Applied So Compu ing 115 (2022) 108176
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Applied So Compu ing
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B ain–compu e in e ace channel selec ion op imiza ion using
me a-heu is ics and e olu iona y algo i hms
Víc o Ma ínez-Cagigal∗, Edua do San ama ía-Vázquez, Robe o Ho ne o
Biomedical Enginee ing G oup, E.T.S.I. Telecomunicación, Uni e si y o Valladolid, Paseo de Belén 15, 47011, Valladolid, Spain
Biomedical Resea ch Ne wo king Cen e in Bioenginee ing, Bioma e ials and Nanomedicine (CIBER-BBN), Spain
g aphical abs ac
a icle in o
A icle his o y:
Recei ed 10 Oc obe 2019
Recei ed in e ised o m 23 Augus 2021
Accep ed 11 No embe 2021
A ailable online 2 Decembe 2021
Keywo ds:
B ain–compu e in e ace (BCI)
Channel selec ion
Mul i-objec i e op imiza ion
E olu iona y algo i hms
P300 e en - ela ed po en ials
abs ac
Many b ain–compu e in e ace (BCI) s udies o e look he channel op imiza ion due o i s inhe en
complexi y. Howe e , a ca e ul channel selec ion inc eases he pe o mance and use s’ com o
while educing he cos o he sys em. E olu iona y me a-heu is ics, which ha e demons a ed hei
use ulness in sol ing complex p oblems, ha e no been ully exploi ed ye in his con ex . The pu pose
o he s udy is wo- old: (1) o p opose a no el algo i hm o ind an op imal channel se o each use
and compa e i wi h o he exis ing me a-heu is ics; and (2) o es ablish guidelines o adap ing hese
op imiza ion s a egies o his amewo k. A o al o 3 single-objec i e (GA, BDE, BPSO) and 4 mul i-
objec i e (NSGA-II, BMOPSO, SPEA2, PEAIL) exis ing algo i hms ha e been adap ed and es ed wi h 3
public da abases: ‘BCI compe i ion III-da ase II’, ‘Cen e Spelle ’ and ‘RSVP Spelle ’. Dual-F on So ing
Algo i hm (DFGA), a no el mul i-objec i e disc e e me hod especially designed o he BCI amewo k,
is p oposed as well. Resul s showed ha all me a-heu is ics ou pe o med he ull se and he common
8-channel se o P300-based BCIs. DFGA showed a signi ican imp o emen o accu acy o 3.9% o e
he la e using also 8 channels; and ob ained simila accu acies using a mean o 4.66 channels. A
opog aphic analysis also ein o ced he need o cus omize a channel se o each use . Thus, he
p oposed me hod compu es an op imal se o solu ions wi h di e en numbe o channels, allowing
he use o selec he mos app op ia e dis ibu ion o he nex BCI sessions.
©2021 The Au ho s. Published by Else ie B.V. This is an open access a icle unde he CC BY-NC-ND
license (h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0/).
∗Co espondence o: Biomedical Enginee ing G oup, E.T.S.I. Telecomuni-
cación, Uni e si y o Valladolid, Paseo de Belén 15, 47011, Valladolid, Spain.
E-mail add esses: [email p o ec ed] (V. Ma ínez-Cagigal),
[email p o ec ed] (E. San ama ía-Vázquez), [email p o ec ed]
(R. Ho ne o).
1. In oduc ion
B ain–Compu e In e aces (BCIs) a e communica ion sys ems
ha allow use s o con ol de ices and applica ions using hei
own b ain signals. These sys ems ha e been success ully applied
h ps://doi.o g/10.1016/j.asoc.2021.108176
1568-4946/©2021 The Au ho s. Published by Else ie B.V. This is an open access a icle unde he CC BY-NC-ND license (h p://c ea i ecommons.o g/licenses/by-
nc-nd/4.0/).
V. Ma ínez-Cagigal, E. San ama ía-Vázquez and R. Ho ne o Applied So Compu ing 115 (2022) 108176
in o de o imp o e he quali y o li e o people wi h mo o dis-
abili ies who su e om a disease ha impai s he neu al pa h-
ways ha con ol muscles o e en he muscles hemsel es [1].
Elec oencephalog am (EEG) is commonly used o moni o he
b ain ac i i y due o i s po abili y, non-in asi eness and low
cos . The e o e, elec ical po en ials a e eco ded by placing elec-
odes on he use ’s scalp [1].
Since decoding use s’ in en ions om he EEG is no s aigh -
o wa d, BCIs ely on con ol signals o handle he con ol o
he sys em. In pa icula , he P300 e oked po en ials, which a e
posi i e peaks p oduced in esponse o in equen and signi ican
s imuli app oxima ely 300 ms a e hei onse , a e he key aspec
o he mos well-known BCI-based spelling sys em [1]. The ‘P300
Spelle ’ gene a es hese signals h ough he odd-ball pa adigm
in o de o spell ce ain wo ds o commands. The applica ion
displays a ma ix con aining cha ac e s o symbols, whose ows
and columns a e andomly lashing. Use s, who ha e o ocus on
a desi ed command, will gene a e a P300 po en ial whene e he
ow o he column ha con ains he command is highligh ed.
Hence, he selec ed command is de e mined by compu ing he
in e sec ion be ween he ow and he column ha p oduced he
po en ial [2].
Due o he low signal- o-noise a io and high in e -session
a iabili y o hese e en - ela ed po en ials, se e al epe i ions
o he same s imulus a e equi ed o de ec a eliable esponse.
Wi hou a p ope p ocessing s age, hese high dimensional da a
can p oduce o e - i ing, esul ing in poo pe o mance [3,4]. The
cu se o dimensionali y can be add essed by means o ea u e
selec ion and ex ac ion me hods [4,5], egula ized classi ie s [6]
o channel selec ion p ocedu es [3,7]. Among hem, only channel
selec ion me hods a e able o educe he cos o he sys em,
educe powe consump ion on EEG caps and inc ease use com-
o [3]. Ne e heless, he selec ion o he mos ele an senso s is
no i ial as he e a e 2Nsubse combina ions o an N-channel
cap, making he exhaus i e sea ch in ac able in p ac ice [3]. Fo
his eason, mos P300-based s udies o e look he op imiza ion
o he mos ele an subse o channels and ake a p ede ined 8-
channel se as a gene al ule o humb [8]. No wi hs anding i s
use ulness as a quick solu ion, an op imiza ion o each use is
bene icial owing o he in insic in e -subjec a iabili y o he BCI
sys ems.
Al hough he e a e many ea u e selec ion me hods ha could
be applied o his p oblem, such as s ep-wise eg ession [9], as
co ela ion based il e s [10], elas ic neu al ne wo ks [11], o ex-
plainable deep lea ning [12], me a-heu is ics ha e demons a ed
high pe o mances sol ing complex op imiza ion p oblems [13].
Heu is ics e e o p oblem-speci ic s a egies ha i e a i ely
imp o e a candida e solu ion, whe eas me a-heu is ics gene alize
hese s a egies o p oblem-independen amewo ks [13,14].
Swa m in elligence echniques and e olu iona y algo i hms, am-
ilies o popula ion-based me a-heu is ics, ha e been p e iously
applied in EEG signals o sol e op imiza ion p oblems [4,7,15–
19,19–30]. Despi e hei popula i y, he con ibu ion o me a-
heu is ics o P300-based BCIs is s ill sca ce. Mos o hese p e-
ious s udies a e ela ed o mo o image y (MI) BCIs [15–23] o
biome ic-o ien ed pe son iden i ica ion sys ems [24,25], whose
signal p ocessing s age is comple ely di e en (e.g., neu al
sou ces, con ol signals, pa adigms, spa ial il e ing and ea u e
ex ac ion) and hus, esul s canno be gene alized o P300-based
BCIs. Rega ding he P300-based s udies, mos o hem ha e used
single-objec i e algo i hms ha op imized he inal classi ica ion
accu acy o he sys em [4,26–28,30]. Howe e , we belie e ha a
channel selec ion p ocedu e should ollow a wo- old objec i e:
(i) o minimize he numbe o selec ed channels, and (ii) o
maximize he sys em’s pe o mance. Some ecen s udies used
a weigh ed agg ega ion app oach o combine bo h objec i es
in o a single one, bu he simul aneous op imiza ion was no
explo ed [7,22,31].
T adi ional mul i-objec i e app oaches, which op imize bo h
objec i es a he same ime, ha e been explo ed in MI-based BCIs,
such as mul i-objec i e pa icle swa m op imiza ion (MOPSO)
[16–18] o non-so ing gene ic algo i hm II (NSGA-II) [20,23].
By con as , mul i-objec i e algo i hms applied o P300-based
BCIs a e mo e limi ed. Kee e al. [19] compa ed he pe o mance
be ween se e al single-objec i e gene ic algo i hms (GA) and
NSGA-II wi h 2 subjec s, whe eas Chau asiya e al. [29] em-
ployed a mul i-objec i e bina y di e en ial-e olu ion algo i hm
wi h 9 subjec s, eaching se e al subse s o channels ha assu ed
sui able classi ica ion pe o mances. Ne e heless, he numbe o
subjec s was limi ed, and bo h da abases we e eco ded using he
ow-col pa adigm (RCP). Nowadays, P300-based BCIs o e a wide
ange o s imula ion pa adigms ha elici di e en e en - ela ed
esponses and hus, he gene aliza ion o hose esul s o o he
se ups is unclea . Fu he mo e, despi e hei sca ce applica ion
in P300-based BCI s udies, swa m in elligence and e olu iona y
compu a ion a e g owing esea ch ields ha in eg a e a la ge
amoun o di e en algo i hms ha could be adap ed o he
channel selec ion p oblem. In ac , he as majo i y o hem
ha e ye o be applied o P300-based BCIs. To he bes o ou
knowledge, he e a e no s udies ha compa e hei e icacy in
selec ing he mos app op ia e subse o channels o e en es ab-
lishing he key aspec s o hei adap a ion o BCI sys ems, which
is no i ial. Fu he mo e, none o he p e ious s udies es ed any
me a-heu is ic wi h pa adigms o he han RCP, es ic ing hei
gene aliza ion. Las ly, i is no ewo hy ha he e is also no s udy
aimed a designing any mul i-objec i e algo i hm cus omized o
he P300-based BCI channel selec ion p oblem.
The objec i e o his s udy is wo- old: (1) o p opose a no el
mul i-objec i e me hod o ind an op imal channel se especially
sui ed o P300-based BCIs and compa e i s use ulness wi h 7
addi ional me a-heu is ics; and (2) o es ablish guidelines o
adap ing hese op imiza ion s a egies o he channel selec ion
p oblem. Al hough he e a e many me a-heu is ics ha could be
adap ed o his p oblem, only hose ha ha e p e iously applied
in BCIs, ha ha e di ec o explici con ibu ion o ou p oposed
me a-heu is ic o ha ha e been ecen ly p oposed we e in-
cluded in his compa ison: GA, BDE and BPSO as single-objec i e;
and NSGA-II, SPEA2, BMOPSO and PEAIL as mul i-objec i e. We
ha e also ied o main ain di e si y in he way hey deal o he
upda ing o he popula ion o each i e a ion. To sum up, he main
con ibu ions o his s udy a e he ollowing: p oposal o a no el
mul i-objec i e algo i hm especially designed o his p oblem,
compa ison o 7 me a-heu is ics o he P300-based BCI channel
selec ion p oblem, enume a ion o a de ailed se o guidelines
o adap any me a-heu is ic o he channel selec ion, and e al-
ua ion wi h h ee da abases ha employ di e en P300-based
pa adigms.
2. Subjec s
In o de o imp o e he gene aliza ion o he esul s, he
algo i hms ha e been es ed wi h h ee public P300-based BCI
da abases ha we e eco ded using di e en s imula ion
pa adigms: ow-col pa adigm (RCP), cen e spelle (CS) and apid
se ial isual p esen a ion (RSVP). Examples o he s imula ion
sequences o hese pa adigms a e depic ed in Fig. 1.
2.1. BCI compe i ion III: da ase II
The ‘BCI compe i ion III: da ase II’ [32] was eco ded om
2 di e en heal hy subjec s (i.e., A and B) ha we e asked o
spell wo ds in 5 RCP sessions. Signals we e eco ded using a
2
V. Ma ínez-Cagigal, E. San ama ía-Vázquez and R. Ho ne o Applied So Compu ing 115 (2022) 108176
Fig. 1. Examples o s imuli in ensi ica ion sequences o he pa adigms: (a) ow-col pa adigm, (b) cen e spelle , (c) apid se ial isual p esen a ion.
64-channel EEG cap wi h a sampling equency o 240 Hz and
band-pass il e ed om 0.1 Hz o 60 Hz. T aining and es ing
se s we e composed o 85 and 100 ials, espec i ely [32]. RCP
is he mos common P300-based spelling pa adigm, which con-
sis s o displaying a ma ix ha con ains cha ac e s o symbols.
Use s ha e o s a e a he a ge command while he ma ix’s
ows and columns a e andomly lashing. Whene e he ow o
column ha con ains he a ge is licke ed, a P300 po en ial
is gene a ed. Hence, he desi ed command can be iden i ied by
compu ing he in e sec ion be ween he ow and he column
ha p oduced hese P300 esponses [2]. In his da ase , he e a e
12 di e en classes (i.e., ows and columns), and 15 sequences
(i.e., epe i ions) we e used. The e o e, a ial is composed by 180
obse a ions [32].
2.2. Cen e spelle da abase
The ‘Cen e Spelle (008-2015)’ da abase [33] was eco ded
om 13 heal hy subjec s (i.e., C01–C13) ha we e asked o pe -
o m spelling asks using he CS pa adigm. Signals we e eco ded
using a 63-channel EEG cap wi h a sampling equency o 250 Hz
and band-pass il e ed om 0.016 Hz o 250 Hz. T aining da a
was composed o 17 ials, whe eas es ing da a a ied be ween
32–49 ials, depending on he subjec [33]. CS was o iginally
designed o a oid eye mo emen s. The pa adigm displays g oups
o commands in he cen e o he sc een, o e laid wi h colo ed
geome ic shapes. The g oups a e andomly licke ed un il he
use selec s one o hem. Then, he commands ha we e included
inside he selec ed g oup a e displayed in he same way, allowing
he use o selec he inal command [33]. In p ac ice, he e a e
12 di e en classes (6 g oups in 2 le els), and 10 sequences we e
used. A ial is composed by 120 obse a ions [33].
2.3. RSVP spelle da abase
The ‘RSVP Spelle (010-2015)’ da abase [34] was eco ded
om 12 heal hy subjec s (i.e., R01–R12) ha we e asked o
pe o m spelling asks using he RSVP pa adigm. Signals we e
eco ded using a 63-channel EEG cap wi h a sampling equency
o 1000 Hz, and hen down-sampled o 200 Hz [34]. Howe e ,
since he i h subjec only used 61 channels, elec odes P8 and
O2 we e excluded om he da abase o he sake o homogenei y.
T aining da a was composed o 24 ials, whe eas es ing da a
(copy and ee spelling) a ied be ween 37–50 ials, depending
on he subjec [34]. RSVP was also de eloped o exploi he o eal
isual ield and a oid eye mo emen s by depic ing symbols in he
cen e o he sc een in a se ial manne . The da abase includes a
ocabula y o 30 cha ac e s (26 le e s and 4 symbols). In o de
o a o he iden i ica ion o he shapes, hal o he le e s we e
uppe case and he o he hal lowe case, using 5 di e en colo s.
The e o e, he e a e 30 classes, and 10 sequences we e used,
esul ing in 300 obse a ions pe ial [34].
3. Me hods
3.1. P e-p ocessing and ea u e ex ac ion
Be o e applying any op imiza ion p ocedu e, ele an ea u es
o he EEG signals should be ex ac ed o each epoch (i.e., s im-
ulus) and channel. In ac , p e-p ocessing, as well as ea u e
ex ac ion and selec ion p ocedu es in luence he inal accu acy
in a high ex en . Due o he pu pose o he s udy, signal p o-
cessing s ages we e composed o a s anda dized amewo k, in-
ended o isola e he channel selec ion p ocedu e. We did no
apply any u he p e-p ocessing s ep besides he a o emen ioned
band-pass il e ing embedded in each da abase [32–34]. Epochs
3
V. Ma ínez-Cagigal, E. San ama ía-Vázquez and R. Ho ne o Applied So Compu ing 115 (2022) 108176
we e ex ac ed using windows in he ange [0,800]ms om he
s imuli onse s, and no malized ia z-sco e o e a [−200,0]ms
baseline. As s a ed in BCI li e a u e, his ange is la ge enough
o cap u e ele an e en - ela ed po en ials, including he P300
wa e [1]. These epochs we e hen decima ed o 25 Hz, keeping
a o al o 20 ea u es pe s imulus and channel. I is no ewo hy
ha he decima ion p ocess encompasses a low-pass il e ing ( o
a oid aliasing), ollowed by a down-sampling p ocedu e [7,35].
He ea e , epochs om di e en da abases and sampling a es
ha e he same numbe o ea u es. No e ha , om he poin o
iew o a subsequen classi ie , epochs a e inpu obse a ions.
3.2. De ining he op imiza ion p oblem
The goal o an op imiza ion algo i hm is o p o ide a sui -
able solu ion ha sa is ies he p oblem cons ain s and op imizes
(ei he maximizing o minimizing) one o mo e objec i e unc-
ions o he g ea es ex en [14]. Since we a e conside ing an
N-channel selec ion p oblem, a possible solu ion may be de ined
as x= [x1,x2,...,xN],xi∈ {0,1}, whe e 1 and 0 ep esen he
selec ion and ejec ion o a channel i, espec i ely. Hence, his
combina o ial p oblem is cons ained o a disc e e N-dimensional
space, whose solu ions a e es ic ed o bina y posi ions. When a
solu ion xis e alua ed, ea u es associa ed wi h he channels ha
sa is y xi=1 a e conca ena ed as an inpu ea u e ec o .
In a BCI channel selec ion p oblem, wo main objec i es mus
be pu sued: (i) maximize sys em pe o mance, and (ii) minimize
he numbe o channels. E en hough he modeling o he la -
e is s aigh o wa d (see Eq. (1)), he sys em pe o mance can
be es ima ed ollowing se e al app oaches. The mos in ui i e
solu ion is o use he ou pu aining accu acy o he classi ie
using a ce ain solu ion x[19,26,29]. Howe e , due o he limi ed
numbe o ials, his me hod usually p o ides a low- esolu ion
sco e [36]. The esolu ion can be imp o ed by using s imuli-
based, a he han cha ac e ial-based. P e ious s udies used
app oaches de i ed om he con usion ma ix o he s imuli
classi ica ion [4,27,28]. Ne e heless, he a ea unde ROC cu e
(AUC) is ecommended because i is able o success ully es ima e
he disc imina i e abili y o a bina y classi ie using only aining
da a [3,36]. The e o e, he objec i es a e modeled as ollows:
min F(x)=⎧
⎪
⎨
⎪
⎩
1(x)=1−AUC(x)
2(x)=
N
∑
n=1
xn
,(1)
whe e 1(x) belongs o he i s objec i e (i.e., minimize he sys-
em e o ) and 2(x) o he second objec i e (i.e., minimize he
numbe o channels). In his s udy, AUC has been de i ed om
a 5- old c oss- alida ed linea disc iminan analysis (LDA) ha is
applied o he solu ion xusing he aining da ase [7,35,37]. Tha
is, he ea u es whose channels sa is y xi=0 a e emo ed om
he obse a ions ma ix, which is he inpu o he LDA classi ie .
T aining se is hen di ided in o 5 subse s and a c oss- alida ion
p ocedu e is applied (i.e., 4 subse s a e used o aining and he
emaining one o es ing), e u ning a o al o 5 AUCs. Finally,
AUC is compu ed as he a e age o all o hem. LDA was used as
classi ie due o i s well-known excellen pe o mances in P300-
based BCIs and he lack o hype pa ame e s o op imize [5–7,19,
20,29,33,38].
3.3. Single-objec i e me a-heu is ics
Me a-heu is ics p oduce accep able solu ions o complex
p oblems in a easonable compu a ion ime [13]. In pa icu-
la , single-objec i e me a-heu is ics i e a i ely p oduce hese so-
lu ions ollowing a ce ain objec i e. Howe e , a BCI channel
selec ion p oblem should ha e a wo- old pu pose. Thus, he
mul i-objec i e p oblem s a ed in Eq. (1) is hen combined in o a
single-objec i e one [39]:
min F(x)=ω1 1(x)+ω2( 2(x)−1
N−1)3
,(2)
whe e ω1+ω2=1, and ω1and ω2a e cons an s ha weigh
he impo ance o each objec i e. Since we conside ha eaching
sui able accu acies is mo e impo an han d as ically educing
he numbe o equi ed channels, coe icien s ha e been heu is-
ically se o ω1=0.7 and ω2=0.3 [7,31,35]. In addi ion,
a e mapping he 2(x) om [1,N]→[0,1], i s ou pu is
aised o he hi d powe o empa hize he sea ch o ligh weigh
solu ions. No e ha he polynomial unc ion punishes he sea ch
o solu ions wi h a high numbe o channels mo e han a simple
linea unc ion. This unc ion was heu is ically chosen a e a
p elimina y es ing [7,31,35]. The h ee single-objec i e me a-
heu is ics ha ha e been adap ed o BCI amewo k a e desc ibed
below.
3.3.1. Gene ic algo i hm
One o he mos well-known me a-heu is ics is he gene ic
algo i hm (GA), o iginally de eloped by Holland [40]. GAs ha e
been modi ied o imp o e hei abili y o ind he global op imum
o complex op imiza ion p oblems in many ways. In sho , GAs
apply he Da winian p inciple o su i al o he i es indi idu-
als in a popula ion using ecombina ion, selec ion and mu a ion
ope a o s [13,14]. In his s udy, a GA wi h eli ism, bina y ou -
namen selec ion, single-poin c osso e and bi s ing mu a ion
has been employed [13,14].
3.3.2. Bina y di e en ial e olu ion
The di e en ial e olu ion (DE) algo i hm, o iginally de eloped
by S o n and P ice [41] o con inuous unc ions, has some simi-
la i ies o GAs in e ms o i s s uc u e, composed by mu a ion,
c osso e and selec ion ope a o . Howe e , ins ead o making
andom mu a ion and c osso e schemes, DE combines he in-
o ma ion o h ee andomly chosen indi iduals. Bina y DE (BDE)
applies a disc e iza ion o he mu a ion o mula in o de o adap
i o bina y p oblems [42]. The mu a ion o he i h channel o an
indi idual xis pe o med as ollows:
x′
i={ui,i and ≤pco i=
xi,o he wise ,(3)
whe e and ∼U(0,1), is a andom in ege be ween [1,N],pcis
he c osso e a e, and uiis he mu a ed channel, compu ed as:
ui={1,i and ≤(1 +e−2b( i+F·(yi−zi)−1/2)
1+2F)−1
0,o he wise ,(4)
whe e and ∼U(0,1); ,yand za e andomly selec ed indi idu-
als o he cu en popula ion; Fis he weigh ing ac o ; and b>0
is he bandwid h ac o .
3.3.3. Bina y pa icle swa m op imiza ion
Kennedy and Ebe ha [43] de eloped he Pa icle Swa m
Op imiza ion (PSO) algo i hm, a na u e-inspi ed me a-heu is ic
based on he social schooling and locking beha io o ishes
and bi ds. The op imiza ion elies on adjus ing he ajec o ies
and posi ions o a se o pa icles (i.e., solu ions) ha ‘‘ ly’’ o e
he sea ch space, whose mo emen ha e bo h de e minis ic and
s ochas ic componen s [13,14,43]. In his s udy, he s anda d
cons ain o Cle c and Kennedy [44] is used, leading o:
′=χ[ +ϵ1C1(l−x)+ϵ2C2(g−x)],(5)
4
V. Ma ínez-Cagigal, E. San ama ía-Vázquez and R. Ho ne o Applied So Compu ing 115 (2022) 108176
χ=2
φ−2+√φ2−4φ
,wi h φ=C1+C2;(6)
whe e ′is he upda ed eloci y o a pa icle x; is he las
eloci y; ϵ1, ϵ2∼U(0,1); χis he cons ain mul iplie ; C1and
C2a e he pe sonal and global con idence cons an s, espec i ely;
lis he bes posi ion ound by he pa icle x; and gis he bes
global posi ion ound so a . I is wo hy o no e ha he s anda d
cons ain equi es ha φ > 4 [44,45]. Since he eloci ies a e
con inuous, he algo i hm should be adap ed o bina y spaces.
Bina y PSO (BSPO) is usually achie ed using a posi ion ans o -
ma ion ia ans e unc ions [46,47]. In his s udy, he adap a ion
has been pe o med ollowing he exp ession:
x′
i={¬xi,i and <T( ′
i)
xi,i and ≥T( ′
i),(7)
whe e and∼U(0,1), and T( )= | /√1+ 2|is a -shaped
ans e unc ion [47].
3.4. Mul i-objec i e me a-heu is ics
In con as o he single-objec i e s a egies, mul i-objec i e
me a-heu is ics in ol e he simul aneous op imiza ion o wo o
h ee objec i es [48]. Since hese objec i es a e usually con lic ing
among hemsel es, he concep o dominance is in oduced o
de e mining he quali y o each solu ion [49]. I is said ha a
solu ion ydomina es a solu ion z(i.e., y≻z) i ∀i: i(y)≤ i(z)
and ∃j: j(y)< j(z). The Pa e o- on , a cu e ha con ains
op imal solu ions (i.e., hose ha a e no domina ed by any o he
solu ions), is es ima ed by he mul i-objec i e algo i hms and
depic s he ade-o among he objec i es [49]. Rega ding he
BCI channel selec ion p oblem, he Pa e o- on e u ns a se o
solu ions ha ha e di e en numbe o channels, allowing he
use o selec one o hem.
3.4.1. Non-So ing Gene ic Algo i hm II
The mos popula app oach o ex ending GAs o mul i-
objec i e op imiza ion p oblems is he Non-So ing Gene ic Al-
go i hm II (NSGA-II), p oposed by Deb e al. [48]. C osso e and
mu a ion ope a o s a e he same as GAs, whe eas he selec ion
ope a o is mo e complex. Fi s ly, in o de o es ima e he quali y
o each ch omosome, he algo i hm es ablishes a hie a chy o
Pa e o- on s acco ding o i s dominance. The i s Pa e o- on
(i.e., ank =1) is composed by he non-domina ed ch omo-
somes o he cu en popula ion. Then, he second Pa e o- on
(i.e., ank =2) is compu ed in he same way, bu igno ing
he ch omosomes o he i s on . This p ocess is epea ed
sequen ially un il he e a e no ch omosomes le [48]. Howe e ,
he selec ion o a pa en popula ion is no only based on he ank
o he ch omosomes, bu also on hei c owding dis ances. These
me ics a e included o sp ead he solu ions along he Pa e o-
on and a oid ge ing apped in local minima. The c owding
dis ance o a ch omosome is compu ed as he a e age dis ance
be ween i s wo adjacen solu ions wi h he same ank. Bounda y
solu ions a e assigned an in ini e dis ance alue. Conside ing wo
ch omosomes, he solu ion wi h lowe ank is p e e ed. Whe he
bo h ha e he same ank, he less c owded solu ion is p e e ed
(i.e., highe dis ance alue). The pa en popula ion is sequen ially
illed wi h he i s s Pa e o- on s un il he numbe o included
solu ions is g ea e o equal han m
/2. Then, pa en solu ions a e
unca ed based on he c owding dis ances un il he numbe o
solu ions is exac ly m
/2. Fu he in o ma ion can be ound in Deb
e al. [48].
3.4.2. Bina y mul i-objec i e PSO
Due o i s use ulness o sol e complex op imiza ion p oblems,
many au ho s ha e ied o adap he PSO algo i hm o mul i-
objec i e en i onmen s [39]. He e, a Bina y Mul i-Objec i e PSO
(BMOPSO) app oach is applied. Since he con lic ing objec i es do
no allow he es ablishing o an op imal global solu ion g, he
majo adap a ion mus eside in he way o selec he leade o
each pa icle. In his s udy, a eposi o y app oach is employed.
Non-domina ed solu ions a e s o ed in an ex e nal eposi o y
wi h ‘‘unlimi ed’’ size. No e ha i s maximum size would be he
maximum numbe o channels (i.e., he esolu ion o he BCI
p oblem). A pa icle’s leade is andomly selec ed om he epos-
i o y, and i is a ached o he pa icle un il he leade is no longe
pa o he eposi o y. In ha case, he leade is subs i u ed by
ano he andomly selec ed one. In addi ion, a h ee- old bi s ing
mu a ion is also used, which consis s on di iding he swa m in
h ee pa s and apply: (1) no mu a ion; (2) uni o m mu a ion
wi h p obabili y pm; (3) non-uni o m mu a ion wi h p obabili y
pn=(1 −gen/ngen)5N[50].
3.4.3. S eng h Pa e o E olu iona y Algo i hm 2
Zi zle e al. [51] p oposed he S eng h Pa e o E olu iona y
Algo i hm 2 (SPEA2), a mul i-objec i e algo i hm ha in eg a es
he concep s o dominance and c owding densi y in a single
me ic: he s eng h. The s eng h Siis compu ed as he numbe
o solu ions ha he i h pa icle domina es. Then, he uni ied
i ness is calcula ed as ollows:
Fi=Ri+1
σk
i+2,(8)
whe e Riis he sum o he s eng hs o he pa icles ha domi-
na es i, and σk
iis he dis ance sough o he pa icle (i.e., dis ance
o he k-nea es neighbo ), whe e k= ⌊√m⌉. No e ha non-
domina ed indi iduals would ha e R=0 and hus, F<1.
SPEA2 also uses a eposi o y wi h ixed size ha is upda ed
ollowing an en i onmen al selec ion p ocedu e. Solu ions a e
so ed acco ding o hei F alues, and he eposi o y is illed
wi h hem. I he numbe o solu ions o he eposi o y is highe
han he maximum size N , a unca ion p ocess is applied. Then,
he algo i hm emo es solu ions om he eposi o y acco ding o
hei σk(i.e., high σk alues a e p e e ed), in o de o p ese e
Pa e o- on sp eading [51].
3.4.4. Pa e o e olu iona y algo i hm based on inc emen al lea ning
Recen ly, Rong-Juan e al. [52] p oposed a disc e e mul i-
objec i e algo i hm ha in oduces he concep o inc emen al
lea ning o upda e solu ions by explo ing p obabili y dis ibu-
ions o p omising sea ch egions. The algo i hm, known as Pa e o
E olu iona y Algo i hm based on Inc emen al Lea ning (PEAIL),
also uses non-domina ed so ing o keep ack o hie a chical
Pa e o on s, as NSGA-II does [48]. The inc emen al lea ning
s age selec s an excellen indi idual, hen es ima es a p oba-
bili y model and p edic s a new child en popula ion using ha
in o ma ion:
xi←(xi+xe·L)/(L+1),(9)
whe e xiis he solu ion being upda ed, xeis a andomly selec ed
solu ion om he i s Pa e o on , and Lis he lea ning a e
pa ame e . Check [52] o u he in o ma ion.
3.5. Ou p oposal: Dual- on gene ic algo i hm
E en hough he e is a g ea a ie y o me a-heu is ics om
single o mul i-objec i e algo i hms, all o hem should be
adap ed o he channel selec ion p oblem. The BCI amewo k
o ces he algo i hms o wo k wi h bina y solu ions, in ol ing he
5

V. Ma ínez-Cagigal, E. San ama ía-Vázquez and R. Ho ne o Applied So Compu ing 115 (2022) 108176
Fig. 2. Summa y o DFGA wi h isual aids o cla i y ope a ions. (a) Flowcha o he algo i hm. No e ha he backwa d elimina ion s ep is only pe o med as an
ini ializa ion. BT: bina y ou namen , F: i ness e alua ion. (b) Dual- on so ing. (c) Pa en selec ion. (d) Single-poin c osso e . (e) Bi -s ing mu a ion. ( ) Eli is
eposi o y upda ing.
use o ans e unc ions in some cases. These unc ions con e
a solu ion al e a ion in o a p obabili y o change, inc easing he
s ochas ici y o he algo i hm. Mo eo e , he con e sion can be
add essed as a mul i alued unc ion o he ype :R→ {0,1},
which means ha he e a e in ini e inpu alues ha p oduce
exac ly he same ou pu , hinde ing he local exploi a ion o new
solu ions. By ex ension, he e is no poin in using ope a o s
based on con inuous dis ances. Since 2(x) al eady es ic s he
size o mul i-objec i e eposi o ies o N, limi a ion s a egies
(e.g., c owding, dis ance sough ) also en ail an unnecessa y com-
pu a ional cos . In o de o o e come hese es ain s, a no el
mul i-objec i e algo i hm is p oposed: he Dual-F on Gene ic Al-
go i hm (DFGA). DFGA is specially designed o he BCI amewo k
by means o i e key aspec s: (i) de e minis ic ini ializa ion, (ii)
dual- on so ing, (iii) gene ic ope a o s, (i ) syn he ic solu ions,
and ( ) eli ism. A de ailed lowcha is depic ed in Fig. 2(a), while
he pseudo-code and a complexi y analysis a e included in he
supplemen a y ma e ial.
De e minis ic ini ializa ion. Heu is ics gene ally ini ialize he
popula ion by gene a ing andom solu ions. Howe e , he use o
de e minis ic ini ializa ion can educe he in e - un a iabili y
due o s ochas ic e ec s and a la ge amoun o compu a ion
ime. Al hough de e minis ic algo i hms a e unlikely o p o ide
a global op imum, DFGA conside s hei ou pu s as in e media e
solu ions. Rega dless o hei quali ies, we hypo hesize ha hese
solu ions a e equi alen o hose ha will be e en ually eached
a e se e al gene a ions o a andomly-ini ialized algo i hm. In
his s udy, backwa d elimina ion (BE) is used o ini ialize he
eposi o y. The algo i hm begins wi h he ull se o channels and
sequen ially emo es he mos i ele an one [9]. The ejec ed
channel in each s ep is he one ha e u ns he minimum 1(x)
alue i emo ed om he model x(i.e., i s inclusion does no
con ibu e o imp o e he sys em’s pe o mance). The algo i hm
con inues emo ing channels un il he se is emp y. No e ha his
ope a ion will ill he eposi o y Rup wi h Nsolu ions.
Dual- on so ing. Due o he de e minis ic ini ializa ion, he
eposi o y should ha e a well-de ined cu e om he e y be-
ginning o he algo i hm. This aspec leads o a Pa e o- on ha
is supposed o include solu ions wi h ew numbe o channels.
T adi ionally, only he Pa e o-op imal solu ions a e conside ed
in he selec ion s age. Despi e hei con enience o e domina ed
solu ions, conside ing only he Pa e o- on would lead o a local
exploi a ion o solu ions wi h ew channels. Howe e , because o
he in insic ixed size o he eposi o y in BCI p oblems (i.e., lim-
i ed o N), he exploi a ion o solu ions wi h a g ea e numbe o
channels is no longe an issue, ins ead i may a o he sp eading
o he Pa e o- on and he global sea ch o DFGA. Acco ding o
his a ionale, DFGA subdi ides he eposi o y in o wo se s: O
(i.e., op imal se ), which includes he non-domina ed solu ions;
and S(i.e., sub-op imal se ), which includes he domina ed so-
lu ions. Dual- on so ing ope a ion is shown in Fig. 2(b). Then,
bina y ou namen selec ion is applied in bo h se s, selec ing
2N/3solu ions om O, and N/3solu ions om S. No e ha a
solu ion may be selec ed mo e han once in he new popula-
ion. Finally, hese solu ions a e combined in he popula ion o
su e ecombina ion (i.e., c osso e ) and mu a ion, as shown
in Fig. 2(c).
Gene ic ope a o s. Owing o he bina y na u e o he sea ch
space, we conside ha adi ional gene ic ope a o s a e he mos
con enien app oach o gene a ing new solu ions om a pa en
6
V. Ma ínez-Cagigal, E. San ama ía-Vázquez and R. Ho ne o Applied So Compu ing 115 (2022) 108176
Table 1
Me hod-speci ic hype pa ame e s.
P m. Value Desc ip ion Algo i hm
m20 No. indi iduals All
pm1/NaMu a ion a e
GA, NSGA-II,
BMOPSO, SPEA2,
PEAIL, DFGA
pc0.90 aC osso e a e GA, NSGA-II,
SPEA2, PEAIL
DFGA
F0.80 bWeigh ing ac o BDE
b N bBandwid h ac o BDE
pde 0.20 bBDE c osso e a e BDE
C12.05 cPe sonal con idence BPSO, BMOPSO
C22.05 cGlobal con idence BPSO, BMOPSO
Vmax 1.00 cMaximum eloci y BPSO, BMOPSO
L0.06 dLea ning a e PEAIL
aDeb e al. [48].
bWang e al. [42].
cCle c and Kennedy [44].
dRong-Juan e al. [52].
popula ion. Fi s , o each solu ion xi, single-poin c osso e is
applied wi h p obabili y pc. Tha is, xiand ano he andomly
picked solu ion xj(i= j) a e combined in o x′
i←xi[1:u]∪xj[u+
1:N], whe e u∼ and ∈ [1,N]. Fo each solu ion, bi -s ing
mu a ion is also compu ed wi h p obabili y pm. In o he wo ds, i
he n h bi o a solu ion x′
ihas o be mu a ed, i s alue is lipped
(i.e., x′′
i[n]←¬x′
i[n]). The p ocedu e is illus a ed in Fig. 2(d–e).
Syn he ic solu ions. When he alues o pco pma e oo high,
he mu a ed popula ion ends o exploi he middle pa o he
eposi o y. In o he wo ds, solu ions wi h ew channels end o
add mo e channels, whe eas c owded solu ions end o dec ease
hei numbe o channels. In o de o main ain a simila exploi a-
ion ac oss he en i e eposi o y spec um, syn he ic solu ions
a e gene a ed apa om he mu a ed popula ion. Howe e , a
andom gene a ion o solu ions ac oss his spec um will unnec-
essa ily inc ease he numbe o e alua ions, slowing down he
algo i hm. DFGA gene a es syn he ic solu ions ying main ain
he mos ele an channels o he cu en eposi o y. The ank
o he i h channel is de ined as he numbe o imes ha he
channel iis p esen in he eposi o y (i.e., i= |i∈R|). DFGA
i e a i ely c ea es solu ions ha ha e om 1 o N−1 channels
by means o a oule e wheel selec ion (i.e., i ness p opo iona e
selec ion) based on he ank alues. I is wo hy o men ion ha
DFGA gene a es a o al o N−1 solu ions, since he N h solu ion
ha con ains all he channels is al eady pa o he eposi o y.
Eli ism. In each gene a ion, he eposi o y is upda ed ollowing
an eli is app oach. As depic ed in Fig. 2( ), o each unique
alue o 2(x) (i.e., o each numbe o channels), he eposi o y
solu ion ha minimizes 1(x) is selec ed. No e ha his ope a ion
is applied in he eposi o y, which includes bo h non-domina ed
and domina ed solu ions, c ea ing a balance be ween local and
global exploi a ion.
4. Resul s
Hype pa ame e s, de ailed in Table 1, we e se ollowing he
ecommenda ions o he li e a u e [42,44,48,52]. In o de
o assu e a ai compa ison among he algo i hms, he numbe
o gene a ions a ied in unc ion o he amoun o e alua ions
ha we e pe o med in a single i e a ion, while he numbe o
indi iduals o e e y single me a-heu is ic was ixed o m=
20 [4,7,25,30]. Table 2 de ails he compu a ional cos , including
he numbe o e alua ions pe gene a ion and he numbe o
Table 2
App oxima e compu a ional cos s o single and mul i-objec i e
me a-heu is ics.
M d. No. e al. E al. ime No. gen.
Single
GA 20 e al./gen. 785 ms/e al. 200 gen.
BDE 20 e al./gen. 810 ms/e al. 200 gen.
BPSO 20 e al./gen. 858 ms/e al. 200 gen.
Mul i
NSGA-II 40 e al./gen. 331 ms/e al. 100 gen.
SPEA2 20 e al./gen. 835 ms/e al. 200 gen.
BMOPSO 20 e al./gen. 852 ms/e al. 200 gen.
PEAIL 40 e al./gen. 415 ms/e al. 100 gen.
DFGA 123 e al./gen. 591 ms/e al. 32 gen.
M d.: me hod, gen.: gene a ion, e al.: e alua ion.
gene a ions o each me hod. In o al, 4000 e alua ions we e
pe o med. Fu he mo e, all he algo i hms we e compu ed 20
imes in o de o a oid local minima. The expe imen s we e
execu ed in an In el Co e i7-7700 CPU @ 3.60 GHz, 32 GB RAM,
Windows 10 P o, using MATLAB®2018b.
A con e gence analysis o single-objec i e me a-heu is ics
is depic ed in Fig. 3. These a e aged con e gence cu es
show he e olu ion o he agg ega ed objec i e unc ion F(x)
ac oss he gene a ions. Thus, hey es ima e he abili y o each
me hod o ind an op imal solu ion in he aining phase. The
de ailed con e gence cu es o each subjec can be ound in he
supplemen a y ma e ial. Conce ning he mul i-objec i e me a-
heu is ics, he e olu ion o he compu ed Pa e o- on s o e he
gene a ions o he algo i hms is depic ed in Fig. 4, also in aining
phase.
Ranks o selec ed channels o bo h single and mul i-objec i e
me a-heu is ics a e displayed in Fig. 5, including he common
K usienski’s 8-channel se . The ank o a channel is de ined as
he no malized numbe o imes ha he channel was selec ed
in he algo i hm epe i ions. Fo mul i-objec i e algo i hms, only
he anks o channels ha belongs o he eposi o y a e in-
cluded. Scalp dis ibu ions o he a e aged ank alues o e he
me a-heu is ics a e depic ed o each subjec as well.
In o de o e alua e he ac ual pe o mance o he single-
objec i e algo i hms using es ing da ase s, i is equi ed o selec
a single solu ion among he epe i ions. The e o e, he solu ion
ha eached he minimal F(x) alue was selec ed o each single-
objec i e me hod. Table 3 summa izes he a e aged es ing accu-
acies and numbe o channels o he selec ed solu ions o each
subjec , in unc ion o he employed me hod, using he maximum
numbe o sequences a ailable in each da abase. Rega ding he
mul i-objec i e algo i hms, he inal Pa e o- on o each subjec
is composed o he non-domina ed solu ions o all epe i ions.
Tes ing accu acies (i.e., a io o co ec ly p edic ed cha ac e s) o
he solu ions ha belongs o he inal Pa e o- on s a e shown
in Fig. 6, again using he maximum numbe o sequences a ail-
able. Finally, compu a ion cos s o all algo i hms a e de ailed
in Table 2.
5. Discussion
5.1. Con e gence analysis
Rega ding he single-objec i e me a-heu is ics, esul s showed
ha he inhe en ly disc e e algo i hms (i.e., GA and BDE) con-
e ge o op imal solu ions as e han BPSO, and we e able o
each he minimal objec i e alue o e e y single subjec . In-
he en disc e e algo i hms a e unde s ood as me a-heu is ics
ha employs bina y me hodologies o imp o e hei solu ions
(i.e., mu a ion, c osso e ). E en hough BPSO showed a slowe
con e gence han GA o BDE, he eached F(x) alues a e almos
7
V. Ma ínez-Cagigal, E. San ama ía-Vázquez and R. Ho ne o Applied So Compu ing 115 (2022) 108176
Fig. 3. A e aged con e gence cu es o single-objec i e me a-heu is ics (GA, BDE and BPSO) o each da abase in unc ion o he F(x) agg ega ed unc ion. Mean
alues a e displayed wi h solid lines, whe eas he 95% con idence in e al o he subjec s’ epe i ions is indica ed by he shaded a ea.
Fig. 4. E olu ion o Pa e o-op imal solu ions o he mul i-objec i e me a-heu is ics o each subjec ac oss all he epe i ions: DFGA ( ed), NSGA-II (blue), SPEA2
(yellow), BMOPSO (g een) and PEAIL (pu ple).
8
V. Ma ínez-Cagigal, E. San ama ía-Vázquez and R. Ho ne o Applied So Compu ing 115 (2022) 108176
Fig. 5. Channel anks o he selec ed and he Pa e o-op imal solu ions o single-objec i e (GA, BDE, BPSO) and mul i-objec i e (NSGA-II, BMOPSO, SPEA2, PEAIL,
DFGA) me a-heu is ics, espec i ely. K usienski’s 8-channel se (KRU) is also included. A e aged scalp dis ibu ions o e he algo i hms a e depic ed as well.
analogous, sugges ing ha BPSO, GA and BDE would show simila
pe o mances in es ing phase. I is also no ewo hy ha , e en
hough he a e aged con e gence o GA was as e han BDE,
he cu e eached a s ands ill o e he 100 h gene a ion, being
o e came by BDE he ea e .
Mul i-objec i e me a-heu is ics esul s showed ha DFGA,
NSGA-II, SPEA2 and PEAIL algo i hms we e able o each simila
Pa e o- on s, ou pe o ming BMOPSO. Besides he p ope pe -
o mance o DFGA, NSGA-II, SPEA2 and PEAIL in aining phase,
hey did no con e ge o hei op imal solu ions in he same
amoun o ime. DFGA con e ged as e han he es , likely due
o i s de e minis ic ini ializa ion, which allows he algo i hm o
a oid e alua ing solu ions ha a e a om eaching he op imal
alue. Among he o he s, NSGA-II con e ged as e han PEAIL and
SPEA2, whose ails sp ead ac oss highe non-op imal 2(x) alues.
9
V. Ma ínez-Cagigal, E. San ama ía-Vázquez and R. Ho ne o Applied So Compu ing 115 (2022) 108176
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i e gene ic algo i hm: NSGA-II, IEEE T ans. E ol. Compu . 6 (2) (2002)
182–197.
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10.1007/0-387-28356-0_10, URL: h p://link.sp inge .com/10.1007/0-387-
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miza ion using c owding, mu a ion and E-dominance, Lec u e No es in
Compu . Sci. 3410 (2005) 505–519.
[51] E. Zi zle , M. Laumanns, L. Thiele, SPEA2: Imp o ing he S eng h Pa e o
E olu iona y Algo i hm, Technical Repo , ETH Zu ich, Zu ich, 2001, pp.
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