Rough Sets and Incomplete Information es Couso 1 Didier Dubois 2 In - - PowerPoint PPT Presentation

Rough Sets and Incomplete Information

In´ es Couso1 Didier Dubois2

• 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

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Introduction

• Rough sets were introduced to cope with the lack of expressivity of descriptions of
• bjects 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.

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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.

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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.

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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.

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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

i=1Ci = 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 |.

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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}

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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!

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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: apprLC(S) = ∪{C ∈ C : C ∩ S = ∅} apprL

C(S)

= [apprLC(Sc)]c = {u ∈ U : ∀ C ∈ C , [u ∈ C ⇒ C ⊆ S]} = ∪{C ∈ C : , C ⊆ S ∧ [ ∃C′ ∈ C, C′ ∩ Sc = ∅ ∧ C ∩ C′ = ∅]}. – The tight pair: apprT

C(S)

= ∪{C ∈ C : C ⊆ S} apprTC(S) = [apprT

C(Sc)]c = {u ∈ U : ∀ C ∈ C , [u ∈ C ⇒ C ∩ S = ∅]}.

= ∪{C ∈ C : C ∩ S = ∅ ∧ [ ∃C′ ∈ C, C′ ⊆ Sc ∧ C ∩ C′ = ∅]}

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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 apprLC(Sc) of its complement.

• The tight outer approximation apprTC(S) of S includes all elements of the covering

intersecting S, but not intersecting the tight inner approximation apprTC(Sc) of its complement. Then: apprL

C(S) ⊂ apprT C(S) ⊆ S ⊆ apprTC(S) ⊂ apprLC(S).

The first approximation pair is looser than the second pair of sets

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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

C(S) ∪C∈M(u) C

• The upper approximation is apprBC(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 F : U → ℘(U) is defined : I F(u) := F ∗(F(u)) = {u′ ∈ U : F(u′) ∩ F(u) = ∅}, ∀ u ∈ U. 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.

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Upper and lower approximations induced by top-class mappings

• in terms of covering : I

F(u) = ∪{C ∈ C : u ∈ C} =

v∈F (u) F ∗({v}).

• ∪{C ∈ M(u)} ⊂ I

F(u) : the latter is wider than the sets of objects having the same minimal description

• Let I

F : U → ℘(U) be the top-class function associated to F. Let apprLC(S) and apprL

C(S) be Y.Y. Yao’s loose upper and lower approximations of S. Then:

– apprLC(S) = I F ∗(S) = ∪u∈SI F ∗(u) = {u, I F(u) ∩ S = ∅} = {u : ∃u′ ∈ S, u′Ru} – apprL

C(S) = I

F∗(S) = ∩u∈SI F ∗(u)c = {u, I F(u) ⊆ S} = {u : ∀u′, u′Ru implies u′ ∈ S}

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

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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 , apprLC(F ∗(A)) does not necessarily coincide with F ∗(A)

(The set of objects defined by a property is not representable by the covering).

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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 :

apprΠf (S) ⊆ S ⊆ apprΠf (S) Then we can express approximations with respect to incomplete mapping F in terms of its selections:

• F ∗({v}) = ∪f∈F f −1({v}); I

F(u) = ∪f∈F [u]f;

• apprLC(S) = ∪f∈F apprΠf (S); apprL

C(S) = ∩f∈F apprΠf (S). 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 apprTC(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 ⊆ apprTC(S) ⊆ apprf(S) ⊆ apprLC(S)

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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. qf(S) = |apprf(S)| |U| and qf(S) = |apprf(S)| |U|

• The upper and lower quality functions of S associated to an imprecise representation

F of f are ill known: qC(S) = [|apprTC(S)| |U| , |apprLC(S)| |U| ] = [| ∩f∈F apprf(S)| |U| , | ∪f∈F apprf(S)| |U| ] and qC(S) = [ |apprL

C(S)|

|U| , |apprT

C(S)|

|U| ] = [ | ∩f∈F apprf(S)| |U| , | ∪f∈F apprf(S)| |U| ]

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Accuracy of an ill-known rough set

• the accuracy of approximation of S by f is the quantity αR(S) =

q(S) q(S) ∈ [0, 1].

• The accuracy of approximation of S by ill-known f is the interval

˜ αR(S) = [ inf

f∈F

qf(S) qf(S), sup

f∈F

qf(S) qf(S)] and not

qC(S) qC(S) = [ |apprL

C(S)|

|apprLC(S)|, |apprT

C(S)|

|apprTC(S)|] (because the latter do not correspond to

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 [ inf

f∈F

|[u]f ∩ S| |[u]f| , sup

f∈F

|[u]f ∩ S| |[u]f| ]

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Imprecise rough probability

Let P be a probability measure on U, and P C(S) := P(apprLC(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 apprLC(S) is the upper inverse image of S via the top-class mapping I F. We get an interval [P C(S), P C(S)] that coincide with Pawlak’s rough probability if C is a partition. Not clear P

T C (S) := P(apprTC(S)),

P T

C (S) := P(apprT C(S)) are plausibility and

belief functions too.

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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, µ ˜

F (u)(v) is the degree of possibility that f(u) = v.

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

• µ ˜

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

• µ ˜

F∗(A) = infv∈A 1 − µ ˜ F (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
• f nested fuzzy sets (apprR(S), apprR(S)):

– µapprR(S)(u) = supu′∈S R(u, u′). : all objects more or less possibly in S. – µapprR(S)(u) = infu′∈S[1 − R(u, u′)] : all objects that surely belong to S.

• 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.

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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 R ˜

F (u, u′) = supv∈V min(µ ˜ F (u)(v), µ ˜ F (u′)(v)).

• Let S ⊆ U : define appr ˜

F (S) and appr ˜ F (S) as follows:

µappr ˜

F (S)(u) = µapprR ˜ F (S)(u) = supu′∈S R ˜

F (u, u′), ∀ u ∈ U,

µappr ˜

F (S)(u) = µapprR ˜ F

(S)(u) = infu′∈S[1 − R ˜ F (u, u′)], ∀ u ∈ U. 21

Interpretation as families of ill-known rough sets

• Consider the crisp multi-mapping : ˜

Fα(u) = { v ∈ V : µ ˜

F (u)(v) ≥ α}.

• Interpretation of fuzzy mapping ˜

F in tems of imprecise probabilities : the probability that f(u) belongs to ˜ Fα(u) = { v ∈ V : µ ˜

F (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 = { ˜ F ∗

α(v), : v ∈ V } the covering induced by ˜

Fα – Consider Rα the tolerance relation defined as uRαu′ as Fα(u) ∩ Fα(u′) = ∅. µapprR ˜

F (S)(u) = sup{α ∈ (0, 1] : u ∈ apprL

Cα}

µapprR ˜

F

(S)(u) = sup{α ∈ (0, 1] : u ∈ apprL Cα},

using the loose upper and lower approximations of Y.Y. Yao wrt covering Cα – Each α-cut of the fuzzy rough set (apprR ˜

F (S), apprR ˜ F (S)) is a pair of sets

bracketing S with probability at least 1 − α.

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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.

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