http://www.imeko.org/publications/wc-2006/PWC-2006-TC7-018u.pdfInternational audienceFuzzy nominal scales were introduced in order to propose a formalism to the representation of empirical quantities by fuzzy subsets of words. This paper presents the results of studies on distances associated to this formalism and proposes a new distance operator

Allevard, Thomas

Benoit, Eric

Foulloy, Laurent

English

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A new distance for fuzzy descriptions of measurements
Thomas Allevard, Eric Benoit, Laurent Foulloy
To cite this version:
Thomas Allevard, Eric Benoit, Laurent Foulloy. A new distance for fuzzy descriptions of measure-
ments. XVIII IMEKO World Congress, Metrology for a Sustainable Development, Sep 2006, Rio de
Janeiro, Brazil. pp.PWC-2006-TC7-018u. ￿hal-00830856￿
XVIII IMEKO WORLD CONGRESS
Metrology for a Sustainable Development
September, 17 – 22, 2006, Rio de Janeiro, Brazil
A NEW DISTANCE FOR FUZZY DESCRIPTIONS OF MEASUREMENTS
Thomas ALLEVARD, Eric BENOIT, Laurent FOULLOY
LISTIC-ESIA, Université de Savoie, Annecy, France, eric.benoit@univ-savoie.frAbstract - Fuzzy nominal scales were introduced in order
to propose a formalism to the representation of empirical
quantities by fuzzy subsets of words. This paper presents the
results of studies on distances associated to this formalism
and proposes a new distance operator.
Keywords: Measurement theory, Fuzzy description,
Fuzzy nominal scale, Fuzzy distance.
1. INTRODUCTION
The introduction of the fuzzy subset theory in the
measurement field takes its origin in 1971 in the Zadeh’s
paper [1] that exposes a mechanism of description of a
quantity by a fuzzy subset of symbols. Since this paper, the
definition of the description process was mainly based on
good practices. Most of description processes had useful
properties but it was sometime difficult to justify them.
Recently, the link between the quantities and their fuzzy
representation was formalized in a scale approach named the
fuzzy nominal scale [2][3].
This approach gives the set of relations and operators that
can be used to define equations on symbols such that these
equations have a meaning on the set of quantity values. When
a fuzzy description is a fuzzy nominal scale, a fuzzy
equivalence relation on quantity values is linked to a fuzzy
equivalence relation on their representation. This last relation
also named similarity relation is used to define a distance
between the fuzzy subset of symbols that represent the
quantity values. This distance had been used to perform signal
processing [4], but is useless to compare symbols that are not
related by the similarity relation.
The purpose of this paper is to propose a distance on fuzzy
representations linked to a metric on the set of quantity
values. With this new distance, a fuzzy nominal scale is
enhanced and the set of authorized operators has now this
distance as member.
2. THE FUZZY SYMBOLISMS
The link between a physical state and its linguistic
representation is characterized by a symbolism defined by the
triplet <E,S,R> where E is the set of physical states, S is the
lexical set used to represent measurement results and R is a
relation on ExS. Two mappings can be extracted from this
relation: The description mapping denoted D associates a
subset of S to any item of E, and the meaning mapping
denoted M associates a subset of E to any item of S. These two
mappings are linked with the following equation.
(1)
The R relation can be a fuzzy relation. Then, the
translation of a physical state into its linguistic representation
is called a fuzzy linguistic description mapping or simply a
fuzzy description mapping. It transforms an object e of the set
of physical states E into a fuzzy subset of linguistic terms
called the fuzzy description of x. The dual mapping, called the
fuzzy meaning mapping, associates a fuzzy subset of E to each
term s of the lexical Set S. This fuzzy subset is the fuzzy
meaning of s. In the paper the fuzzy subsets of linguistic terms
also named lexical fuzzy subsets are denoted LFS. This two
mappings are also linked:
(2)
In [5] it is defined that <E,S,R> is a φ-symbolism if the set
of the meanings of the elements of S is a φ-partition of E and
if each meaning is normalized. This paper restricts its
investigation to id-symbolisms based on id-partition i.e. on
Ruspini partition. The set of all possible LFS obtained by a
fuzzy description based on id-symbolism is denoted Fid(S).
Any LFS respects then the condition:
(3)
A fuzzy equivalence relation on the physical states can be
associated to any id-symbolism.
(4)
From this fuzzy equivalence relation and from the relation
R, the following relation can be simply defined.
(5)
The symbolism <E,S,R> is then considered as a fuzzy
nominal scale.
3. CHOICE OF A DISTANCE OPERATOR
The relation used in the id-symbolism can define a
distance between LFSs [4].
(6)
e∀ E∈ s∀ S∈ e, , M s( )∈ s D e( )∈⇔
e s,( )∀ E S×∈ µM s( ) e( ) µD e( ) s( )=,
A∀ Fid S( )∈ µA s( )
s S∈
, 1=
x y,( )∀ E2∈ µ, ~ x y,( ) min µM s( ) x( ) µM s( ) y( ),( )
s S∈
=
A B,( )∀ Fid S( )
2
∈ µ,
~
A B,( ) min µA s( ) µB s( ),( )
s S∈
=
d
~
A B,( ) 1 - µ
~
A B,( )=
This distance is discriminant for LFSs that are at least
partially equivalent but is equal to 1 when 2 LFS have empty
intersection. This result is consistent with the absence of
distance on the lexical set. This mean that the definition of a
metric on the set Fid(S) needs the definition of a metric on the
set S. Let dS be a distance defined on S.
3.1. Required properties
The fuzzy subset theory proposes a large set of distance
operators and the best way to select a distance operator is to
list the properties that must be verified.
• The first property is the singleton coincidence: If two
LFSs are singleton {s1} and {s2} then the distance
between them is equal to the distance between symbols s1
and s2. This property supposes that the distance dS on S
exists.
• The continuity property is verified when the distance is a
continuous mapping from SxS to the set of positive num-
bers.
• The precision property simply imposes that the distance
between two LFSs must be a positive real number, and
not a fuzzy subset of positive real numbers.
• The consistency property is usually verified by distances
on crisp subsets: If A,B,C,E are four subsets of a metric
space, d is the distance on this space, and dg is a distance
that generalize d on subsets, it verifies:
(7)
Fig. 1. Consistency property
The extension of a distance defined on a finite space to a
distance defined on the set of the fuzzy subsets of this space
was widely studied but, as shown below, no existing distance
can be applied to Fid(S).
Distances on fuzzy subsets can by classified into the
following categories.
• The distances that generalise an existing distance.
• The distances defined from a similarity measure.
• The distances defined with subset operators.
• The distances computed from a symbolic approach.
In our approach, a distance dS is supposed to be defined on
S. Then only the first cathegory is investigated.
The generalisation of a distance dS defined on a finite set
S, to a distance dF(S) defined on the set of fuzzy subsets of S
is a recurrent subject of study. In [6] Bloch proposes four
types of generalisation.
3.2. The geometrical approach
In this approach, fuzzy subsets in a ndimentional space are
considered as crisp subsets in a (n+1)dimentional space. This
means that the distance between membership degrees has the
same semantic than distance on the ndimentional space. Such
hypothesis can not be justified in our problem and this
approach is not kept.
3.3. The fuzzification approach
In an other approach a distance DS between crisp subsets
is defined from the distance dS. Then the distance DS is
fuzzyfied. In [7] three fuzzyfications of the Hausdorff
distance are proposed.
(8)
(9)
(10)
Where Fα and Gα are the alpha-cuts of F and G, and HS is
the Hausdorff distance:
(11)
and do not verify the continuity
property, but does. It also verifies the singleton
coincidence, but not the consistency property.
3.4. The weighting approach
The distance dS can be generalized with a weighting of
membership degrees.
(12)
Where T is a continuous triangular norm.
Such operator does not respect the separation axiom that
imposes:
then it cannot be considered as a distance.
3.5. A new approach
So a new distance that respects the four properties had
been created. This distance is named the transportation
distance dtp. Its calculation is equivalent to the solution of a
mass transportation problem [8]. It is similar to the
Wasserstein distance used in probability theory, and can also
be considered as a fuzzy version of the Levenshtein distance
used to compare strings [10].
sup
u A v B∈,∈
d u v,( ) inf
x C y E∈,∈
d x y,( )≤
dg A B,( ) dg C E,( )≤
inf
x C y E∈,∈
d x y,( )
E
C sup
u A v B∈,∈
d u v,( )
A B
dg A B,( ) dg C E,( )≤
H1FId S( )
F G,( ) HS Fα Gα,( ) αd
0
1
=
H∞FId S( ) F G,( ) supα 0≥ HS Fα Gα,( )=
H*FId S( ) F G,( ) HS F1.0 G1.0,( )=
HS A B,( ) max maxa A∈ minb B∈ dS a b,( ) maxb B∈ mina A∈ dS a b,( ),( )=
H∞FId S( ) H
*
FId S( )
H1FId S( )
dT F G,( ) T µF s1( ) µG s2( ),( )dS s1 s2,( )
s2 S∈

s1 S∈
=
dT F G,( ) 0 F⇔ G= =
4. THE TRANSPORTATION DISTANCE
The transportation distance between two LFSs is based on
the cost calculation of a set of transformations needed to
transform the first LSF to the other. First a family of
transformation mappings is defined:
Let be a mapping on a set Fid(S) such that:
(13)
Fig. 2. The mapping with S={s1,s2,s3,s4}.
We demonstrate that any element of Fid(S) can be
transformed into any other element of Fid(S) with the use of a
sequence of such transformation mapping.
Let the following sets:
Proposition 1: Let S be a finite set. Let F and G be 2
elements of the set Fid(S). Let ∆s = µF(s) - µG(s). The
following equality is verified:
(14)
Then the definition of the sequence of transformation
mapping is equivalent to the well known linear programming
problem named the transportation problem [8]. The problem
is to bring a product from a set of n1 sources to a set of n2
destinations. Each source i gives a quantity si of product, and
each destination receives a quantity sj of product. The total
given quantity must be equal to the total received quantity:
(15)
A solution is to associate to each displacement from a
source i to a destination j a quantity xij of transported product
and a displacement unity cost cij. The aim is to find a solution
that minimises the total cost:
(16)
Considering the membership degrees as the transported
product, the set as the set of sources and the set
as the set of destinations, a distance can be computed as the
total cost of the optimal solution for the transportation of
membership degrees.
The transportation distance dtp is defined on Fid(S) from
distance dS on the lexical set S. The distance dtp is the sum of
the costs of each transformation mapping. And the cost of a
transformation mapping is equal to: .
It is now shown that the transportation distance is a
distance, and it verifies the 4 constraints presented before.
For any , dtp must verify:
(17)
(18)
(19)
• The relation is equivalent to
that is equivalent to
.
• The symmetry of is deduced from the symmetry of
the transportation problem.
• Finally, is by definition the distance corre-
sponding to the optimal sequence of transformations
that changes F into H. Then Adding a constraint
in order to include G in the set of transformation steps
will increase the distance.
• Calculating the distance between singletons and
using the transportation problem is equivalent to
finding the cheaper solution to bring a unity quantity of
product from source i to destination j. The solution is
made of only one transformation mapping . The
cost of this solution is that is equal to the distance
then the singleton coincidence is verified.
• The precision and the continuity properties are deduced
from the definition of the distance
• The consistency property of dtp is demonstrated below:
Let four elements of and
such that:
(20)
where:
(21)
and supp(A) is the support of A i.e. the set of lexical terms
s such that .
It must be shown that
. (22)
Tsi sj x, ,
G Tsi sj x, , F( )
µG si( ) µF si( ) x–=
µG sj( ) µF sj( ) x+=
⇔=
Ts1 s2 0.2, ,
SF G> s S∈ µF s( ) µG s( )>,{ }=
SF G= s S∈ µF s( ) µG s( )=,{ }=
∆s
s SF G>∈
 ∆s
s SG F>∈
–=
si
i 1=
n1
 sj′
j 1=
n2
–=
xijcij
i j,

SF G> SF G<
Tsi sj x, ,
x dS si sj,( )⋅
F G H, , FId S( )∈
dtp F G,( ) 0= F⇔ G= ,
dtp F G,( ) dtp G F,( )= ,
dtp F G,( ) dtp G H,( )+ dtp F H,( )≥ .
F G=
SF G> SF G< ∅= =
dtp F G,( ) 0=
dtp
dtp F H,( )
Tsi sj x, ,
li{ }
lj{ }
Tsi sj 1, ,
cij
dS si sj,( )
F G H I, , , FId S( )
s1 s2 s3 s4, , ,( ) S
4
∈
dFG dHI≤ .
dHI inf
s3 s4,( ) supp H( ) supp I( )×∈
d
S
s3 s4,( )= ,
dFG sups1 s2,( ) supp F( ) supp G( )×∈
d
S
s1 s2,( )= .
µA s( ) 0≠
dtp F G,( ) dtp H I,( )≤
If , then and . F and G are
the same singleton and . (22) is trivially
verified.
If , The brought quantity associated to the
calculation of is equal to 1 and
because the transportation distance is then a weighted average
of distances that are greater or equal to . With the same
reasoning, . Then, (20) induces
.
5. RESULTS
In this section, the transportation distance is applied on the
hand posture recognition. More details on the application can
be found in [9].
Fig. 3. : The 18 sensors of the Cyberglove®.
The hand posture is acquired with the 18 angle sensor of a
CyberGlove® (Fig. 3.). The finger flexion (except for the
thumb) is aquired with two angle sensors : MCP (métacarpal-
phalanx angle) et IP (interphalanx angle). The linguistic
description of a finger uses the set Sflexion = {Folded, Claw,
Round, Square, Straight} see Fig. 4.
Fig. 4. Words used to describe the finger flexion.
The dataglove gives a numeric representation of the finger
flexion as a couple . The definition of the
fuzzy linguistic description is performed through the
definition of the fuzzy meaning of each lexical term. This
meanings are fuzzy subsets in as shown in example in
Fig. 5..
.
Fig. 5. Meanings of the items of Sflexion.
The illustration of this new distance is presented with the
exemple of a finger flexion. Considering 2 numeric values of
finger flexion and their
fuzzy description are shown in fig. 6.
Fig. 6. Descriptions of f1 and f2.
The distance is arbitrary chosen as shown in
Table 1. It it represents a human knowledge about the
description of a finger flexion.
The distances to the two finger flexions f1 and f2 are
calculated for any other finger posture (fig. 7 and 8). On both
figs, five plates corresponding to the distance between each
term and f1 or f2. The value on each plate is directly connected
to the distance .
H I∩ ∅≠ dHI 0= dFG 0=
dtp F G,( ) 0=
H I∩ ∅=
dtp H I,( ) dtp H I,( ) dHI≥
dHI
dtp F G,( ) dFG≤
dtp F G,( ) dtp H I,( )≤
base sensorsmedial sensors
abduct sensors
thumb tip sensor
thumb medial sensor
thumb base sensor
palm arch
wrist yaw
wrist pitch
1
2
3
4
5
Folded Square Round Claw Straight
mcp ip,( ) ℜ2∈
ℜ2
Table 1: Distance défined on Sflexion.
Folded Claw Round Square Straight
Folded 0 1 2 3 4
Claw 1 0 1 2 3
Round 2 1 0 1 2
Square 3 2 1 0 1
Straight 4 3 2 1 0
IP
1
0.5
MCP0.5
1
1.5
2
0.5
1
1.5
2
Straight
Square
Round
Claw Folded
f1 0.5 0.5,( )= f2 1.2 1.0,( )=
Folded
1.0
Claw
Round
Square
Straight
D f1( )
0.25
Folded
Claw
Round
Square
Straight
D f2( )
0.250.5
dSflexion
dSflexion
dSflexion
dSflexion
Fig. 7. with .
In Fig. Fig. 8. the values of the plates corresponding to the
terms Square and Round are identical. This means that:
(23)
even if as shown in Fig.
Fig. 6.. This result is consistent because the transportation
distance takes all the terms into account. In this case f2 is a
little bit Straight: .
Fig. 8. with .
6. CONCLUSION
The formalization of the fuzzy subsets based description
process of quantities is a slow process that needs to be
performed in order to be able to manage this specific quantity
representations. With the transportation distance, this paper
gives a new tool for processing lexical fuzzy subsets issued
from a measurement process. With its four properties:
singleton coincidence, continuity, precision and constancy,
the transportation is a good candidate to perform signal
processing on this particular kind of representation issued
from a fuzzy description of measurement. In this paper the
distance betwen lexical term was issued from human
knowledge, but it will be possible to extract it from a metric
on physical states and from the fuzzy nominal scale. Then the
scale will be enhanced in order to bring a metric from the set
of physical states to the set of lexical fuzzy subsets.
REFERENCES
[1] Zadeh L.A., “Quantitative fuzzy semantics”, Information
Sciences, Vol. 3, 1971, pp. 159-176.
[2] Benoit E., Foulloy L., “Towards fuzzy nominal scales”,
Measurement, ISSN 0263-2241, Vol. 34, No. 1 (Fundamental of
Measurement), 2003, pp. 49-55.
[3] Benoit E., Foulloy L., Mauris G., “Fuzzy approaches for
measurement”, Handbook of Measuring Systems Design, ISBN:
0-470-02143-8, Vol. 1, Eds. Peter Sydenham and Richard Thorn,
John Wiley & Sons, 2005, pp. 60-67.
[4] Allevard T., Benoit E., Foulloy L., “Fuzzy nominal scale
approach for signal processing”, 10th IMEKO TC 7 Int. Symp.,
Vol. 2, Saint-Petersburg, Russia, June 2004, pp. 478-483.
[5] Allevard T., “Représentation symbolique de la configuration de
la main - application à la reconnaissance de signes et au contrôle
d'un robot mobile”, Phd Thesis, Université de Savoie, 2005.
[6] I. Bloch, “On fuzzy distances and their use in image processing
under imprecision”, Pattern Recognition, Vol. 32, 1999, pp.
1873-1895.
[7] R. Zwick, E. Carlstein, D.V. Budescu, “Measures of similarity
among fuzzy concepts: a comparative analysis”, International
Journal of Approximate reasoning, Vol. 1, 1987, pp. 351-356.
[8] Rachev S.T., Rüschendorf L., “Mass transportation problems”,
Springer-Verlag, New-York, 1998.
[9] T. Allevard, E. Benoit, L. Foulloy, Hand posture
recognition with the fuzzy glove, IPMU’04, Juillet 2004,
pp. 2139-2146.
[10]Wagner A.R., Fischer J.M., “The string to string correction
problem”, Journal of the Association for Computing Machinery,
Vol. 21, N° 1, 1974, pp. 168-173.
IP
4
MCP0.5
1
1.5
2
0.5
1
1.5
2
3
2
1
Straight
Square
Round
Claw
Folded
dtp D f1( ) D g( ),( ) f1 0.5 0.5,( )=
d D f2( ) Square{ },( ) d D f2( ) Round{ },( )=
µD f2( ) Round( ) µD f2( ) Square( )≥
µD f2( ) Straight( ) 0.25=
IP
0.5
1
1.5
2
0.5
1
1.5
2
3
2
1
MCP
1.2
Straight
Square
Round
Claw
Folded
dtp D f2( ) D g( ),( ) f2 1.2· 1.0,( )=


A new distance for fuzzy descriptions of measurements

Hal - Université Grenoble Alpes

HAL Université de Savoie

XVIII IMEKO WORLD CONGRESSMetrology for a Sustainable DevelopmentSeptember, 17 – 22, 2006, Rio de Janeiro, BrazilA NEW DISTANCE FOR FUZZY DESCRIPTIONS OF MEASUREMENTSThomas ALLEVARD, Eric BENOIT, Laurent FOULLOYLISTIC-ESIA, Université de Savoie, Annecy, France, eric.benoit@univ-savoie.frAbstract - Fuzzy nominal scales were introduced in orderto propose a formalism to the representation of empiricalquantities by fuzzy subsets of words. This paper presents theresults of studies on distances associated to this formalismand proposes a new distance operator.Keywords: Measurement theory, Fuzzy description,Fuzzy nominal scale, Fuzzy distance.1. INTRODUCTIONThe introduction of the fuzzy subset theory in themeasurement field takes its origin in 1971 in the Zadeh’spaper [1] that exposes a mechanism of description of aquantity by a fuzzy subset of symbols. Since this paper, thedefinition of the description process was mainly based ongood practices. Most of description processes had usefulproperties but it was sometime difficult to justify them.Recently, the link between the quantities and their fuzzyrepresentation was formalized in a scale approach named thefuzzy nominal scale [2][3].This approach gives the set of relations and operators thatcan be used to define equations on symbols such that theseequations have a meaning on the set of quantity values. Whena fuzzy description is a fuzzy nominal scale, a fuzzyequivalence relation on quantity values is linked to a fuzzyequivalence relation on their representation. This last relationalso named similarity relation is used to define a distancebetween the fuzzy subset of symbols that represent thequantity values. This distance had been used to perform signalprocessing [4], but is useless to compare symbols that are notrelated by the similarity relation.The purpose of this paper is to propose a distance on fuzzyrepresentations linked to a metric on the set of quantityvalues. With this new distance, a fuzzy nominal scale isenhanced and the set of authorized operators has now thisdistance as member.2. THE FUZZY SYMBOLISMSThe link between a physical state and its linguisticrepresentation is characterized by a symbolism defined by thetriplet <E,S,R> where E is the set of physical states, S is thelexical set used to represent measurement results and R is arelation on ExS. Two mappings can be extracted from thisrelation: The description mapping denoted D associates asubset of S to any item of E, and the meaning mappingdenoted M associates a subset of E to any item of S. These twomappings are linked with the following equation.(1)The R relation can be a fuzzy relation. Then, thetranslation of a physical state into its linguistic representationis called a fuzzy linguistic description mapping or simply afuzzy description mapping. It transforms an object e of the setof physical states E into a fuzzy subset of linguistic termscalled the fuzzy description of x. The dual mapping, called thefuzzy meaning mapping, associates a fuzzy subset of E to eachterm s of the lexical Set S. This fuzzy subset is the fuzzymeaning of s. In the paper the fuzzy subsets of linguistic termsalso named lexical fuzzy subsets are denoted LFS. This twomappings are also linked:(2)In [5] it is defined that <E,S,R> is a φ-symbolism if the setof the meanings of the elements of S is a φ-partition of E andif each meaning is normalized. This paper restricts itsinvestigation to id-symbolisms based on id-partition i.e. onRuspini partition. The set of all possible LFS obtained by afuzzy description based on id-symbolism is denoted Fid(S).Any LFS respects then the condition:(3)A fuzzy equivalence relation on the physical states can beassociated to any id-symbolism.(4)From this fuzzy equivalence relation and from the relationR, the following relation can be simply defined.(5)The symbolism <E,S,R> is then considered as a fuzzynominal scale.3. CHOICE OF A DISTANCE OPERATORThe relation used in the id-symbolism can define adistance between LFSs [4].(6)e∀ E∈ s∀ S∈ e, , M s( )∈ s D e( )∈⇔e s,( )∀ E S×∈ µM s( ) e( ) µD e( ) s( )=,A∀ Fid S( )∈ µA s( )s S∈, 1=x y,( )∀ E2∈ µ, ~ x y,( ) min µM s( ) x( ) µM s( ) y( ),( )s S∈=A B,( )∀ Fid S( )2∈ µ,~A B,( ) min µA s( ) µB s( ),( )s S∈=d~A B,( ) 1 - µ~A B,( )=This distance is discriminant for LFSs that are at leastpartially equivalent but is equal to 1 when 2 LFS have emptyintersection. This result is consistent with the absence ofdistance on the lexical set. This mean that the definition of ametric on the set Fid(S) needs the definition of a metric on theset S. Let dS be a distance defined on S.3.1. Required propertiesThe fuzzy subset theory proposes a large set of distanceoperators and the best way to select a distance operator is tolist the properties that must be verified.• The first property is the singleton coincidence: If twoLFSs are singleton {s1} and {s2} then the distancebetween them is equal to the distance between symbols s1and s2. This property supposes that the distance dS on Sexists.• The continuity property is verified when the distance is acontinuous mapping from SxS to the set of positive num-bers.• The precision property simply imposes that the distancebetween two LFSs must be a positive real number, andnot a fuzzy subset of positive real numbers.• The consistency property is usually verified by distanceson crisp subsets: If A,B,C,E are four subsets of a metricspace, d is the distance on this space, and dg is a distancethat generalize d on subsets, it verifies:(7)Fig. 1. Consistency propertyThe extension of a distance defined on a finite space to adistance defined on the set of the fuzzy subsets of this spacewas widely studied but, as shown below, no existing distancecan be applied to Fid(S).Distances on fuzzy subsets can by classified into thefollowing categories.• The distances that generalise an existing distance.• The distances defined from a similarity measure.• The distances defined with subset operators.• The distances computed from a symbolic approach.In our approach, a distance dS is supposed to be defined onS. Then only the first cathegory is investigated.The generalisation of a distance dS defined on a finite setS, to a distance dF(S) defined on the set of fuzzy subsets of Sis a recurrent subject of study. In [6] Bloch proposes fourtypes of generalisation.3.2. The geometrical approachIn this approach, fuzzy subsets in a ndimentional space areconsidered as crisp subsets in a (n+1)dimentional space. Thismeans that the distance between membership degrees has thesame semantic than distance on the ndimentional space. Suchhypothesis can not be justified in our problem and thisapproach is not kept.3.3. The fuzzification approachIn an other approach a distance DS between crisp subsetsis defined from the distance dS. Then the distance DS isfuzzyfied. In [7] three fuzzyfications of the Hausdorffdistance are proposed.(8)(9)(10)Where Fα and Gα are the alpha-cuts of F and G, and HS isthe Hausdorff distance:(11)and do not verify the continuityproperty, but does. It also verifies the singletoncoincidence, but not the consistency property.3.4. The weighting approachThe distance dS can be generalized with a weighting ofmembership degrees.(12)Where T is a continuous triangular norm.Such operator does not respect the separation axiom thatimposes:then it cannot be considered as a distance.3.5. A new approachSo a new distance that respects the four properties hadbeen created. This distance is named the transportationdistance dtp. Its calculation is equivalent to the solution of amass transportation problem [8]. It is similar to theWasserstein distance used in probability theory, and can alsobe considered as a fuzzy version of the Levenshtein distanceused to compare strings [10].supu A v B∈,∈d u v,( ) infx C y E∈,∈d x y,( )≤dg A B,( ) dg C E,( )≤infx C y E∈,∈d x y,( )EC supu A v B∈,∈d u v,( )A Bdg A B,( ) dg C E,( )≤H1FId S( )F G,( ) HS Fα Gα,( ) αd01=H∞FId S( ) F G,( ) supα 0≥ HS Fα Gα,( )=H*FId S( ) F G,( ) HS F1.0 G1.0,( )=HS A B,( ) max maxa A∈ minb B∈ dS a b,( ) maxb B∈ mina A∈ dS a b,( ),( )=H∞FId S( ) H*FId S( )H1FId S( )dT F G,( ) T µF s1( ) µG s2( ),( )dS s1 s2,( )s2 S∈s1 S∈=dT F G,( ) 0 F⇔ G= =4. THE TRANSPORTATION DISTANCEThe transportation distance between two LFSs is based onthe cost calculation of a set of transformations needed totransform the first LSF to the other. First a family oftransformation mappings is defined:Let be a mapping on a set Fid(S) such that:(13)Fig. 2. The mapping with S={s1,s2,s3,s4}.We demonstrate that any element of Fid(S) can betransformed into any other element of Fid(S) with the use of asequence of such transformation mapping.Let the following sets:Proposition 1: Let S be a finite set. Let F and G be 2elements of the set Fid(S). Let ∆s = µF(s) - µG(s). Thefollowing equality is verified:(14)Then the definition of the sequence of transformationmapping is equivalent to the well known linear programmingproblem named the transportation problem [8]. The problemis to bring a product from a set of n1 sources to a set of n2destinations. Each source i gives a quantity si of product, andeach destination receives a quantity sj of product. The totalgiven quantity must be equal to the total received quantity:(15)A solution is to associate to each displacement from asource i to a destination j a quantity xij of transported productand a displacement unity cost cij. The aim is to find a solutionthat minimises the total cost:(16)Considering the membership degrees as the transportedproduct, the set as the set of sources and the setas the set of destinations, a distance can be computed as thetotal cost of the optimal solution for the transportation ofmembership degrees.The transportation distance dtp is defined on Fid(S) fromdistance dS on the lexical set S. The distance dtp is the sum ofthe costs of each transformation mapping. And the cost of atransformation mapping is equal to: .It is now shown that the transportation distance is adistance, and it verifies the 4 constraints presented before.For any , dtp must verify:(17)(18)(19)• The relation is equivalent tothat is equivalent to.• The symmetry of is deduced from the symmetry ofthe transportation problem.• Finally, is by definition the distance corre-sponding to the optimal sequence of transformationsthat changes F into H. Then Adding a constraintin order to include G in the set of transformation stepswill increase the distance.• Calculating the distance between singletons andusing the transportation problem is equivalent tofinding the cheaper solution to bring a unity quantity ofproduct from source i to destination j. The solution ismade of only one transformation mapping . Thecost of this solution is that is equal to the distancethen the singleton coincidence is verified.• The precision and the continuity properties are deducedfrom the definition of the distance• The consistency property of dtp is demonstrated below:Let four elements of andsuch that:(20)where:(21)and supp(A) is the support of A i.e. the set of lexical termss such that .It must be shown that. (22)Tsi sj x, ,G Tsi sj x, , F( )µG si( ) µF si( ) x–=µG sj( ) µF sj( ) x+=⇔=Ts1 s2 0.2, ,SF G> s S∈ µF s( ) µG s( )>,{ }=SF G= s S∈ µF s( ) µG s( )=,{ }=∆ss SF G>∈ ∆ss SG F>∈–=sii 1=n1 sj′j 1=n2–=xijciji j,SF G> SF G<Tsi sj x, ,x dS si sj,( )⋅F G H, , FId S( )∈dtp F G,( ) 0= F⇔ G= ,dtp F G,( ) dtp G F,( )= ,dtp F G,( ) dtp G H,( )+ dtp F H,( )≥ .F G=SF G> SF G< ∅= =dtp F G,( ) 0=dtpdtp F H,( )Tsi sj x, ,li{ }lj{ }Tsi sj 1, ,cijdS si sj,( )F G H I, , , FId S( )s1 s2 s3 s4, , ,( ) S4∈dFG dHI≤ .dHI infs3 s4,( ) supp H( ) supp I( )×∈dSs3 s4,( )= ,dFG sups1 s2,( ) supp F( ) supp G( )×∈dSs1 s2,( )= .µA s( ) 0≠dtp F G,( ) dtp H I,( )≤If , then and . F and G arethe same singleton and . (22) is triviallyverified.If , The brought quantity associated to thecalculation of is equal to 1 andbecause the transportation distance is then a weighted averageof distances that are greater or equal to . With the samereasoning, . Then, (20) induces.5. RESULTSIn this section, the transportation distance is applied on thehand posture recognition. More details on the application canbe found in [9].Fig. 3. : The 18 sensors of the Cyberglove®.The hand posture is acquired with the 18 angle sensor of aCyberGlove® (Fig. 3.). The finger flexion (except for thethumb) is aquired with two angle sensors : MCP (métacarpal-phalanx angle) et IP (interphalanx angle). The linguisticdescription of a finger uses the set Sflexion = {Folded, Claw,Round, Square, Straight} see Fig. 4.Fig. 4. Words used to describe the finger flexion.The dataglove gives a numeric representation of the fingerflexion as a couple . The definition of thefuzzy linguistic description is performed through thedefinition of the fuzzy meaning of each lexical term. Thismeanings are fuzzy subsets in as shown in example inFig. 5...Fig. 5. Meanings of the items of Sflexion.The illustration of this new distance is presented with theexemple of a finger flexion. Considering 2 numeric values offinger flexion and theirfuzzy description are shown in fig. 6.Fig. 6. Descriptions of f1 and f2.The distance is arbitrary chosen as shown inTable 1. It it represents a human knowledge about thedescription of a finger flexion.The distances to the two finger flexions f1 and f2 arecalculated for any other finger posture (fig. 7 and 8). On bothfigs, five plates corresponding to the distance between eachterm and f1 or f2. The value on each plate is directly connectedto the distance .H I∩ ∅≠ dHI 0= dFG 0=dtp F G,( ) 0=H I∩ ∅=dtp H I,( ) dtp H I,( ) dHI≥dHIdtp F G,( ) dFG≤dtp F G,( ) dtp H I,( )≤base sensorsmedial sensorsabduct sensorsthumb tip sensorthumb medial sensorthumb base sensorpalm archwrist yawwrist pitch12345Folded Square Round Claw Straightmcp ip,( ) ℜ2∈ℜ2Table 1: Distance défined on Sflexion.Folded Claw Round Square StraightFolded 0 1 2 3 4Claw 1 0 1 2 3Round 2 1 0 1 2Square 3 2 1 0 1Straight 4 3 2 1 0IP10.5MCP0.511.520.511.52StraightSquareRoundClaw Foldedf1 0.5 0.5,( )= f2 1.2 1.0,( )=Folded1.0ClawRoundSquareStraightD f1( )0.25FoldedClawRoundSquareStraightD f2( )0.250.5dSflexiondSflexiondSflexiondSflexionFig. 7. with .In Fig. Fig. 8. the values of the plates corresponding to theterms Square and Round are identical. This means that:(23)even if as shown in Fig.Fig. 6.. This result is consistent because the transportationdistance takes all the terms into account. In this case f2 is alittle bit Straight: .Fig. 8. with .6. CONCLUSIONThe formalization of the fuzzy subsets based descriptionprocess of quantities is a slow process that needs to beperformed in order to be able to manage this specific quantityrepresentations. With the transportation distance, this papergives a new tool for processing lexical fuzzy subsets issuedfrom a measurement process. With its four properties:singleton coincidence, continuity, precision and constancy,the transportation is a good candidate to perform signalprocessing on this particular kind of representation issuedfrom a fuzzy description of measurement. In this paper thedistance betwen lexical term was issued from humanknowledge, but it will be possible to extract it from a metricon physical states and from the fuzzy nominal scale. Then thescale will be enhanced in order to bring a metric from the setof physical states to the set of lexical fuzzy subsets.REFERENCES[1] Zadeh L.A., “Quantitative fuzzy semantics”, InformationSciences, Vol. 3, 1971, pp. 159-176.[2] Benoit E., Foulloy L., “Towards fuzzy nominal scales”,Measurement, ISSN 0263-2241, Vol. 34, No. 1 (Fundamental ofMeasurement), 2003, pp. 49-55.[3] Benoit E., Foulloy L., Mauris G., “Fuzzy approaches formeasurement”, Handbook of Measuring Systems Design, ISBN:0-470-02143-8, Vol. 1, Eds. Peter Sydenham and Richard Thorn,John Wiley & Sons, 2005, pp. 60-67.[4] Allevard T., Benoit E., Foulloy L., “Fuzzy nominal scaleapproach for signal processing”, 10th IMEKO TC 7 Int. Symp.,Vol. 2, Saint-Petersburg, Russia, June 2004, pp. 478-483.[5] Allevard T., “Représentation symbolique de la configuration dela main - application à la reconnaissance de signes et au contrôled'un robot mobile”, Phd Thesis, Université de Savoie, 2005.[6] I. Bloch, “On fuzzy distances and their use in image processingunder imprecision”, Pattern Recognition, Vol. 32, 1999, pp.1873-1895.[7] R. Zwick, E. Carlstein, D.V. Budescu, “Measures of similarityamong fuzzy concepts: a comparative analysis”, InternationalJournal of Approximate reasoning, Vol. 1, 1987, pp. 351-356.[8] Rachev S.T., Rüschendorf L., “Mass transportation problems”,Springer-Verlag, New-York, 1998.[9] T. Allevard, E. Benoit, L. Foulloy, Hand posturerecognition with the fuzzy glove, IPMU’04, Juillet 2004,pp. 2139-2146.[10]Wagner A.R., Fischer J.M., “The string to string correctionproblem”, Journal of the Association for Computing Machinery,Vol. 21, N° 1, 1974, pp. 168-173.IP4MCP0.511.520.511.52321StraightSquareRoundClawFoldeddtp D f1( ) D g( ),( ) f1 0.5 0.5,( )=d D f2( ) Square{ },( ) d D f2( ) Round{ },( )=µD f2( ) Round( ) µD f2( ) Square( )≥µD f2( ) Straight( ) 0.25=IP0.511.520.511.52321MCP1.2StraightSquareRoundClawFoldeddtp D f2( ) D g( ),( ) f2 1.2· 1.0,( )=

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A new distance for fuzzy descriptions of measurements

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