CLASSIFICATION AND CLUSTERING FOR KNOWLEDGE DISCOVERY
2, X1X2, X2X1) ?F = R4 THIS PROPERTY OF THE KERNEL FUNCTION DOES NOT A?ECT THE ALGORITHMS DISCUSSED IN THIS CHAPTER CLASSIFICATION AND CLUSTERING FOR KNOWLEDGE DISCOVERY ?I?(XI) ?I?J?(XI, XJ) CLASSIFICATION AND CLUSTERING FOR KNOWLEDGE DISCOVERY DISTANCE BETWEEN FEATURE VECTORS THE LENGTH OF THE LINE JOINING TWO IMAGES ?(X) AND ?(Z) CAN BE COMPUTED AS ??(X) ??(Z)?2 = ??(X) ??(Z), ?(X) ??(Z)? = ??(X), ?(X)??2 ??(X), ?(Z)?+ ??(Z), ?(Z)? = ?(X, X) ?2?(X, Z) + ?(Z, Z) (1 1) IT IS EASY TO ?ND OUT THAT THIS IS A SPECIAL CASE OF THE NORM THE ALGORITHMS DEMONSTRATED AT THE END OF THIS CHAPTER ARE BASED ON DISTANCE NORM AND DISTANCE FROM THE CENTER OF MASS CONSIDER NOW THE CENTER OF MASS OF THE SET ?(S) THIS IS THE VECTOR ?S = 2 = ??S, ?S?= ?(XI), ?(XJ) = ??(XI), ?(XJ)?= ?(XI, XJ) HENCE, THE SQUARE OF THE NORM OF THE CENTER OF MASS IS EQUAL TO THE AVERAGE OF THE ENTRIES IN THE KERNEL MATRIX THIS IMPLIES THAT THIS SUM IS EQUAL TO ZERO IF THE CENTER OF MASS IS AT THE ORIGIN OF THE COORDINATE SYSTEM AND GREATER THAN ZERO OTHERWISE THE DISTANCE OF THE IMAGE OF A POINT X FROM THE CENTER OF MASS ?S IS: ??(X) ??S?2 = ??(X), ?(X)?+ ??S, ?S??2??(X), ?S? = ?(X, X) + ?(X, XI) (1 2) LINEAR CLASSI?CATION CLASSI?CATION, ALSO CALLED CATEGORIZATION IN TEXT ANALYSIS, IS ONE OF THE POSSIBLE TASKS THAT CAN BE SOLVED USING KERNEL APPROACH THE AIM IS TO ASSIGN ANY INPUT OF OUR TRAINING SET TO ONE OF A ?NITE SET OF CATEGORIES; THE CLASSI?CATION IS BINARY IF THERE ARE TWO CATEGORIES, OTHERWISE WE ARE CONSIDERING A MULTI-CLASS PROBLEM CLASSIFICATION AND CLUSTERING FOR KNOWLEDGE DISCOVERY WHERE ?+ W IS THE MEAN OF THE ELEMENTS IN CLASS +1 ONTO THE DIRECTION W, ?? W THE MEAN OF THE ELEMENTS IN CLASS ?1, AND THE (?+ W)2, (?? W)2 THE CORRESPONDING CLASSIFICATION AND CLUSTERING FOR KNOWLEDGE DISCOVERY