Rough Sets and Incomplete Information es Couso 1 Didier Dubois 2 In´ 1. Department of Statistics, Universty of Oviedo, Spain e-mail: couso@uniovi.es 2. IRIT - CNRS Universit´ e Paul Sabatier - Toulouse, France, e-mail: dubois@irit.fr May, 2009 0-0
Introduction • Rough sets were introduced to cope with the lack of expressivity of descriptions of objects by means of attributes in databases (indiscernibility). • Another source of uncertainty is the lack of information about objects (incompleteness). Both situations lead to upper and lower approximations of sets of objects. • Independently, formal definitions of rough sets have been extended to relations other than equivalence relations – Fuzzy similarity relations (fuzzy rough sets induced by fuzzy partitions) – Tolerance relations (rough sets induced by coverings) Goal : define approximations of sets when both indiscernibility and incompleteness are present, and bridge the gap with coverings-based rough sets. 1
Pawlak’s Rough sets • Let f : U → V be an attribute function from a finite set of objects to some domain V = { v 1 , . . . , v m } ( f may represent a collection of attributes). • Let C = f − 1 ( { v } ) be collection of objects associated to v • Non-empty C ’s form a partition Π = { C 1 , . . . , C m } of U . • Upper and lower approximations of an arbitrary set of objects S induced by f : appr Π ( S ) = ∪{ C ∈ Π : C ∩ S � = ∅} ; appr Π ( S ) = ∪{ C ∈ Π : C ⊆ S } . (1) – S is an exact set when appr Π ( S ) = S = appr Π ( S ) . – If not, it is called a rough set . Then appr Π ( S ) � S � appr Π ( S ) is the best we can do to describe S with attribute function f . For instance, in a classification problem, the partition induced by a decision function d : U → V , will be approximated by the partition induced by an attribute function f . 2
Ill-known sets • A one-to-many mapping F : U → ℘ ( V ) represents an imprecise attribute function f : U → V . • How to describe the set f − 1 ( A ) of objects that satisfy a property A ⊆ V , namely f − 1 ( A ) ⊆ U . • Because incomplete information, the subset f − 1 ( A ) is an ill-known set . NOTE: F is NOT a set-valued attribute: For each object u ∈ U , all that is known about the attribute value f ( u ) is that it belongs to the set F ( u ) ⊆ V . f − 1 ( A ) can be approximated by upper and lower inverses of A via F : • F ∗ ( A ) = { u ∈ U : F ( u ) ∩ A � = ∅} : all objects that possibly belong to f − 1 ( A ) . • F ∗ ( A ) = { u ∈ U : F ( u ) ⊆ A } : all objects that surely belong to f − 1 ( A ) . The pair ( F ∗ ( A ) , F ∗ ( A )) is such that F ∗ ( A ) ⊆ f − 1 ( A ) ⊆ F ∗ ( A ) . Mappings F ∗ and F ∗ : 2 V → 2 U are Dempster ’s upper and lower inverses of F . 3
Ill-known rough sets • In the rough set construction, it is impossible to precisely describe sets defined in extension by means of attribute values, subsets thereof (properties) etc... : insufficient language . • In the ill-known set construction, it is impossible to give an explicit list of objects defined by means of properties : incompletely informed attributes . This paper : the case when both sources of imperfection are combined. When a set cannot be described perfectly : neither in extension in terms of properties, neither in intension. 4
Covering induced by an ill-known attribute function Again the multimapping F between U and V . For each value v ∈ V , let us consider its upper inverse image, the subset of objects of U for which it is possible that f ( u ) = v : F ∗ ( { v } ) = { u ∈ U : v ∈ F ( u ) } ⊆ U. In other words, if u �∈ F ∗ ( { v } ) , we are sure that f ( u ) � = v. COVERING INDUCED BY F : C = { F ∗ ( { v 1 } ) , . . . , F ∗ ( { v m } ) } = { C 1 , . . . , C k } . Then, it is obvious that: 1. If F ( u ) � = ∅ , ∀ u ∈ U then C is a covering of U , i.e. ∪ m i =1 C i = U. 2. v ∈ F ( u ) if and only if u ∈ F ∗ ( v ) , the set attached to attribute value v in the covering. 3. If F ∗ is injective then, the covering C determines F only up to a possible permutation of elements of V , i.e. |C| = | V | . 5
Example • Let U = { u 1 , u 2 , u 3 , u 4 } . Let V = { v 1 , v 2 , v 3 } and F ( u 1 ) = { v 1 , v 2 } , F ( u 2 ) = { v 1 , v 3 } , F ( u 3 ) = { v 2 , v 3 } , F ( u 4 ) = { v 3 } . • The covering associated to F , C = { C 1 , C 2 , C 3 } , is given by: C 1 = F ∗ ( { v 1 } ) = { u 1 , u 2 } , C 2 = F ∗ ( { v 2 } ) = { u 1 , u 3 } , C 3 = F ∗ ( { v 3 } ) = { u 2 , u 3 , u 4 } . • If we only know the covering C = { C 1 , C 2 , C 3 } , F can then be retrieved (up to a renaming of elements in V ) as follows: F ( u 1 ) = { v k : u 1 ∈ C k } = { v 1 , v 2 } F ( u 2 ) = { v k : u 2 ∈ C k } = { v 1 , v 3 } F ( u 3 ) = { v k : u 3 ∈ C k } = { v 2 , v 3 } F ( u 4 ) = { v k : u 4 ∈ C k } = { v 3 } 6
Interpretation of coverings • C i is the class of objects that are possibly in one equivalence class induced by the real information on objects • The covering C encodes an ill-known partition. According to the information provided by F , we know that f induces one of the 7 following partitions of U : Π 1 = {{ u 1 , u 2 } , { u 3 } , { u 4 }} ; Π 2 = {{ u 1 , u 2 } , { u 3 , u 4 }} Π 3 = {{ u 1 } , { u 2 , u 4 } , { u 3 }} ; Π 4 = {{ u 1 } , { u 2 , u 3 , u 4 }} Π 5 = {{ u 1 , u 3 } , { u 2 } , { u 4 }} ; Π 6 = {{ u 1 } , { u 2 } , { u 3 , u 4 }} Π 7 = {{ u 1 , u 3 } , { u 2 , u 4 }} . Note : • There are at most � u ∈ U | F ( u ) | partitions • the covering could be a family of nested sets! 7
Covering based rough sets: Y.Y. Yao The same definitions of rough sets as for a partition can be used, but the duality between upper and lower approximations is lost. • Y.Y. Yao (1998) considers the following two pairs of approximations – The loose pair: appr L C ( S ) = ∪{ C ∈ C : C ∩ S � = ∅} [appr L C ( S c )] c = { u ∈ U : ∀ C ∈ C , [ u ∈ C ⇒ C ⊆ S ] } appr L C ( S ) = ∪{ C ∈ C : , C ⊆ S ∧ [ � ∃ C ′ ∈ C , C ′ ∩ S c � = ∅ ∧ C ∩ C ′ � = ∅ ] } . = – The tight pair: appr T ∪{ C ∈ C : C ⊆ S } C ( S ) = C ( S c )] c = { u ∈ U : ∀ C ∈ C , [ u ∈ C ⇒ C ∩ S � = ∅ ] } . [appr T appr T C ( S ) = ∪{ C ∈ C : C ∩ S � = ∅ ∧ [ � ∃ C ′ ∈ C , C ′ ⊆ S c ∧ C ∩ C ′ � = ∅ ] } = 8
Covering based rough sets: Y.Y. Yao • The loose inner approximation appr L C ( S ) of S includes all elements of the covering included in S , but not intersecting the loose outer approximation appr L C ( S c ) of its complement. • The tight outer approximation appr T C ( S ) of S includes all elements of the covering intersecting S , but not intersecting the tight inner approximation appr T C ( S c ) of its complement. Then: appr L C ( S ) ⊂ appr T C ( S ) ⊆ S ⊆ appr T C ( S ) ⊂ appr L C ( S ) . The first approximation pair is looser than the second pair of sets 9
Covering -based rough sets : Bonikowski Bonikowski et al. (1998) rely on the duality between intensions (properties) and extensions (sets of objects) along the line of formal concept analysis. Then, a covering is a set of known concepts or properties. • The minimal description M ( u ) of object u is the set of minimal elements in the covering C , that contain u . • The lower approximation of a subset S of objects is chosen as appr T C ( S ) • The boundary of S is Bn ( S ) = ∪ u ∈ S \ appr T C ( S ) ∪ C ∈ M ( u ) C • The upper approximation is appr B C ( S ) = appr T C ( S ) ∪ Bn ( S ) . 10
The top-class mapping • Based on the multi-valued mapping F : U → ℘ ( V ) , another multi-valued mapping F : U → ℘ ( U ) is defined : I F ( u ) := F ∗ ( F ( u )) = { u ′ ∈ U : F ( u ′ ) ∩ F ( u ) � = ∅} , ∀ u ∈ U. I I F is called the top-class function associated to F . • I F ( u ) is the set of objects that could be in the same equivalence class as u if attribute function were better known : a kind of neighborhood of u . • Associated tolerance relation R : uRu ′ if and only if u ′ ∈ I F ( u ) . Orlowska & Pawlak (1984) interpret uRu ′ as a similarity between u and u ′ , but this is misleading as it is only potential similarity. 11
Upper and lower approximations induced by top-class mappings v ∈ F ( u ) F ∗ ( { v } ) . • in terms of covering : I F ( u ) = ∪{ C ∈ C : u ∈ C } = � • ∪{ C ∈ M ( u ) } ⊂ I F ( u ) : the latter is wider than the sets of objects having the same minimal description F : U → ℘ ( U ) be the top-class function associated to F . Let appr L C ( S ) and • Let I appr L C ( S ) be Y.Y. Yao’s loose upper and lower approximations of S . Then: F ∗ ( S ) = ∪ u ∈ S I F ∗ ( u ) = { u, I – appr L C ( S ) = I F ( u ) ∩ S � = ∅} = { u : ∃ u ′ ∈ S, u ′ Ru } F ∗ ( u ) c = { u, I – appr L C ( S ) = I F ∗ ( S ) = ∩ u �∈ S I F ( u ) ⊆ S } = { u : ∀ u ′ , u ′ Ru implies u ′ ∈ S } • These definitions are thus the natural ones in the setting of incomplete information. 12
Recommend
More recommend
