Panel Discussion on Granular Computing at RSCTC2004

1. Panel Discussion on Granular Computing at RSCTC2004 J. T. Yao University of Regina Email: jtyao@cs.uregina.ca Web: http://www2.cs.uregina.ca/~jtyao

2. What is Granular Computing? • “There are three basic concepts that underline human cognition: granulation, organization and causation. • Informally, granulation involves decomposition of whole into parts; • Organization involves integration of parts into whole; • Causation involves association of causes with effects. • Granulation of an object A leads to a collection of granules of A, with a granule being a clump of points (objects) drawn together by indistinguishability, similarity, proximity or functionality” (Zadeh 1997)

3. What is Granular Computing • An umbrella term to cover any theories, methodologies, techniques, and tools that make use of granules in problem solving. • A subset of the universe is called a granule in granular computing. • Basic ingredients of granular computing are subsets, classes, and clusters of a universe.

4. Granule Computing and Data Mining • A concept is understood as a unit of thoughts that consists of two parts, the intension and extension of the concept. • The intension of a concept consists of all properties or attributes that are valid for all those objects to which the concept applies. • The extension of a concept is the set of objects or entities which are instances of the concept. • A rule can be expressed in the form, φ=>ψ where φ and ψ are intensions of two concepts. • Rules are interpreted using extensions of the two concepts.

5. How do Rough Sets Contribute to Granular Computing? • Zadeh define Granular Computing in BISC/SIG on GrC as “a superset of the theory of fuzzy information granulation, rough set theory and interval computations, and is a subset of granular mathematics”.

6. Zadeh’s Fuzzy GrC Model • Granules are constructed and defined based on the concept of generalized constraints. Relationships between granules are represented in terms of fuzzy graphs or fuzzy if then rules. • A granule is defined by a fuzzy set G = {X | X isr R}

7. Pawlak’s Rough Set Model • Granulation: Universe => granules • Some granules can only be approximately described. • Rough sets can deal with approximation of information granulation. • In the case one cannot describe X using E

8. Information Tables • U: a finite nonempty set of objects. • At: a finite nonempty set of attributes. • L: a language defined using attributes in At. • Va: a nonempty set of values for a ∊ At • Ia : U → Va is an information function.

9. Concept Formation • Atomic formula: a=v (a ∊ At, v ∊ Va ) • If φ, ψ are formulas, so is φ∧ ψ • If a formula is a conjunction of atomic formulas we call it a conjunctor. • Meaning of a formula: • m(φ)={x ∊ U | x ⊨φ} • x ⊨ a=v iff Ia(x)=v • A definable concept is a pair (φ, m(φ)) • φ is the intension of the concept • m(φ) is the extension of the concept

10. Classification Problems • Assume that each object is associated with a unique class label. • Objects are divided into disjoint classes which form a partition of the universe. • The set of attributes is expressed as At = F ∪ {class}, where F is the set of attributes used to describe the objects. • To find classification rules of the form, φ⇒ class = ci, where φ is a formula over F and ci is a class label.

11. Solution to Classification Problems • The partition solution to a consistent classification problem is a conjunctively definable partition π such that π≼πclass. • The covering solution to a consistent classification problem is a conjunctively definable covering  such that ≼πclass.

12. A construction algorithm • Construct the family of basic concept with respect to atomic formulas: • BC(U) = (a=v, m (a = v)) | a  F, v  Va} • Set the unused basic concepts to the set of basic concepts: • UBC(U) = BC(U). • Set the granule network to GN = ({U},), which is a graph consists of only one node and no arc. • While the set of smallest granules in GN is not a covering solution of the classification problem do the following: • Compute the fitness of each unused basic concept. • Select the basic concept C=(a=v, m(a=v)) with maximum value of fitness. • Set UBC(U) = UBC(U) - {C}. • Modify the granule network GN by adding new nodes which are the intersection of m(a=v) and the original nodes of GN; connect the new nodes by arcs labelled by a = v.

13. References • Pawlak, Z., Granularity of knowledge, indiscernibility and rough sets, IEEE International Conference on Fuzzy Systems, 106-110, 1998. • Yao, J.T., Yao, Y.Y. A granular computing approach to machine learning, FSKD'02, 732-736, 2002. • Yao, Y.Y. Granular computing: basic issues and possible solutions, JCIS (I), 86-189, 2000. • Zadeh, L.A. Towards a theory of fuzzy information granulation and its centrality in human reasoning and fuzzy logic, Fuzzy Sets and Systems, 19, 111-127, 1997.