Investment Decision Support Based on Interval Type-2 Fuzzy Expert System

-118- 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 - 119 - 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 - 120 - 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 - 122 - 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 - 123 - 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 - 124 - 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 - 125 - 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 Re e ence Alhassan, M. S. E., & Hag as, H. (2018). Towa ds Conges ion Con ol App oach Based on Weigh ed Random Ea ly De ec ion and Type-2 Fuzzy Logic Sys em. 10 h Compu e Science and Elec onic Enginee ing (CEEC). IEEE, 2018, 71–74. A enas, M., Bilbao, & Rod iguez, M. V. (2001). A uzzy goal p og amming app oach o po olio selec ion. Eu opean Jou nal o Ope a ional Resea ch, 133, 287–297. h ps://doi.o g/10.1016/S0377-2217(00)00298-8 Be na do, D., Hag as, H., & Tsang, E. (2012). An in e al ype-2 Fuzzy Logic based sys em o model gene a ion and summa iza ion o a bi age oppo uni ies in s ock ma ke s. 12 h UK Wo kshop on Compu a ional In elligence (UKCI). IEEE, 1–7. h ps://doi.o g/10.1109/UKCI.2012.6335765 B oz, Z., & Dos al, P. (2013). Fuzzy Logic Decision Suppo o Long-Te m In es ing in he Financial Ma ke s. Nos adamus: Mode n Me hods o P edic ion, Modeling and Analysis o Nonlinea Sys ems. Be lin, Heidelbe g: Sp inge . h ps://doi.o g/10.1007/978-3-642-33227-2_14 Cama a, R. C., Cuzzoc ea, A., G asso, G. M., Leung, C. K., Powell, S. B., Souza, J., & Tang, B. (2018). Fuzzy Logic- Based Da a Analy ics on P edic ing he E ec o Hu icanes on he S ock Ma ke . In: 2018 IEEE In e na ional Con e ence on Fuzzy Sys ems (FUZZ-IEEE). IEEE, 2018. 1–8. h ps://doi.o g/10.1109/FUZZ-IEEE.2018.8491523 Cas illo, O., Melin, P., & Cas o, R. J. (2013). Compu a ional in elligence so wa e o in e al ype-2 uzzy logic. Compu e Applica ions in Enginee ing Educa ion, 21(4), 737–747. h ps://doi.o g/10.1002/cae.20522