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Inzine ine Ekonomika-Enginee ing Economics, 2021, 32(2), 118–129
In es men Decision Suppo Based on In e al Type-2 Fuzzy Expe Sys em
Zuzana Janko a1,*, Dipak Kuma Jana2, Pe Dos al1
1B no Uni e si y o Technology, Facul y o Business and Managemen , Ins i u e o In o ma ics
Kolejní 2906/4, K alo o Pole, 612 00 B no, Czech Republic
E-mail. zuzana.janko a@ u b .cz; dos al@ bm. u b .cz
* Co esponding au ho
2Haldia Ins i u e o Technology, School o Applied Science & Humani ies
Haldia, Pu ba Midnapu -721657, Wes Bengal, India
E-mail. [email protected]
h p://dx.doi.o g/10.5755/j01.ee.32.2.24884
The decision-making p ocess on in es ing in inancial ma ke s is a e y complex and di icul ask, mainly due o he chao ic
beha io and high unce ain y in he de elopmen o he p ices o in es men ins umen s. Fo his eason, inancial ma ke s
a e inc easingly using means o a i icial in elligence, namely uzzy logic, which is able o cap u e he nonlinea
beha io .Fuzzy logic p o ides a way o d aw de ini i e conclusions om ague, ambiguous, o inaccu a e
in o ma ion.Howe e , he e a e some d awbacks associa ed wi h ype-1 uzzy logic, so he ype-2 uzzy logic comes o wa d,
which can wo k wi h g ea e unce ain y. Type-2 uzzy logic wo ks wi h a new hi d dimension uzzy se ha p o ides
addi ional deg ees o eedom and allows o model and p ocess nume ical and linguis ic unce ain ies di ec ly. The pape
applies ype-2 uzzy logic o he s ock ma ke wi h he aim o c ea e a simple and unde s andable model o deciding on
in es ing in in es men ins umen s, which is impo an o in es o s in his a ea. The p oposed ype-2 uzzy model uses
e u n, isk, di idend and o al expense a io o ETF as inpu a iables. The c ea ed sys em is able o gene a e agg ega ed
models om a ce ain numbe o language ules, which allows he in es o o unde s and he c ea ed inancial model. Using
ype-2 uzzy logic can lead o mo e ealis ic and accu a e esul s han ype-1 uzzy logic.
Keywo ds: Fuzzy Logic; In e al Type-2 Fuzzy Logic; In es men Analysis; So Compu ing; T2FLS.
In oduc ion
Cu en ly, linea models a e widely used o
o ecas ing, bu hese models a e g ea ly limi ed, especially
when applied o seasonal and nonlinea unce ain y issues.
Hence, nonlinea me hods such as neu al ne wo ks, uzzy
logic, and gene ic algo i hms a ac mo e and mo e
a en ion. Fuzzy Logic p o ides a way o d aw de ini i e
conclusions om ague, ambiguous, o inaccu a e
in o ma ion. An a i icial neu al ne wo k is widely accep ed
mainly because o i s abili y o lea n and e eal ela ionships
be ween non-linea a iables. Many esea che s ag ee ha
a i icial in elligence su passes adi ional models based on
s a is ical eg essions, as epo ed by Tung & Le (2017). In
pa icula , uzzy logic is able o wo k wi h inaccu a e da a
and in o ma ion in a ela i ely simple way as well as o
unde s and he meaning o wo ds in na u al language.
Su p isingly, uzzy logic, unlike o he echniques, is able o
use such aguely de ined expe ise o i s ad an age. Zadeh
(1965) called his ac he p inciple o incompa ibili y,
because uzzy logic is able o cap u e he ela ionship
be ween he ele ance and accu acy o in o ma ion. Zadeh
(1965) adds ha in a numbe o si ua ions a pe son decides
on he basis o inaccu a e o inde e mina e in o ma ion ha
is ga he ed om ou side, ye he esul o he ac i i y
ob ained om hese ague da a is s ill su icien . The
po en ial o uzzy logic o imp o e o ecas ing models can
be ound in a ious applica ions (such as Jana & Ghosh,
2018) due o i s known abili y o b idge he gap be ween
nume ical da a (quan i a i e in o ma ion) and language
exp ession (quali a i e in o ma ion),
In pa icula , inancial ma ke s a e in luenced by
de e minis ic and andom ac o s. Fu he mo e, Dos al &
Lin (2018) add ha he ime se ies o s ock i les,
commodi ies, cu ency a es, e c. a e in luenced by complex
economic and psychological phenomena ha con ain a high
p opo ion o chaos, hence uzzy logic and o he so
compu ing ools he bes ha cu en ly exis o p ocessing
and e alua ing economic and inancial in o ma ion and
da a. Simila ly, Rajab & Sha m (2019) epo ha s ock
p ice p edic ion is a complex and di icul ask due o
chao ic beha io and high unce ain y in he de elopmen o
equi y ma ke p ices. The design o a highly accu a e,
simple and unde s andable p edic i e model is o pa amoun
impo ance in his a ea. Yu & Yan (2019) add ha since
inancial da a con ains complex, incomple e and uzzy
in o ma ion, an icipa ing hei de elopmen al ends is an
ex emely di icul challenge. Fluc ua ions in inancial da a
depend on innume able co ela ed, cons an ly changing
ac o s. The e o e, p edic ing and analyzing inancial da a is
a non-linea and ime-dependen p oblem. Chang e al.
(2011) conclude and s a e ha s ock ma ke o ecas s can
only be success ul wi h he use o ools and echniques ha
can o e come he p oblem o p ice unce ain y and non-
linea i y. Wang &Wang (2015) epo ha uzzy logic and
neu al ne wo ks a e inc easingly being used in inancial
Zuzana Janko a, Dipak Kuma Jana, Pe Dos al. In es men Decision Suppo Based On In e al Type-2 Fuzzy Expe Sys em
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ma ke s, and hei a o p edic ion is widely acknowledged
in pa icula because o he abili y o cap u e non-linea
beha io . The same esul s a e p o ided by Janko a (2019).
Rao e al. (2017) a gue ha in es men decisions based on a
uzzy model can be pa icula ly use ul o in es o s looking
o minimize isk in sol ing hei long- e m in es men
po olio.The au ho s ecommend uzzy logic as a sui able
me hod o sol e he complexi y o s ock ma ke . O hman &
Schneide (2010) conside uzzy logic o be easie and mo e
bene icial o in es o s.
Howe e , as Melin & Cas illo (2014) and Cas illo e al.
(2007) s a ed, i is no easonable o use he exac unc ions
o uzzy logic belonging o some hing unce ain. As u he
desc ibed by Tang e al. (2020) ype-1 uzzy logic may be
unsui able o sol ing cases in he eal wo ld due o he
g owing unce ain y o he p oblem. In his case, i is
necessa y o use ano he ype o uzzy logic ha can handle
hese unce ain ies, namely he ype-2 uzzy logic (T2FLS)
sys em. Cas illo e al. (2013) epo ha T2FLS a e
essen ially " uzzy uzzy" se s ha gene alize ype-1 uzzy
se s and sys ems o wo k wi h g ea e unce ain y. Alhassan
& Hag as (2018) epo ha ype-2 uzzy se is educed o a
ype-1 uzzy se i he e is no unce ain y ha is analogous
o he p obabili y o dec easing o de e minism when
unp edic abili y disappea s. The membe ship unc ion o
he gene al ype-2 uzzy se is h ee-dimensional and
includes he oo p in o unce ain y (FOU). I is a new hi d
dimension o he uzzy se ha p o ides addi ional deg ees
o eedom and allows you o di ec ly model and p ocess
nume ical and linguis ic unce ain ies. Chen e al. (2018)
no e ha T2FLS ha e a highe app oxima ion capabili y
han neu al ne wo ks. Howe e , esea che s had o wai o
some ime o de elop he heo y. P og ess T2FLS p ima ily
limi ed ha dwa e equipmen .Simila ly, Wang e al. (2018)
and Liu e al. (2019) no e ha ype 2 uzzy se s a ac much
mo e in e es om scien is s because hey a e able o handle
unce ain and inaccu a e in o ma ion be e han ype-1
uzzy se s.
The decision-making p ocesses a e e y complica ed in
he economy and inance because hey include poli ical,
social, psychological, economic, inancial, and o he
ac o s.The e a e housands o heo ies and me hodologies
wi h a ious success o applica ion in eal wo ld. The heo y
and applica ion o uzzy p ocessing a e e y p omising,
especially newly disco e ed ype-2 uzzy me hod, which is
able o include addi ional highe le el o unce ain y
esul ing om unclea , unce ain o inaccu a e da a and,
hus enables be e desc ip ion o he economy and inancial
phenomena in hei solu ion o eal wo ld p oblems. We a e
sea ching o a new app oach o he ype-2 uzzy model o
suppo decision-making, which allows in es o s o inc ease
he quali y and speed o he decision-making p ocesses in
eal wo ld. The mo i a ion o his esea ch is he ac ha
he issue o in es men unds on he s ock ma ke has no
ye been su icien ly explo ed and es ed. We belie e ha
his issue is especially impo an o in es o s, as a sui able
and accu a e model can se e as a suppo o deciding in
which in es men und o in es .
The aim o he pape is o apply a highe deg ee o uzzy
logic, speci ically ype-2 uzzy logic as a suppo ing ool o
in es men decision making. T2FLS is used o make
decisions abou in es ing in exchange- aded unds on he
US s ock ma ke , he la ges ETFs o ganize s in he wo ld.
The s ock ma ke is cha ac e ized by chao ic beha iou as
desc ibed abo e, so i is p e e able o use ype-2 uzzy logic
a he han ype-1 uzzy logic because T2FLS is able o
co e he highe deg ee o unce ain y a ising om he
ypical ea u es o inancial ma ke s. T2FLS is designed o
inc ease he cla i y o he gene a ed model and achie e
be e pe o mance.
Li e a u e Re iew
Jilani& Bu ney (2008) p esen ed a simple p ognos ic
me hod o uzzy ime se ies. Dou a e al. (2002) used uzzy
in o ma ion echnology in hei wo k h ough echnical
analysis and simula e human beha io in s ock ading.
Escoba e al. (2013) p oposed an indica o o echnical
analysis based on uzzy logic, which includes he subjec i e
ea u es o he in es o . The au ho s highligh he app oach
o uzzy logic because o he abili y o ep esen a "human"
way o decision making ha a non-in es o in he eal
ma ke has. Wang (2018) p oposed a da a ame o p edic
s ock p ice p ices using uzzy ime se ies. His me hod uses
wo key echnologies, he heo y o uzzy se s and he
classical me hod o o ecas ing ime se ies. Expe imen al
esul s sugges ed ha he p oposed p edic i e amewo k
p o ides be e pe o mance. Na anjo e al. (2018) p oposed
a me hodology o de ec ing candles ick pa e ns in a s ock
ading sys em using uzzy logic. Candles ick based ules
a e mo e na u al and ealis ic han s anda d c isp ules. The
pe o mance o he sma s ock ading sys em is es ed in
wo di e en s ock ma ke po olios. The c ea ed model is
mo e s able and p o i able han o he ading sys ems.
Rus am e al. (2018) appliedsuppo ec o machines and
uzzy ke nel c-means o p edic he p ice mo emen o
Indonesian s ock ma ke s ock p ices by ocusing on he
banking subsec o . Based on his o ical s ock da a, eigh
echnical indica o s o model en y we e calcula ed. In a
pa icula case, he bes model is he en i e FKCM
obse a ion. Cama a e al. (2018) used a compu ing
in elligen ool ha uses uzzy logic da a analysis o p edic
he e ec s o hu icanes on he s ock ma ke . Liu & Zhang
(2019) used uzzy ime se ies o analyze and o ecas s ock
p ices o S a e Bank o India and Dow-Jones Indus ial
A e age (DJIA). The au ho s' expe imen al esul s show
ha he p oposed model o e comes o he ime se ies models
and can handle la ge amoun s o da a. Chen e al. (2019)
deal wi h he p icing o a Eu opean call op ion and s udying
G eek le e s o op ions in a uzzy en i onmen . The au ho s
deal wi h he de elopmen o a uzzy pa e n o Eu opean
call op ion p o ided ha he s ock e u n is a Gaussian uzzy
numbe . The esul s show ha uzzy op ions a e close o
heo e ical op ions de i ed om he Black-Scholes model.
Gau am &Abhishekh (2019) de eloped a new mo ing
a e age based p ognos ic app oach on he uzzy ime se ies
da a se . The de eloped mo ing a e age me hod o uzzy
ime se ies p o ides imp o ed p ognos ic ou pu wi h he
smalles RMSE, which shows ha he new me hod is much
be e han o he exis ing models a ailable in he li e a u e.
Recen ly, ype-2 uzzy logic has gained popula i y in a
wide ange o applica ions, mainly due o i s abili y o
handle a highe deg ee o unce ain y. Linag & Mendel
(2000) poin ou ha he knowledge ha is used o cons uc
Inzine ine Ekonomika-Enginee ing Economics, 2021, 32(2), 118–129
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ules in ype-1 uzzy logic (T1FLS) is unce ain. The e a e
h ee ways in which such a ule's unce ain y can appea : (1)
wo ds ha a e used in an eceden s and consequences o ules
can mean di e en hings o di e en people; (2)
consequences o o ing in a panel o expe s o en di e o
he same ules; (3) noise in aining da a. Unce ain y o
an eceden s o consequen s is ans o med in o an an eceden
and consequen membe ship unc ion. Type-1 uzzy logic
sys ems a e unable o di ec ly inco po a e hese unce ain ies,
while ype-2 uzzy logic sys ems can handle his unce ain y.
Fu he mo e, acco ding o Za andi e al. (2009) i should be
no ed ha ype-2 uzzy se s can model and minimize he
e ec o unce ain y in ules-based uzzy sys ems. The e ec s
o unce ain y can be minimized by op imizing he ype-2
uzzy se pa ame e s du ing he aining p ocess. Type-2
uzzy logic is pa icula ly use ul when i is di icul o
de e mine he exac unc ions o uzzy se .
Due o he highe deg ee o unce ain y, T2FLS is
applied o inancial ma ke issues. These s udies include
Jianga e al. (2018), who designed in e al T2FLS o p edic
s ock index in Taiwan, DJIA and NASDAQ using uzzy
ime se ies. The esul s o uzzy logic a e compa ed wi h
non-pa ame ic s a is ical es ing, s a ing ha hei p oposed
model exceeds o he me hods. Za andi e al. (2009) applied
a ype-2 uzzy model o analyze he p ices o au omo i e
equi y ins umen s in Asia using echnical and undamen al
indica o s. Thei esul s a e e y encou aging and can be
implemen ed o o ecas ins umen p ices in ading
sys ems. Hua ng & Yu (2005) ha e designed a ype-2 uzzy
model o p edic he TAIEX index ha achie es be e
esul s han he ype-1 uzzy model. Liue e al. (2012)
in oduced a ype-2 neu o uzzy model o TAIEX
p edic ion. Thei esul s showed ha his me hod showed
highe p edic ion accu acy wi hou he use o addi ional
in o ma ion. Hasuike &Ishii (2009) discussed he p oblem
o choosing a ype-2 uzzy po olio wi h expec ed e u ns
wi h espec o in es o 's subjec i i y. Be na do e al. (2012)
designed T2FLS, which is able o gene a e agg ega e
models om a p ede e mined numbe o language ules,
allowing he use o unde s and he gene a ed models o
p edic ing s ock ma ke oppo uni ies. Janko a & Dos al
(2019) applied ype-2 uzzy logic o he Czech s ock ma ke
and c ea e a model ha is used o decide on in es ing in PX
index s ocks. Runkle e al. (2017) p esen ed a new
app oach o using ype-2 uzzy se s in decision-making wi h
ega d o he isk associa ed wi h his decision. Conside ing
he le el o isk inc eases he scope o decision-making o
allow be e decision making. Vella & Lon Ng (2016)
in es iga ed he abili y o highe o de uzzy sys ems o
cope wi h inc eased unce ain y, which is mos ly due o
ma ke mic os uc u e noise. The au ho s p oposed an
inno a i e app oach o design he ype-2 in e al model,
which is based on a gene alized ype-1 ANFIS app oach.
The p oposed model achie es signi ican pe o mance
imp o emen s o e s anda d ANFIS and Buy and Hold
me hods. Zhang e al. (2017) used T2FLS o alida e be e
axa ion pe o mance me hodology a he Shanghai S ock
Exchange Composi e Index and Taiwan S ock Capi al
Capi alized Weigh ed Index. The expe imen al esul s show
ha he p oposed me hod o e comes o he basic me hods.
Pulido & Melin (2016) ep esen ed he op imiza ion o
uzzy ype-1 and ype-2 ile neu al ne wo ks o p edic ing
complex ime se ies on he Taiwanese S ock Exchange
(TAIPEX).
Me hodology
The T2FLS s uc u e is e y simila o he T1FLS
s uc u e. Figu e 1 shows he s uc u e o T2FLS. The
measu ed eal a iables a e i s ans o med in a
uzzi ica ion block in o linguis ic a iables, wi h he
linguis ic a iables based on he basic linguis ic a iables.
Dos al (2011) s a es ha h ee o se en a ibu es o his
basic a iable a e usually used. The deg ee o a ibu e o a
gi en a iable in a se is ep esen ed by a ma hema ical
unc ion. Th ee ypes o uzzi ica ion a e a ailable in
T2FLS. I he measu ed da a is pe ec , modeled as a sha p
se , da a wi h noise and da a wi h s a iona y noise a e
modeled as ype-1 uzzy se s, wi h non-s a iona y noise
modeled as ype-2 uzzy se s. The la e ype o uzzi ica ion
canno be pe o med in T1FLS.
Figu e 1. S uc u e o Type-2 Fuzzy Logic Sys em
x
x
x
Membe ship Func ion
An eceden s
Fuzzi ica ion
In e ence
ules
Membe ship Func ion
Consequen
Ou pu
De uzzi ica ion
An eceden s
An eceden
deg ees o
u h
Zuzana Janko a, Dipak Kuma Jana, Pe Dos al. In es men Decision Suppo Based On In e al Type-2 Fuzzy Expe Sys em
- 121 -
As epo ed by Medasani e al. (1998), all exis ing
T2FLS membe ship unc ions a e modi ied e sions o
con en ional T1FLS membe ship unc ions. In o he wo ds,
he basis o he membe ship unc ions o he o iginal ype-1
is blu ed i he p ac i ione is unce ain abou he alue o
he membe ship unc ion o a pa icula poin . The e a e a
numbe o ype-2 uzzy membe ship unc ions, such as
iangula , Gaussian, apezoidal, sigmoidal, e c.as no ed by
Wang e al. (2018). Gaussian membe ship unc ion is
widely used in li e a u e in which unce ain y is associa ed
wi h mean and s anda d de ia ion.
Kayacan e al. (2018) adds ha in applica ions o he
heo y o uzzy se s, he membe ship unc ions a e chosen
based on he subjec i e pe cep ion o ague o inaccu a e
ca ego ies. Fu he mo e, he e a e no c i e ia o assess he
app op ia eness o co ec ness o he chosen membe ship
unc ion. Type-2 uzzy se is deno ed 𝐴󰆻 is cha ak e ized
ype-2 membe ship unc ion 𝜇𝐴(𝑥,𝑢), whe e 𝑥∈𝑋,∀ 𝑢∈
𝐽𝑥𝑢⊆[0,1]and 0≤𝜇𝐴(𝑥,𝑢)≤1is ma hema ically de ined
(Jana &Ghosh, 2018):
𝐴󰆻={(𝑥, 𝜇𝐴(𝑥))|𝑥∈𝑋}
(1)
𝐴󰆻={(𝑥,𝑢, 𝜇𝐴(𝑥,𝑢))|𝑥∈𝑋,∀𝑢
∈𝐽𝑥𝑢[0,1]}
(2)
I ype-2 uzzy se 𝐴󰆻 is a con inuous a iable, he
exp ession has he o m:
𝐴󰆻={∫[∫𝑓𝑥(𝑢)/𝑢
𝑢∈𝐽𝑥𝑢]/𝑥
𝑥∈𝑋 }
(3)
whe eʃʃdeno es a connec ion be ween 𝑥 and 𝑢. I ype-
2 uzzy se 𝐴󰆻 is disc e e, hen he exp ession has he o m:
𝐴󰆻={∑𝜇𝐴(𝑥) / 𝑥
𝑥∈𝑋 }
(4)
𝐴󰆻={∑[∑𝑓𝑥𝑖(𝑢𝑗)/ 𝑢𝑖𝑗
𝑚𝑖
𝑗=1 ]/𝑥𝑖
𝑛
𝑖=1 }
(5)
whe e ∑∑deno es he connec ion be ween 𝑥 and 𝑢.
Assuming 𝑓𝑥(𝑥)= 1,∀𝑢∈[𝐽−𝑥
𝑢,𝐽𝑥−𝑢]⊆[0,1], is ype-2
membe ship unc ion 𝜇𝐴(𝑥,𝑢)exp essed by ype-1 in e io
membe ship unc ion𝐽−𝑥
𝑢=𝜇𝐴(𝑥) and ype-1
supe io ,𝐽𝑥−𝑢=𝜇𝐴(𝑥)s hen called he ype-2 in e al uzzy
se deno ed by he ollowing ma hema ical o mula:
𝐴󰆻={(𝑥,𝑢,1)|∀𝑥∈𝑋,∀𝑢
∈[𝜇−𝐴(𝑥),𝜇𝐴(𝑥)]
⊆[0,1]}
(6)
o
𝐴󰆻={∫[ ∫ 1/𝑢
𝑢∈[𝐽−𝑥
𝑢,𝐽𝑥−𝑢]⊆[0,1]]/𝑥
𝑥∈𝑋 }
(7)
𝐴󰆻={∫[ ∫ 1/𝑢
𝑢∈[𝜇−𝐴(𝑥),𝜇𝐴(𝑥)]⊆[0,1]]/𝑥
𝑥∈𝑋 }
(8)
The unce ain y is hen de e mined by combining all he
membe ship ha a e labeled as oo p in o unce ain y
(FOU). The size o he FOU depends di ec ly on he
unce ain y ha he ype-2 uzzy se media es. FOU is as
ollows:
𝐹𝑂𝑈(𝐴󰆻)=⋃𝐽𝑥
∀𝑥∈𝐴 ={(𝑥,𝑢):𝑢∈𝐽𝑥
⊆[0,1]}
(9)
The uppe and lowe membe ship unc ions o he ype-
2 uzzy 𝐴󰆻 a e wo ype-1 membe ship unc ions.
𝐽𝑥= [𝜇𝑥(𝑥),𝜇𝑥(𝑥)]
(10)
Using he FOU, a gi en o mula can also be exp essed
as:
𝐹𝑂𝑈(𝑋)=⋃[𝜇𝑥(𝑥),𝜇𝑥(𝑥)]
𝑥∈𝐴
(11)
The uzzi ica ion is ollowed by a uzzy in e ence ha
de ines he beha io o he sys em using he IF-THEN ules
and he language le el e alua ing he s a us o membe ship
o e aci y o he a iable. Each combina ion o he
a ibu es o he a iables en e ing he sys em and occu ing
in he condi ion exp esses one ule.Consequen ly, o each
ule, he deg ee o suppo o he weigh o he ule in he
sys em needs o be de e mined. The esul o uzzy
in e ence, as men ioned by Dos al e al. (2005) is a language
a iable. In mos applica ions, howe e , a inal ou pu is
equi ed as a speci ic numbe , and no a uzzy se . As a
esul , he ou pu uzzy se mus be con e ed o a numbe .
Fuzzy ules de ine he connec ion be ween inpu and
ou pu uzzy a iables. T2FLS ules can o e an al e na i e
i he e is a need o model he unce ain y o he p oblem.
T2FLS ules a e be e a no using he exac le els o
membe ship, o example, when aining da a is a ec ed by
noise. Fuzzy ule has, acco ding o Cas illo e al. (2007), he
ollowing o m, whe e he an eceden and he consequen
a e now ype 2:
𝑅𝑛:𝐼𝐹 𝑥1 𝑖𝑠 𝑋1𝑛 𝑎𝑛𝑑 ….𝑎𝑛𝑑 𝑥𝑙 𝑖𝑠 𝑋𝑙𝑛 𝑇𝐻𝐸𝑁 𝑦 𝑖𝑠 𝑌𝑛
(12)
whe e 𝑋𝑙𝑛 is he T2FLS an eceden and 𝑌𝑛= [𝑦𝑛,𝑦𝑛]
is he T2FLS consequen . He e 𝑦𝑛 and 𝑦𝑛, as s a ed by
Taskin & Kumbasa (2015), he e may be consequences o
linea unc ions:
𝑦𝑛=𝑎1𝑛𝑥1+⋯+ 𝑎𝑙𝑛𝑥𝑙+𝑏𝑛
(13)
𝑦𝑛=𝑎1𝑛𝑥1+⋯+ 𝑎𝑙𝑛𝑥𝑙+𝑏𝑛
(14)
whe ein he an eceden is a compound uzzy logical
exp ession o one o mo e simple uzzy exp essions
associa ed wi h uzzy ope a o s, and he consequen is an
exp ession ha assigns uzzy alues o an ou pu a iable.
The in e ence sys em e alua es all ules and combines he
weigh s o he consequence o all ele an ules in o one
uzzy se using a summa y ope a ion.
Inzine ine Ekonomika-Enginee ing Economics, 2021, 32(2), 118–129
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In T1FLS, he con e sion p ocess o a speci ic numbe
is called de uzzi ica ion. The e a e many ways o achie e
he esul , o example, o calcula e he cen e o g a i y o
he membe ship unc ion o a uzzy se , calcula e he
weigh ed a e age o he cen e o g a i y o each
membe ship unc ion, e c. Howe e , he ma e is much
mo e complica ed o T2FLS because mo ing om ype-2
uzzy o a c isp se equi es wo s eps. The i s s ep is
named as a ype educe in which he ype-2 uzzy se is
educed o he ype-1 uzzy se . The e a e as many ype
educe s as he T1FLS de uzzi ica ion me hods. The mos
commonly used algo i hm de eloped by Ka nik & Mendel
(2001) and Mendel (2001) is i e a i e and as . The ype
educe gene a es a uzzy se T1FLS, which is hen
ans o med in o an ou pu by de uzzi ica ion. In he case o
using he cen e -o -sum me hod (cos) o he ype educe , i s
ma hema ical exp ession, acco ding o Taskin & Kumbasa
(2015), is as ollows:
𝑌𝑐𝑜𝑠(𝑥)=[𝑦𝑡,𝑦𝑟]=⋃∑𝑦𝑛𝑓𝑛𝑁
𝑛=1
∑𝑓𝑛
𝑁
𝑛=1
𝑓𝑛∈𝐹𝑛(𝑥)
(15)
whe e 𝑦𝑡 and 𝑦𝑟 de ined as:
𝑦𝑡=∑𝑦𝑛𝑓−𝑛+∑𝑦𝑛 𝑓𝑛𝑁
𝑛=𝐿+1
𝐿𝑛=1
∑𝑓−𝑛+∑𝑓𝑛
𝑁
𝑛=𝐿+1
𝐿𝑛=1
(16)
𝑦𝑟=∑𝑓𝑛𝑦−𝑛+∑𝑦−𝑛𝑓−𝑛𝑁
𝑛=𝑅+1
𝑅𝑛=1∑𝑓𝑛+∑𝑓−𝑛
𝑁
𝑛=𝑅+1
𝑅𝑛=1
(17)
whe ein R and L a e poin s ha can be ound using he
i e a i e KM algo i hm.
The second s ep o p ocessing he ou pu ha ollows
he ype educe is s ill called de uzzi ica ion. In Figu e 1 i
can be seen ha he e may be wo nume ical ou pu s o
T2FLS designa ed as c isp ou pu s and a ype educed se .
The la e exp esses he deg ee o unce ain y ha T2FLS
has due o unce ain inpu measu emen s (Mendel, 2007;
Za andi e al., 2009). A e age alues a e ob ained om he
ype educe , and he calcula ion o he de uzzi ica ion
ou pu is as ollows:
𝑦=𝑦𝑡+𝑦𝑟
2
(18)
Expe imen al Resul s and Analysis
This pape uses T2FLS o make decision making
p ocess on in es ing in exchange aded unds. Fo he ime
being, he applica ion o highe deg ee o uzzy logic has
been unde used o he s ock ma ke o ecas , as eco ded
by he li e a u e e iew. The ollowing pa o he pape
p esen s no only selec ed da a sample bu also he equi ed
ou pu s. Then a model is c ea ed based on he T2FLS. The
model ou pu is a signal o buy o sell he Exchange T aded
Funds (ETF) s ock.
Da a Sample
Fo he T2FLS model c ea ion, 10 ETFs in he eal
es a e sec o a e selec ed. The mos impo an cha ac e is ic
ea u e o he ETF, as he name sugges s, is ha i is aded
simila ly o s ock exchange. They a e alued and aded
con inuously du ing he ading day, allowing in es o s o
buy o sell wi hou delay. S ock exchange unds in es in a
de ined index o he baske o asse s, he in es o s a e
allowed o in es in he en i e po olio wi h a single sha e.
These a e open-ended, passi e- unded unds ha aim o
copy he benchma k as accu a ely as possible. ETFs ha e
become e y popula o e he pas decade, mainly due o
hei e y low cos , high liquidi y and lowe isk due o good
po olio di e si ica ion.
ETFs a e a ela i ely new in es men ins umen ha
p o ides ce ain ad an ages o e adi ional mu ual unds.
Abo e all, ETFs a e e y low-cos unds, hei goal is he
mos ai h ul eplica ion o he unde lying index, which is
e e ed o as a benchma k. The la ges and mos liquid
s ock ma ke , he US s ock ma ke , is selec ed o his s udy,
which, acco ding o he ICI Fac book (2018), o e s 1,832
ETFs and $ 3.4 illion o asse s unde managemen . Table
1 shows summa y s a is ics o inpu a iables o selec ed
ETFs en e ing he model.
Financial Indica o s
The key ac o s o in es men decisions ha in es o s
ake in o accoun a e usually he e u n and isk o he
in es men , as con i med by Fang e al. (2006). Some a e
also inclined o include he di idend esul ing om he
in es men acco ding o Gup e al. (2008). O he au ho s
p e e in es men liquidi y (A enas e al., 2001). Li e al.
(2000) o Khayamim e al. (2018) a gues ha igno ing he
cos o an in es men leads o inaccu a e and ine icien
models ha can lead o a loss-making in es men . Based on
he abo e, he ollowing a e selec ed as inpu a iables o
he p esen ed model: e u n, di idend, isk and o al expense
a io (TER). The ou pu o he model is he decision whe he
o in es in he ETF o no .
Table 1
Summa y S a is ic o Inpu s Va iables
S a is ics
Mean
Max
Min
Sd
Skewness
Ku ois
Inpu s
Risk
0,037
0,055
0,015
0,009
-0,826
0,061
Re u n
0,010
0,032
-0,006
0,009
0,910
0,118
Di idend
0,039
0,049
0,029
0,007
0,260
-1,426
TER
0,004
0,006
0,001
0,002
-0,660
-0,976

Zuzana Janko a, Dipak Kuma Jana, Pe Dos al. In es men Decision Suppo Based On In e al Type-2 Fuzzy Expe Sys em
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Table 1 summa izes he basic s a is ics o inpu
a iables. On a e age, he isk o selec ed ETFs was a ound
3.7 % and e u n was a ound 1.0 %. The amoun o
di idends paid and he TER indica o a e almos iden ical
o he moni o ed unds. The able abo e shows he ac ha
ETFs a e e y low-cos unds wi h an a e age cos o 0.4 %,
which is a signi ican di e ence compa ed o con en ional
mu ual unds. ETFs a e also sui able o in es o s seeking
s able di idend income, as he ETF's minimum di idend is
2.9 %.
Expe imen al Resul s
Fuzzy logic is able o accep an explana ion o
ambiguous and ague hinking. Fuzzy Logic is a e y
popula ool o sol ing a ious p oblems. The wo mos
common ypes o uzzy in e ence sys em a e Mamdani and
Sugeno. The ollowing s udy uses T2FLS ype Sugeno.
Sugeno T2FLS model design consis s o a o al o ou inpu
a iables di ided in o a o al o h ee a ibu es and one
ou pu a iable. The s uc u e o he model is shown in
Figu e 2.
.
Figu e 2.S uc u e o he Sugeno T2FLS Model
Al hough he choice o membe ship unc ion is
subjec i e and depends on he choice o he expe and he
sample o he da a se , se e al s udies a e ca ied ou
ocusing on compa ing he di e en ypes o hese unc ions
in o de o ind he mos app op ia e one. Mayil aganan &
Naidu (2011) ound ha he bes pe o mance and esul s
we e achie ed by he Gaussian membe ship unc ion. Bell
and apezoidal membe ship unc ion wo se han Gaussian.
The abo e s udy is ollowed by Talpu e al. (2017), who
ocused on compa ing Gaussian, iangula , apezoidal, and
bell membe ship unc ions. Thei s udy shows ha he
Gaussian membe ship unc ion is mos app op ia e in he
ANFIS model. In hei esea ch, Es ahanipou &Aghami i
(2010) used he Gaussian membe ship unc ion in he
ANFIS model o es da a on in es men ins umen s. Fo
his eason, in he p oposed T2FLS model, he Gaussian
membe ship unc ion is chosen. Th ee a ibu es (LOW,
MEDIUM, HIGH) o he membe ship unc ion a e used,
which a e u he di ided in o UPPER and LOWER
unc ion in o de o cap u e a highe deg ee o unce ain y.
The isk can be cha ac e ized as he possibili y ha he
expec ed e u n de ia es om he ac ual e u n. This is a
ce ain deg ee o unce ain y associa ed wi h he expec ed
e u n. The Gaussian T2F membe ship unc ion o isk is
shown in Figu e 3. Re u n can be unde s ood as an in es o 's
emune a ion o he isk incu ed. The goal o in es o s is o
achie e he highes possible e u n while minimizing isk.
The Gaussian T2F membe ship unc ion o e u n is shown
in Figu e 4. ETFs paying di idends gene a e egula income
o in es o s. All ETFs selec ed pay di idends on a qua e ly
basis. In es o s can bene i no only om po en ially ising
ETF sha e p ices bu also om di idend income. The ype-2
uzzy membe ship unc ion o di idend is shown in Figu e
5. TER is a measu e o he o al cos o managing and
ope a ing he und. These cos s include managemen ees and
addi ional expenses, such as ading ees, legal ees, audi o
ees, and o he ope a ional expenses. The o al cos o he und
is di ided by he o al asse s o he und o each he
pe cen age ha ep esen s he TER. Fo ETFs, TERs a e a a
e y low le el compa ed o con en ional mu ual unds. The
ype-2 uzzy membe ship unc ion o TER is shown in
Figu e 6.
Figu e 3. Risk T2F Membe ship Func ion
Figu e 4. Re u n T2F Membe ship Func ion
Inzine ine Ekonomika-Enginee ing Economics, 2021, 32(2), 118–129
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Figu e 5. Di idend T2F Membe ship Func ion
Figu e 6. TER T2F Membe ship Func ion
In he c ea ed T2FLS model, 81 IF-THEN ules a e
c ea ed wi h ANFIS, as shown in Figu e 7. Ve bal
in e p e a ion o uzzy ules is as ollows:
IF isk is low AND e u n is high AND di idend is high
AND TER is low, THEN ETF is ecommended o BUY.
IF isk is low AND e u n is high AND di idend is high
AND TER is medium, THEN ETF is ecommended o BUY. …
IF isk is high AND e u n is low AND di idend is low
AND TER is high, THEN ETF is ecommended o SELL.
IF isk is high AND e u n is low AND di idend is low
AND TER is medium, THEN ETF is ecommended o SELL.
Simila ly, addi ional ules a e selec ed. The ules a e se
in such a numbe , which desc ibes he p oblem.
The c ea ed T2FLS model simula es in es men
decisions in ETF lis ed s ocks on he US s ock ma ke . The
model can be demons a ed on a case s udy based on he
membe ship unc ions and Sugeno ules in he Figu e 7. I
he ETF sha eholding isk is 2.5 %, he e u n is 1.45 %, he
qua e ly di idend is 3 % and he ETF's o al expense a io
is 0.34 %, acco ding o T2FLS i is app op ia e o pu chase
he ETF in he in es men po olio as he alue is exac ly
equal o 1.
Figu e 7. Fuzzy Model Con olle ules
The g aphs gene a ed om he c ea ed IT2FLS a e used
o pe o m he sensi i i y analysis. Figu e8 shows he
sensi i i y analysis o isk and TER. A i s glance, i is
clea ha he highe he isk o in es ing in he ETF, he
mo e i is ecommended no o in es in he und.
Con e sely, a e y low le el o isk is posi i e o in es o s
wi h a clea ecommenda ion o buy low isk ETF sha es.
F om he same g aph you can ead he sensi i i y o he o al
cos o he und o he o e all ecommenda ion whe he o
in es in he ETF o no . As men ioned abo e, ETFs a e low
cos unds wi h an a e age cos o 0.4 %, wi h a maximum
cos o 0.6 % and a minimum cos o 0.1 %. I ollows ha
he TER has no signi ican in luence on he
ecommenda ions gene a ed by T2FLS whe he o in es o
no o in es in he und, as he TER is e y low o all
moni o ed unds.
Zuzana Janko a, Dipak Kuma Jana, Pe Dos al. In es men Decision Suppo Based On In e al Type-2 Fuzzy Expe Sys em
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Figu e 8. Sensi i i y Analysis o Risk and TER
Figu e 9. Sensi i i y Analysis o Re u n and Di idend
Figu e 9 depic s he sensi i i y analysis o he e u n
and amoun o ETF di idends paid. The g aph shows ha
he highe he e u n om he ETF, he s onge i is o in es
in he und. Basically, i can be seen i he e u n is posi i e,
i.e. he in es o does no ealize he loss om holding ETF
sha es, so i is ecommended o buy he ETF. The sensi i i y
analysis o he di idend is also shown in he same g aph.
Selec ed ETFs a e ai ly consis en in di idend payou a io.
On a e age, ETFs pay 3.9 %, wi h a minimum ETF payou
o 2.9 % and a maximum di idend o 4.9 %. I ollows ha ,
like TER, he amoun o di idends paid does no ha e a
majo impac on in es men decisions, as indeed he
indica o s a e almos he same o selec ed ETFs o e he
epo ing pe iod.
Compa ison o Resul s
Based on he inpu and ou pu pa ame e s se , a T2FLS
model is c ea ed o make ETF in es men decisions.
S a is ical pa ame e s a e de ined o he assessmen and
con ol o he conclusions eached in he T2FLS model. The
p edic i e abili y o he c ea ed model is de e mined by
compa ing he o iginal da a and ou pu s ob ained om he
model. Fo his pu pose, he oo -mean-squa e e o
(RMSE) poin e is used, he ma hema ical no a ion o which
is as ollows:
𝑅𝑀𝑆𝐸=√1𝑛∑(𝑦𝑡−𝑦󰇗𝑡)2
𝑛
𝑡=1
(19)
The p edic i e abili y o he c ea ed model is u he
e alua ed using he de e mina ion coe icien (R2)
calcula ed acco ding o he ollowing ma hema ical
o mula:
𝑅2=1−∑(𝑦𝑡−𝑦󰇗𝑡)2
𝑦󰇗𝑡2
𝑛
𝑡=1
(20)
In addi ion o he abo e s a is ical pa ame e s, mean
absolu e e o (MAE) can also be used o e alua e model
pe o mance as ollows:
𝑀𝐴𝐸=1𝑛∑|𝑦𝑡−𝑦󰇗𝑡|
𝑛
𝑡=1
(21)
In pe cen age e ms, he mean absolu e pe cen age e o
(MAPE) can be used o modi y he abo e o mula as
ollows:
𝑀𝐴𝑃𝐸=1𝑛∑|𝑦𝑡−𝑦󰇗𝑡|
𝑦𝑡
𝑛
𝑡=1 ×100
(22)
whe ein 𝑛 is he numbe o obse a ions o he da a se ,
y indica es he ou pu p edic ed by he model o he - h
alue, y󰇗 indica es he o iginal measu ed ou pu o he da a
se o he - h alue.
Table 2
S a is ical Da a Analysis o FLS
Model
RMSE
R^2
MAE
MAPE
T2FLS Sugeno
0.08286
0.93581
0.01478
15.22 %
T1FLS Sugeno
0.08179
0.92568
0.01493
15.43 %
T1FLS Mamdani
0.12909
0.88889
0.01667
17.26 %
The c ea ed T2FLS model is compa ed, based on he
abo e s a is ical pa ame e s, wi h he T1FLS model o bo h
Sugeno and Mamdani. Compa ison esul s a e p esen ed in
Table 2. F om he poin o iew o T1FLS, he Sugeno
model wi h RMSE 0.08179 compa ed o he Mamdani
0.12909 model. Be e esul s achie ed T2FLS Sugeno wi h
0.08286. Also, he de e mina ion coe icien p o ides e y
p omising esul s in e ms o Sugeno's abili y. I can be
s a ed ha T2FLS is sui able o in es men analysis
because i can be e deal wi h he unce ain y and chaos ha
p e ails in he inancial ma ke and p o ides cons an
esul s. An indispu able ad an age o T2FLS in he
inancial ma ke s is he abili y o accommoda e a highe
deg ee o unce ain y han T1FLS. T2FLS has demons a ed
compu a ional lexibili y and sui abili y o modeling
complex, dynamic and nonlinea ela ionships ha a e
common in inancial ma ke s.
Inzine ine Ekonomika-Enginee ing Economics, 2021, 32(2), 118–129
- 126 -
Discussion
I is well known ha he s ock ma ke is a e y complex
sys em ha exhibi s dynamic and highly nonlinea beha io .
Fo his eason, i is no easy o p edic o a leas de e mine
he di ec ion o i s u u e de elopmen . The aim o he
p esen ed pape was o c ea e a sui able and simple model
ha will se e as a suppo o deciding on p o i able
oppo uni ies in he s ock ma ke and hus educe in es o
unce ain y. The model applied an in e al ype-2 uzzy
model, which is able o con ain a highe deg ee o
unce ain y esul ing om he na u e o he examined
ma ke s. The de eloped model is e y limi ed; howe e , i
illus a es he possibili y o u he use o T2FLS, which is
applied o a dominan ex en in echnical indus ies. The
p esen ed model could be u he ex ended by o he
impo an inpu a iables and c ea e a mo e complex e ised
model, howe e , he aim o his pape was o keep he se o
inpu a iables and he model as simple as possible so ha
i can be used by an inexpe ienced in es o . As Doskočil &
Dos ál (2017) u he s a e, only a p o en model can be used
as a ool o in es men decisions. Fo his eason, he
p oposed neu o- uzzy model canno be conside ed inal and
only co ec . I is also necessa y o discuss some limi s and
limi a ions o uzzy logic. Fullé (1995) s a es ha i is no
gua an eed ha any uzzy model c ea ed will emain s able
and obus . A majo limi a ion is he ac ha uzzy logic has
no memo y and, mo eo e , is no able o lea n om he
p esen ed da a. In addi ion, de e mining he shape and size
o he membe ship unc ion is a e y complex and
subjec i e ma e ha can ul ima ely a ec he pe o mance
o he en i e model. Ve i ica ion o he uzzy model also
equi es loop es ing, which can cause di icul ies.
Conclusion
The decision-making p ocess on in es men
oppo uni ies is a widely discussed opic oday. A i icial
in elligence models used in many ields can be used o his
pu pose. This pape in oduces he implemen a ion o ype-
2 uzzy logic o he in es men decision p oblem. The
applica ion o T2FLS is s ill inadequa e especially in he
a ea o inancial ma ke s, al hough he e is a high deg ee o
unce ain y, chaos and non-linea i y especially in inancial
ma ke s. In o he wo ds, equi y ma ke s in pa icula
p o ide su icien scope o examining he pe o mance o a
highe deg ee o uzzy logic.
The pape is an easy- o-use model ha con ains simple
inpu a iables ha undamen ally in luence he decision o
in es in exchange- aded unds in he US s ock ma ke .
Speci ically, he accu acy o he T2FLS Sugeno model is
almos 93.6 % compa ed o he T1FLS Sugeno 92.6 %,
espec i ely T1FLS Mamdani 88.9% measu ed by
de e mina ion coe icien . Simila ly, e o a es such as
RMSE, MAE and MAPE sound in a o o T2FLS on he
equi y ma ke s. The highe pe o mance o T2FLS o
in es men decisions is mainly due o he use o he
membe ship unc ion o he gene al ype-2 uzzy se which
is h ee-dimensional and includes he oo p in o
unce ain y (FOU). I is a new hi d dimension o he uzzy
se ha p o ides addi ional deg ees o eedom and allows
you o di ec ly model and p ocess nume ical and linguis ic
unce ain ies
Using T2FLS can lead o mo e ealis ic and accu a e
esul s han T1FLS. Fo u he esea ch, i would be
app op ia e o ex end he p oposed model o include o he
signi ican unde lying indica o s o o include echnical and
psychological indica o s and o moni o he s eng h o he
e ised model in o he s ock ma ke s
Acknowledgemen s
This pape was suppo ed by p ojec No. FP-J-20-6246 ‘The Use o A i icial In elligence in Business IV’ and No. FP-S-20-
6376 ‘Modeling and op imiza ion o business p ocesses in condi ions o digi al ans o ma ion’ om he In e nal G an
Agency a B no Uni e si y o Technology
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