# 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

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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).

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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

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