Thing Complex Fuzzy Sys ems by Supe ised Lea ning Algo i hms
F.
J.
Mo eno-Velo,
I.
Ba u one,
R.
Senhadji,
S.
Sbnchez-Solano
Ins i u o de Mic oelec 6nica de Se illa (IMSE-CNh4)
Cen o Nacional de Mic oelec 6nica
-
CSIC
A da. Reina Me cedes,
s/n.
Edi . CICA,
E-41012,
Se illa, Spain
x uzzy- eam@ imse.cnm.es
Abs ac
Tuning a uzzy sys em o mee a gi en se o inpu ou pu
pa e ns is usually a di icul ask ha in ol es many pa am-
e e s.
This
pape p esen s an s udy o di e en app oaches
ha can be applied o pe o m his uning p ocess au oma -
ically, and desc ibes a CAD ool, named x sl, which allows
applying
a
wide se o hese app oaches: (a) a la ge numbe
o supe ised lea ning algo i hms; (b) di e en p ocesses
o
simpli y he lea ned sys em; (c) uning only speci ic pa am-
e e s o he sys em; (d) he abili y o une hie a chical uzzy
sys ems, sys ems wi h con inuous ou pu (like uzzy con-
olle ) as well
as
wi h ca ego ical ou pu (like uzzy classi-
ie s), and e en sys ems ha employ use -de ined uzzy
unc ions; and, inally, (e) he abili y
o
employ his uning
wi hin he design low o a uzzy sys em, because x sl
is
in-
eg a ed in o he uzzy sys em de elopmen en i onmen
X uzzy
3.0.
1.
In oduc ion
Tuning he sys em beha io is o en one o he mos di -
icul ask in he design low o a uzzy sys em. Much
o
he
uning e o
is
dedica ed o sea ch a p ope con igu a ion o
he sys em pa ame e s, because he e
is
usually a la ge
numbe o pa ame e s. To con on his ask, supe ised
lea ning algo i hms a e commonly used
as
au oma ic uning
me hods. In supe ised lea ning echniques, he desi ed sys-
em beha io
is
desc ibed by a se o inpu ou pu pa e ns
and he objec i e
is
o
minimize he e o be ween he de-
si ed and he cu en sys em beha io .
The pape
is
s uc u ed
as
ollows. Sec ion
2
p esen s he
di e en e o unc ions ha can be employed in supe ised
lea ning. Sec ion
3
desc ibes b ie ly some amilies o supe -
ised lea ning algo i hms. The p oblem
o
uning unde
cons ain s
is
add essed in Sec ion
4,
while Sec ion
5
sum-
ma izes some simpli ica ion p ocesses ha can be easily
pe o med a e uning.
Mos
o hese possibili ies has been
included in o a CAD
ool,
named x sl,
so
as
o
au oma e he
uning p ocess o complex uzzy sys ems. This
ool
is b ie -
ly desc ibed in Sec ion
6.
Finally, Sec ion
7
and
8
show se -
e al examples
o
illus a e he uning
o
sys ems wi h ei he
con inuous o ca ego ical ou pu s.
2.
The
e o
unc ion
The i s s ep in elabo a ing a supe ised lea ning p oc-
ess
supposes desc ibing sys em de ia ion by means o a
unc ion, known
as
e o unc ion.
A e y commonly used
e o unc ion
is
he mean squa e e o (MSE):
whe e
N
is
he numbe o da a pa e ns,
M
is
he numbe o
ou pu a iables in he sys em,
y,
is
he
j- h
ou pu gene a -
ed by he sys em o he i- h pa e n,
j..
is he co ec ou -
pu exp essed by
he
aining pa e n, and
j
is he ange o
he j- h ou pu
ha
is
used
o
no malize he de ia ions.
I can be use ul o he designe o selec he ela i e in-
luence o e e y ou pu a iable on he global de ia ion
om
i s
in ended beha io .
The
ollowing unc ion can be
used
in
his case:
V
whe e
wj
is he weigh o he j- h ou pu a iable on he
glo-
bal sys em e o . These weigh s should be no malized
so
as
o
sum
1.
I can be also use ul o employ he absolu e alue ins ead
o he quad a ic e o , wi h he co esponding op ions
o
a iable no maliza ion and weigh accoun ing:
The abo e exp essions assume a nume ical ou pu om
he uzzy sys em. Howe e ,
i
is
possible o de ine uzzy
sys ems whose ou pu s
a e
linguis ic labels, as
is
he case o
classi ie s. In hese sys ems, he ou pu alue
is
he linguis-
ic label p esen ing
he
highes ac i a ion deg ee as a esul
o he in e ence p ocess.
A
common de ini ion o he de i-
This
wo k
has
been
padally
suppo ed
by
he Spanish
CICYT
P ojec
TIC2001-1726.
0-7803-7810-5/03/517.00
WO03
IEEE
226
The
IEEE
In e na ional Con e ence on Fuzzy Sys ems
a ion in he beha io
o
his kind o sys em
is
he numbe o
classi ica ion e o s:
11
CE
=
-. -.
z
6..
N
M
i,
II
(4)
whe e
Si
is
1
when he classi ica ion o he pa e n has been
inco ec and
0
o he wise. This ype
o
unc ion conside s
equally all classi ica ion ailu es, wi hou aking in o ac-
coun he dis ance om a co ec classi ica ion.
To
conside
his in o ma ion i
is
necessa y o add a new e m like he
ollowing:
ACE
=
-'
Z6..
NM+I
i,i
J
(5)
wi h
whe e
Gj
is
he ac i a ion deg ee o he co ec label, and
ai
is
he ac i a ion deg ee o he label selec ed by
he
sys-
em.
Ano he way o aking in o accoun he dis ances om
igh classi ica ions
is
o
conside he ollowing classi ica-
ion squa e e o , which
is
a di e en iable unc ion:
wi h
1
i
y..=j..
0
'I
0
i y.. j..
{
'J
[J
p..
=
The choice o an adequa e e o unc ion o he lea ning
p ocess depends bo h on he ype o uzzy sys em o be
uned and on he algo i hm selec ed o pe o ming he
p ocess.
Fo
example, i
he
lea ning algo i hm belongs o
he amily o g adien descen algo i hms, he e o unc ion
mus be de i able,
so
ha he classi ica ion e o s
CE
and
ACE
can no be used.
3.
Supe ised lea ning algo i hms
Since he objec i e o supe ised lea ning algo i hms
is
o minimize an e o unc ion, hey can be conside ed
as
al-
go i hms o unc ion op imiza ion. Some supe ised lea n-
ing algo i hms ha can be used o une uzzy sys ems a e
b ie ly desc ibed in he ollowing.
3.1.
G adien descen algo i hms
The equi alence be ween uzzy and neu al ne wo ks led
o
apply he neu al lea ning p ocesses
o
uzzy in e ence
sys ems. In his sense,
a
well-known algo i hm employed in
uzzy sys ems
is
he
BackP opaga ion
algo i hm, which
modi ies he pa ame e alues p opo ionally
o
he g adien
o he e o unc ion in o de o each
a
local
minimum.
Since he con e gence speed o his algo i hm
is
slow, se -
e al modi ica ions we e p oposed like using
a
di e en
lea ning a e o each pa ame e o adap ing heu is ically
he con ol a iables o he algo i hm, hus leading o
Back-
P opaga ion wi h Momen um, Adap i e Lea ning Ra e,
Adap i e S ep Size
o
Manha an
algo i hms. An in e es ing
modi ica ion ha imp o es g ea ly he con e gence speed is
o ake in o accoun he g adien alue o wo successi e i -
e a ions. This idea
is
ollowed by he algo i hms
Quickp op
and
RP op
[l].
3.2.
Conjuga e g adien algo i hms
Since he g adien indica es he di ec ion o maximum
unc ion a ia ion, i may be con enien o gene a e no only
one s ep bu se e al s eps which minimize he unc ion e o
in ha di ec ion. This idea, which
is
he basis
o
he's eep-
es -descen
algo i hm, has he d awback o p oducing a zig-
zag ad ancing because he op imiza ion in one di ec ion
may de e io a e p e ious op imiza ions. The solu ion is o
ad ance by conjuga e di ec ions ha do no in e e e each
o he . The se e al conjuga e g adien algo i hms epo ed in
he li e a u e (like
Polak-Ribie e, Fle che -Ree es,
Hes enes-S ie el,
and
One-s ep Secan )
di e in he equa-
ions used o gene a e he conjuga e di ec ions. The main
d awback o he conjuga e g adien algo i hms
is
he imple-
men a ion o a linea sea ch in each di ec ion, which may be
cos ly in e ms o unc ion e alua ions. The line sea ch can
be a oided by using second-o de in o ma ion,
as
done by
he
scaled conjuga e g adien
[2].
3.3.
Second-o de algo i hms
A o wa d s ep owa ds speeding up he con e gence o
lea ning algo i hms is o make
use
o second-o de in o ma-
ion o he e o unc ion. Since
he
calculus o he second
de i a i es
is
complex, one solu ion
is
o app oxima e he
Hessian by means o he g adien alues o successi e i e -
a ions. This is he idea o he algo i hms o
B oyden-Fle ch-
e -Golds@-Shanno
and
Da idon-Fle che -Powell
[3].
An
special case
is
when he unc ion o minimize
is
a quad a ic
e o because, in his case, he Hessian can be app oxima ed
by only he i s de i a i es o he e o unc ion, as done by
he
Gauss-New on
algo i hm. Since his algo i hm can lead
o ins abili y when he app oxima ed Hessian is no de ined
posi i e,
he
Ma qua d -Le enbe g
algo i hm sol es his
p oblem by in oducing an adap i e e m.
3.4.
Algo i hms wi hou de i a i es
The g adien o
he
e o unc ion can no be always cal-
cula ed because i can be oo cos ly o no de ined. In hese
cases, op imiza ion algo i hms wi hou de i a i es can be
227
The
IEEE
In e na ional Con e ence
on
Fuzzy Sys ems
employed. An example is he
Downhill Simplex
algo i hm,
which conside s
a
se o unc ion e alua ions o decide
a
pa-
ame e change. Ano he example is
Powell’s me hod,
which implemen s linea sea ches by
a
se o di ec ions ha
e ol e o be conjuga e
141.
These algo i hms a e oo much
slowe han he p e ious ones, A bes solu ion can be o es-
ima e he de i a i es om he secan s
o
o
employ no he
de i a i e alue bu i s sign (as
RP op
does), which can
be
es ima ed om small pe u ba ions o he pa ame e s.
3.5.
S a is ical algo i hm
All he abo e commen ed algo i hms do no each he
global bu
a
local minimum o he e o unc ion. The s a is-
ical algo i hms can disco e he global minimum because
hey gene a e di e en sys em con igu a ions ha sp ead
he sea ch space. One way o b oadening he space explo ed
is o gene a e andom con igu a ions and choose he bes o
hem. This is done by he
blind sea ch
algo i hm whose
con e gence speed is ex emely slow. Ano he way is o
pe o m small pe u ba ions in he pa ame e s
o
ind a be -
e con igu a ion as done by he
algo i hm
o
i e a i e im-
p o emen s.
A be e solu ion is o employ
simula ed
annealing algo i hms
[4].
They a e based on an analogy be-
ween he lea ning p ocess, which is in ended
o
minimize
he e o unc ion, and he e olu ion o
a
physical sys em,
which ends o lowe i s ene gy
as
i s
empe a u e dec eases.
Se e al annealing schemes (like linea , exponen ial, classic,
as
o
adap i e) ha e been p oposed, p oducing di e en
e sions o he simula ed annealing algo i hm.
4.
Tuning uzzy sys ems unde cons ain s
The pa ame e s
o
adjus
in
a
uzzy sys em usually ha e
o
mee
se e al
cons ain s.
Fo
ins ance,
when
uning
he
pa ame e s o a Gaussian membe ship unc ion, he lea ning
algo i hms should always ejec
a
nega i e alue o he pa-
ame e ep esen ing he wid h o he unc ion. The con-
s ain s ha usually appea when uning
a
uzzy sys em
pa ame e .pi, a e he ollowing:
“pi
<=
cons an ”,
<
con-
s an ”,
‘pi
>=
cons an ”, “pi
>
cons an ”,
o
‘>i
<
pj”.
The
la e ones appea , o ins ance, be ween he h ee poin s
ha can de ine a iangula membe ship unc ion. They a e
he mos di icul cons ain s
o
main ain and should be
a oided as much
as
possible.
In
his
sense,
a
iangula unc-
ion is be e de ined by i s cen e , wid h, and slope.
S a is ical algo i hms manage cons ain s easily, by di-
ec ly ejec ing he andom con igu a ion gene a ed i i
does
no mee he cons ain s. On he o he hand, he algo-
i hms based on any kind o g adien descen gene a e he
same (de enninis ic) displacemen a
a
gi en i e a ion,
so
ha he solu ion is no o ejec i i i is o bidden (i would
be again gene a ed in he nex i e a ion) bu o change i in o
ano he displacemen accep ed by he sys em. The
usual
so-
P2
P2
I
I
c
I
I
Figu e
1:
Lea ning unde cons ain s.
lu ion is o y an smalle displacemen in he same di ec-
ion,
as
shown in Figu e
la.
The p oblem is ha his solu ion
does no dis inguish be ween pa ame e s,
so
ha he
con-
s ain on one pa ame e can limi no only he displacemen
o ha pa ame e bu also o o he ones (as shown in Figu e
la).
This p oblem can be a oided by managing he pa ame-
e s independen ly. This means, in he example o Figu e
I,
ha he sys em could
be
mo ed o he
inal
poin shown in
Figu e Ib. Fo hose cons ain s ela ing se e al pa ame e s
(as hose o he iangle shown in Figu e IC),
a
good solu ion
is o conside hem
as
pa icles subjec
o
inelas ic collisions
in
a
one dimensional space (Figu e Id).
Ano he in e es ing poin o ema k is ha he s able
poin eached by he sys em a e lea ning unde cons ain s
will p o ide
o
no he minimum possible e o depending
on he uning algo i hm employed. When he p oposed dis-
placemen ollows he g adien di ec ion, he sys em
e ol es o
a
poin whe e he g adien componen s o e he
non cons ained pa ame e s a e null, ha is, o he poin o
he on ie whe e he e o
is
minimum
( he
poin
“b”
in
Figu e
2).
On he o he hand, i he lea ning algo i hm gen-
e a es a di ec ion which is no pa allel
o
he g adien (like
ha shown wi h
a
dashed line in Figu e
3).
he s able poin
o he sys em will no p o ide
a
minimum
e o
( he poin
“
1,
.
a
in Figu e
2).
5.
Simpli ica ion p ocesses
Supe ised leaming can
be
used no only o une uzzy
sys ems bu
also
o help ob aining uzzy models in iden i i-
o bidden
egion
Figu e
2:
S able poin s depending on he algo i hm.
228
The
IEEE
In e na ional Con e ence
on
Fuzzy Sys ems
ca ion p oblems. Simpli ying he uzzy sys em ob ained
a e a uning p ocess allows ex ac ing a aluable in o ma-
ion abou he logical s uc u e o he sys em. One
o
hese
simpli ica ion p ocess consis s in de ec ing and dele ing
hose uzzy ules and membe ship unc ions ha a e ne e
ac i a ed su icien ly by any
o
he aining inpu lou pu
pa e ns.
A
ypical esul o he uning p ocess is ha he membe -
ship unc ions co e ing he ou pu a iables o e lap each
o he in a high deg ee. A clus e ing p ocess o e hese unc-
ions d i es o a good and simple uzzy sys em. This clus-
e ing can be made au oma ically by means o he
Ha d
C-
means
algo i hm, whe e he numbe o clus e can be ixed
manually,
o
can be selec ed by some clus e e alua ion
unc ions
[5].
6.
The
X sl
ool
In o de o au oma e he uning p ocess o a uzzy sys em
we ha e de eloped he ool
x sl.
X l
includes all he
e o
unc ions and supe ised lea ning algo i hms desc ibed in
Sec ions
2
and 3, and apply he solu ions desc ibed in Sec-
ion 4 o allow lea ning e icien ly unde cons ain s. Sim-
pli ica ion me hods desc ibed in Sec ion
5
a e also included
and can be execu ed p io o
o
a e he lea ning algo i hms.
In addi ion, he sys em pa ame e s o une can be selec ed
by a g aphical in e ace.
Xjsl
allows he use o apply supe ised lea ning algo-
i hms o uzzy sys ems speci ied wi h he
XFL3
language
[6],
he o mal language
o
X uzzy 3.0
[7].
XFL3 pe mi s
he desc ip ion o complex uzzy sys ems wi h hie a chical
ule bases. Besides, he e
is
no limi a ion in he numbe o
ules wi hin a ule base, linguis ic a iables, o linguis ic
labels co e ing he a iables. The ules suppo complex
logic ela ions in he p emise pa (wi h conjunc ions, dis-
junc ions, and linguis ic hedges), and hese ope a o s as
well
as
he implica ion ope a o s, membe ship unc ions,
o
de uzzi ica ion me hods can be de ined eely by he use .
The language
XFL3
is he nexus be ween
he
di e en
Figu e
3:
Main window
o
he ool
xjsl.
229
X uzzy 3.0 ools (dedica ed o desc ip ion, lea ning, e i i-
ca ion,
o
syn hesis
o
uzzy sys ems).
Figu e
3
illus a es he main window o
x sl.
This win-
dow
is
di ided in o ou pa s. The le uppe co ne
is
he
a ea
o
con igu e he lea ning p ocess. The p ocess s a e
is
shown a he igh uppe pa . The cen al a ea illus a es he
e olu ion o he lea ning, and he bo om pa con ains se -
e al con ol bu ons o un
o
s op he p ocess,
o
sa e he e-
sul s, and o exi .
7.
Fuzzy sys ems wi h con inuous ou pu
Since uzzy sys ems wi h con inuous ou pu can be seen
as in e pola o s, le
us
conside , as example, he p oblem o
app oxima ing he unc ion shown a Figu e 4a. We con-
side a wo-inpu uzzy sys em wi h
7
Gaussian membe -
ship unc ions pe inpu , hus con aining 49 ules. The
Weigh ed Fuzzy Mean is selec ed as de uzzi ica ion
me hod. Ini ially, inpu membe ship unc ions a e homoge-
neously dis ibu ed in hei uni e se o discou se, while he
49 ou pu unc ions a e equal and cen e ed in hei uni-
e se.
All
he pa ame e s o hese membe ship unc ions as
well as he weigh s a e going o be uned, which means
126
pa ame e s.
Figu es 4b, c, and d show he e olu ion
o
he sys em
beha io while being uned by he di e en lea ning algo-
i hms p o ided by
x sl.
The g adien descen algo i hms
a e shown in Figu e 4b. I can be seen ha modi ica ions
o
he
BackP opaga ion
algo i hm no o iously inc ease he
con e gence speed. Figu e 4c
is
dedica ed o he conjuga e
g adien and second o de algo i hms. As i
is
shown on
his igu e, hese algo i hms a e signi ican ly as e han he
s eepes
descen
algo i hm, especially
BFGS
and
Ma -
qua d -Le enbe g
algo i hms. Algo i hms wi hou de i a-
i es and s a is ical algo i hms a e shown
in
Figu e
4d.
These algo i hms a e se e al o de s
o
magni ude slowe
han he p e ious ones,
so
hei use
is
only ecommended
when hose a e disca ded (in non-de i able sys ems, o
ins ance). I can be seen ha
Powell's
algo i hm
is
much
as e han
Downhill Simplex
algo i hm. Conce ning
he
s a is ical algo i hms,
Simula ed Annealing
inc eases he
con e gence speed wi h espec
o
Blind Sea ch
o
I e a i e
Imp o emen
algo i hms.
The exis ence o non-linea pa ame e s gene a es he
p esence o se e al local minima in
he
uning p ocess.
The e o e,
i
is no possible o asse wha
is
he bes algo-
i hm, since a e y as algo i hm may be some imes d i en
o a local minimum a away om he op imum beha io .
A
solu ion
o
his
p oblem
is
o
make se e al uning p ocesses
wi h di e en andom ini ial con igu a ions, selec ing he
bes
o
he lea ning esul s.
Wi hin he lea ning p ocess. he membe ship unc ions
o
he ou pu a iable end
o
g oup a ound some common
The
IEEE
In e na ional
Con e ence
on
Fuzzy Sys ems
3
Adap i e Lea ning
Ra e
'I
lo3
lo4
CPUTime
1
ob
RMSE
(b)
100
10'
d
1
U-
10.'
1
0-2
10-3
104
10-5
1
S eepes Descen
2
Ccnjuga eG adien
3
Scaled Conjuga e-
G adien
4
BFGS
5
1
5DFP
1
IO*$
10
10'
le
lo3
IO'
CPU
Time
RMSE
(C)
1
Downhill Simplex
2
Powell
3
Blind
Sea ch
4
I e a i e Impm e-
5
Sim. Annealing
men
o*
10
10'
le
10'
IO4
10
CPUTime
(d)
Figu e
4:
Compa ison o he di e en algo i hms.
o ms (Figu e 5a). Applying he clus e ing p ocess
sup-
po ed by
x l,
he
49
membe ship unc ions a e educed o
6 (Figu e 5b). This educ ion leads o he simpli ied ule
base shown in Figu e 5c.
The use o hie a chical s uc u es allows simpli ying he
desc ip ion o
a
sys em
because complex beha io s can be
gene a ed by composing simple
ule
bases.
A
ele an ad-
an age o he
x s/
ool
is i s abili y
o
adjus hie a chical
sys ems. To illus a e his ype
o
lea ning p ocess, we will
conside again he p oblem o app oxima ing he beha iou
a Figu e 4a, bu now using a hie a chical uzzy sys em wi h
wo cascaded ule bases, like ha shown in Figu e 6a. The
ini ial desc ip ion we ha e aken o he i s ule base
is
e y simple: wo uzzy se s o each inpu a iable and ou
single on alues o he ou pu , hus gi ing
4
ules (Figu e
6b). The second ule base employs only one uzzy
se
o he
21
21
22
23
zS
26
26
zl
zl
21
B
23
z5
26
Figu e
5:
Rule
base ob ained a e lea ning and clus e ing.
inpu because wo o he ones a e gene a ed by using linguis-
ic hedges, and h ee single on alues o he ou pu (Figu e
6c). The de uzzi ica ion me hod pe o med by bo h ule
bases is he Fuzzy Mean me hod. Since ini ially
all
he ou -
pu alues a e equal, he inpu -ou pu ela ion p o ided by
his sys em is la .
The in luence o he pa ame e s on he global beha io
o
a
hie a chical sys em is complex. In e ms o lea ning, his
means he exis ence
o
a
lo
o local minima which make
no
use ul he applica ion
o
g adien based lea ning algo i hms.
Con a y o he p e ious example, he s a is ical algo i hms
p o ide now be e esul s.
In
pa icula , we ha e used he
Blind
Sea ch
algo i hm ollowed by
Ma qua d -Le en-
be g's
algo i hm.
In
he la e algo i hm,
x sl
does no com-
pu e he de i a i es (i is no possible in hie a chical
sys ems) bu es ima es hem om small pa ame e modi i-
ca ions.
X
Y
(b)
(4
Figu e 6. Tuning
a
hie a chical sys em
230
The
IEEE
In e na ional Con e ence
on
Fuzzy
Sys ems

A e lea ning, he global sys em beha io app oxima es
he a ge beha io wi h an RMSE o 0.41%. Wha is e y
in e es ing is ha he ule bases ha e lea n he in insic
composi ion
o
he a ge unc ion. The i s ule base iden-
i ies
a
sub ac ing ela ion be ween y and x (wi h a ce ain
scaling ac o ), while he second ule base iden i ies an s ep-
wise ela ion. Be ween
25
and 36 ules a e equi ed by
a
g id-based uzzy sys em (using he Fuzzy Mean de uzzi i-
ca ion me hod) o pe o m as well as a hie a chical sys em
wi h only 7 ules, hus showing he impo ance o uning hi-
e a chical desc ip ions o
a
CAD ool.
8. Fuzzy sys ems
wi h
disc e e ou pu
A
uzzy classi ie can be seen
as
a
uzzy sys em in which
he membe ship unc ions o he ou pu a iable ep esen
he di e en ca ego ies o which he ou pu may belong. The
ou pu p o ided by
a
uzzy classi ie is gene ally he ou pu
membe ship unc ion wi h he highes ac i a ion deg ee.
Since he ou pu alues
o
hese sys ems a e ca ego ies, he
lea ning p ocess aces an addi ional obs acle. I should
modi y he pa ame e s o he membe ship unc ions associ-
a ed
o
he inpu a iables o imp o e he success a e o he
classi ica ion. Con a y o he case o uzzy in e pola o s,
uzzy classi ie s can pe o m be e when educing hei ule
bases since classi ica ion bounda ies no pa allel o he g id
pa i ion can be ob ained.
As an example o uning a uzzy classi ie , le
us
consid-
e he p oblem illus a ed in Figu e
7.
I shows a se o
80
da a g ouped in o 4 di e en ca ego ies wi h
20
da a each
one (Cl, C2, C3, and C4). The uzzy classi ie o
be
uned
by
~ sl
con ains
9
ules ini ially, wi h
3
membe ship unc-
ions co e ing each inpu a iable,
as
shown a he op and
le pa s o Figu e
7.
Figu e 7: Example
o
a
classi ica ion p oblem.
Applying he p uning p ocess
o
x sl,
he
9
ules a e e-
duced o
6,
and he membe ship unc ions a e lea ned as
shown a he bo om and igh pa s o Figu e 7. The classi-
ica ion bounda ies implemen ed by he
6
ules each
a
clas-
si ica ion a e o
100%.
as
shown in Figu e
7.
9. Conclusions
The ool
x sl
p esen ed he ein ep esen s an impo an e -
o owa ds he.au oma iza ion o he lea ning p ocess in
he design o uzzy sys ems. The wide se
o
algo i hms in-
cluded ( om g adien -based o s a is ical) allows sol ing
many applica ion p oblems. The inco po a ed me hods o
clus e ing and p uning pe mi s he simpli ica ion o he
uzzy sys em conside ed. I s capabili y o uning hie a chi-
cal uzzy sys ems makes i also possible o simpli y he de-
sc ip ion o
a
sys em because complex beha io s can be
usually gene a ed by composing simple ule bases. I s abil-
i y
o
wo k wi h sys ems ha employ linguis ic hedges al-
lows adjus ing he sys em as well as main aining i s
linguis ic meaning, which is e y in e es ing when ex ac -
ing knowledge om da a. Since he ool is in eg a ed in o
he en i onmen X uzzy 3.0, i is possible no only o une a
sys em bu
also
using o he ools
o
g aphically de ine i , o
ep esen i s beha io by 2-D o 3-D plo s, and o simula e,
moni o
o
syn hesize so wa e desc ip ions
o
i .
As
pa
o
X uzzy 3.0,
x sl
is dis ibu ed eely unde he
GNU
Gene al
Public License om he X uzzy o icial web page (h p://
www.imse.cnm.es/X uzzy/).
Re e ences
[I]
S.
Haykin, “Neu al Ne wo ks. A Comp ehensi e Founda-
ion”, IEEE P ess Macmillan, 1994.
[2] Mglle , M.F.,
“A
Scaled Conjuga e G adien Algo i hm o
Fas Supe ised Leaming”, Neu al Ne wo ks, Vol.
6,
pp. 525-
533.1993.
[3] Scales,
L.E..
“In oduc ion
o
Non-Linea Op imiza ion”,
Sp inge -Ve lag New Yo k Inc., 1985.
[4] P ess,
W.H.,
Teukolsky,
S.A.,
Ve e ling,
W.T.,
Flmne y,
B.P., Nume ical Recipes
in
C,
Camb idge Uni e si y
P ess,
1997.
[SI
Bezdek, J.C., Pal,
N.R.,
“Clus e alida ion
wi h
gene alized
Dum’s indices”, P oc. 2nd
NZ
In . Two-S eam Con .
on
ANNES, pp. 190-193, Dunedin, 1995.
[6]
F.
I.
Mo eno-Velo,
S.
Sinchez-Solano,
A.
Baniga,
1.
Ba u one, D. R. Lopez,
“XFL3:
A
New Fuzzy Sys em Speci-
ica ion Language”, Ma hwa e
&
So Compu ing, pp. 239-
253, Decembe
2M)I.
[7] F.
J.
Mo eno-Velo, I. Ba u one,
S.
Sinchez-Solmo,
A.
Ba i-
ga, “Rapid Design o Fuzzy Sys ems
wi h
XFUZZY, sub-
mi ed
o
FUZZ-lEEE2003.
231
The
IEEE
In e na ional Con e ence
on
Fuzzy Sys ems