Fuzzy modeling by hie a chically buil
uzzy ule bases
Osca Co d
on
a
, F ancisco He e a
a,*
, Igo Zwi
b
a
Depa men o Compu e Science and A i®cial In elligence, ETS de Ingenie a In o ma ica,
Uni e si y o G anada, A da. Andalucia 38, 18071 G anada, Spain
b
Depa men o Compu e Science, Uni e si y o Buenos Ai es, 1428 Buenos Ai es, A gen ina
Recei ed 1 Augus 2000; accep ed 1 Ma ch 2001
Abs ac
Al hough Mamdani- ype uzzy ule-based sys ems (FRBSs) became success ully
pe o ming clea ly in e p e able uzzy models, hey s ill ha e some lacks ela ed o hei
accu acy when sol ing complex p oblems. A a ian o hese kinds o sys ems, which
allows o pe o m a mo e accu a e model ep esen a ion, a e he so-called app oxima e
FRBSs. This al e na i e ep esen a ion s ill canno a oid he p oblems conce ning he
uzzy ule lea ning me hods, which as p o o ype iden i®ca ion algo i hms, y o ex ac
hose app oxima e ules om he objec p oblem space. In his pape we deal wi h he
p e ious p oblems, iewing uzzy models as a class o local modeling app oaches which
a emp o sol e a complex p oblem by decomposing i in o a numbe o simple sub-
p oblems wi h smoo h ansi ions be ween hem. In o de o de elop his class o
models, we ® s p opose a common amewo k o cha ac e ize a ailable app oxima e
uzzy ule lea ning me hods, and la e we modi y i by in oducing a uzzy ule base
hie a chical lea ning me hodology (FRB-HLM). This me hodology is based on he
ex ension o he simple building p ocess o he uzzy ule base o FRBSs in a hie a -
chical way, in o de o make he sys em mo e accu a e. This ¯exibiliza ion will allow us
o ha e uzzy ules wi h die en deg ees o speci®ci y, and hus o imp o e he mod-
eling o hose p oblem subspaces whe e he o me models ha e bad pe o mance, as a
e®nemen . This app oach allows us no o ha e o assume a ®xed numbe o ules and
o in eg a e he good local beha io o he hie a chical model wi h he global model,
ensu ing a good global pe o mance. Ó2001 Else ie Science Inc. All igh s ese ed.
In e na ional Jou nal o App oxima e Reasoning 27 (2001) 61±93
www.else ie .com/loca e/ija
*
Co esponding au ho . Tel.: +34-58-24-40-19; ax: +34-58-24-33-17.
E-mail add esses: [email p o ec ed] (O. Co d
on), [email p o ec ed] (F. He e a),
[email p o ec ed] (I. Zwi ).
0888-613X/01/$ - see on ma e Ó2001 Else ie Science Inc. All igh s ese ed.
PII: S 0 8 8 8 - 613X(01)00034-2
Keywo ds: Fuzzy modeling; Mamdani- ype uzzy ule-based sys ems; Fuzzy ule base;
Gene ic algo i hms; Hie a chical uzzy clus e ing; App oxima e uzzy ules
1. In oduc ion
Nowadays, one o he mos impo an a eas o he applica ion o uzzy se
heo y as de eloped by Zadeh [35] a e uzzy ule-based sys ems (FRBSs). These
kinds o sys ems cons i u e an ex ension o classical ule-based sys ems, be-
cause hey deal wi h uzzy ules ins ead o classical logic ules. Thanks o his,
hey ha e been success ully applied o a wide ange o p oblems om die en
a eas p esen ing unce ain y and agueness in die en ways [3,21,24,26].
The e a e a leas wo die en kinds o FRBSs in he li e a u e, he
Mamdani and Takagi±Sugeno±Kang (TSK), which die on he composi ion
o he ule consequen . The use o one o he o he depends on he ac ha he
main equi emen is he in e p e abili y o he accu acy o he model, espec-
i ely.
Al hough he Mamdani- ype FRBS p esen s he maximum desc ip ion le-
el, i is no as accu a e as desi ed in some cases. The e o e, a leas wo hings
could be done o imp o e he accu acy o his model ype. On he one hand,
we can p ese e he linguis ic ep esen a ion o his model and pe o m suc-
cessi e e®nemen s on i , imp o ing i s accu acy wi hou losing in e p e abili y
o a high deg ee [13,15]. On he o he hand, we can imp o e he model by
using a mo e accu a e ep esen a ion. To do so, we ocus ou a en ion on a
a ian o Mamdani- ype, he so-called app oxima e FRBSs [1,2]. These kinds
o FRBSs a e he ones ha ha e uzzy ules composed o uzzy a iables ±
wi h a uzzy se associa ed de®ning hei meaning ± ha do no ake as a alue
a linguis ic e m, like in he case o linguis ic a iables [36±38], bu a eal uzzy
se .
E en hough a g ea deal o esea ch ac i i y has ocused on he de elop-
men o me hods o build o e®ne app oxima e FRBSs om nume ical da a,
hey s ill p esen some p oblems. To deal wi h hese kinds o models and
me hods, in his pape we ® s p opose a common amewo k o g oup and
cha ac e ize uzzy ule gene a ion me hods (FRG-me hods), i.e., me hods o
lea ning app oxima e ules. La e , we in oduce a modi®ca ion o his ame-
wo k in o de o sol e many o he o me p oblems by designing a uzzy ule
base hie a chical lea ning me hodology (FRB-HLM). The main pu pose o
his me hodology is o au oma ically gene a e mo e accu a e app oxima e
uzzy models by pe o ming successi e e®nemen s o ini ial models gene a ed
by FRG-me hods.
To do so, we in oduce he concep o laye s, which was p e iously applied
o desc ip i e models in [13,15]. In his ex ension, he uzzy ule base (FRB) is
cons uc ed by he de elopmen o se o laye s o FRBs, each one con aining
62 O. Co don e al. / In e na . J. App ox. Reason. 27 (2001) 61±93
uzzy ules wi h a die en speci®ci y le el, i.e. die en uzzinesses. These
kinds o ules a e called hie a chically gene a ed uzzy ules.
In o de o do ha , his pape is se up as ollows. In Sec ion 2, a desc ip ion
o he app oxima e FRBS model is in oduced, as well as i s ad an ages and
d awbacks. In his sec ion, we also conside some p oblems associa ed wi h he
FRG-me hods, and p opose a common amewo k o dealing wi h hem. In
Sec ion 3, we in oduce he FRB-HLM as a solu ion o many o he p e ious
p oblems and pe o m a desc ip ion o he hie a chically buil FRB philosophy
and he ela ion be ween i s componen s. Nex , he algo i hm is explained
in de ail. In Sec ion 4, he uzzy modeling p ocess ob ained om FRB-
HLM and well-known induc i e FRG-me hods is applied o sol e h ee di -
e en applica ions. Finally, in Sec ion 5, some concluding ema ks a e poin ed
ou .
2. App oxima e FRBSs
In his sec ion we ® s compa e he app oxima e FRBSs wi h he linguis ic
ones, highligh ing hei ad an ages and lacks. Nex , we cha ac e ize he FRG-
me hods, which buil app oxima e FRBSs, p o iding a common amewo k o
deal wi h hem. Finally, he d awbacks o he FRG-me hods a e also discussed.
2.1. App oxima e e sus linguis ic FRBSs
As we ha e said, he e a e a leas wo die en o ms o uzzy modeling:
Mamdani- ype and TSK FRBSs. The o me p esen s he maximum desc ip-
ion and in e p e abili y le el, bu i is no as accu a e as desi ed in some
complex p oblems. In opposi e, he second app oach pe o ms he mo e ac-
cu a e app oxima ion wi h he d awback o losing in e p e abili y in hei
consequen s.
The lack o accu acy o Mamdani- ype models is due o some p oblems
ela ed o he linguis ic ule s uc u e conside ed, which a e a consequence o
he in¯exibili y o he concep o linguis ic a iable [36±38]. A summa y o
hese p oblems may be ound in [1,4,8], and is b ie¯y enume a ed as ollows:
·The e is a lack o ¯exibili y in he FRBS because o he igid pa i ioning o
he inpu and ou pu spaces.
·When he sys em inpu a iables a e dependen hemsel es, i is e y ha d o
uzzy pa i ion he inpu spaces.
·The homogenous pa i ioning o he inpu and ou pu spaces when he in-
pu ±ou pu mapping a ies in complexi y wi hin he space is inecien
and does no scale o high-dimensional spaces.
·The size o he FRB di ec ly depends on he numbe o a iables and linguis-
ic e ms in he sys em. Ob aining an accu a e FRBS equi es a signi®can
O. Co don e al. / In e na . J. App ox. Reason. 27 (2001) 61±93 63
g anula i y amoun , i.e., i needs he c ea ion o new linguis ic e ms. This
g anula i y inc ease causes he numbe o ules o ise signi®can ly, which
may ake he sys em o lose he capabili y o being in e p e able o human
beings.
A a ian o hese Mamdani- ype FRBS-based modeling app oaches has
been p oposed in he las ew yea s, he app oxima e FRBS [1,4]. I is based on
he o me app oach bu conside s he lack o accu acy as a majo d awback.
While he o me desc ip i e FRBSs ha e associa ed a knowledge base com-
posed o a da abase ± con aining linguis ic pa i ions ± and a ule base ±
composed o linguis ic ules which make use o hese linguis ic pa i ions ±, he
app oxima e ones only ha e o de®ne an FRB. This happens because hei
ap oxima e uzzy ules con ain a iables which a e die en locally de®ned
uzzy alues.
In o de o dis inguish be ween he ype o modeling pe o med o ob ain
Mamdani- ype and app oxima e FRBSs, we a e going o e e o he o me as
linguis ic modeling, and o he la e as uzzy modeling. Tha is, linguis ic models
a e pe o med by desc ip i e Mamdani- ype ules o linguis ic ules, and uzzy
models a e de eloped by app oxima e Mamdani- ype ules o uzzy ules. In he
ollowing some dis inc ions be ween bo h ypes o modeling a e gi en:
·Linguis ic modeling makes use o uzzy ules composed o linguis ic a iables
ha ake alues in a e m se wi h eal-wo ld meaning (linguis ic ules).
These kinds o models a e cha ac e ized by he ac ha hei main equi e-
men is he sys em in e p e abili y.
·In uzzy modeling, he uzzy ules a e composed o uzzy p edica es wi hou a
linguis ic meaning, i.e., he a iables o ming he ules do no ake as a alue
a linguis ic e m wi h a uzzy se associa ed de®ning hei meaning, bu a eal
uzzy se . These models p e end o be mo e accu a e han he o me ones.
The choice be ween how in e p e able and how accu a e he model mus be,
usually depends on he use 's needs o a speci®c p oblem and will condi ion
he kind o FRBS selec ed o model i . As well as ha , in his pape we will
ocus on de eloping mo e accu a e uzzy models by an FRB-HLM, which
p o ides app oxima e solu ions o die en p oblems, especially eal-wo ld
p oblems.
2.2. App oxima e FRBS ea u es
App oxima e FRBSs ha e some in e es ing ad an ages ha ge hem o be
e y sui able o uzzy modeling pu poses in many cases [8]:
·The exp essi e powe o he ules, ha p esen hei own speci®ci y in e ms
o he uzzy se s in ol ed in hem, hus in oducing addi ional deg ees o
eedom in he sys em.
·The numbe o ules is adap ed o he complexi y o he p oblem, needing
less ules in simple p oblems, and being able o use mo e ules i i is neces-
64 O. Co don e al. / In e na . J. App ox. Reason. 27 (2001) 61±93
sa y. This is likely o be o bene® in ackling he cou se o dimensionali y
when scaling o mul idimensional sys ems.
These ac s, which allow app oxima e FRBSs o be mo e accu a e in com-
plex p oblems, ha e un o una ely some d awbacks associa ed [1]:
·The FRB eadabili y is los because he e is no global in e p e a ion o he
a iables conside ed. In spi e o his, app oxima e FRBSs locally desc ibe
he sys em beha io in a simila way o o he models like neu al ne wo ks,
bu in a mo e desc ip i e way.
·The app oxima ion capabili y causes an excessi e speci®ci y wi h bad gene -
aliza ion, some imes ob aining an unwan ed o e ® ing.
Al hough uzzy and linguis ic modeling a e no incompa ible, bu comple-
men a y, in his pape we ocus ou a en ion in he o me , and consequen ly
on he model accu acy.
2.3. App oxima e FRBSs lea ning me hods (FRG-me hods) as p o o ype-
iden i®ca ion algo i hms
Some au oma ic echniques ha e been p oposed o lea n a p ope FRB o
an app oxima e FRBS o sol e a speci®c p oblem. The accu acy o he FRBS
in sol ing his p oblem will depend on he in insic cha ac e is ics o he
p oblem and on he men ioned lea ning asks. In spi e o hese dependences, we
will a emp o cha ac e ize hese lea ning me hods which we ha e labeled as
FRG-me hods.
Rega ding [29,40], we can say ha basically an FRG-me hod does i s job as
a p o o ype-iden i®ca ion algo i hm, which pe o ms he op imiza ion o a
unc ional QF;Modelc ha measu es he ex en by which he pa ame e -
ized model Modelc® s he subse Fo he objec being desc ibed (see Fig. 1).
F om his pe spec i e, he p oblem is o mula ed as a clus e ing p oblem in
he sense ha ex ac ed subse s mee , o some ex en , he equi emen s imposed
by he model collec ion in he same way ha elemen s o a clus e ing pa i ion
sa is y he cons ain ha hei membe s be as simila as possible [29,40]. This
poin o iew ollows he o iginal ideas o Ruspini [28], la e expanded by
Bezdek in oducing a ious me hods cen e ed upon he no ion o p o o ype [5].
The basic idea o summa izing a da ase by a numbe o ep esen a i e p o o-
ypes ± objec s lying in he same space as he sample poin s ± was la e ex ended
in many signi®can di ec ions by elaxing his concep in a a ie y o ways, o
example, line segmen s, ellipsoids, e c. [7]. In his pape we pa icula ize his
concep by conside ing hese p o o ypes as being uzzy ules [2,17,18].
Ha ing hese concep s in mind, FRG-me hods can be seen as iden i®ca ion
algo i hms wi h uzzy ule p o o ypes, i.e., uzzy model builde s whose main
pu pose is o ex ac he mos sui able se o uzzy ules om an objec (inpu ±
ou pu da a) acco ding o an op imiza ion measu e, which e alua es he
quali y o he app oxima ion. Addi ionally, hey o ganize esul s and
O. Co don e al. / In e na . J. App ox. Reason. 27 (2001) 61±93 65

summa ize hem by an in e es ingness c i e ion, in o de o p o ide a mo e
compac and use ul ep esen a ion o he salien s uc u es.
In o de o illus a e his si ua ion, conside o example he Weigh ed
Coun ing Algo i hm in oduced by Ba dossy and Ducks ein [3] which, as can
be seen in Appendix A, iden i®es app oxima e uzzy ules om a se o inpu ±
ou pu da a (objec F) o an app oxima e uzzy model Modelc. The quali y
o iden i®ed candida e subs uc u es ( ule p emises) is measu ed in base o i s
deg ee o ul®llmen , i.e., a co e ing c i e ion (QF;Modelc). These ex ac ed
ules could also be summa ized by, o example, a use -based ela ion o in-
e es which imposes a maximum accep able numbe o ules.
2.4. D awbacks o he FRG-me hods
All o hese models gene a ed by FRG-me hods ha e he same d awbacks
ha p o o ype iden i®ca ion me hods ha e, and all o hem y o gi e hei
die en own solu ions, which become pa icula o he co esponding me hod:
·Simple o mula ion o he p o o ype-iden i®ca ion p oblem as an op imiza-
ion o a unc ional would simply esul in a la ge collec ion o e y speci®c
ules wi h small ex en and high accu acy, bu wi h poo gene aliza ion.
Smalle a he han la ge signi®ca i e se s wi h high gene aliza ion powe
would be p e e ed.
·The de e mina ion o a comple e clus e ing o a pa i ion o he da ase in o
a ®xed numbe o p o o ypes becomes a big deal o a long ime.
App oxima e Fuzzy Rules
:
3
R IF x is THEN y
is
:
2
R IF x is THEN y
is
:
1
R IF x is THEN y
is
MODEL
STRUCTURE
PROTOTYPE
OBJECTS
RELATIONS
OF INTEREST
SUMMARIZATION
MODEL IDENTIFIED (Rules)
Final Rule Se
Model
FRB-HLM
IDENTIFICATION
ALGORITHM
Fig. 1. FRG-me hods as p o o ype-iden i®ca ion algo i hms.
66 O. Co don e al. / In e na . J. App ox. Reason. 27 (2001) 61±93
To deal wi h hese p oblems, in he ollowing sec ion we will p esen an FRB-
HLM in o de o build a FRB wi h he pu pose o sol ing some o he abo e
d awbacks.
3. Fuzzy ule base hie a chical lea ning me hodology
To o e come some o he d awbacks o he app oxima e FRBSs (Sec ion
2.4) and o he FRG-me hods (Sec ion 2.3), we p opose a FRB-HLM which, as
a me a-me hod, modi®es he amewo k shown in Fig. 1 and conside s he
ollowing poin s:
·On he one hand, we would like o implemen a so o ade-o be ween he
ex ensionali y and he accu acy o he models gene a ed, ha ing in mind ha
ules which pe o m good explana ions end o be limi ed in ex en while
hose ha , con e sely, a e capable o desc ibing la ge subse s o he da ase ,
do i so poo ly.
·On he o he hand, we will adop a mo e gene al ea men han ha o a
ypical clus e ing p oblem, emphasizing he sequen ial isola ion o indi id-
ual clus e s [23] a he han de e mina ion o a ull clus e ing. Fu he mo e,
we do no wan o assume a p io i knowledge o he o al numbe o clus e s
± ule p o o ypes ± equi ing ha he se o all clus e s be an exhaus i e pa -
i ion o he comple e objec .
To do so, he FRB-HLM will modi y he ini ial model iden i®ed by a FRG-
me hod in an i e a i e way, pe o ming a g adual e®nemen o i . Mo eo e , i
will also modi y he summa iza ion p ocess seen in Fig. 1, by adding a ule
selec ion p ocess o ob ain a compac se o ules ha ha e good coope a ion
be ween hem and o emo e he unnecessa y ones.
3.1. Keypoin s o he FRB-HLM
Ou app oach owes much o hose clus e ing gene aliza ions men ioned in
Sec ion 2.3 and o he no ion o hie a chical clus e s [17,19]. I is also closely
ela ed o o he modeling echniques which wo k wi h uzzy g anules, uzzy
g aphs, e c. [27,39]. As hem, we should also answe some impo an ques ions
which conce n he s uc u es (clus e s, ules, g anules, e c.) used by he o me
echniques:
How can ules, g anules, clus e s, e c. be pa i ioned?
This ques ion is ela ed o some aspec s like he compac ness o a clus e ,
measu es inside he clus e , measu es be ween clus e s, g anule pe ime e s, ule
scopes, e c.
O. Co don e al. / In e na . J. App ox. Reason. 27 (2001) 61±93 67
How many ules, g anules, clus e s, e c. could exis ?
This is conce ned wi h sys em comp ehension, accu acy, o e ® ing, e c.
In o de o gi e an answe o hese ques ions, and o p e ious p oblems, in he
ollowing we lis some keypoin s o he FRB-HLM:
·Dynamic ule expansion o pa i ion depending on he FRG-me hod used, in
o de o ake ad an age o i s in insic capabili ies. These me hods some-
imes pe o m his ask in a mo e s a ic o dynamic way acco ding o hei
philosophy.
·I e a i e me hodology which, as is done by hie a chical clus e ing echniques,
emphasizes he sequen ial isola ion o clus e s a he han a ull clus e ing.
Hence, we do no ha e o assume an a p io i ®xed numbe o ules.
·G adual localized e®nemen s on bad modeled zones a he han in he whole
p oblem domain, as a egula ion among ex ensionali y and p ecision. This
ask is con olled by an expansion ac o , which also ac s as an o e ® ing
o esee .
·Summa iza ion by ule selec ion, in o de o in eg a e he local beha io o
he hie a chically buil model wi h he global one o he whole model, ensu -
ing a good pe o mance.
We should no e ha he e a e many p oposals in o de o answe each one o
he o me ques ions o o sol e he said p oblems [2,27,29,39]. Some o hei
skills will be conside ed as ex ensions o he p esen me hodology in u u e
wo ks. In he ollowing subsec ions, he composi ion o he hie a chically buil
FRB and he me hodology will be desc ibed in de ail.
3.2. Hie a chically buil uzzy ule base
In his sec ion we p esen a ¯exible hie a chical p ocess o de®ne he FRB
s uc u e ± based on p e ious keypoin s ± ha allows us o sol e some o he
lacks desc ibed in Sec ions 2.2 and 2.4, and consequen ly o imp o e he ap-
p oxima e uzzy models pe o mance/accu acy. The hie a chical p ocess is
based on he gene a ion o a se o laye s, each one becoming an FRB which
con ains uzzy ules wi h a die en deg ee o speci®ci y o ex en , i.e., uzziness
laye  FRB [
i
R
i
wi h FRB being he FRB buil in i e a ion o med by app oxima e uzzy
ules R
i, acco ding o he p esen me hodology. F om now on and o he sake
o simplici y, we a e going o e e o he componen s o a FRB as - uzzy
ules.
These - uzzy ules a e o ganized as a hie a chy, whe e he o de is gi en by
inc easingly mo e speci®c inpu subspaces, i.e., he inpu suppo ex en co -
e ed by he an eceden s o he app oxima e uzzy ules. This can be ega ded as
68 O. Co don e al. / In e na . J. App ox. Reason. 27 (2001) 61±93
a kind o in o ma ion quan iza ion included in he uzzy a iables o he ules,
i.e., hei uzziness. Fo example, gi en wo successi e laye s and 1, he
uzzy inpu subspace co e ed by a - uzzy ule is mo e gene al (la ge ) han he
ones emb aced by each one o he ( 1)- uzzy ules de i ed om i [25]. F om
his poin o iew, he successi e app oxima e uzzy ules gene a ed can be seen
as an inpu subspace e®nemen o p e ious laye uzzy ules. This s uc u e is
illus a ed in Fig. 2.
As is seen, he ep esen a ion akes he o m o a ee, whe e he oo ep-
esen s he en i e p oblem domain space, and he nodes ep esen good pe -
o mance - uzzy ules (ligh g ey ec angles), which model uzzy inpu
subspaces ha do no equi e u he decomposi ion, o mo e speci®c ( 1)-
uzzy ules ha model decomposed subspaces om a bad pe o mance - uzzy
ule (black ec angles). This p ocess is pe o med in an i e a i e way. Thus,
good - uzzy ules and new gene a ed ( 1)- uzzy ules compose he new le el
o he ee gene a ed in i e a ion 1, i.e., laye ( 1).
How can we de elop a FRB 1 om a FRB in o de o c ea e he ®nal
mo e accu a e FRB?
Each FRB is o med by a collec ion o app oxima e Mamdani- ype uzzy ules
R
i:IF x1is S
i1and ... and xmis S
im THEN yis B
i
wi h x1;...;xmand ybeing he inpu uzzy a iables and he ou pu one, e-
spec i ely; and wi h S
i1;...;S
im,B
ibeing he uzzy se s. As has been said, each
indi idual uzzy ule di ec ly con ains he meaning desc ibing i , i.e., each
a iable has a uzzy se associa ed as shown in Fig. 3.
Laye 1 - 1
FRB
Laye 2 - 2
FRB
Laye 3 - 3
FRB
. . .
Laye -
FRB
+ Fuzziness - Speci ici y
- Fuzziness + Speci ici y
1
i
R
2
2,
i
R
3
1,1,
i
R3
2,1,
i
R3
2,
i
R
i
R1,...,1,1,
Li
R,...,2,1,
i
R1,...,2,1,
i
R2,..,2,
i
R1,..,2,
2
1,
i
R
Fig. 2. Hie a chically buil FRB.
O. Co don e al. / In e na . J. App ox. Reason. 27 (2001) 61±93 69
wi h CFRR
ibeing he image o he expanded uzzy ule R
i, i.e., he
candida e uzzy ules o be in he FRB 1 om ule R
i,ANTR
ibeing
he p oduc o he suppo se o each an eceden uzzy e m o he ule
R
i
ANTR
isuppS
i1suppS
im
and CONR
ibeing he suppo se o he consequen s which in case o
using CCS is de®ned as
CONR
isuppB
i
and in case o using UCS as
CONR
iV
wi h Vbeing he domain whe e he consequen is de®ned.
S ep 2. Summa iza ion p ocess. Ob ain a joined se o candida e uzzy ules
(JCFR) pe o ming he union o he g oup o he new gene a ed ( 1)- uzzy
ules and he o me good pe o mance - uzzy ules:
JCFR FRB
good [[
i
CFRR
i
!
wi h R
i2FRB
bad.
S ep 3. FRB selec ion p ocess. Simpli y he se JCFR by emo ing he un-
necessa y ules om i , in o de o gene a e an FRB 1wi h good coope a ion.
In his pape we conside a gene ic p ocess [11,20,22] o pu his ask in o eec ,
bu any o he echnique could be conside ed
FRB 1Selec ionJCFR:
In he JCFR ± whe e he e a e coexis ing ules gene a ed in die en laye s ± i
may happen ha a comple e se o ( 1)- uzzy ules, which eplaces an ex-
panded - uzzy ule, does no p oduce good esul s. Howe e , a subse o his
se o ( 1)- uzzy ules may wo k p ope ly, wi h less ules ha ha e good
coope a ion be ween hem and wi h he good ules om he p e ious laye .
Thus, he JCFR se o ules gene a ed may p esen edundan o unnecessa y
ules making he model using his FRB less accu a e.
The gene ic ule selec ion p ocess [11,20] is based on a bina y coded gene ic
algo i hm (GA) in which he selec ion o he indi iduals is pe o med using he
s ochas ic uni e sal sampling p ocedu e oge he wi h an eli is selec ion
scheme, and he gene a ion o he osp ing popula ion is pu in o eec by
using he classical bina y mul ipoin c osso e (pe o med a wo poin s) and
uni o m mu a ion ope a o s.
76 O. Co don e al. / In e na . J. App ox. Reason. 27 (2001) 61±93

The coding scheme gene a es ®xed-leng h ch omosomes. Conside ing he
ules con ained in JCFR coun ed om 1 o z,anz-bi s ing Cc1;...;cz
ep esen s a subse o ules o he FRB 1, such ha
IF ci1 THEN Ri2FRB 1ELSE Ri62 FRB 1:
The ini ial popula ion is gene a ed by in oducing a ch omosome ep esen ing
he comple e p e iously ob ained ule se , i.e., wi h all ci1. The emaining
ch omosomes a e selec ed a andom.
As ega ds he ® ness unc ion FCj, i is based on a global e o measu e
ha de e mines he accu acy o he FRBS encoded in he ch omosome, which
depends on he coope a ion le el o he ules exis ing in he JCFR. We usually
wo k wi h he MSE o e a aining da a se , as was p e iously de®ned, al-
hough o he measu es may be used.
S ep 4. Model alida ion p ocess. The ®nal model is ei he accep ed as
p ope o he gi en pu pose o i is ejec ed gene a ing ano he i e a ion o he
p ocess. Al hough many indexes can be used o measu e he quali y o linea o
nonlinea sys ems a e an iden i®ca ion loop [2,7], we conside a mono onic
MSE measu e on he aining and es se s, combined wi h a p e iously de®ned
maximum numbe o i e a ions Tmax, which is based on a ade-o be ween
he complexi y and he accu acy o he model gene a ed.
This measu e is compu ed as:
IF MSEFRB 1ETDS 6MSEFRB ETDS and
MSEFRB 1ETST 6MSEFRB ETST and  <Tmax
THEN 1;
Go o S ep 1
ELSE
FRB inal FRB o FRB 1:
We should no e ha o he sake o simplici y in he p esen implemen a ion,
we only keep he las wo laye s o he FRB in o de o allow he alida ion o
he algo i hm ou pu model. The e o e, a las we only selec as FRB inal one o
hese wo FRBs , i.e., FRB o FRB 1, he one wi h be e pe o mance.
4. Examples o applica ion: expe imen s and analysis o esul s
Wi h he aim o analyzing he beha io o he p oposed me hodology, wo
die en FRG-me hods aligned wi h he cha ac e is ics p esen ed in his pape
ha e been chosen. The ® s is he one p oposed by Ba dossy and Ducks ein [3],
Weigh ed Coun ing Algo i hm (WCA), and he second is a well-known uzzy
clus e ing me hod based on Bezdek's wo k, he Fuzzy C-means (FCM) [5±7].
Bo h me hods a e b ie¯y desc ibed in Appendix A.
O. Co don e al. / In e na . J. App ox. Reason. 27 (2001) 61±93 77
In his sec ion, FRB-HLM will be combined wi h he o me FRG-me hods
o model h ee die en applica ions: wo- h ee-dimensional unc ions [9,11]
and a eal-wo ld elec ical enginee ing dis ibu ion p oblem in Spain [12,30,31].
In o de o do his, we ha e o ganized his Sec ion in h ee pa s: a ® s pa
o no a ion and pa ame e s, a second o expe imen s and a ®nal one wi h an
analysis o esul s o he expe imen al s udy.
4.1. No a ion and pa ame e s
Fo he sake o simplici y, in he ollowing applica ions we a e going o e e
o hose expe imen s p oduced by he FRB-HLM by he ollowing no a ion:
FRB-HLMFRG-me hod;CS; ;
whe e is he numbe o laye s o i e a ions wi h ule expansions pe o med by
he me hodology, and CS ep esen s he ype o consequen scope selec ed, i.e.,
CCS o UCS, e.g., FRB-HLM (FRG-me hod, CCS, 3).
Two FRG-me hods, WCA and FCM, will be used o he expe imen a ion.
The WCA is conside ed wi h wo die en in e al ini ializa ions, bo h based
on he ex ac ion o he suppo se om uzzy pa i ions [1]. The ® s one,
s a ic ini ializa ion (S-WCA), is buil based on a symme ical and uni o mly
dis ibu ed uzzy pa i ion o h ee and ® e uzzy e ms o ini ial and subse-
quen i e a ions o he algo i hms [13], espec i ely. The second one, dynamic
ini ializa ion (D-WCA), is pe o med by he use o a uzzy clus e ing-de el-
oped uzzy pa i ion. To do so, in his pape we use FCM combined wi h a
alida ion index which measu es he pa i ion quali y and i e a i ely de ec s a
good numbe o clus e s [32]. Bo h me hods a e b ie¯y desc ibed in Appendix
A and summa ized in Table 1.
The gene al pa ame e s used in all o hese expe imen s a e lis ed in Table 2.
Table 1
Summa iza ion o WCA and FCM asks
WCA FCM
S-WCA D-WCA
Symme ical,
uni o mly dis ibu ed
uzzy pa i ion
Inc emen al applica ion o
FCM and alida ion index
Inc emen al applica ion o
FCM and alida ion index
Clus e cen e p ojec ions
on he an eceden domains
Clus e s p ojec ion on
an eceden and consequen
a iable domains
Pa i ion cons uc ion wi h a 0.5 c oss le el be ween
adjacen uzzy se s
Suppo se ex ac ion and in e als de®ni ion
Rule cons uc ion by WCA philosophy Rule cons uc ion
78 O. Co don e al. / In e na . J. App ox. Reason. 27 (2001) 61±93
In his con ibu ion, we will use he minimum -no m in he ole o con-
junc i e and implica ion ope a o and he cen e o g a i y weigh ed by he
ma ching deg ee [10] as de uzzi®ca ion s a egy.
The esul s ob ained in he expe imen s de eloped a e collec ed in ables
whe e MSE a and MSE s s and o he alues ob ained in he MSE measu e
compu ed o e he aining and es da a se s, espec i ely, and % indica es he
pe cen age in which he FRG-me hod models in he able a e imp o ed by he
hie a chically buil FRB-based models. #Rs ands o he numbe o ules o
he co esponding FRB.
4.2. Expe imen s
We will show esul s compa ing he eec i eness o he hie a chical lea ning
me hodology wi h bo h o iginal FRG-me hods (WCA and FCM) on he said
h ee p oblems. We should no e ha he hie a chically buil FRBs a e ini-
ialized wi h he FRG-me hods ou pu , in o de o allow he o me compa -
ison. In all cases we show examples applying CCS and UCS consequen scope.
Addi ionally, in he eal-wo ld elec ical p oblem we will also compa e he
esul s ob ained by FRB-HLM wi h o he echniques: classical eg essions,
mul ilaye pe cep on and a linguis ic modeling hie a chical lea ning me h-
odology HSLR-LM [13,16].
Finally, we will analyze he eec o using die en expansion ac o s a,
showing he obus ness o he FRB-HLM and i s ole as an accu acy-com-
plexi y egula o and o e ® ing con olle .
4.2.1. Fuzzy modeling o an in e media e complexi y mul imodal h ee-dimen-
sional unc ion (F1)
The exp ession o he selec ed unc ion is shown as ollows, along wi h he
uni e ses o discou se conside ed o he a iables [9]. I s g aphical ep esen-
a ion is shown in Fig. 7. As may be seen, F1is an in e media e complexi y
mul imodal unc ion whose exp ession is shown as ollows:
Table 2
Pa ame e alues
Pa ame e Decision
Gene a ion
a ule expansion ac o 0:5;0:9;1:1
sposi i e examples 0:5;0:3
GA selec ion
Numbe o gene a ions 500, 1500
Popula ion size 61, 81
Mu a ion p obabili y 0:1, 0:21
C osso e p obabili y 0:6
O. Co don e al. / In e na . J. App ox. Reason. 27 (2001) 61±93 79
F1x1;x2ex1sin2x2ex2sin2x1;
x1;x228;8;F1x1;x220;5836:
In o de o model he F1 unc ion, a aining da a se composed o 1089 da a
uni o mly dis ibu ed in he h ee-dimensional de®ni ion space has been ob-
ained expe imen ally. On he o he hand, ano he da a se has been gene a ed
o i s use as a es se o e alua ing he pe o mance o he design me hods,
a oiding any possible bias ela ed o he da a in he aining se . The size o his
da a se is a pe cen age o he aining se one, en pe cen o be p ecise. The
da a a e ob ained by gene a ing a andom he s a e a iable alues in he
speci®c uni e ses o discou se o each one o hem, and compu ing he asso-
cia ed ou pu a iable alue. Hence he es se , o med by 108 da a, is used o
measu e he accu acy o he die en models designed by compu ing he MSE
o hem.
4.2.1.1. Expe imen s wi h FRG-me hods. The esul s ob ained wi h ou FRB-
HLM o WCA and FCM a e shown in Table 3 and a g aphical illus a ion o
he modeling ob ained can be seen in Fig. 8 (FRB-HLM (S-WCA, UCS, 3)).
Table 3
Resul s ob ained in he uzzy modeling o he unc ion F1
Me hod a#RMSE a MSE s % a % s
S-WCA 9 244,632 257,047
FRB-HLM (S-WCA, CCS, 3) 0.5 316 3876 6140 98.41 97.61
FRB-HLM (S-WCA, UCS, 3) 0.5 201 2406 3634 99.01 98.58
D-WCA 570 116,992 59,878
FRB-HLM (D-WCA, CCS, 2) 0.5 718 60,142 27,947 48.59 53.32
FRB-HLM (D-WCA, UCS, 2) 0.5 838 4984 4036 95.73 93.25
FCM 6 447,584 430,713
FRB-HLM (FCM, CCS, 1) 0.5 5 123,984 88,494 72.29 79.45
FRB-HLM (FCM, UCS, 1) 0.5 9 110,210 55,404 75.37 87.13
-5
0
5-5
0
5
0
1000
2000
3000
4000
5000
6000
Fig. 7. Exac g aphical ep esen a ion o he unc ion F1.
80 O. Co don e al. / In e na . J. App ox. Reason. 27 (2001) 61±93
4.2.1.2. Expe imen s wi h die en alues o he expansion ac o a. The esul s
ob ained wi h FRB-HLM wi h die en alues o he expansion ac o aa e
shown in Table 4. We should no e ha we only p esen expe imen s wi h alues
o a ha become ep esen a i e, e en hough o he alues could ha e been
used. Besides his, Table 4 shows expe imen s ha ha e been done up o he
same i e a ions in o de o allow he compa ison among die en alues o a.
Anyway, almos all o hese alues can be o e come in mo e i e a ions.
4.2.2. Fuzzy modeling o a e y complex mul imodal h ee-dimensional unc ion
(F2)
In he ollowing, we p esen a e y complex mul imodal h ee-dimensional
unc ion, F2[9,11]. I s g aphical ep esen a ion is shown in Fig. 9. I s exp es-
sion, along wi h he uni e ses o discou se conside ed o he a iables, is also
shown as ollows:
Table 4
Resul s ob ained in he uzzy modeling o he unc ion F1conside ing die en alues o he ex-
pansion ac o a
aCCS UCS
#RMSE a MSE s #RMSE a MSE a
(a) S-WCA
0.5 316 3876 6140 201 2406 3634
0.9 214 5012 6248 239 3897 3123
1.1 218 3780 5979 168 3614 4036
(b) D-WCA
0.5 718 60,142 27,944 838 4984 4036
0.9 424 73,408 79,082 668 27,476 39,463
1.1 371 94,682 69,347 571 28,771 43,508
(c) FCM
0.5 5 123,984 88,494 9 110,210 55,404
0.9 5 123,984 88,494 9 110,210 55,404
1.1 4 112,871 90,068 9 110,210 55,404
-5
0
5-5
0
5
0
1000
2000
3000
4000
5000
6000
Fig. 8. F1modeled wi h 201 ules.
O. Co don e al. / In e na . J. App ox. Reason. 27 (2001) 61±93 81

F2x1;x2x2
1x2
2cos18x1cos18x2;
x1;x221;1;F2x1;x222;3:5231:
The second unc ion, F2, has been modeled using a aining da a se composed
o 1681 da a uni o mly dis ibu ed in he h ee-dimensional de®ni ion space. A
es se o 167 da a, gene a ed in he same way ha was done in unc ion F1,
was selec ed o e alua ing he pe o mance o he design me hods.
4.2.2.1. Expe imen s wi h FRG-me hods. The esul s ob ained wi h ou FRB-
HLM o WCA and FCM me hods a e shown in Table 5 and also illus a ed in
Fig. 10 (FRB-HLM (S-WCA, CCS, 3)).
4.2.2.2. Expe imen s wi h die en alues o he expansion ac o a. The esul s
ob ained wi h ou FRB-HLM wi h die en alues o he expansion ac o a
a e shown in Tables 6 and 7. The assump ions made in he p e ious expe imen
emain o he p esen one. Emp y boxes mean ha he alue o ais oo high o
expand ules.
Table 5
Resul s ob ained in he uzzy modeling o he unc ion F2
Me hod a#RMSE a MSE s % a % s
S-WCA 9 0.580 0.660
FRB-HLM (S-WCA, CCS, 3) 0.9 486 0.106 0.129 81.72 80.45
FRB-HLM (S-WCA, UCS, 3) 0.9 379 0.178 0.195 69.31 70.45
D-WCA 49 0.516 0.578
FRB-HLM (D-WCA, CCS, 2) 0.5 1657 0.094 0.084 81.78 85.46
FRB-HLM (D-WCA, UCS, 2) 0.5 2296 0.073 0.072 85.85 87.54
FCM 7 0.553 0.579
FRB-HLM (FCM, CCS, 4) 0.5 146 0.324 0.331 41.41 42.83
FRB-HLM (FCM, UCS, 3) 0.5 15 0.506 0.541 8.49 6.56
-1
-0.5
0
0.5
1-1
-0.5
0
0.5
1
-2
-1
0
1
2
3
4
Fig. 9. Exac g aphical ep esen a ion o he unc ion F2.
82 O. Co don e al. / In e na . J. App ox. Reason. 27 (2001) 61±93
4.2.3. The elec ical enginee ing dis ibu ion p oblems
Some imes, he e is a need o measu e he amoun o elec ici y lines ha an
elec ic company owns. This measu emen may be use ul o se e al aspec s
such us he es ima ion o he main enance cos s o he ne wo k, which was he
main goal o he p oblem p esen ed in Spain [12,31]. High and medium ol age
lines can be easily measu ed, bu low ol age line is con ained in ci ies and
Table 6
Resul s ob ained in he uzzy modeling o he unc ion F2conside ing die en alues o he ex-
pansion ac o a
aCCS UCS
#RMSE a MSE s #RMSE a MSE a
(a) S-WCA
0.5 710 0.100 0.121 699 0.171 0.195
0.9 486 0.106 0.129 379 0.178 0.195
1.1
(b) D-WCA
0.5 2296 0.094 0.084 1657 0.073 0.072
0.9 697 0.163 0.169 587 0.138 0.184
1.1 213 0.297 0.299 210 0.290 0.275
(c) FCM
0.5 127 0.374 0.388 15 0.506 0.541
0.9 66 0.385 0.421
1.1
-1
-0.5
0
0.5
1-1
-0.5
0
0.5
1
-2
-1
0
1
2
3
4
Fig. 10. F2modeled wi h 486 ules.
Table 7
Resul s ob ained in he uzzy modeling o he unc ion F2compa ing esul s wi h wo die en
alues o aand a die en numbe o i e a ions
Me hod a#RMSE a MSE s
FRB-HLM (D-WCA, CCS, 2) 0.5 1657 0.094 0.084
FRB-HLM (D-WCA, CCS, 4) 1.1 750 0.129 0.133
O. Co don e al. / In e na . J. App ox. Reason. 27 (2001) 61±93 83
illages, and i would be e y expensi e o measu e i . This kind o line used o
be e y con olu ed and, in some cases, one company may se e mo e han
10,000 small nuclei. An indi ec me hod o de e mine he leng h o line is
needed.
The e o e, a ela ionship mus be ound be ween some cha ac e is ics o he
popula ion and he leng h o line ins alled on i , making use o some known
da a, ha may be employed o p edic he eal leng h o line in any o he
illage. We will y o sol e his p oblem by gene a ing die en kinds o
models de e mining he unknown ela ionship: uzzy, classical eg ession and
neu al models. To do so, we we e p o ided wi h he measu ed line leng h, he
numbe o inhabi an s and he mean dis ance om he cen e o he own o he
h ee u hes clien s, conside ed as he adius o popula ion iin a sample o
495 u al nuclei [30,31]. Ou a iables a e named as shown in Table 8.
To design he die en models we ha e andomly di ided he sample in o
wo se s comp ising 396 and 99 samples, labeled aining and es , espec i ely.
4.2.3.1. Expe imen s wi h die en p ede®ned ypes o FRG-me hods. The esul s
ob ained wi h ou FRB-HLM wi h he wo said FRG-me hods a e shown in
Table 9.
4.2.3.2. Expe imen s wi h die en alues o he expansion ac o a. The esul s
ob ained wi h ou FRB-HLM wi h die en alues o he expansion ac o a
Table 8
No a ion conside ed o he p oblem a iables
Symbol Meaning
x1Numbe o clien s in popula ion
x2Radius o i h popula ion in he sample
yLine leng h, popula ion i
Table 9
Resul s ob ained in he low ol age elec ical applica ion
Me hod a#RMSE a MSE s % a % s
S-WCA 9 777,306 717,472
FRB-HLM (S-WCA, CCS, 2) 0.5 60 154,109 184,178 80.17 74.32
FRB-HLM (S-WCA, UCS, 2) 0.5 40 158,879 186,819 79.56 73.96
D-WCA 25 192,818 202,095
FRB-HLM (D-WCA, CCS, 1) 1.1 115 97,187 144,865 49.59 28.31
FRB-HLM (D-WCA, UCS, 1) 1.1 136 87,675 146,155 54.52 27.68
FCM 5 508,426 464,130
FRB-HLM (FCM, CCS, 4) 0.5 27 181,196 158,312 64.36 55.01
FRB-HLM (FCM, UCS, 2) 0.9 8 208,777 171,379 58.93 63.07
84 O. Co don e al. / In e na . J. App ox. Reason. 27 (2001) 61±93
a e shown in Table 10. The assump ions made in p e ious expe imen s emain
o he p esen one.
4.2.3.3. Expe imen compa ing models om FRB-HLM wi h o he echniques.
Once we ha e analyzed he beha io o he uzzy models designed indi idually,
we a e going o compa e hei accu acy wi h he emaining echniques con-
side ed. Table 11 shows he esul s ob ained by hem and he bes ones ob-
ained by ou FRB-HLM as well. To apply classical eg ession, he pa ame e s
o he polynomial models we e ® by Le enbe g±Ma qua d , while exponen ial
and linea models we e ® by linea leas squa es. The mul ilaye pe cep on
was ained wi h he QuickP opaga ion algo i hm. The numbe o neu ons in
he hidden laye was chosen o minimize he es e o [12,31]. We also compa e
he ob ained esul s wi h a linguis ic model ob ained by means o a hie a chical
app oach desc ibed in [13,15,16], i.e., a hie a chical sys em o linguis ic ules
Table 10
Resul s ob ained in he low ol age elec ical applica ion conside ing die en alues o he ex-
pansion ac o a
aCCS UCS
#RMSE a MSE s #RMSE a MSE a
(a) S-WCA
0.5 60 154,109 184,178 40 158,879 186,819
0.9 38 186,343 207,585 39 168,988 189,795
1.1 23 359,169 320,839 31 228,388 235,460
(b) D-WCA
0.5 205 80,840 206,961 209 70,903 158,572
0.9 132 87,091 163,687 151 78,857 169,366
1.1 115 97,187 144,865 136 87,675 146,155
(c) FCM
0.5 14 191,441 181,375 11 211,719 177,138
0.9 10 205,221 189,842 8 208,777 171,379
1.1 7 229,559 206,363 8 231,280 207,120
Table 11
Resul s ob ained in he low ol age elec ical applica ion compa ed wi h o he echniques
Me hod MSE a MSE s Complexi y
Linea 287,775 209,656 7 nodes, 2 pa .
Exponen ial 232,743 197,004 7 nodes, 2 pa
Second-o de polynomial 235,948 203,232 25 nodes, 2 pa .
Thi d-o de polynomial 235,934 202,991 49 nodes, 2 pa .
Th ee-laye pe cep on 2-25-1 169,399 167,092 102 pa .
HSLR-LM 154,411 156,197 25 ules
FRB-HLM (D-WCA, UCS, 1) 97,187 144,865 115 ules
FRB-HLM (D-WCA, CCS, 1) 87,675 146,155 136 ules
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