possibilistic evaluation of fuzzy temporal intervals
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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:
jpons@decsai.ugr.es
http://decsai.ugr.es/˜ jpons
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