One tries to find a cluster with maximum inner summary similarity at a similarity matrix preprocessed according to the uniform and modularity approaches considered in Sect. 4.6.3 The other applies the encoder-decoder least-squares approach to modeling data by a one-cluster structure. It appears, criteria emerging within the latter approach are much akin to those described earlier, the summary and semi-average similarities, although parameters now can be adjusted according to the least-squares approach. This applies to a distinct direction, the so-called additive clustering approach, which can be usefully applied to the analysis of similarity data.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
L. Breiman, J.H. Friedman, R.A. Olshen, C.J. Stone, Classification and Regression Trees (Wadswarth, Belmont, Ca, 1984)
B. Mirkin, Mathematical Classification and Clustering (Kluwer Academic Press, 1996)
B. Mirkin, Clustering: A Data Recovery Approach (Chapman & Hall/CRC, 2012)
F. Murtagh, Multidimensional Clustering Algorithms (Physica-Verlag, Vienna, 1985)
Articles
O. Boruvka, Příspěvek k řešení otázky ekonomické stavby elektrovodních sítí (Contribution to the solution of a problem of economical construction of electrical networks)” (in Czech). Elektronický Obzor 15, 153–154 (1926)
D.H. Fisher, Knowledge acquisition via incremental conceptual clustering. Mach. Learn. 2, 139–172 (1987)
S. Guattery, G. Miller, On the quality of spectral separators. SIAM J. Matrix Anal. Appl. 19(3), 701–719 (1998)
C. Klein, M. Randic, Resistance distance. J. Math. Chem. 12, 81–95 (1993)
J.B. Kruskal, On the shortest spanning subtree of a graph and the traveling salesman problem. Proc. Am. Math. Soc. 7(1), 48–50 (1956)
G.N. Lance, W.T. Williams, A general theory of classificatory sorting strategies: 1. Hierarchical Systems. Comput. J. 9, 373–380 (1967)
U. Luxburg, A tutorial on spectral clustering. Stat. Comput. 17, 395–416 (2007)
B. Mirkin, Additive clustering and qualitative factor analysis methods for similarity matrices. J. Classif. 4, 7–31 (1987); Erratum 6, 271–272 (1989)
B. Mirkin, R. Camargo, T. Fenner, G. Loizou, P. Kellam, Similarity clustering of proteins using substantive knowledge and reconstruction of evolutionary gene histories in herpesvirus. Theor. Chem. Acc.: Theory, Comput., Model. 125(3–6), 569–582 (2010)
F. Murtagh, G. Downs, P. Contreras, Hierarchical clustering of massive, high dimensional data sets by exploiting ultrametric embedding. SIAM J. Sci. Comput. 30, 707–730 (2008)
M.E.J. Newman, Modularity and community structure in networks. PNAS 103(23), 8577–8582 (2006)
R.C. Prim, Shortest connection networks and some generalizations. Bell Syst. Tech. J. 36, 1389–1401 (1957)
J. Shi, J. Malik, Normalized cuts and image segmentation. IEEE Trans. Pattern Anal. Mach. Intell. 22(8), 888–905 (2000)
R.N. Shepard, P. Arabie, Additive clustering: Representation of similarities as combinations of discrete overlapping properties. Psychol. Rev. 86, 87–123 (1979)
S.K. Tasoulis, D.K. Tasoulis, V.P. Plagianakos, Enhancing principal direction divisive clustering. Pattern Recogn. 43, 3391–3411 (2010)
J.H. Ward Jr., Hierarchical grouping to optimize an objective function. J. Am. Stat. Assoc. 58, 236–244 (1963)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Mirkin, B. (2019). Divisive and Separate Cluster Structures. In: Core Data Analysis: Summarization, Correlation, and Visualization. Undergraduate Topics in Computer Science. Springer, Cham. https://doi.org/10.1007/978-3-030-00271-8_5
Download citation
DOI: https://doi.org/10.1007/978-3-030-00271-8_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-00270-1
Online ISBN: 978-3-030-00271-8
eBook Packages: Computer ScienceComputer Science (R0)