Consider an M by N data matrix X whose elements may assume values 0, 1, 2, . . ., H. Denote the rows of X by (alpha)(,1), (alpha)(,2), . . ., (alpha)(,M). A tree on the rows of X is a sequence of distinct partitions {P(,1)}(,i=1) such that: (a) P(,1) = {((alpha)(,1)), . . ., ((alpha)(,M))}, (b) P(,i) is a refinement of P(,i+1) for i = 1, . . ., k-1, and (c) P(,k) = {((alpha)(,1), . . ., (alpha)(,M))}. The two-way clustering problem consists of simultaneously constructing trees on the rows, columns, and elements of X. A generalization of a two-way joining algorithm (TWJA) introduced by J. A. Hartigan (1975) is used to construct the three trees. / The TWJA requires the definition of measures of dissimilarity between row clusters and column clusters respectively. Two approaches are used in the construction of these dissimilarity coefficients--one based on intuition and one based on a formal prediction model. For matrices with binary elements (0 or 1), measures of dissimilarity between row or column clusters are based on the number of mismatching pairs. Consider two distinct row clusters R(,p) and R(,q) containing m(,p) and m(,q) rows respectively. One measure of dissimilarity, d(,0)(R(,p), R(,q)), between R(,p) and R(,q), is / (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) / where b(,p(beta)) and b(,q(beta)) are the number of ones in column (beta) of clusters R(,p) and R(,q) respectively. Two additional intuitive dissimilarity coefficients are also defined and studied. / For matrices containing nominal level data, dissimilarity coefficients are based on a formal prediction model. Analogous to the procedure of Cleveland and Relles (1974), for a given data matrix, the model consists of a scheme for random selection of two rows (or columns) from the matrix and an identification rule for distinguishing between the two rows (or columns). A loss structure is defined for both rows and columns and the expected loss due to incorrect row or column identification is computed. The dissimilarity between two (say) row clusters is then defined to be the increase in expected loss due to joining those two row clusters into a single cluster. / Stopping criteria are suggested for both the intuitive and prediction model approaches. For the intuitive approach, it is suggested that joining be stopped when the dissimilarity between the (say) row clusters to be joined next exceeds that expected by chance under the assumption that the (say) column totals of the matrix are fixed. For the prediction model approach the stopping criterion is based on a cluster prediction model in which the objective is to distinguish between row or column clusters. A cluster identification rule is defined based on the information in the partitioned data matrix and the expected loss due to incorrect cluster identification is computed. The expected cluster loss is also computed when cluster identification is based on strict randomization. The relative decrease in expected cluster loss due to identification based on the partitioned matrix versus that based on randomization is suggested as a stopping criterion. / Both contrived and real data examples are used to illustrate and compare the two clustering procedures. Computational aspects of the procedure are discussed and it is concluded that the intuitive approach is less costly in terms of computation time. Further, five admissibility properties are defined and, for certain intuitive dissimilarity coefficients, the trees produced by the TWJA are shown to possess three of the five properties. / Source: Dissertation Abstracts International, Volume: 41-05, Section: B, page: 1822. / Thesis (Ph.D.)--The Florida State University, 1980.
| Identifer | oai:union.ndltd.org:fsu.edu/oai:fsu.digital.flvc.org:fsu_74194 |
| Contributors | COOPER, PAUL GAYLORD., Florida State University |
| Source Sets | Florida State University |
| Detected Language | English |
| Type | Text |
| Format | 198 p. |
| Rights | On campus use only. |
| Relation | Dissertation Abstracts International |
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