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The study of fuzzy intervals is of particular interest in temporal database research. In order to optimize the storage of fuzzy temporal intervals, some transformations have been proposed. In this paper we analyze the possibilistic evaluation of the ill-known temporal intervals. We propose a framework to deal with the evaluation of ill-known temporal intervals. It is shown how the reasoning behind the transformations implies a possibilistic information lost.
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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:
http://decsai.ugr.es/˜ jpons