1. Adaptive fuzzy clustering Kharkiv National University of Radio Electronics Control Systems Research Laboratory

2. Crisp Approach to Clustering

3. Fuzzy Approach to Clustering

4. Input Data Set Goal Function of Clustering Batch Approaches

5. Batch Fuzzy Clustering Algorithms: Probabilistic Approach Fuzzy clustering as a constrained optimization problem Divide a data set into clusers. (1) (2) (3) Lagrange function: (4)

6. Batch Fuzzy Clustering Algorithms: Probabilistic Approach Degrees of membership: (5) prototypes: (6)

7. Batch Fuzzy Clustering Algorithms: Possibilistic Approach Unconstrained optimization Modified objective function: Scalar parameters determine the distance at which the degrees of membership are equal to 0.5: (8) (7)

8. Batch Fuzzy Clustering Algorithms: Possibilistic Approach Possibilistic algorithm Degrees of membership: prototypes: (10) (9)

9. Adaptive Fuzzy Probabilistic Algorithms One-step Largange function: Degrees of membership: prototypes: (13) (11) (12) Unsurprised fuzzy competitive learning (UFCL) algorithm

10. Adaptive Fuzzy Probabilistic Algorithms UFCL is an extension of the Kohonen competitive learning rule: where When, UFCL transforms into the Gradient-based fuzzy c-means (GBFCM) algorithm prototypes: (16) (14) (15) is a number of the “winning” prototype. Degrees of membership:

11. Adaptive Fuzzy Possibilistic Algorithms One-step objective function: Adaptive possibilistic algorithm Degrees of membership: prototypes: (19) (18) (17)

12. Adaptive Fuzzy Possibilistic Algorithms When β=2 Degrees of membership: prototypes: (20) (21)

13. Adaptive Fuzzy Possibilistic Algorithms Possibilistic membership function for

14. Matrix Fuzzy C-means Algorithm Input Data Set (22). Goal Function of Clustering. Batch Approaches (25).(26) (23) (24)

15. Matrix Fuzzy C-means Algorithm Karush-Kuhn-Tucker equation system: (27)

16. Matrix Fuzzy C-means Algorithm Fuzzy clustering algorithm: (28)

17. Matrix Fuzzy C-means Algorithm J. Bezdek’s FCM-algorithm:.(29)

18. Experiments Classification system based on fuzzy clustering (the number of the rule nodes is equal to the number of classes)

19. Experiments DataFuzzy c-meansBatch possibilistic Park-DagherRecursive possibilistic Iris7,4%7,6%6,9%7,8% Wine3,8%4,4%3,9%4,3% DataFuzzy c-meansBatch possibilistic Park-DagherRecursive possibilistic Iris, class 333,3%14,0%33,3%15,3% Iris, class 138,0%6,0%38,0%6,7% Wine, class 329,0%19,6%29,0%19,6% Wine, class 135,9%23,6%35,9%23,4% Thyroid, class 3 18,1%20,9%18,6%11,6% Thyroid, class 1 15,4%17,2%15,5%5.6% Table 1: Classification with known number of classes (error rate on testing data) Table 2: Classification with one unknown class (error rate on testing data)

20. Robust Probabilistic Fuzzy Clustering Algorithms Subject to constraints (31) (32) where areth components of -vectors respectively. (33) (30)

21. Robust Probabilistic Fuzzy Clustering Algorithms (34) where and are the parameters that define the center and width of the distribution respectively.,(35) (36) (37)

22. Robust Probabilistic Fuzzy Clustering Algorithms Objective function for robust clustering Lagrange function The system of Kuhn-Tucker equations (40) (39) (38)

23. Robust Probabilistic Fuzzy Clustering Algorithms Local Lagrange function (42) (41)

24. Robust Probabilistic Fuzzy Clustering Algorithms Arrow-Hurwitz-Uzawa procedure where is the-th component of the -th prototype vector calculated at the -th step. (43)

25. Robust Possibilistic Fuzzy Clustering Algorithms The criterion The system of Kuhn-Tucker equation (45) (44)

26. Robust Possibilistic Fuzzy Clustering Algorithms Local Lagrange function Adaptive possibilistic fuzzy clustering algorithm (47) (46)

27. Robust Possibilistic Fuzzy Clustering Algorithms Experiments (48) Complete data set (left) and its central part (right)

28. Robust Possibilistic Fuzzy Clustering Algorithms Cluster prototypes layout obtained by different algorithms

29. Robust Possibilistic Fuzzy Clustering Algorithms Table 3: Classification results Algorithm Classification error rate TrainingChecking Bezdek’s fuzzy c-means17.1 % (1229 samples)16.6 % (299 samples) Probabilistic robust clustering (41)15.6 % (1127 samples)15.6 % (281 samples) Possibilistic robust clustering (47)15.2 % (1099 samples)14.6 % (263 samples)