DM and Cluster Identification Algorithm
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1 DM and Cluster Identification Algorithm Andrew Kusiak, Professor oratory Seamans Center Iowa City, Iowa - Tel: 9-9 Fax: 9- andrew-kusiak@uiowa.edu Homepage: Cluster Analysis Decomposition Aggregation Grouping Models Cluster Representation Matrix formulation Mathematical programming formulation Graph formulation B A C E G D F H Mutually exclusive regions Cluster Representation Cluster Representation G B A C D H E F A B C E F D G H Overlapping regions Hierarchy Page
2 Cluster Representation Clusters A... B... C... D... Objects E... F... G... H... Two types of data Object with decisions Object without decisions Fuzzy clusters Clustering Data with Decisions Matrix Formulation Method Feature GF - GF- Ignore decisions Treat the objects as data without decisions Method Object Yes Red Yes Red GO- GF- Red Red Yes Yes Group objects according to the decisions Treat each group with decisions as a separate data set Matrix Formulation Solving the Clustering Problem: Binary Matrix Formulation GO- { GO- { Overlapping feature Similarity coefficient methods Sorting based algorithms Bond energy algorithm Cost-based method Cluster identification algorithm Extended cluster identification algorithm Page
3 Cluster Identification Algorithm Cluster Identification Algorithm Step 0. Set iteration number k =. Step. Select row i of incidence matrix [aij](k) and draw a horizontal line hi through it ([aij](k) is read: matrix [aij] at iteration k ). Step. For each entry of crossed by the horizontal line hi draw a vertical line vj. Step. For each entry of crossed-once by the vertical line vj draw a horizontal line hk. Step. Repeat steps and until there are no more crossed-once entries of in [aij](k). All crossed-twice entries in [aij](k) form row cluster RC-k and column cluster CC-k. Step. Transform the incidence matrix [aij](k) into [aij](k+) by removing rows and columns corresponding to the horizontal and vertical lines drawn in steps through. Step. If matrix [aij](k+) = 0 (where 0 denotes a matrix with all empty elements ), stop; otherwise set k = k + and go to step. Example Incidence matrix Object Feature 8 8 Next Step 8 Delete all double-crossed elements Page
4 Resultant Matrix Iteration Iteration v v v v h h h h RC- RC- RC- Final decomposition result CC- CC- CC- 8 (a) Three types of matrices (b) (c) a b c d e f g h a b c d e f g h a b c d e f (a) decomposable matrix (b) non-decomposable matrix with overlapping features (c) non-decomposable matrix with objects Other algorithms CI Clustering with DM Algorithms F F F F F F Transformed Data Set Dummy decision F F F F F F D One One One One One One One 8 Two F F F F F F Dummy object Page
5 Decision Rules F F F F F F D One One One One One One Rule. (F = 0) THEN (D = One); [, 8.%, 00.00%][,,,,, ] Rule. (F = 0) THEN (D = One); [,.%, 00.00%][,,, ] Rule. (F = ) AND (F = ) THEN (D = Two); [, 00.00%, 00.00%][8] One 8 Two Clustered Matrix F F F F F F Rule. (F = 0) THEN (D = One); [,.%, 00.00%][,,, ] CI Algorithm Reference F F F F F F F F F F F F A. Kusiak, Computational Intelligence in Design and Manufacturing, John Wiley, New York, 000. Page
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