Remainde Subse Awa eness o Fea u e Subse Selec ion
Gab iel P a -Mas amon Llu´ıs A. Belanche-Mu˜noz
Facul y o Compu e Science Facul y o Compu e Science
Poly echnical Uni e si y o Ca alonia Poly echnical Uni e si y o Ca alonia
Ba celona, Spain Ba celona, Spain
[email p o ec ed]c.edu [email p o ec ed]
2007-05-29
Abs ac
Fea u e subse selec ion has become mo e and
mo e a common opic o esea ch. This popu-
la i y is pa ly due o he g ow h in he num-
be o ea u es and applica ion domains. The
amily o algo i hms known as plus-l-minus-
and i s immedia e de i a i es (like o wa d
selec ion) a e e y popula and o en he only
iable al e na i e when used in w appe mode.
In consequence, i is o he g ea es impo ance
o ake he mos o e e y e alua ion o he in-
duce , which is no mally he mo e cos ly pa .
In his pape , a echnique is p oposed ha
akes in o accoun he induce e alua ion bo h
in he cu en subse and in he emainde sub-
se (i s complemen a y se ) and is applicable
o any sequen ial subse selec ion algo i hm a
a easonable o e head in cos . I s easibili y is
demons a ed on a se ies o benchma k da a
se s.
1 In oduc ion
In he las ew yea s ea u e selec ion has be-
come a mo e and mo e common opic o e-
sea ch, a ac p obably due o he in oduc ion
o new applica ion domains and he g ow h o
he numbe o ea u es in ol ed. An exam-
ple o hese new domains is web page ca ego-
iza ion, a domain cu en ly o much in e es
o in e ne sea ch engines whe e housands
o e ms can be ound in a documen . An-
o he example is ound in appea ance-based
image classifica ion me hods which may use
e e y pixel in he image. Classifica ion p ob-
lems wi h many ea u es a e also e y common
in medicine and biology; e.g. molecule classi-
fica ion, gene selec ion o medical diagnos ics.
Fea u e selec ion can help in sol ing a clas-
sifica ion p oblem wi h i ele an and/o e-
dundan ea u es o many easons. Fi s i
can make he ask o da a isualiza ion and
unde s anding easie by elimina ing i ele an
ea u es which can mislead he in e p e a ion
o he da a. I can also educe cos s since he
measu emen o eco ding o some o he ea-
u es can be a oided; his is especially impo -
an in domains whe e some ea u es a e e y
expensi e o ob ain, e.g., a cos ly o in asi e
medical es . In addi ion, a big benefi o ea-
u e selec ion is in de ying he cu se o dimen-
sionali y o help he induc ion o good classi-
fie s om he da a. When many unuse ul,i.e.
i ele an o edundan , ea u es a e p esen
in he aining da a, classifie s a e p one o
finding alse egula i ies in he inpu ea u es
and lea n om ha ins ead o lea ning om
he ea u es ha eally de e mine he ins ance
class ( his is also alid when p edic ing he in-
s ance a ge alue in he case o eg ession).
This wo k add esses he p oblem o selec -
ing a subse o ea u es om a gi en se by in-
oducing a gene al-pu pose modifica ion o
ea u e subse selec ion algo i hms which i -
e a i ely selec and disca d ea u es. An im-
po an amily o algo i hms o ea u e sub-
se selec ion pe o m an explici sea ch in he
space o subse s by i e a i ely adding and/o
emo ing ea u es one a a ime un il some
s op condi ion is me . The idea is hen o
1
use he e alua ion o he induce in he so-
called emainde se ( he se complemen a y
o he cu en subse o selec ed ea u es) as
an addi ional sou ce o in o ma ion. This in-
o ma ion is used in conjun ion o con en ional
algo i hms, ha only use he e alua ion on
he cu en subse o selec ed ea u es. In ou
expe imen al esul s, such simple modifica ion
achie es significan imp o emen s in he max-
imiza ion o he objec i e unc ion o classifi-
ca ion asks. The modified algo i hms ob ain
simila numbe s o selec ed ea u es in gene al,
o a u he educ ion i pu ely o wa d algo-
i hms a e used.
The es o he pape is o ganized as ollows:
fi s we b iefly e iew he ea u e subse selec-
ion p oblem (Sec ion 2). Sec ion 3 in oduces
he concep o he emainde se o ea u es
and sugges s a modifica ion o he p e iously
p esen ed algo i hms. In he nex sec ion he
expe imen al design is defined and he esul s
a e exposed and discussed. Finally, sec ion 5
ex ac s conclusions abou hese esul s and
in oduces some u u e wo k.
2 Fea u e Subse Selec ion
The e a e wo main app oaches o ea u e se-
lec ion: fil e me hods and w appe me hods.
These wo amilies o me hods only diffe in
he way hey e alua e he candida e se s o
ea u es. While he o me uses a p oblem in-
dependen c i e ion, he la e uses he pe -
o mance o he final classifie o e alua e he
quali y o a ea u e subse . The basic idea o
he fil e me hods is o selec he ea u es ac-
co ding o some p io knowledge o he da a.
Fo example, selec ion o ea u es based on he
condi ional p obabili y ha an ins ance is a
membe o a ce ain class gi en he alue o
i s ea u es [1]. Ano he c i e ion commonly
used by fil e me hods is he co ela ion o a
ea u e wi h he class, i.e. selec ing ea u es
wi h high co ela ion [3]. In con as , w appe
me hods sugges a se o ea u es ha is hen
supplied o a classifie , which uses i o classi y
he aining da a and e u ns he classifica ion
accu acy o some o he measu e he eo [5].
I is common o see ea u e subse selec ion
in a se Yo size nas an op imiza ion p ob-
lem whe e he sea ch space is P(Y)[6]. In
his se ing, he ea u e selec ion p oblem is o
find an op imal subse X∗∈P(Y)whichmax-
imizes a gi en c i e ion J:P(Y)→[0,1] as
seen in eq. (1). Ha ing J(X) as he e alua ion
c i e ion is a common cha ac e is ic o many
ea u e selec ion algo i hms. The c i e ion J
may be p oblem-independen o may be he
classifie ha will be used o sol e a classifi-
ca ion p oblem, and hus his se ing is alid
ei he o fil e o w appe algo i hms. In any
case, we will e e o J(X)as heuse ulness o
ea u e subse X.
X∗= a g max
X∈P(Y)
J(X)(1)
In he li e a u e, se e al subop imal algo-
i hms ha e been p oposed o doing his.
Among hem, a wide amily is o med by hose
algo i hms which, depa ing om an ini ial so-
lu ion, i e a i ely add o dele e ea u es by lo-
cally op imizing he objec i e unc ion. The
sea ch s a s wi h an a bi a y se o ea u es
(e.g. he ull se o he emp y se ) and mo es
i e a i ely o neighbo solu ions by adding
o emo ing ea u es. These sequen ial algo-
i hms wi h no back acking can be cas in he
gene al class o hill-climbing algo i hms. They
lea e a sequence o isi ed s a es Xk1,...X
km,
whe e he size o e e y Xkiis kiand mis he
numbe o isi ed s a es. The difficul y o he
ea u e selec ion p oblem can be illus a ed by
he ollowing ac s:
1. In gene al, i is no he case ha
J(Xki+1 )≥J(Xki)(no e en o hesim-
ples algo i hms like SFG o SBG).
2. The e may exis “ugly” ea u e subse s
along he way Xkisuch ha J(Xki)<
J(∅), o some i≥0.
3. Take Xkiand Xki+1 such ha Xki⊂
Xki+1 and ki+1 =ki+ 1. In gene al, i is
no he case ha J(Xki+1 )≥J(Xki).
4. Calling X∗
k+1 a cu en op imal solu ion
wi h k+ 1 ea u es, i is no he case ha
J(X∗
k+1) con ains J(X∗
k). This is known
as he nes ing p oblem.
32
II Cong eso Español de In o má ica
Exp essed mo e succinc ly, any pa ial o -
de wi h espec oJin he se P(X)iscon-
cei able. Among he p oposed algo i hms o
a acking his p oblem a e he sequen ial o -
wa d gene a ion (SFG) and sequen ial back-
wa d gene a ion (SBG), he plus l- ake away
o PTA(l, ) p oposed by S ea ns [10] o he
floa ing sea ch me hods [9]. They bo h in o-
duce me hods o he gene a ion o he se s o
ea u es by combining s eps o SFG wi h s eps
o SBG bu keep using a ce ain J(X)ase al-
ua ion c i e ion.
X0←∅//Ini ial subse
i←0
epea
//Subse gene a ion

Si+1 ←{X|X=Xi∪{x}∧x∈Y Xi}
//Subse e alua ion
Xi+1 ←a g max
X∈
Si+1
J(X)
i←i+1
//S opping c i e ion
un il J(Xi)≤J(Xi−1)∨i=n
e u n Xi−1//Selec ed subse
Algo i hm 1: SFG
X0←Y//Ini ial subse
i←0
epea
//Subse gene a ion

Si+1 ←{X|X=Xi {x}∧x∈Xi}
//Subse e alua ion
Xi+1 ←a g max
X∈
Si+1
J(X)
i←i+1
//S opping c i e ion
un il J(Xi)≤J(Xi−1)∨i=0
e u n Xi−1//Selec ed subse
Algo i hm 2: SBG
Algo i hm 1 and Algo i hm 2 below de-
sc ibe wo o he classic ea u e selec ion algo-
i hms using his poin o iew: sequen ial o -
wa d gene a ion (SFG) and sequen ial back-
wa d gene a ion (SBG). In hese algo i hms
X0is he s a ing se o ea u es o he algo-
i hm, 
Sk he se o se s o ea u es gene a ed
du ing he subse gene a ion phase and Xk he
selec ed se o ea u es a i e a ion k.I can
be seen ha he subse e alua ion phase in he
wo algo i hms is exac ly he same while he
ini ializa ion and he subse gene a ion phases
change. No e ha a all imes he size o Xk
is kand hus Xn=Yand X0=∅.
3 The Remainde Se o Fea u es
As he goal o ea u e selec ion is o find an op-
imal subse X∗as seen in (1), i seems plausi-
ble o choose an Xk o each i e a ion as in (2)
in a s epwise and g eedy way, which is exac ly
wha he p e iously desc ibed ea u e selec ion
algo i hms do:
Xk= a g max
X∈
Sk
J(X),k=1,...,n (2)
In eal p oblems, ea u es a e a om in-
dependen , hus no always he bes ea u e
se in e e y i e a ion has o be he bes op-
ion. Qui e possibly he e is some combina-
ion o ea u es ha would be a be e choice
ha he ea u e which maximizes J(X) in his
i e a ion. In his ain, he o wa d s eps in
he p e ious algo i hms a e no aking in o ac-
coun some in o ma ion hey could use. Only
he use ulness o e e y gene a ed subse o ea-
u es is measu ed, as in (2). Howe e , by con-
side ing he cu en se o ea u es Xkano he
se is implici ly c ea ed, he se o emaining
ea u es o emainde se Yk=Y Xk. This
se can also gi e in o ma ion abou he new
a iable o be added o emo ed a e e y s ep.
I is ou conjec u e ha a way o enhance he
de ec ion o ea u e in e ac ions is o see how
he addi ion o a ea u e o Xk(a emo al,
om he poin o iew o Yk)affec s heuse-
ulness o he emainde se . The idea is o
add ha ea u e mos use ul o Xkand whose
emo al is mos ha m ul o Yk. An analogous
easoning can be made in backwa d s eps by
in e changing he oles o he cu en and e-
mainde subse s. The gene al idea is called
Remainde Subse Awa eness o ob ious ea-
sons.
Wi h his o mula ion we ha e a mul i-
objec i e p oblem, since no always he sub-
IV Talle de Mine ía de Da os y Ap endizaje
33
se wi h maximum J(Xk) will coincide wi h
he subse wi h minimum J(Yk), so i will no
be possible o sa is y bo h objec i es wi h he
same single solu ion. In his case, ei he he
wo solu ions ha e o be explo ed o a ade-
off has o be ound ha pa ly op imizes bo h
objec i es. I bo h solu ions a e chosen o
u he explo a ion, hen he sea ch space is
highly inc eased o e he o iginal e sion o
he algo i hm, and he complexi y o he algo-
i hm g ows om polynomial o exponen ial,
which is un easible. A easonable al e na i e
is o choose he subse which maximizes some
p edefined unc ion o he wo c i e ia among
he wo candida e subse s, as exp essed by:
a g max
X∈
Sk
[J(X),J(Y X)],k=1,...,n
(3)
The unc ion :(0,1)2→(0,1) has o be
chosen o be con inuous in bo h a gumen s, in-
c easing in he fi s and dec easing in he sec-
ond and o pe mi con ol on he ela i e im-
po ance o he wo a gumen s ( hus i is non-
symme ical). Following his al e na i e, an
algo i hm o he sequen ial kind can be modi-
fied by eplacing he e alua ion unc ion J(X)
wi h he one in Eq. 3. As an example, he
ollowing Algo i hm 3 shows he s aigh -
o wa d Remainde Subse Awa e e sion o
he o iginal SFG p esen ed in Algo i hm 1.
O he o wa d/backwa d algo i hms would be
modified analogously.
X0←∅//Ini ial subse
i←0
epea
//Subse gene a ion

Si+1 ←{X|X=Xi∪{x}∧x∈Y Xi}
//Subse e alua ion
Xi+1 ←a g max
X∈
Si+1
[J(X),J(Y X)]
i←i+1
//S opping c i e ion
un il J(Xi)≤J(Xi−1)∨i=n
e u n Xi−1//Selec ed subse
Algo i hm 3: Remainde se awa e SFG
The chosen e alua ion unc ion ,which
combines he use ulness o he selec ed subse
o ea u es wi h ha o he emaining subse
is shown in Eq. 4.
(x, y)=xk×(1 −y)1−k, k,x,y ∈[0,1] (4)
No e ha k= 1 eco e s he con en ional
algo i hms and k=0.5 co esponds o he ge-
ome ical mean be ween xand 1 −y.Ingen-
e al, lowe alues o kgi e mo e weigh o he
e alua ion o he induce in he emainde se .
4 Expe imen al wo k
The p e ious idea is fi s illus a ed using he
Co Al p oblem, a small da ase wi h some
specific cha ac e is ics ha make i use ul o
es ea u e subse selec ion algo i hms in a
known en i onmen [4]. This da ase has wo
classes and six boolean ea u es (A0;A1;B0;
B1;I;C). Fea u e Iis i ele an , ea u e Cis
co ela ed o he class label 75% o he ime,
and he o he ou ea u es a e ele an o he
boolean a ge concep : (A0∧A1)∨(B0∧B1).
SFG will choose Cfi s as i is he bes ea u e
when aken all alone [4]. The hypo hesis is
ha he use ulness o he emainde se would
be so high i Cwas chosen ha he modified
e sion o SFG would no choose i . A e un-
ning he expe imen s wi h Co Al, he hypo h-
esis was confi med: a con en ional SFG chose
he ea u es in he o de {C, I, A0,A
1,B
0,B
1},
whe eas he modified emainde se awa e e -
sion chose he o de {A0,A
1,B
0,B
1,C,I}.
4.1 Expe imen al se ings
Expe imen al wo k is now p esen ed in o -
de o assess he desc ibed modifica ion wi h
a g oup o ou sequen ial algo i hms, using
some well known da ase s om he UCI epos-
i o y o machine lea ning da abases [2]. The
amily PTA(l, ) has been selec ed o ca y
ou he expe imen s, compa ing hei o igi-
nal e sions and he modified ones, which a e
awa e o he emainde se o non-selec ed ea-
u es. Fou diffe en combina ions o alues
o land ha e been es ed as seen on Table
1. No e SFG can be seen as a pa icula case o
PTA(l, )wi hl=1and = 0 and e e ed
34
II Cong eso Español de In o má ica
Table 1: Tes ed alues o he PTA(l, )pa-
ame e s ( o wa d and backwa d s eps)
Algo i hm Fwd s . Bwd s .
PTA(0,1) ≡SBG 01
PTA(1,0) ≡SFG 10
PTA(1,2) 1 2
PTA(2,1) 2 1
o as PTA(1,0). The same can be done o
SBG calling i PTA(0,1). The alue o kin
eq. (4) was se o 0.8 a e some p elimina y
expe imen s and should be aken only as an
educa ed guess.
Each pai o algo i hms has been es ed wi h
he ollowing da ase s:
Ionosphe e Classifica ion o ada e u ns
om he ionosphe e. The e a e 2 classes,
351 ins ances, 34 nume ic ea u es. The
a ge s we e ee elec ons in he iono-
sphe e. ”Good” ada e u ns a e hose
showing e idence o some ype o s uc-
u e in he ionosphe e. ”Bad” e u ns
a e hose ha do no : hei signals pass
h ough he ionosphe e.
Mammog am Mammog aphy da a dona ed
by he Pa e n Recogni ion and Image
Modeling Labo a o y a Uni e si y o
Cali o nia, I ine. The e a e 86 cases wi h
65 ea u es each and a bina y class indi-
ca ing benign o malignan .
Spec The da ase desc ibes diagnosing o
ca diac Single P o on Emission Com-
pu ed Tomog aphy (SPECT) images.
Each o he pa ien s is classified in o wo
ca ego ies: no mal and abno mal. The e
a e 22 bina y ea u es ex ac ed om
he o iginal SPECT images and 267 in-
s ances.
Spec The same da a as he p e ious da ase
bu his ime a con inuous ea u e pa e n
o size 44 was c ea ed o each pa ien .
The same bina y class and he same 267
ins ances.
Sona The e a e 208 pa e ns ob ained by
bouncing sona signals off a me al cylin-
de and ocks a a ious angles and un-
de a ious condi ions. Each pa e n is
a se o 60 numbe s in he ange 0.0 o
1.0. Each numbe ep esen s he ene gy
wi hin a pa icula equency band, in e-
g a ed o e a ce ain pe iod o ime. The
class is bina y indica ing whe he he ob-
jec was a ock o a me al cylinde .
Wa e o m A ificial da ase whe e each
class is gene a ed om a combina ion o
2 o 3 ”base” wa es. The e a e 5000 in-
s ances wi h 21 ea u es each, all o which
include noise, and 3 classes.
Wdbc B eas cance da abases ob ained
om he Uni e si y o Wisconsin Hospi-
als, Madison om D . William H. Wol-
be g [7]. Fea u es 2 h ough 10 ha e been
used o ep esen ins ances. The e a e 699
ins ances wi h 10 ea u es, each has one o
2 possible classes: benign o malignan .
The expe imen s we e ca ied ou by ex-
ending he YALE lea ning en i onmen [8] in
o de o implemen a con en ional PTA and
he modified emainde se awa e e sion o i .
Each expe imen consis ed o a ea u e selec-
ion chain wi h a 1-nea es neighbo lea ne
(using Euclidean dis ance) and 5- old c oss-
alida ion o es ima ing ea u e use ulness.
The quan i y epo ed is he mean classifica-
ion e o in he fi e es olds. I is impo -
an o men ion ha he e was no s opping
c i e ion in he expe imen s: o wa d me h-
ods un un il all he ea u es we e selec ed and
backwa d ones un il all o hem we e emo ed.
Then he bes o he ob ained sequence o sub-
se s was e u ned. The esul s a e displayed
in Table 4. The able also shows he s an-
da d de ia ion o hese alues ound in he
c oss- alida ion uns and he size o he final
selec ed subse s.
IV Talle de Mine ía de Da os y Ap endizaje
35

Table 2: Summa y esul s o he expe imen s.
The esul s a e om he poin o iew o he
modified algo i hms.
#Fea .
E o =><To al
be e 3% 28% 25% 56%
equal 19% 3% 22%
wo se 8% 14% 22%
To al 22% 39% 39% 100%
Table 3: Summa y esul s o he expe imen s
o o wa d algo i hms. The esul s a e om
he poin o iew o he modified algo i hms.
#Fea .
E o =><To al
be e 44% 17% 61%
equal 17% 6% 22%
wo se 6% 11% 17%
To al 17% 56% 28% 100%
4.2 Expe imen al esul s
Tables 2 and 3 show he summa y esul s o
he expe imen s o all he 8 da a se s ans 4
algo i hms, and o he o wa d e sions o he
algo i hms only, espec i ely. The ables c oss
he numbe o ea u es selec ed wi h he clas-
sifica ion e o .
Upon looking a he summa y ables, he
fi s ac o no e om he expe imen esul s
is ha he emainde awa e e sion o he algo-
i hms ou pe o med he con en ional e sion
in mos o he cases. I is seen ha pe o -
mance is in gene al inc eased (as exp essed by
he chosen J) while keeping he numbe o se-
lec ed ea u es oughly equal (Table 2). The
imp o emen is e en g ea e i we only look a
he o wa d e sions o he algo i hms (Table
3). In his pa icula case, pe o mance is in-
c eased while lowe ing he numbe o selec ed
ea u es. This is mainly due o he ac ha
he o wa d me hods can easyly make w ong
decisions a ea ly i e a ions as (almos ) no ea-
u e in e ac ion is aken in o accoun when
e alua ing indi idual ea u es. A clea exam-
ple o his has been exposed a he beginning
o his sec ion wi h he Co Al da ase . The e
SFG selec ed he co ela ed ea u e while SBG
co ec ly disca ded i as he in e ac ion wi h
he o he mo e ele an ea u es was aken in o
accoun . Whene e he con en ional and he
modified algo i hm a e in ies o e y close o,
he modified e sions offe a solu ion wi h a
lowe numbe o ea u es,whichisalsoin e -
es ing om he poin o iew o ea u e se-
lec ion ( he e is an excep ion o his ule o
he pa icula case o he Sona da a se and
PTA (2,1)). The de ailed expe imen esul s
a e displayed in Table 4.
5 Conclusions
This pape has p esen ed a modifica ion o
ea u e subse selec ion algo i hms ha i e a-
i ely e alua e subse s o ea u es, by making
hem compu e no only he use ulness o he
selec ed se bu also he use ulness o he e-
mainde se . A se o expe imen s ha e been
conduc ed in o de o compa e he modified
e sions o he algo i hms wi h hei o iginal
e sions. Ou expe imen al esul s indica e
a gene al imp o emen in pe o mance while
keeping he size o he final subse oughly
equal o lowe . The ac ha he modified e -
sion does no always imp o e he esul s o he
o iginal should no be a su p ise. Acco ding
o he No ee lunch heo ems,i analgo i hm
achie es supe io esul s on some p oblems, i
mus pay wi h in e io i y on o he p oblems.
Howe e , i is possible o modi y a sea ch al-
go i hm o ob ain a e sion ha is gene ally
supe io in pe o mance o he o iginal e sion
[11]. In he p esen si ua ion his ac can be
explained by he way he modified e sion se-
lec s subse s o ea u es. Fo ins ance, gi en
wo ea u es: One ha makes a significan e-
duc ion o he pe o mance o he emainde
se and no a big change on he pe o mance
o he selec ed se . And one ha inc eases he
pe o mance o he selec ed se a bi mo e han
he fi s one bu does no make a big change on
he emainde one. A con en ional algo i hm
would always selec he la e while he mod-
36
II Cong eso Español de In o má ica
Table 4: De ailed Expe imen Resul s. The e o shown is he a e age o es se e o in he
5 olds, while σis he s anda d de ia ion o hese alues. Figu es in bold ace co espond o
imp o emen s.
Da ase S eps Con en ional Algo i hm Modified Algo i hm
w bw e o σ#Fea u es e o σ#Fea u es
co Al 01 0,00% 0,00% 4 0,00% 0,00% 4
co Al 10 3,20% 6,40% 5 0,00% 0,00% 4
co Al 12 0,00% 0,00% 4 0,00% 0,00% 4
co Al 21 0,00% 0,00% 4 0,00% 0,00% 4
Ionosphe e 01 9,68% 2,41% 9 7,12% 2,85% 13
Ionosphe e 106,27% 2,15% 11 7,70% 1,95% 7
Ionosphe e 127,11% 2,33% 12 7,42% 2,12% 7
Ionosphe e 21 6,26% 1,09% 14 5,11% 2,44% 7
Mammog am 0115,03% 5,65% 4 16,27% 2,29% 29
Mammog am 1012,75% 6,74% 17 12,68% 5,29% 15
Mammog am 1214,97% 7,41% 13 11,57% 5,05% 13
Mammog am 2111,57% 3,42% 26 8,10% 4,61% 22
Spec 0120,59% 1,07% 1 20,22% 1,81% 4
Spec 1023,23% 3,11% 6 20,59% 1,07% 1
Spec 1223,21% 1,38% 10 20,60% 1,68% 7
Spec 2121,35% 1,92% 5 22,09% 2,68% 6
Spec 0120,98% 4,04% 11 18,72% 5,52% 13
Spec 1019,48% 4,98% 10 17,97% 3,85% 6
Spec 1220,58% 3,00% 22 16,48% 4,34% 27
Spec 2120,59% 3,53% 8 17,64% 5,07% 24
Sona 0113,94% 5,72% 39 10,10% 4,12% 28
Sona 10 8,19% 4,70% 42 7,21% 2,61% 29
Sona 1212,02% 3,99% 22 9,11% 3,11% 26
Sona 217,20% 5,42% 36 10,56% 4,87% 42
Wa e o m 0120,60% 0,83% 18 21,32% 1,26% 17
Wa e o m 1021,16% 0,85% 16 20,60% 0,83% 18
Wa e o m 1221,00% 1,22% 15 21,00% 1,22% 15
Wa e o m 2121,62% 1,11% 15 21,14% 0,48% 17
Wdbc 01 5,27% 2,41% 11 4,39% 2,66% 25
Wdbc 10 3,86% 2,39% 24 3,34% 2,17% 13
Wdbc 123,51% 3,04% 13 4,04% 1,89% 27
Wdbc 21 3,86% 1,80% 27 3,86% 2,26% 14
IV Talle de Mine ía de Da os y Ap endizaje
37
ified e sion would maybe selec he o me .
Tha could lead he modified e sion o a oid
local maxima by no selec ing he bes ea u e
in his i e a ion ea u e and end wi h a be e
subse ; bu when he algo i hm has selec ed a
se close o op imal subse , he modifica ion
may cause he algo i hm o loose p ecision in
choosing ea u es. This loss o p ecision can
be g ea e when he emainde se o ea u es
is e y small compa ed wi h he selec ed se
so ew ea u e in e ac ions a e aken in o ac-
coun in his emainde se . Thus a u u e line
o wo k is o make he weigh o he emainde
se pe o mance a y wi h i s size o compen-
sa e o his ac .
Re e ences
[1] M. Ben-Bassa . Use o Dis ance Mea-
su es, In o ma ion Measu es and E o
Bounds in Fea u e E alua ion, olume 2,
pages 773–791. No h Holland, 1982.
[2] C.L. Blake D.J. Newman, S. He ich and
C.J. Me z. UCI eposi o y o machine
lea ning da abases, 1998.
[3] Ma k A. Hall. Co ela ion-based Fea u e
Selec ion o Machine Lea ning.PhD he-
sis, Uni e si y o Waika o, 1999.
[4] Geo ge H. John, Ron Koha i, and Ka l
Pflege . I ele an ea u es and he sub-
se selec ion p oblem. In William W. Co-
hen and Haym Hi sh, edi o s, Machine
Lea ning, P ocs. o he 11 h In l. Con .,
pages 121–129, Ru ge s Uni e si y, New
B unswick, NJ, USA, July 10-13 1994.
Mo gan Kau mann.
[5] Ron Koha i and Geo ge H. John. W ap-
pe s o ea u e subse selec ion. A i .
In ell., 97(1-2):273–324, 1997.
[6] P. Langley. Selec ion o ele an ea-
u es in machine lea ning. In P ocs. o
he AAAI Fall Symposium on Rele ance,
pages 140–144, New O leans, LA, USA,
1994. AAAI P ess.
[7]O.L.Mangasa ianandW.H.Wolbe g.
Cance diagnosis ia linea p og amming.
23(5):1–18, 1990.
[8] Ingo Mie swa, Michael Wu s , Ral
Klinkenbe g, Ma in Scholz, and Timm
Eule . Yale: apid p o o yping o com-
plex da a mining asks. In Tina Eliassi-
Rad, Lyle H. Unga , Ma k C a en, and
Dimi ios Gunopulos, edi o s, P oceed-
ings o he Twel h ACM SIGKDD In e -
na ional Con e ence on Knowledge Dis-
co e y and Da a Mining, Philadelphia,
PA, USA, Augus 20-23, 2006, pages 935–
940. ACM, 2006.
[9] Pa el Pudil, Jana No o ico ´a, and Jose
Ki le . Floa ing sea ch me hods in ea-
u e selec ion. Pa e n Recogni ion Le -
e s, 15(11):1119–1125, 1994.
[10] S.D. S ea ns. On selec ing ea u es o
pa e n classifie s. In P ocs. o he 3 d
In l. Con . on Pa e n Recogni ion (ICPR
1976), pages 71–75, Co onado, CA, 1976.
[11] Da id Wolpe and William G. Mac eady.
No ee lunch heo ems o op imiza ion.
IEEE T ans. E olu iona y Compu a ion,
1(1):67–82, 1997.
38
II Cong eso Español de In o má ica