Type-1 OWA Unbalanced Fuzzy Linguis ic Agg ega ion
Me hodology. Applica ion o Eu obonds C edi Risk E alua ion
F ancisco Chiclana∗
Cen e o Compu a ional In elligence, De Mon o Uni e si y, Leices e , UK
Email: [email protected]
F ancisco Ma a, Luis G. P´e ez
Depa men o Compu e Science, Uni e si y o Ja´en, Spain
Email: [email p o ec ed]; [email p o ec ed]
En ique He e a-Viedma
Depa men o Compu e Science, Uni e si y o G anada, Spain
Email: [email p o ec ed]
June 1, 2017
Abs ac
In decision making, a widely used me hodology o manage unbalanced uzzy linguis ic in o ma ion is
he linguis ic hie a chy (LH), which elies on a linguis ic symbolic compu a ional model based on o dinal
2- uple linguis ic ep esen a ion. Howe e , he o dinal 2- uple linguis ic app oach does no exploi all
ad an ages o Zadeh’s uzzy linguis ic app oach o model unce ain y because he membe ship unc ion
shapes a e igno ed. Fu he mo e, he LH me hodology is an indi ec app oach ha elies on he uni o m
dis ibu ion o symme ic linguis ic assessmen s. These d awbacks a e o e come by applying a uzzy
me hodology based on he implemen a ion o he Type-1 O de ed Weigh ed A e age (T1OWA) ope a o .
The T1OWA ope a o is no a symbolic ope a o and i allows o di ec ly agg ega e membe ship unc ions,
which in p ac ice means ha he T1OWA me hodology is sui able o bo h balanced and unbalanced
linguis ic con ex s and wi h he e ogeneous membe ship unc ions. Fu he mo e, he inal ou pu o he
T1OWA me hodology is always uzzy and de ined in he same domain o he o iginal unbalanced uzzy
∗Co esponding au ho
1
linguis ic labels, which acili a es i s in e p e a ion ia a isual join ep esen a ion. A case s udy is
p esen ed whe e he T1OWA ope a o me hodology is used o assess he c edi wo hiness o Eu opean
bonds based on eal c edi isk a ings o indi idual Eu ozone membe s a es modelled as unbalanced
uzzy linguis ic labels.
Keywo ds: T1OWA ope a o , Linguis ic hie a chy, Unbalanced uzzy linguis ic assessmen s, C edi quali y.
1 In oduc ion
In mos decision making p ocesses he e exis s unce ain y conce ning he sui abili y o each one o al e na i es
o choose om. Ma hema ically, unce ain y has been ackled using p ecise nume ic assessmen alues o
using linguis ic assessmen alues in bo h i s ep esen a ion and measu emen . The second app oach, hough,
happens when expe s’ sensa ions and eelings pe ades he decision making p oblem.
The uzzy linguis ic me hodology, in oduced by Zadeh in his seminal pape ,28 has p o ed o be use ul
in p o iding a ma hema ical s uc u ed amewo k o deal wi h decision making p oblems wi h agueness
and imp ecise pe ading he in o ma ion a ailable, i.e. when p ecise nume ic assessmen s a e no a ailable
bu linguis ic assessmen s a e ins ead. Fo hese ype o decision making p oblems, adi ionally ca ego ised
as uns uc u ed, can indeed be applied an s uc u ed me hodology based on he implemen a ion Zadeh’s
concep o linguis ic a iable and i s seman ics o desc ibe he meaning o each one o he elemen s o he
conside ed uni e se o discou se, which is done using uzzy se s membe ship unc ions. An impo an aspec
o be aken in o conside a ion wi hin a linguis ic me hodology is ca dinali y o he co esponding linguis ic
e m se ,1as he highe he ca dinali y is, he highe he unce ain y disc imina ion among he elemen s o
he uni e se o discou se is achie ed.
I is a common p ac ice in decision making p oblems wi h linguis ic assessmen s o assume linguis ic
e m se s wi h uni o m dis ibu ion o symme ic linguis ic assessmen on he discou se domain. Clea ly
his app oach may be app op ia e o p oblems whe e he dis inc ion o unce ain y is p opo ional and equal
among he se o linguis ic e ms, bu no whe e his may no be he case. A ypical example o his la e case
is gi en in14 o desc ibing he UK educa ional g ading sys em (see Fig. 1). Clea ly, he igh -hand side o
he scale has mo e e ms han he le -hand side and consequen ly a iangula uzzy se ep esen a ion o he
seman ic o each assessmen can only be cap u ed wi h non-uni o m dis ibu ion o non-symme ic linguis ic
labels, which in li e a u e has been named as unbalanced linguis ic ep esen a ion o in o ma ion.2,14,18
A me hodology p oposed in li e a u e and widely used o add ess decision making p oblems wi h un-
balanced linguis ic in o ma ion is he linguis ic hie a chy me hodology (LH) in oduced in6and la e ap-
plied o imp o e he p ecision in p ocesses o compu ing wi h wo ds in mul i-g anula linguis ic con ex s in
2
FDCBA
FDABC
Figu e 1: Seman ic ep esen a ion o he UK educa ional g ading sys em
in.9–11,17, 21 The agg ega ion o unbalanced linguis ic in o ma ion using he LH me hodology was p esen ed
in.14 In summa y, he LH me hodology consis s o building a ep esen a ion s uc u e wi h se e al le els,
each one ep esen ing a di e en g anula i y se o uni o m and symme ic linguis ic e ms ha keeps he
p eceden le el modal poin s in o de o achie e a smoo h ansi ion be ween successi e le els. T ans o ma-
ion unc ions a e in oduced o map linguis ic labels o a le el o linguis ic labels a a di e en le el wi hou
loss o in o ma ion. Doing his, he unbalanced linguis ic labels a e mapped wi h i s app op ia e symme ic
linguis ic labels wi hin he s uc u e, and a e ans o med o a common domain wi h maximum g anula i y,
which ul ima ely a e agg ega ed using he 2- uple linguis ic compu a ional model. Thus, he LH me hod-
ology deals wi h unbalanced linguis ic in o ma ion using an indi ec app oach ia he al eady common and
known uni o m dis ibu ion o symme ic linguis ic assessmen on he uni e se o discou se.
The LH me hodology elies on a linguis ic symbolic compu a ional model based on o dinal scales and
indexes, he 2- uple linguis ic ep esen a ion, and he e o e i does no exploi he ad an ages o Zadeh’s
uzzy linguis ic app oach. To a oid his issue, an al e na i e app oach o p ocess unbalanced linguis ic
in o ma ion is possible by using he Type-1 O de ed Weigh ed A e age (T1OWA) ope a o ,31 which was
de eloped applying Zadeh’s ex ension p inciple o Yage ’s OWA ope a o .27 The T1OWA ope a o is no
a symbolic ope a o ; i allows o di ec ly agg ega e he whole linguis ic e ms because i s compu a ion
in ol es he whole membe ship unc ions o he uzzy se s used o app op ia ely ep esen he meaning o
he linguis ic e ms, which in p ac ice means ha i can be sui able o bo h balanced and unbalanced
linguis ic con ex s and wi h he e ogeneous ypes o membe ship shapes ( iangula , apezoidal, gaussian,
e c.). As a consequence, he ou pu o he T1OWA ope a o is o he same ype han he linguis ic e ms,
i.e. a uzzy se on he same uni e se o discou se. The T1OWA ope a o has been success ully applied o
agg ega e uzzy linguis ic in o ma ion wi h uzzy linguis ic weigh s5, 32 and o add ess consensus eaching
p ocesses in mul i-g anula uzzy linguis ic con ex s.3,20 Thus, he T1OWA ope a o is mos app op ia e
o be implemen ed in decision making p oblems whe e unce ain y is linked o uzzy se heo y a he han
p obabili y heo y,24–26 and in pa icula o con ex s wi h unbalanced uzzy linguis ic in o ma ion. This is
3
he ocus o he p esen pape , which aims o p esen a T1OWA based me hodology o deal wi h decision
making p oblems wi h unbalanced uzzy linguis ic in o ma ion by using as example he s udy o c edi isk
on a po en ial Eu obonds a ing based on eal c edi isks o Eu ozone membe s a es as opposed o p e ious
e o based on mock examples.32
C edi isk e alua ion o co po a ions o he deb issuance o a s a e o go e nmen is usually ca ied ou
by a ing agencies, wi h he h ee big ones being S anda d & Poo ’s (S&P), Moody’s and Fi ch G oup. Each
agency u ilises i s own me hodology and c i e ia o measu e he c edi wo hiness o co po a ions e alua ed and
i s own scale based on a combina ion o le e s, numbe s and/o posi i e and nega i e signs o assess he o e all
c edi isk le el o he co po a ion o s a e. Ra ing agencies ely on economic expe s, ma hema ical models
o a combina ion o bo h o a i e a hei inal c edi isk assessmen . Financial in o ma ion is ob ained om
bo h public and p i a e ins i u ions as well as om expe s wi h g ea knowledge and expe ience. Al hough
a ing agencies wo k wi h in o ma ion ha is qui e p ecise, i is ob ious ha he e also exi economic
ac o s ou side hei con ol ha gene a e unce ain y ega ding hei ecommenda ions and e alua ions.
The unce ain y ha a ises when expe s analyse all he a ailable economic in o ma ion may make mo e
di icul he p ecise assessmen o c edi isk. Indeed, c edi isk assessmen s end o include in ui ions and
eelings o expe s ha emana e om he men ioned unce ain y. Thus, he e a e well ounded g ounds
o suppo he use o uzzy linguis ic app oaches in his con ex . The in o ma ion can be associa ed o
unbalanced linguis ic labels whose meanings can be ep esen ed using uzzy se membe ship unc ions.
The s uc u e o he es o he pape is as ollows: Sec ion 2 e iews succinc ly he basic concep a
linguis ic a iable and i s seman ics as well as he 2- uple LH me hodology. Sec ion 3 p esen s a new uzzy
al e na i e o manage unbalanced uzzy linguis ic in o ma ion based on he T1OWA ope a o . A de ailed
desc ip ion o i s exp ession o agg ega ion uzzy se s is gi en in Sec ion 3.2, while Sec ion 3.3 p esen s an
example o agg ega ion o unbalanced linguis ic labels using he T1OWA ope a o and applied o assess he
c edi wo hiness o Eu opean bonds based on eal c edi isk a ings o indi idual Eu ozone membe s a es.
The pape is closed wi h Sec ion 4 whe e conclusions a e d awn.
2 Unbalanced linguis ic labels: he indi ec o dinal 2- uple LH
app oach
In his seminal pape published in 1996,29 Zadeh explici ly s a ed ha he a ionale o compu ing wi h wo ds
(CWW) migh be suppo ed by a necessi y when numbe s a e no able o be used o quan i y he imp ecision
o he in o ma ion a ailable; o by a ole ance o imp ecision ha allows o wo ds ins ead o numbe s, which
4
migh be cos ly o ge . La e in30 an addi ional a ionale was added when wo ds a e simply used o summa ise
nume ical in o ma ion.
In CWW, he wo ds a e modelled in o well-de ined ma hema ical objec s, which in u n a e manipula ed
wi h sound ma hema ical compu a ional ools. Indeed, wo ds in CWW a e conside ed labels o uzzy se s
wi h speci ied membe ship unc ions, which a e compu a ionally manipula ed using uzzy a i hme ics, i.e.
adi ional ma hema ical a i hme ics ans o med ia he ex ension p inciple.
Linguis ic a iables a e employed ex ensi ely in applica ions o uzzy logic, and hey a e o mally ep e-
sen ed as a 5- uple hL, T(L), U, S, Mi28 whe e: (i) Lis he name o he a iable; (ii) T(L) is a ini e e m
se o (p ima y) labels o wo ds (a collec ion o linguis ic alues); (iii) Uis a uni e se o discou se o base
a iable; (i ) Sis he syn ac ic ule which gene a es he e ms in T(L); and ( ) Mis a seman ic ule which
associa es wi h each linguis ic alue Xi s meaning M(X) : U→[0,1]. Usually, T(L) is deno ed as Lwhen
he e is no isk o con usion. A ‘compa ibili y unc ion’28 o seman ic ule associa es wi h each elemen o he
base a iable i s compa ibili y wi h each linguis ic alue. This in e p e a ion o he meaning o a linguis ic
label coincides wi h ha o a uzzy se , and as men ioned abo e linguis ic labels a e o mally ep esen ed as
uzzy subse s o hei base a iable.
A e y popula app oach o ep esen and agg ega e linguis ic in o ma ion is using a linguis ic symbolic
compu a ional model based on indexes,4,7, 8 which is based on an o dinal in e p e a ion o he linguis ic label
meaning. In,15 a mo e gene al symbolic app oach was in oduced: he 2– uple linguis ic model, which up
o now has been used as he LH me hodology compu a ional model o unbalanced linguis ic in o ma ion.
Sec ions 3.2 and 3.3 will p esen a uzzy compu a ional app oach o unbalanced linguis ic in o ma ion based
on he T1OWA ope a o .
2.1 O dinal linguis ic ep esen a ion using he 2- uple linguis ic model
This linguis ic model adds he concep o symbolic ansla ion o he symbolic ep esen a ion model based on
indexes, which is used o ep esen he ou pu o symbolic agg ega ion ope a o s by means o a pai o alues
called linguis ic 2– uple: (si, αi), wi h sibeing one o he o iginal linguis ic e ms (i.e. si∈S={s0, ..., sg})
and αi∈[−.5, .5) is he symbolic ansla ion. The aim o his ep esen a ion s uc u e is o achie e ha he
symbolic agg ega ion ou pu is iden ical o he one ob ained using he symbolic ep esen a ion model based
on indexes while a he same ime p e en ing loss o in o ma ion by making use o in o ma ion p e iously
disca ded by such symbolic ep esen a ion model.
Fo mally, le β∈[0, g] be he esul o a symbolic agg ega ion o he indexes o a se o labels in a
linguis ic e m se S={s0, ..., sg}, and i= ound(β)∈ {0, . . . , g}. The alue αi=β−i∈[−0.5,0.5) is
5

called a symbolic ansla ion, and he pai o alues (si, αi) is called he 2– uple linguis ic ep esen a ion
model. Thus, he ollowing isomo phism can be es ablished be ween he 2- uple se associa ed wi h S,
hSi=S×[−0.5,0.5), and he closed in e al [0, g]:
∆(β) = (si, α),wi h 




i= ound(β),
α=β−i,
The in e se unc ion is ∆−1(si, α) = i+α, and he co esponding symbolic compu a ional model was p esen ed
in.16
2.2 The o dinal 2- uple linguis ic hie a chy
A LH may be seen as a hie a chy s uc u e o di e en le els o linguis ic e m se s wi h di e en g anula i y,
which a e deno ed as l( ,n( )) wi h ep esen ing he LH le el and n( ) he g anula i y o he linguis ic e m
se a ha le el. Assump ion a e ha he ca dinali y o all linguis ic e m se s is odd, and g aphically a e
ep esen ed using symme ical and uni o m dis ibu ed iangula membe ship unc ions on he domain [0,1]
as Fig. 2 shows. Bo h he p ocess o build a LH and i s compu a ional model a e explained below.
Building linguis ic hie a chies. The g anula i y o each linguis ic e m se is inc eased om one le el ( )
o he nex ( +1) using he ollowing exp ession17 and as illus a ed in Fig. 2:
l( , n( )) →l( + 1,2·n( )−1).
Figu e 2: LH wi h ou le els o g anula i y 3, 5, 9 and 17, espec i ely
An issue associa ed o his app oach is ha he g anula i y o le els inc eases e y apidly, which has
6
been pa ially esol ed applying he leas common mul iple app oach o all g anula i ies o he LH as
i was p oposed in.12,13
Compu a ional model. The LH compu a ional model is based on he ollowing symbolic 2- uple ans o -
ma ion,17
TF
0:l( , n( )) −→ l( 0, n( 0))
TF
0(sn( )
i, αn( ))=∆ ∆−1(sn( )
i, αn( ))·(n( 0)−1)
n( )−1!.
The aim o such ans o ma ion unc ion is ha linguis ic e ms, independen ly o i s shape and seman-
ic, can be mapped o a unique exp ession domain, and consequen ly be amenable o be manipula ed
wi h he 2- uples compu a ional model. Ob iously, his app oach dis ega d he membe ship unc ions
and so linguis ic labels a e modelled ia hei co esponding symbolic o dinal 2- uple ep esen a ions
and no ea ed as uzzy se s.
Unbalanced linguis ic in o ma ion is managed wi hin he LH me hodology and 2- uple compu a ional
model by i s di iding he unbalanced linguis ic e m se in o h ee e m subse s, he one con aining all
labels below he cen al one (le la e al se ), he one con aining all he labels abo e he cen al one ( igh
la e al se ) and he one con aining he cen al label (cen al se ). In a second s ep, he g anula i ies o he
le la e al se and he igh la e al se a e compa ed agains he (hal ) he g anula i y alue o each LH
le el so ha he closes symme ical and uni o m dis ibu ed LH labels is ound o ep esen he unbalanced
linguis ic in o ma ion. A e his mapping has been comple ed, he symbolic agg ega ion based on he 2- uple
compu a ional model is applied o p ocess he in o ma ion, wi h i s ou pu being inally e ansla ed in o he
o iginal unbalanced linguis ic e m se .
3 Unbalanced uzzy linguis ic labels: he di ec T1OWA app oach
In his sec ion, a uzzy app oach o manage unbalanced uzzy linguis ic in o ma ion will be p esen ed based
on he use o T1OWA ope a o . The ad an ages o using his ou e a e: (i) i is uzzy and no o dinal
because he membe ship unc ion cha ac e ising he uzzy linguis ic labels a e ully used in he compu a ion
p ocess; (ii) he shape o he membe ship unc ion is no es ic ed o be iangula ype bu could be o any
ype; (iii) he e is no need o ansla e and e ansla e unbalanced in o ma ion using an indi ec balanced
amewo k, i.e. i is a di ec ca dinal app oach o dealing wi h unbalanced in o ma ion compa ed o he
indi ec o dinal 2- uple LH me hodology; (i ) he inal ou pu will be a uzzy se on he same domain han
he o iginal unbalanced uzzy linguis ic labels and i can be in e p e ed easily when compa ed wi h hem
7
g aphically. I necessa y, a de uzzi ica ion p ocess could be applied, o example by compu ing he cen oid
o he solu ion uzzy se o using he 2- uple ep esen a ion model. Anyway i is p o ed ha an equi alen
se o alues o he co esponding 2- uple ep esen a ion app oach is ob ained.
3.1 Fuzzy linguis ic ep esen a ion model
The ep esen a ion o linguis ic in o ma ion using uzzy numbe s, i.e con ex no mal uzzy subse s o he
eal line, is commonly e e eed o as he ca dinal ep esen a ion in con as o he o dinal ep esen a ion
co e ed abo e. In his amewo k, a linguis ic label is cha ac e ised by a membe ship unc ion on he uni
in e al [0,1] ha maps each alue in [0,1] o a deg ee o pe o mance ep esen ing i s compa ibili y wi h
he linguis ic assessmen ,28 examples o which a e gi en in Fig. 1.
I is no di icul o see ha he e exis s a one- o-one mapping be ween he o dinal app oach based on he
2- uple ep esen a ion o a e m se o linguis ic labels and he se o cen oid elemen s o he uzzy numbe s
used in he ca dinal ep esen a ion o he same e m se o linguis ic labels.22 Indeed, deno ing he cen oid
o he linguis ic e m sh∈Sby (sh), he seman ic o he linguis ic labels unde lies a anking ela ion ha
implies (lh)> (lk) when h > k. Wi hou loss o gene ali y i can be assumed ha (s0) = 0 and (sg) = 1,
o he wise he cen oids a e eplaced by he alues (sh)− (s0)
(sg)− (s0). Deno ing he symbolic 2– uple ep esen a ion
o shby ah= ∆−1((sh,0)), he mapping
δ(ah) = (lh).(1)
is he es ic ion o a con inuous and s ic ly inc easing unc ion δ: [0, s]−→ [0,1] such ha δ(0) = 0 and
δ(s) = 1, i.e. a bijec i e unc ion δexis s and i can be used o de i e he o dinal 2- uple ep esen a ion o
a linguis ic e m se om he se o cen oids o he uzzy numbe s used in a ca dinal ep esen a ion o he
same linguis ic e m se . Ob iously, i is no possible o de i e a ca dinal ep esen a ion o a linguis ic e m
se om an o dinal 2- uple ep esen a ion model. Fu he mo e, he ype o membe ship unc ion used in he
ca dinal ep esen a ion model is no es ic ed o iangula ype bu could be apezoidal o gaussian ype,
i.e. i could be o any ype as long as i is con ex and no mal e i ying ha (lh)> (lk) when h>k. Thus,
he ca dinal uzzy app oach o linguis ic in o ma ion ep esen a ion is gene al, lexible and app op ia e o
cap u e unce ain y, which is no he case wi h he o dinal app oach.
3.2 The T1OWA ope a o
In con as o Yage ’s OWA ope a o 27 ha is able o agg ega e c isp numbe s wi h c isp weigh s, he T1OWA
ope a o was in oduced in31 o di ec ly agg ega e uzzy se s wi h unce ain y weigh s. Thus, gi en a se
{A1,· · · , An}o ype-1 uzzy se s on R ha a e o be agg ega ed using he se o ype-1 uzzy weigh s se s
8
de ined on he domain o discou se [0,1], {W1,· · · , W n}, he T1OWA ope a o ou pu is a uzzy se Y:
Φ(A1,· · · , An) = Y
wi h membe ship unc ion
µY(y) = sup
n
X
k=1
¯wiaσ(i)=y
wi∈U, ai∈X
µW1(w1)∧ · · · ∧ µWn(wn)∧µA1(a1)∧ · · · ∧ µAn(an)(2)
whe e ¯wi=wi
Pn
i=1 wi
;σis a pe mu a ion unc ion such ha aσ(i)≥aσ(i+1),∀i= 1,· · · , n −1.
Exp ession (2) has been p o ed o be oo expensi e om a compu a ional poin o iew, wha ine i ably
implied ha i s p ac ical applica ion in eal wo ld decision making p oblems was cu ailed. This issue,
howe e , was o e come wi h he de elopmen o a as app oach o T1OWA ope a ions based on he ho izon al
ep esen a ion o a uzzy se s ia hei co esponding amily o c isp α-le el se s, in wha i is known as he
ep esen a ion heo em o uzzy se s.28
Fo each α∈[0,1], he α-le el T1OWA ope a o o agg ega es he α-le el se s A1
α,· · · , An
αwi h α-le el
weigh se s W1
α, . . . , Wn
αis is gi en as
ΦαA1
α,· · · , An
α=






n
P
i=1
wiaσ(i)
n
P
i=1
wi
wi∈Wi
α, ai∈Ai
α,∀i






(3)
whe e Wi
α={w|µWi(w)≥α},Ai
α={x|µAi(x)≥α}, and σis a pe mu a ion unc ion such ha aσ(i)≥
aσ(i+1),∀i= 1,· · · , n −1.
Acco ding o he Rep esen a ion Theo em o ype-1 uzzy se s, he ollowing ype- uzzy se on Rcan be
cons uc ed:
G=∪
0<α≤1αΦαA1
α,· · · , An
α(4)
wi h membe ship unc ion
µG(x) = ∨
α:x∈Φα(A1
α,··· ,An
α)α
α(5)
Fuzzy se s Yand G, which appa en ly seem o be di e en , we e p o ed in32 o ha e he same membe ship
unc ions and consequen ly a e equal. This undamen al esul is known as he Rep esen a ion Theo em o
Type-1 OWA Ope a o s. Fu he mo e, his α-le el app oach was p o ed o be much as e han (2),32 which
9
4 Conclusions
A uzzy app oach o di ec ly use unbalanced linguis ic in o ma ion based on he T1OWA ope a o has been
p esen ed. In compa ison o he exis ing app oach o unbalanced linguis ic in o ma ion, he o dinal 2- uple
LH me hodology, i is wo h no ing he ollowing: (i) i allows o a so in e p e a ion o he linguis ic
in o ma ion, implemen s and makes use o he whole membe ship unc ions cha ac e ising he linguis ic label
as uzzy se s; (ii) he shape o he membe ship unc ion is no es ic ed o be iangula ype; (iii) he e is
no need o ansla e and e ansla e unbalanced in o ma ion as he 2- uple LH me hodology does; (i ) he
inal ou pu is always uzzy and de ined in he same domain han he o iginal unbalanced uzzy linguis ic
labels, which acili a es i s in e p e a ion ia hei isual join ep esen a ion; ( ) de uzzi ica ion could be
applied i necessa y, and indeed his p ocess will always de i e in an equi alen esul o he 2- uple LH
me hodology. The applica ion o he T1OWA unbalanced uzzy linguis ic me hodology has been illus a ed
in he e alua ion o he c edi wo hiness and c edi isk quali y o a po en ial issuance o bonds a Eu opean
Communi y le el, ha we e he ocus o many discussion wi hin he EU du ing he ha des and mos di icul
yea s o he p esen economic c isis and ha we e known as Eu obonds. In he u u e, he T1OWA app oach
he e p esen ed will be compa ed wi h al e na i e linguis ic ools ha could be use ul o manage unbalanced
linguis ic in o ma ion, an example o which migh de i e om he wo k p esen ed in.19
Acknowledgemen
This esea ch wo k has been suppo ed by he esea ch p ojec s g an s (TIN2013-40658-P and TIN2016-
75850-R) om he FEDER unds, and he Uni e si y o G anada ‘S eng hening h ough Sho -Visi s’ (Re .
GENIL-SSV 2015) p og amme.
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