Rough Sets and Incomplete Information

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					Rough Sets and Incomplete Information
               In´ s Couso1 Didier Dubois2
                 e

   1. Department of Statistics, Universty of Oviedo, Spain
              e-mail: couso@uniovi.es
                          e
2. IRIT - CNRS Universit´ 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 = {v1 , . . . , vm }(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 = {C1 , . . . , Cm } of U .

  • Upper and lower approximations of an arbitrary set of objects S induced by f :

        apprΠ (S) = ∪{C ∈ Π : C ∩ S = ∅}; apprΠ (S) = ∪{C ∈ Π : C ⊆ S}.                  (1)

     – S is an exact set when apprΠ (S) = S = apprΠ (S).
     – If not, it is called a rough set. Then apprΠ (S)   S    apprΠ (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∗ : 2V → 2U 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 ∗ ({v1 }), . . . , F ∗ ({vm })} = {C1 , . . . , Ck }.
Then, it is obvious that:

 1. If F (u) = ∅, ∀ u ∈ U then C is a covering of U , i.e. ∪m Ci = U.
                                                            i=1

 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 = {u1 , u2 , u3 , u4 }. Let V = {v1 , v2 , v3 } and
     F (u1 ) = {v1 , v2 },   F (u2 ) = {v1 , v3 },   F (u3 ) = {v2 , v3 },   F (u4 ) = {v3 }.

 • The covering associated to F , C = {C1 , C2 , C3 }, is given by:
                 C1 = F ∗ ({v1 }) = {u1 , u2 }, C2 = F ∗ ({v2 }) = {u1 , u3 },
                             C3 = F ∗ ({v3 }) = {u2 , u3 , u4 }.

 • If we only know the covering C = {C1 , C2 , C3 }, F can then be retrieved (up to a
   renaming of elements in V ) as follows:
                             F (u1 ) = {vk : u1 ∈ Ck } = {v1 , v2 }
                             F (u2 ) = {vk : u2 ∈ Ck } = {v1 , v3 }
                             F (u3 ) = {vk : u3 ∈ Ck } = {v2 , v3 }
                             F (u4 ) = {vk : u4 ∈ Ck } = {v3 }

                                                                                                6
Interpretation of coverings


 • Ci 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 :

             Π1 = {{u1 , u2 }, {u3 }, {u4 }}; Π2 = {{u1 , u2 }, {u3 , u4 }}
             Π3 = {{u1 }, {u2 , u4 }, {u3 }} ; Π4 = {{u1 }, {u2 , u3 , u4 }}
             Π5 = {{u1 , u3 }, {u2 }, {u4 }} ; Π6 = {{u1 }, {u2 }, {u3 , u4 }}
             Π7 = {{u1 , u3 }, {u2 , u4 }}.
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:

         apprL C (S) = ∪{C ∈ C : C ∩ S = ∅}
         apprL C (S) =       [apprL C (S c )]c = {u ∈ U : ∀ C ∈ C , [u ∈ C ⇒ C ⊆ S]}
                         = ∪{C ∈ C : , C ⊆ S ∧ [ ∃C ∈ C, C ∩ S c = ∅ ∧ C ∩ C = ∅]}.

     – The tight pair:

         apprT C (S)     = ∪{C ∈ C : C ⊆ S}
         apprT C (S)     =   [apprT C (S c )]c = {u ∈ U : ∀ C ∈ C , [u ∈ C ⇒ 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 apprL C (S) of S includes all elements of the covering
    included in S, but not intersecting the loose outer approximation apprL C (S c ) of its
    complement.

  • The tight outer approximation apprT C (S) of S includes all elements of the covering
    intersecting S, but not intersecting the tight inner approximation apprT C (S c ) of its
    complement.

Then: apprL C (S) ⊂ apprT C (S) ⊆ S ⊆ apprT C (S) ⊂ apprL 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 apprT C (S)

  • The boundary of S is Bn(S) = ∪u∈S\apprT          (S)   ∪C∈M (u) C
                                                 C


  • The upper approximation is apprB C (S) = apprT C (S) ∪ Bn(S).



                                                                                        10
The top-class mapping


 • Based on the multi-valued mapping F : U → ℘(V ), another multi-valued mapping
   I : U → ℘(U ) is defined :
    F

             I (u) := F ∗ (F (u)) = {u ∈ U : F (u ) ∩ F (u) = ∅}, ∀ u ∈ U.
              F

    F
   I is called the top-class function associated to F .

 • I (u) is the set of objects that could be in the same equivalence class as u if attribute
    F
   function were better known : a kind of neighborhood of u.

 • Associated tolerance relation R: uRu if and only if u ∈ I (u).
                                                            F
   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


 • in terms of covering : I (u) = ∪{C ∈ C : u ∈ C} =
                           F                                v∈F (u)   F ∗ ({v}).

 • ∪{C ∈ M (u)} ⊂ I (u) : the latter is wider than the sets of objects having the same
                     F
   minimal description

 • Let I : U → ℘(U ) be the top-class function associated to F . Let apprL C (S) and
        F
   apprL C (S) be Y.Y. Yao’s loose upper and lower approximations of S. Then:
    – apprL C (S) = I ∗ (S) = ∪u∈S I ∗ (u) = {u, I (u) ∩ S = ∅}
                     F              F             F
      = {u : ∃u ∈ S, u Ru}
    – apprL C (S) = I ∗ (S) = ∩u∈S I ∗ (u)c = {u, I (u) ⊆ S}
                     F              F              F
      = {u : ∀u , u Ru implies u ∈ S}

 • These definitions are thus the natural ones in the setting of incomplete information.


                                                                                       12
Differences with pure rough sets


 • The covering C provides more information than R and I .
                                                        F
   Example :F : U → ℘(V ) and F : U → ℘(V ) defined as follows:

                 F (u1 ) = {v1 , v2 }, F (u2 ) = {v2 , v3 }, F (u3 ) = {v1 , v3 }.

                        F (u1 ) = F (u2 ) = F (u3 ) = {v1 , v2 , v3 },
   but they induce the same binary relation R = U × U.
   But different coverings C = {{u1 , u3 }, {u1 , u2 }, {u2 , u3 }} and C = {{u1 , u2 , u3 }}

 • For a property A ⊂ V , apprL C (F ∗ (A)) does not necessarily coincide with F ∗ (A)
   (The set of objects defined by a property is not representable by the covering).



                                                                                          13
The selection function approach


 • The multiple valued mapping F represents a set of attribute functions f such that
   ∀u, f (u) ∈ F (u) (f is a selection of F .)

 • each selection f is associated with a possible partition Πf of U , with equivalence
   classes [u]f = f −1 (f (u)).

 • Each subset S ⊆ U can be approximated with respect to f :
   apprΠ (S) ⊆ S ⊆ apprΠf (S)
          f


Then we can express approximations with respect to incomplete mapping F in terms of
its selections:

 • F ∗ ({v}) = ∪f ∈F f −1 ({v}); I (u) = ∪f ∈F [u]f ;
                                  F

 • apprL C (S) = ∪f ∈F apprΠf (S); apprL C (S) = ∩f ∈F apprΠ (S).
                                                                f




                                                                                         14
Ill-known rough sets as nested 4-tuples of sets


 • The tight pair of upper and lower approximations of S by covering C in the sense of
   Y.Y. Yao, as induced by F is
    apprT C (S) = ∪{C ∈ C : C ⊆ S} = ∪f ∈F ∪ {f −1 ({v}) ⊆ S}
    = ∪f ∈F apprf (S) ⊆ S (union of all possible lower approximations).

 • Hence, by duality apprT C (S) = ∩f ∈F apprf (S). It contains S.

BASIC CLAIM : An ill-known rough set is the description of a subset S ⊆ U of
ill-known objects by means of an imprecise and incomplete attribute function described
by a multimapping F , and it consists of four subsets

                     apprL C (S) ⊆ apprf (S) ⊆ apprT C (S) ⊆ S

                     S ⊆ apprT C (S) ⊆ apprf (S) ⊆ apprL C (S)


                                                                                     15
Quality functions of an ill-known rough set


 • The upper and lower quality functions of S reflect how well S is described by
   attribute function f .

                         |apprf (S)|                              |apprf (S)|
               q f (S) =                   and        q f (S) =
                            |U |                                     |U |

 • The upper and lower quality functions of S associated to an imprecise representation
   F of f are ill known:

                |apprT C (S)| |apprL C (S)|     | ∩f ∈F apprf (S)| | ∪f ∈F apprf (S)|
    q C (S) = [              ,              ]=[                   ,                   ]
                    |U |          |U |                  |U |               |U |
   and
                |apprL C (S)| |apprT C (S)|     | ∩f ∈F apprf (S)| | ∪f ∈F apprf (S)|
    q C (S) = [              ,              ]=[                   ,                   ]
                    |U |          |U |                  |U |               |U |

                                                                                     16
Accuracy of an ill-known rough set


                                                                                                q(S)
 • the accuracy of approximation of S by f is the quantity αR (S) =                             q(S)   ∈ [0, 1].

 • The accuracy of approximation of S by ill-known f is the interval
                                                                q f (S)           q f (S)
                                         ˜
                                         αR (S) = [ inf                   , sup             ]
                                                       f ∈F     q f (S)    f ∈F   q f (S)
             q (S)          |apprL (S)| |apprT (S)|
   and not     C
             q C (S)   =[            C
                                              ,            C
                                                                    ] (because the latter do not correspond to
                            |apprL   C (S)|       |apprT   C (S)|
   the same f in numerator and denominator. )

 • Imprecise rough membership function : The Laplacean probability P (S|u) that
   an object u belongs to S is only known to lie in interval
                                                |[u]f ∩ S|       |[u]f ∩ S|
                                          [ inf            , sup            ]
                                           f ∈F    |[u]f | f ∈F |[u]f |

                                                                                                                   17
Imprecise rough probability


Let P be a probability measure on U , and
P C (S) := P (apprL C (S)), P C (S) := P (apprL C (S)), ∀ S ⊆ U.
Theorem : P C and P C are respectively a plausibility and a belief function.
Proof: This is because apprL C (S) is the upper inverse image of S via the top-class
         F
mapping I .
We get an interval [P C (S), P C (S)] that coincide with Pawlak’s rough probability if C is a
partition.
            T
Not clear P C (S)  := P (apprT C (S)),   P T (S) := P (apprT C (S)) are plausibility and
                                           C
belief functions too.



                                                                                           18
Ill-known sets from fuzzy attribute mappings


                    ˜
  • A fuzzy mapping F : U → [0, 1]V represents an ill-known attribute function f .

  • How to describe the set f −1 (A) ⊆ U of objects that satisfy a crisp property A.

  • Because incomplete information, the subset f −1 (A) is an ill-known set bracketed by
    a pair of fuzzy sets.

For each object u ∈ U , µF (u) (v) is the degree of possibility that f (u) = v.
                         ˜

                 ˜           ˜
Define fuzzy sets F ∗ (A) and F∗ (A) as:

  • µF ∗ (A) = supv∈A µF (u) (v) : all objects more or less possibly in f −1 (A).
     ˜                 ˜

  • µF∗ (A) = inf v∈A 1 − µF (u) (v) : all objects that surely belong to f −1 (A).
     ˜                     ˜

           ˜       ˜                            ˜                         ˜
The pair (F∗ (A), F ∗ (A)) is such that Support(F∗ (A)) ⊆ f −1 (A) ⊆ core(F ∗ (A)) and is
called a twofold fuzzy set (Dubois - Prade, 1987).

                                                                                       19
Fuzzy rough sets


 • A fuzzy relation R on U that is symmetric and reflexive and min-transitive
   (Similarity)

 • Any subset S ⊆ U of objects can be described by a fuzzy rough set defined as a pair
   of nested fuzzy sets (apprR (S), apprR (S)):
    – µapprR (S) (u) = supu ∈S R(u, u ). : all objects more or less possibly in S.
    – µappr       (S)(u)   = inf u ∈S [1 − R(u, u )] : all objects that surely belong to S.
              R

 • Again, Support(apprR (S)) ⊆ S ⊆ core(apprR (S))

 • It is in the spirit of the loose approximation pairs of Y.Y. Yao, taking uRu as
   ∃C, c ∈ C, u ∈ C, u ∈ C and C ∩ C = ∅ in the crisp case. It is the tolerance
   relation induced by the covering.


                                                                                              20
Fuzzy rough set from a fuzzy attribute mapping


We show that:
                                                   ˜
 1. A fuzzy–valued imprecise attribute function F induces a fuzzy rough set in a natural
    way. But now, R will not be a similarity relation. It will be reflexive and symmetric,
    but it will not necessarily be min-transitive.

 2. The fuzzy rough set expresses loose fuzzy upper and lower approximations of a
    crisp rough set.
     • Define RF (u, u ) = supv∈V min(µF (u) (v), µF (u ) (v)).
              ˜                       ˜           ˜

     • Let S ⊆ U : define apprF (S) and apprF (S) as follows:
                             ˜             ˜


                  µapprF (S) (u) = µapprR
                       ˜                           (S) (u)   = supu ∈S RF (u, u ), ∀ u ∈ U,
                                                                        ˜
                                               ˜
                                               F
                µappr ˜ (S) (u) = µappr        (S) (u)   = inf u ∈S [1 − RF (u, u )], ∀ u ∈ U.
                                                                          ˜
                     F                    R˜
                                           F




                                                                                                 21
Interpretation as families of ill-known rough sets


                                      ˜
 • Consider the crisp multi-mapping : Fα (u) = { v ∈ V : µF (u) (v) ≥ α}.
                                                          ˜

                                    ˜
 • Interpretation of fuzzy mapping F in tems of imprecise probabilities : the probability
                         ˜
   that f (u) belongs to Fα (u) = { v ∈ V : µF (u) (v) ≥ α} is greater or equal to 1 − α.
                                              ˜

 • The ill-known fuzzy rough set approximating S can be retrieved as follows:
                       ˜∗
    – Consider Cα = {Fα (v), : v ∈ V } the covering induced by Fα˜
    – Consider Rα the tolerance relation defined as uRα u as Fα (u) ∩ Fα (u ) = ∅.

                       µapprR       (S) (u)   = sup{α ∈ (0, 1] : u ∈ apprLα }
                                                                         C
                                ˜
                                F


                      µappr        (S) (u)    = sup{α ∈ (0, 1] : u ∈ apprL },
                                                                         C
                              R˜                                          α
                               F

       using the loose upper and lower approximations of Y.Y. Yao wrt covering Cα
    – Each α-cut of the fuzzy rough set (apprR (S), apprRF (S)) is a pair of sets
                                                         ˜
                                                       ˜
                                                       F
       bracketing S with probability at least 1 − α.
                                                                                      22
Conclusion

 • We proposed an interpretation of covering-based rough sets as an ill-known rough
   set induced both by the ill-observation of attribute values and the lack of
   discrimination of the set of attributes.
 • A upper and a lower approximation is not enough: in fact the rough approximations
   are themselves bracketed from above and from below since ill-known.
 • Our choice of covering-based generalization of rough sets is justified by the
   incomplete information semantics.
 • Perspectives
   1. Relate the definition of ill-known rough sets to incomplete information database
      research (Nakata, especially).
   2. Complete the fuzzy extension by the study of tight pairs of approximations,
      fuzzy top-class function etc.
   3. fuzzy quality indices and fuzzy rough probabilities
   4. Connection with formal concept analysis.
                                                                                      23