possibilistic evaluation of fuzzy temporal intervals

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Motivation

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Proposal

Comparison

Conclusions and . . .

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Possibilistic evaluation offuzzy temporal intervals

Jose Enrique Pons1 Antoon Bronselaer2 Olga Pons Capote1

Guy De Tre2

1 Department of Computer Science and Artificial IntelligenceUniversity of Granada, Spain{jpons,opc}@decsai.ugr.es

2 Department of Telecommunications and Information ProcessingGhent University, Belgium

{Antoon.Bronselaer,Guy.De.Tre}@telin.ugent.be

February 2, 2012

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1. Contents

The structure of the presentation is:

2 Motivation.

3 Context:

3.1 Temporal databases.

3.2 Possibilistic variables and fuzzy numbers.

4 Proposal: Interval evaluation by ill-known con-straints.

5 Analysis of proposed transformations.

6 Conclusions and future work.

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2. Motivation

• The study of fuzzy intervals is of particular in-terest in temporal database research.

• To optimize the storage of fuzzy temporal inter-vals, some transformations have been proposed.⇒ Information Lost.

• The proposal is a framework to deal with theevaluation of ill-known temporal intervals.

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I Before J I

J

I Equal J

-Time

J

I Meets J J

I Overlaps J J

I During J J

I Starts J

I Finishes J

J

J

Relations

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3. Context

3.1 Temporal databases

3.2 Possibilistic variables and fuzzy numbers

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3.1. Temporal Databases:

A temporal database is a database that managesthe time in its schema.

• The time is usually represented as an interval inthe database.

X Y

• The user provides a crisp temporal interval in thequery specification.

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3.2. Possibilistic variables and fuzzy

numbers

Two different natures for a fuzzy set:

• Conjunctive nature: The fuzzyfication of aregular set. This interpretation corresponds withthe following two semantics: Degree of prefer-ence and degree of similarity.

• Disjunctive nature: In this case, the disjunc-tive nature indicates a description of incompleteknowledge. This interpretation corresponds withthe semantics for the degree of uncertainty.

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Possibilistic Variable:A possibilistic variable X over a universe U is de-fined as a variable taking exactly one value in U ,but for which this value is (partially) unknown.The possibility distribution πX gives the availableknowledge about the value that X takes. For eachu ∈ U , πX(u) represents the possibility that Xtakes the value u.

-

6

1

N1 2 3 4

r

r

πX

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It is important to understand the difference betweenthe following two concepts:

• A possibilistic variable X is bounded to takeonly one value , but this value is not known dueto incomplete knowledge.

• An ill-known set : a possibilistic variable definedover the universe P(U).

Note that while a possibilistic variable refers to one(partially) unknown value, an ill-known set is a crispset but, for some reason, (partially) unknown.

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Fuzzy numbers and fuzzy intervalsA fuzzy interval is a fuzzy set M on the set of realnumbers R such that:

∀(u, v) ∈ R2 :

∀w ∈ [u, v] : µM(w) ≥ min(µM(u), µM(v))

∃m ∈ R : µM(m) = 1

If m is unique, then M is referred to as a fuzzynumber, instead of a fuzzy interval.

1

0

possibility

values

D-a D D+b

1

0

possibility

α β γ δ

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4. Interval evaluation by ill-known constraints

4.1 Constraint

4.2 Ill-known constraint

4.3 Example

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4.1. Constraint:

Given a universe U , a constraint C on a set A ⊆ Uis specified by means of the binary relation R ⊆ R2

and a fixed value x ∈ U :

C4= (R, x)

It is said that a set A satisfies the constraint C ifand only if:

∀a ∈ A : (a, x) ∈ R.

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4.2.

Ill-known constraint:

Given a universe U , an ill-known constraint C on a set A ⊆ U is specified bymeans of a binary relation R ⊆ U2 and an ill-known value X, i.e.:

C4= (R,X) .

The uncertainty that a set A ⊆ U satisfies C is given by:

Pos(C(A)) = mina∈A

(Pos(a,X) ∈ R

)= min

a∈A

(sup

(a,w)∈R

πX(w)

)Nec(C(A)) = min

a∈A

(Nec(a,X) ∈ R

)= min

a∈A

(inf

(a,w)/∈R1− πX(w)

)

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4.3.

Consider the two ill-known values X and Y .

X Y

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Allen Relation Constraints B(C1(I), ..., Cn(I)

)I before J C1

4= (<,X) C1(I)

I equal J

C14= (≥, X) C1(I) ∧ ¬C2(I) ∧ C3(I) ∧ ¬C4(I)

C24= (6=, X)

C34= (≤, Y )

C44= (6=, Y )

I meets JC1

4= (≤, X) C1(I) ∧ ¬C2(I)

C24= (6=, X)

I overlaps JC1

4= (<, Y ) C1(I) ∧ ¬C2(I) ∧ ¬C3(I)

C24= (≤, X)

C34= (≥, X)

I during J

C14= (>,X)

(C1(I) ∧ C2(I)

)∨(C3(I) ∧ C4(I)

)C2

4= (≤, Y )

C34= (≥, X)

C44= (<, Y )

I starts JC1

4= (≥, X) C1(I) ∧ ¬C2(I)

C24= (6=, X)

I finishes JC1

4= (≤, Y ) C1(I) ∧ ¬C2(I)

C24= (6=, Y )

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5. Analysis of proposedtransformations

Optimize storage⇒ Transformation from two fuzzynumbers to a fuzzy interval.

2 main proposals:

• Transf. Preserving the imprecision.

• Transf. based on the convex-hull.

Drawbacks:

• (Dubois and Prade): the fuzzy interval is apossibility distribution on R while the twofuzzy numbers are a set that belong to P(R).

• The lack of the necessity measure, used forranking purposes.

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5.1. Transformation that preserves the

imprecision

1

0

possibility

ds-as ds

1

0

possibilityds+bs

de-ae de de+be

S1

S2

S3

S4

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5.2. Transformation based on the convex

hull

1

0

possibility

1

0

possibility

ds deds-as ds+bs de-ae de+be

ds-as ds de de+be

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ComparativeConsider two ill-known points representing a timeinterval: X = [3, 2, 1] and Y = [7, 2, 3]The value for I = [a, b] is [3, 6]The relation R: I is inside X :

Method Possibility NecessityIll-known constraint 1 0.5

Preserving the imprecision 0.667 -Convex hull 1 -

Nec (C (A)) > 0⇐⇒ Pos (C (A)) = 1

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Pos+Nec

0

1

2

Poss Nec

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6. Conclusions

• The necessity measure is lost when dealing witha transformation.

• The possibility measure in the transformations is(w.r.t. the ill-known evaluation):

– Convex hull returns the same value as possi-bility.

– The preserving the imprecision approach re-turns a different value.

• If the support for the ill-known values do overlap,it is not possible to compute any transformations.

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Future work:

• A new theoretical model for valid-time databases.

• Extension of the Allen’s relations for the compar-ison between two ill-known values.

• Implementation of the theoretical model in a re-lational database.

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Thank you!

Questions?

Contact:

[email protected]

http://decsai.ugr.es/˜ jpons

possibilistic evaluation of fuzzy temporal intervals

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