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REES MATRIX CONSTRUCTIONS FOR CLUSTERING OF DATA

Part of: Semigroups Rings and algebras arising under various constructions

Published online by Cambridge University Press:  15 December 2009

A. V. KELAREV ,
P. WATTERS  and
J. L. YEARWOOD
A. V. KELAREV*
Affiliation:
Graduate School of Information Technology and Mathematical Sciences, University of Ballarat, PO Box 663, Ballarat, Victoria 3353, Australia (email: [email protected])
P. WATTERS
Affiliation:
Graduate School of Information Technology and Mathematical Sciences, University of Ballarat, PO Box 663, Ballarat, Victoria 3353, Australia (email: [email protected])
J. L. YEARWOOD
Affiliation:
Graduate School of Information Technology and Mathematical Sciences, University of Ballarat, PO Box 663, Ballarat, Victoria 3353, Australia (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
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Abstract

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This paper continues the investigation of semigroup constructions motivated by applications in data mining. We give a complete description of the error-correcting capabilities of a large family of clusterers based on Rees matrix semigroups well known in semigroup theory. This result strengthens and complements previous formulas recently obtained in the literature. Examples show that our theorems do not generalize to other classes of semigroups.

Keywords

Rees matrix semigroupsclusteringdata mining

MSC classification

Secondary: 16S36: Ordinary and skew polynomial rings and semigroup rings 20M35: Semigroups in automata theory, linguistics, etc.
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 87 , Issue 3 , December 2009 , pp. 377 - 393
Copyright
Copyright © Australian Mathematical Publishing Association, Inc. 2009

Footnotes

The first author was supported by Discovery Grant DP0449469 from the Australian Research Council. The second author was supported by Linkage Grant LP0776267 from the Australian Research Council. The third author was supported by a Queen Elizabeth II Fellowship and Discovery Grant DP0211866 from the Australian Research Council.

References

[1]Araújo, I. M., Kelarev, A. V. and Solomon, A., ‘An algorithm for commutative semigroup algebras which are principal ideal rings with identity’, Comm. Algebra 32(4) (2004), 1237–1254.CrossRefGoogle Scholar
[2]Ash, C. J., Hall, T. E. and Pin, J.-E., ‘On the varieties of languages associated with some varieties of finite monoids with commuting idempotents’, Inform. and Comput. 86(1) (1990), 32–42.CrossRefGoogle Scholar
[3]Asuncion, A. and Newman, D. J., UCI Machine Learning Repository (University of California, School of Information and Computer Science, Irvine, CA, 2009), http://www.ics.uci.edu/∼mlearn/MLRepository.html.Google Scholar
[4]Bagirov, A. M., Rubinov, A. M. and Yearwood, J. L., ‘A global optimization approach to classification’, Optim. Eng. 3 (2002), 129–155.CrossRefGoogle Scholar
[5]Bagirov, A. M. and Yearwood, J. L., ‘A new nonsmooth optimization algorithm for minimum sum-of-squares clustering problems’, European J. Oper. Res. 170 (2006), 578–596.CrossRefGoogle Scholar
[6]Batten, L. M., Coulter, R. S. and Henderson, M., ‘Extending abelian groups to rings’, J. Aust. Math. Soc. Ser A. 82(3) (2007), 297–314.CrossRefGoogle Scholar
[7]Boslaugh, S. and Watters, P., Statistics in a Nutshell: A Desktop Quick Reference (O’Reilly & Associates, Sebastopol, CA, 2008).Google Scholar
[8]Cazaran, J. and Kelarev, A. V., ‘Generators and weights of polynomial codes’, Arch. Math. (Basel) 69 (1997), 479–486.CrossRefGoogle Scholar
[9]Cazaran, J., Kelarev, A. V., Quinn, S. J. and Vertigan, D., ‘An algorithm for computing the minimum distances of extensions of BCH codes embedded in semigroup rings’, Semigroup Forum 73 (2006), 317–329.CrossRefGoogle Scholar
[10]Clifford, A. H. and Preston, G. B., The Algebraic Theory of Semigroups (American Mathematical Society, Providence, RI, 1961).Google Scholar
[11]Crabb, M. J., McGregor, C. M. and Munn, W. D., ‘A property of the complex semigroup algebra of a free monoid’, J. Aust. Math. Soc. Ser A. 81(1) (2006), 97–104.CrossRefGoogle Scholar
[12]Davey, B. A., Jackson, M., Maróti, M. and McKenzie, R. N., ‘Principal and syntactic congruences in congruence-distributive and congruence-permutable varieties’, J. Aust. Math. Soc. 85(1) (2008), 59–74.CrossRefGoogle Scholar
[13]Easdown, D., East, J. and FitzGerald, D. G., ‘A presentation of the dual symmetric inverse monoid’, Internat. J. Algebra Comput. 18 (2008), 357–374.CrossRefGoogle Scholar
[14]Easdown, D. and Munn, W. D., ‘Trace functions on inverse semigroup algebras’, Bull. Aust. Math. Soc. 52(3) (1995), 359–372.CrossRefGoogle Scholar
[15]Goberstein, S. M., ‘Inverse semigroups determined by their partial automorphism monoids’, J. Aust. Math. Soc. Ser A. 81(2) (2006), 185–198.CrossRefGoogle Scholar
[16]Hall, T. E., ‘The radical of the algebra of any finite semigroup over any field’, J. Aust. Math. Soc. Ser. A 11 (1970), 350–352.CrossRefGoogle Scholar
[17]Hall, T. E., ‘Biprefix codes, inverse semigroups and syntactic monoids of injective automata’, Theoret. Comput. Sci. 32(1–2) (1984), 201–213.CrossRefGoogle Scholar
[18]Howie, J. M., Fundamentals of Semigroup Theory (Clarendon Press, Oxford, 1995).CrossRefGoogle Scholar
[19]Jackson, M., ‘The embeddability of ring and semigroup amalgams is undecidable’, J. Aust. Math. Soc. Ser. A 69 (2000), 272–286.Google Scholar
[20]Jackson, M., ‘On the finite basis problem for finite Rees quotients of free monoids’, Acta Sci. Math. (Szeged) 67 (2001), 121–159.Google Scholar
[21]Jackson, M. and Volkov, M., ‘Undecidable problems for completely 0-simple semigroups’, J. Pure Applied Algebra, to appear.Google Scholar
[22]Kambites, M., ‘The loop problem for Rees matrix semigroups’, Semigroup Forum 76(2) (2008), 204–216.CrossRefGoogle Scholar
[23]Kang, B. H., Kelarev, A. V., Sale, A. H. J. and Williams, R. N., ‘A new model for classifying DNA code inspired by neural networks and FSA’, in: Pacific Knowledge Acquisition Workshop, PKAW2006, Guilin, China, Lecture Notes in Computer Science, 4303 (Springer, Berlin, Heidelberg, 2006), pp. 187–198.Google Scholar
[24]Kelarev, A. V., Ring Constructions and Applications (World Scientific, River Edge, NJ, 2002).Google Scholar
[25]Kelarev, A. V., Graph Algebras and Automata (Marcel Dekker, New York, 2003).CrossRefGoogle Scholar
[26]Kelarev, A. V., Göbel, R., Rangaswamy, K. M., Schultz, P. and Vinsonhaler, C., Abelian Groups, Rings and Modules, Contemporary Mathematics, 273 (American Mathematical Society, Providence, RI, 2001).CrossRefGoogle Scholar
[27]Kelarev, A., Kang, B. and Steane, D., ‘Clustering algorithms for ITS sequence data with alignment metrics’, in: AI06: Advances in Artificial Intelligence, 19th Australian Joint Conference on Artificial Intelligence, Lecture Notes in Computer Science, 4304 (Springer, Berlin, Heidelberg, 2006), pp. 1027–1031.Google Scholar
[28]Kelarev, A. V. and Passman, D. S., ‘A description of incidence rings of group automata’, Contemp. Math. 456 (2008), 27–33.CrossRefGoogle Scholar
[29]Kelarev, A. V., Yearwood, J. L. and Mammadov, M. A., ‘A formula for multiple classifiers in data mining based on Brandt semigroups’, Semigroup Forum 78(2) (2009), 293–309.CrossRefGoogle Scholar
[30]Kelarev, A. V., Yearwood, J. L. and Vamplew, P. W., ‘A polynomial ring construction for classification of data’, Bull. Aust. Math. Soc. 79 (2009), 213–225.CrossRefGoogle Scholar
[31]Lawson, M. V., Margolis, S. W. and Steinberg, B., ‘Expansions of inverse semigroups’, J. Aust. Math. Soc. Ser A. 80(2) (2006), 205–228.CrossRefGoogle Scholar
[32]López-Permouth, S. R., Shum, K. P. and Sanh, N. V., ‘Kasch modules and pV-rings’, Algebra Colloq. 12(2) (2005), 219–227.CrossRefGoogle Scholar
[33]McCombie, S., Watters, P., Ng, A. and Watson, B., ‘Forensic characteristics of phishing—petty theft or organized crime?’, in: Proc. 4th Internat. Conf. on Web Information Systems and Technologies, WEBIST (INSTICC Press, Setúbal, Portugal, 2008).Google Scholar
[34]Reilly, N. R., ‘Varieties generated by completely 0-simple semigroups’, J. Aust. Math. Soc. 84(3) (2008), 375–403.CrossRefGoogle Scholar
[35]Watters, P., Martin, F. and Stripf, H. S., ‘Visual detection of LSB-encoded natural image steganography’, ACM Trans. Appl. Perception 5(1) (2008).CrossRefGoogle Scholar
[36]Witten, I. H. and Frank, E., Data Mining: Practical Machine Learning Tools and Techniques (Elsevier/Morgan Kaufmann, Amsterdam, 2005).Google Scholar
[37]Wolpert, D. H., ‘The lack of a priori distinctions between learning algorithms’, Neural Comput. 8(7) (1996), 1341–1390.CrossRefGoogle Scholar
[38]Yearwood, J. L., Bagirov, A. M. and Kelarev, A. V., ‘Optimization methods and the k-committees algorithm for clustering of sequence data’, Appl. Comput. Math. 8(1) (2009), 92–101.Google Scholar
[39]Yearwood, J. L., Kang, B. H. and Kelarev, A. V., ‘Experimental investigation of classification algorithms for ITS dataset’, in: PKAW 2008, Pacific Rim Knowledge Acquisition Workshop, Hanoi, Vietnam, 15–16 December 2008 (part of PRICAI-08, Tenth Pacific Rim Internat. Conf. Artificial Intelligence), Lecture Notes in Computer Science, 5465 (Springer, Berlin, Heidelberg, 2009), pp. 262–272.Google Scholar
[40]Yearwood, J. L. and Mammadov, M., Classification Technologies: Optimization Approaches to Short Text Categorization (IGI Global, Hershey, PA, 2007).Google Scholar