The University of Iowa Intelligent Systems Laboratory The University of Iowa. f1 f2 f k-1 f k,f k+1 f m-1 f m f m- 1 D. Data set 1 Data set 2

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1 Decomposition in Data Mining Basic Approaches Andrew Kusiak 4312 Seamans Center Iowa City, Iowa Direct mining of data sets Mining of transformed data sets: - decomposition - discretization - filling missing values Data Set Decomposition Feature Set Decomposition Feature set decomposition Object set decomposition Content-based decomposition Intermediate-decision decomposition Feature type decomposition Feature relevance decomposition Object Set Decomposition Data Set Decomposition Object content decomposition (e.g., context) Decision value decomposition Feature value decomposition (e.g., context) f1 f2 f k-1 f k,f k+1 f m-1 f m f m- 1 D Data set 1 Data set 2 Data set 3 Function F 1 Function F 2 Rule set 1 Stage 1 fk =F 1 (f k-1 ) Rule set 2 f m =F 2 (f m-1 ) Rule set 3 Stage 2 Stage 3 Stage 4 Stage 5 Hybrid system 1

2 Data Set Decomposition Rule set 1 IF f 1 = 2 AND f 2 = Low THEN f 3 = 4 IF f 2 in [2.1, 4] THEN f 3 = 5 Function F 1 f 4 = (f 3-3.1) 3 Rule set 2 IF f 4 < 8.5 AND f 5 = High THEN f 6 = 8.4 IF f 5 = Low THEN f 6 = 12.4 Function F 2 f 7 = ln (f ) 1/2 Rule set 3 IF f 7 < 1.3 THEN D = Good IF f 8 >= 3.3 AND f 9 = Positive THEN D = Bad Domain Decomposition Ease of model construction and understanding Support of the evolutionary computation concept Increased model structural stability Ease of data acquisition and model maintenance Reuse of known models and dependencies Representation of alternative solutions Rule Structuring Possible generalization to f 3 >=2 and f 5 <=8 Rule Structuring Feature independence f1 f2 f3 f4 f5 D Rule Algorithm {B, C, D} a Low R1 A1 <2 (2, 5] Low R7 A3 {E, F} b Low R4 A3 {C, F} <4 Medium R8 A1 >9 Medium R5 A1 Unstructured rule-feature incidence matrix f3 f5 f2 f1 f4 D Rule Algorithm Equivalent a {B, C, D} Low R1 A1 Independent b {E, F} Low R4 A3 >9 Medium R5 A1 Identical Structured rule-feature incidence matrix Rule Structuring f1 f2 f3 f4 f5 D RuleAlgorithm {B, C,D}a Low R1 A1 <2 (2, 5] Low R7 A3 {E, F} b Low R4 A3 {C, F} <4 Medium R8 A1 >9 Medium R5 A1 Structured Removed due to overlap with feature value ranges of other rules f 3 f 5 f 2 f 1 f 4 D Rule Algorithm a {B, C, D} Low R1 A1 b {E, F} Low R4 A3 >9 Medium R5 A1 Matrix Types Mutually separable matrix Non-decomposable matrix Matrix with conflicting outcomes 2

3 Mutually Separable Matrix Mutually Separable Matrix D One One One One One One One 8 1 Two D One One Rules One One One One One 8 1 Two Rule 1. (F6 = 0) THEN (D = One); [6, 85.71%, %][6, 0][1, 2, 3, 5, 6, 7] Rule 2. (F5 = 0) THEN (D = One); [4, 57.14%, %][1, 2, 4, 5] Rule 3. (F1 = 1) AND (F6 = 1) THEN (D = Two); [1, %, %][8] Rule 1 Clusters Rule 1. (F6 = 0) THEN (D = One); [6, 85.71%, %][6, 0][1, 2, 3, 5, 6, 7] Rule 2 Cluster Rule 2. (F5 = 0) THEN (D = One); [4, 57.14%, %][1, 2, 4, 5] Non-Decomposable Matrix F7 F8 F9 F10 3

4 Non-Decomposable Matrix F1 F2 F3 F4 F5F6 F7 F8 F9 F10 F7 F8 F9 F10 D One One One One One 9 Two Rules Rule 8. (F8 = 0) THEN (D = One); [7, 87.50%, %][2, 3, 4, 5, 6, 7, 8] Rule 9. (F5 = 0) THEN (D = One); [5, 62.50%, %][1, 2, 5, 6, 8] Rule 10. (F1 = 1) AND (F8 = 1) THEN (D = Two); [1, %, %][0, 1][9}] F1F2F3F4F5F6 F7 F8 F9F10 D One One One One One 9 Two Rule 8 Clusters F5 F10 F1 F4 F2 F8 F3 F6 F7 F9 Rule 8. (F8 = 0) THEN (D = One); [7, 87.50%, %][2, 3, 4, 5, 6, 7, 8] Rule 9 Clusters F5 F10 F1 F4 F2 F8 F3 F6 F7 F9 Rule 9. (F5 = 0) THEN (D = One); [5, 62.50%, %][1, 2, 5, 6, 8] Data Set with Ill-Defined Outcomes F7 F8 F9 F10 D Two Three Two Three Three Two Three Decision Rules Exact rules Rule 1. (F1 = 1) AND (F9 = 0) THEN (D = One) Rule 2. (F4 = 0) AND (F7 = 0) AND (F10 = 0) THEN (D = One) Rule 3. (F8 = 1) THEN (D = Two) Rule 4. (F7 = 1) THEN (D = Three) Approximate rules Rule 5. (F1 = 0) AND (F10 = 1) THEN (D = One) OR (D = Two) Rule 6. (F3 = 0) AND (F4 = 1) THEN (D = Two) OR (D = Three) 4

5 Clusters F5 F10 F1 F4 F2 F8 F3 F6 F7 F9 Clusters F7 F8 F9 F10 D Two Three Two Three Three Two Three Conflicts detected Rule 1. (F1 = 1) AND (F9 = 0) THEN (D = One); [1, 33.33%, %][3] Rule 2. (F4 = 0) AND (F7 = 0) AND (F10 = 0) THEN (D = One); [1, 33.33%, %][7] Rule 3. (F8 = 1) THEN (D = Two); [1, 33.33%, %][1] Rule 4. (F7 = 1) THEN (D = Three); [3, 75.00%, %[2, 6, 8] Rule 5. (F1 = 0) AND (F10 = 1) THEN (D = One) OR (D = Two); [2, %, %][4, 9] Rule 6. (F3 = 0) AND (F4 = 1) THEN (D = Two) OR (D = Three); [2, %, %][5, 10] Data Set with Ill-Defined Outcomes About 100 objects About 80 features Case 1 N Z P None N Z P Correct Incorrect None N 12.50% 57.50% 10.00% Z 50.83% 44.17% 5.00% P 79.17% 10.69% 10.12% Av 51.39% 38.33% 10.28% As-is Data Set N Z P None N 9 1 Z P Correct Incorrect None N 36.37% 40.83% 2.50% Z 50.00% 32.50% 17.50% P 86.48% 9.60% 3.92% Av 61.67% 30.00% 8.33% Case 2 The outcome of one of the two conflicting object of Case 1 changed from D = P to D = Z Case 1 vs Case 2 Case 1 Case 2 N 2 9 Z 9 11 P Case 1 Case 2 Correct 46.25% 66.67% Incorrect 45.97% 30.00% None 7.78% 8.33% 5

6 References A. Kusiak, Decomposition in Data Mining: An Industrial Case Study, IEEE Transactions on Electronics Packaging Manufacturing, Vol. 23, No. 4, 2000, pp A. Kusiak and C. Kurasek, Data Mining Analysis of Printed-Circuit Board Defects, IEEE Transactions on Robotics and Automation, Vol. 17, April A. Kusiak, Decomposition in Data Mining: A Medical Case Study, Proceedings of the Conference on Data Mining and Knowledge Discovery: Theory, Tools, and Technology III, SPIE, Orlando, FL, April