A generalization of multi-source fusion-based framework to stock selection

In o ma ion Fusion 102 (2024) 102018 A ailable online 17 Sep embe 2023 1566-2535/© 2023 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/). Con en s lis s a ailable a ScienceDi ec In o ma ion Fusion jou nal homepage: www.else ie .com/loca e/in us Full leng h a icle A gene aliza ion o mul i-sou ce usion-based amewo k o s ock selec ion Václa Snášel a,1, Juan D. Velásquezc,e,1, Millie Pan b, ,1, Dimi ios Geo giou d,1, Lingping Kong a,∗,1 aFacul y o Elec ical Enginee ing and Compu e Science, VŠB-Technical Uni e si y o Os a a, Czech Republic bMeh a Family School o Da a Science and A i icial In elligence, Indian Ins i u e o Technology, Roo kee, U a akhand, 247667, India cDepa men o Indus ial Enginee ing, Uni e si y o Chile, San iago, Chile dFacul y o Elec ical Enginee ing and Compu e Science, Na ional Technical Uni e si y o A hens, G eece eIns i u o Sis emas Complejos de Ingenie ía (ISCI), Chile Depa men o Applied Ma hema ics and Scien i ic Compu ing, Indian Ins i u e o Technology, Roo kee, U a akhand, 247667, India ARTICLE INFO Keywo ds: S ock selec ion Mul i-sou ce usion Mul i-c i e ia decision-making Time-se ies p edic ion Dynamic ime wa ping ABSTRACT Selec ing ou s anding echnology s ocks o in es men is challenging. Speci ically, he esea ch o he in es men o academic pu poses is no ma u e enough due o he disa ay o he publica ions and he o e whelming in o ma i e expe ience om p o i -making websi es. O en, some au ho s en i le he s ock p ice p edic ion as a s ock selec ion p oblem; bo h p edic ion and selec ion a e jus sub-sec ions o po olio managemen . Mo eo e , s ock websi es p o ide nume ous po en ial c i e ia showing a ious e alua ions o simula e and moni o he s ock ma ke p o essionally, which inc eases he di icul y o academic s udies on s ock selec ion. The pape gene alizes a no el amewo k wi h a use -in e ac i e in e ace o s ock selec ion p oblems based on mul i-sou ce da a usion and decision-le el usion o enhance eliabili y limi ed by he na ow c i e ia pe o mance and he s eng h o a model o e coming he weakness o a single-pe o med model. This amewo k bene i s he ime-se ies p edic ion and decision-making s udy. Besides, we p opose adop ing dynamic ime wa ping o assis a ask-lea ning p ocess by cus omizing a loss unc ion ha imp o es he accu acy o da a p edic ion. The expe imen shows ha he p oposed me hod educes he p edic ion log e o by 6.3% on a e age and dec eases he wa ping cos by 5.6% on a e age o e all cases o eal-si ua ion da a. Finally, we illus a e he p oposed amewo k by implemen ing i in a eal-wo ld s ock da a selec ion. The esul s a e p ac ical and e ec i e, u he jus i ied h ough a de ailed abla ion s udy. The sou ce code will be a ailable a h ps://gi hub.com/lingping- uzzy. 1. In oduc ion This sec ion s a s wi h he in oduc ion o in es men and explains why we chose echnology s ock (Tech s ock) as an example o pe - o m he selec ion. Then we enume a e se e al po en ial ac o s ha make he s ock selec ion esea ch challenging. A las , we s a e ou mo i a ion o ou wo k and he con ibu ion om i . 1.1. S ock in es men In es ing, as pe [1], is b oadly a anging money o ope a e o a while in some p ojec o unde aking o yield posi i e e u ns (i.e., p o - i s exceeding he ini ial in es men ) [2]. I alloca es esou ces, usually capi al (i.e., money), o yield an income, p o i , o gain. The ques ion o ∗Co esponding au ho . E-mail add esses: [email p o ec ed] (V. Snášel), [email p o ec ed] (J.D. Velásquez), [email p o ec ed] (M. Pan ), [email p o ec ed] (D. Geo giou), [email p o ec ed] (L. Kong). 1All au ho s con ibu e he same. ‘how o in es ’ depends on which ype o in es men one p e e s: Do-I - You sel (DIY) [3] o p o essionally managed in es ing. DIY in es o s choose o manage hei capi al and ades on hei pla o ms, also called sel -di ec ed in es ing, which equi es a ai amoun o ins uc ion, ime dedica ion, e c., [4]. Howe e , isk and gains a e always he wo sides o a coin; low isk gene ally means low expec ed e u ns and ice- e sa. Hence, p e e ences and isk ole ance a e also ac o s ha in luence in es men choices. DIY in es ing needs a s a egically ou lined plan o he in es men ’s goal, amoun , and equency be o e alloca ing esou ces. An eme ging ield o obo -ad iso [5] o s ock ecommen- da ion is a ac ing huge a en ion om esea che s, equi ing use s o p o ide hei in o ma ion o he ecommenda ions acco dingly. h ps://doi.o g/10.1016/j.in us.2023.102018 Recei ed 10 Ma ch 2023; Recei ed in e ised o m 8 Sep embe 2023; Accep ed 10 Sep embe 2023 In o ma ion Fusion 102 (2024) 102018 2 V. Snášel e al. Fig. 1. A simple iew o po en ial c i e ia o s ock selec ion om he sou ce o YahooFinance.com. Na ional Associa ion o Secu i ies Deale s Au oma ic Quo a ion Sys- em (NASDAQ) [6] is he s ock exchange based in New Yo k and he wo ld’s i s -e e elec onic s ock ma ke . The echnology h ps: //indexes.nasdaqomx.com/Index/B eakdown/COMP akes a la ge p o- po ion o he NASDAQ composi e composi ion, ising om 46.4% in 2018 o 51.13% by his yea o 2022. Mos Tech companies a e new and unp o en, and he use mus be cau ious be o e op ing o hese com- panies. Mul i-c i e ia Decision Making (MCDM) [7] is a b anch o op- e a ions esea ch ha explici ly assesses mul iple con lic ing decision- making c i e ia, such as bene i , cos , o p ice. MCDM may help de- e mine a p e e ence o de based on ins an aneous c i e ia among al e na i es. S ill, his o de ing mechanism canno wo k well when he gi en aw da a is na ow o less accu a e, leading o undesi ed biased esul s. When i comes o he s ock da a, his phenomenon o ine iciency appea s. In de ail, we will show his p oblem in Sec ion 5. As his wo k ocuses on he s ock opic, we ecommend he eade s e e o mo e in o ma ion abou MCDM om [8]. This wo k will ocus on he s ock selec ion p oblem and p opose a cus omized DIY s ock selec ion amewo k wi h a use -in e ac i e in e ace. We will illus a e he amewo k while expe imen ing wi h Tech s ocks as an example. 1.2. Di icul ies and challenges In es men is a complex discipline in ol ing mul iple changeable ac o s. Making decisions abou any s ock sec o is challenging o any s ock in es o because o he complexi ies and ola ile na u e [9] ha indi ec ly impac a sec o ’s g ow h. In addi ion, hese dynamic o public policy complexi ies also a ec he s ock ma ke s, making in es ing challenging. Speci ically, s udying he in es men o aca- demic pu poses is also no ma u e enough due o he disa ay o he publica ions and he o e whelming in o ma i e expe ience om p o i -making websi es. F om he public channel iew, we could ind ons o s ock- ela ed p o i -making websi es p o iding ‘o e ’ comp ehensi e in o ma ion. He e we say i is ‘o e ’ as mos da a a e ough o unde s and and ‘ha dly’ use ul o DIY use s. I is easy o sc een/sea ch op lis s o po en ial s ocks (wi h he highes ea nings in ela ed companies) ega ding hei ne p o i . Bu we know we canno jus go ahead and buy hose s ocks a hei cu en ansac ion p ice. The e a e ons o ela ed c i e ia [10] ha a ec s di ec ly o indi ec ly o he selec ion o s ock h ps://business.unl.edu/ and h ps://gi hub.com/ Las Ancien One/S ock_Analysis_Fo _Quan . E en he expe s canno calcula e he in insic alue o all s ocks h ps://ge money ich.com/ s ock-selec ion-c i e ia/. Wha do hey do? They apply s ock selec- ion c i e ia o sho lis po en ial s ocks. Fo example, om Yahoo- Finance.com, one can c ea e a leas 100 c i e ia in a iew lis o moni o ing s ock ends. Fig. 1 p esen s pa ially moni o ed/e alua ed c i e ia om YahooFinance h ps:// inance.yahoo.com/po olio/p_0/ Fig. 2. Pape s dis ibu ion on po olio managemen in he Yea 2018 o 2022 om Web o Science. iew/ 1,2including 16 sub-c i e ia o ‘Fundamen als’, 13 sub-c i e ia o ‘Basic’, 17 sub-c i e ia o ‘De ails‘, and ‘Es ima es’ ‘Mo ing A e ages’ o main c i e ia, and so on. Mo eo e , om In es ing.com, one can loca e he main c i e ia o ‘Financials’ wi h sub-c i e ia o ‘Financial Summa y’, ‘Income S a emen ’, ‘Balance Shee ’, ‘Cash Flow’, e . cl. ‘To al Re enue’, ‘G oss P o i ’, and ‘G oss ma gin’ is shown unde a sub-c i e ia o ‘Income S a emen ’. Fu he , o mo e de ailed sub-sub- c i e ia, please e e o he si e h ps://www.in es ing.com/equi ies/ esla-mo o s- inancial-summa y. To a DIY use , i is no easy o sum- ma ize ‘ex ensi e’ in o ma ion om mul iple sou ces o e ons o c i e ia. Pa o he complexi ies o s ock in es men in academic ields s ems om he disa ay o a icle ca ego ies and he ambiguous sou ce in o- duc ion. The inco ec classi ica ion o s ock- ela ed pape s misleads he eade s on he main subjec . In addi ion, he ague da a sou ce in oduc ion makes i di icul o e-implemen he selec ion p ocess and lea es eade s wi h ques ion ma ks. S ock p edic ion, s ock selec ion, and po olio op imiza ion cons i- u e he po olio managemen p oblem, as shown in Fig. 2. In Fig. 2, he size o an ellipse implies he numbe o ela ed a icles and he scope co e ing he opics. Fo example, he e a e 1679 a icles ela ed o he po olio managemen p oblem, whe eas only 49 a icles in ol e s ock selec ion and p edic ions. S ock p edic ion ocuses on he s ock p ice ends o ecas ing ep esen ing he po en ial gain/ e u n. The ask o s ock selec ion is o e alua e s ock s icke s om con lic mul i- c i e ia and selec a op lis . A gi en dis ibu ion o esou ces among he chosen s ocks is known as he s ock po olio. De ining he mos con enien dis ibu ion o esou ces is known as po olio op imiza ion. O en, some a icle ope a es on he p ice p edic ion opic wi h he i le including ‘s ock selec ion’. Fo example, Fig. 3, ob ained by ee open so wa e VOS iewe h ps://www. os iewe .com/, p esen s he 2To access his link, one should c ea e ‘My Po olio’ by an accoun . In o ma ion Fusion 102 (2024) 102018 3 V. Snášel e al. Fig. 3. Diag am o keywo ds dis ibu ion ha ela es o s ock selec ion/managemen (plo ed by VOS iewe ). Fig. 4. Web o science ca ego ies on a icles wi h s ock selec ion as keywo ds in he yea 2018 o 2022. dis ibu ion o he keywo ds ela ed o a icles wi h s ock selec ion o s ock managemen om 2018 o 2022. In Fig. 3, keywo ds wi h a bigge on deno e hei la ge p opo ion in he whole dis ibu ion, and he links be ween keywo ds show hei connec ions o appea ance in he same a icles. Despi e being wo subjec s, we can obse e ha he ‘p edic ion’ is hea ily associa ed wi h s ock selec ion opics. The e a e a o al o 1672 pape esul s om he Web o Science Co e Collec ion ela ed o TS(Topic) =(‘‘s ock selec ion’’ OR ‘‘s ock managemen ’’ OR ‘‘po olio managemen ’’) in all disciplines. The e a e 69 esul s, aking a po ion o 4.127% o esul s om Compu e Science In e disciplina y Applica ions and 143 wi h 8.553% p opo - ion in Compu e Science A i icial In elligence ields, cons i u ing 182 esul s ( ime e ie al 19. Jan. 2023). Howe e , he e a e only 59 publica ions selec ed om Web o Science Co e Collec ion TS(Topic) = (‘‘s ock selec ion’’ OR ‘‘s ock managemen ’’) in all disciplines, only 49 esul s accoun o 83.051% in o al om Compu e Science A i icial In elligence ield, as in Fig. 4.Fig. 4 shows ha he opic o s ock selec- ion appea s in ele en disciplines, such as compu e science, a i icial, and elecommunica ions, and he size o colo ed ec angles a ea implies he quan i ies o he appea ance. Mo eo e , due o he weak subjec cons ain , he publica ions may no ocus on he selec ion p oblem bu ins ead on he p edic ion p oblem. The blu ed in oduc ion o sou ce da a gi es no de ails on how he au ho s il e ou he undesi ed s ock c i e ia, making i ha d o o he esea che s o s udy. Table 1 selec se e al wo ks ha applied a ious c i e ia and ocused on p edic ion, selec ion, and op imiza ion p oblems. The main highligh s o ou p o- posed me hod include he managemen o s ock p ice p edic ion and s ock selec ion p oblems, and i is scalable o an a bi a y numbe o c i e ia o decision-making. 1.3. Mo i a ion In es men is a complex p oblem in ol ing many ac o s a ec ing he gains. The s ock selec ion p oblem can be as easy wi h limi ed componen s and sophis ica ed wi h ully designed a chi ec u e, which includes known da a c i e ia and p edic ed inancial ends and poli y. In his wo k, we aim o a simple ye comple e amewo k ha helps DIY use s o selec s ock wi h ull con ol o he p ocess o s udy pu poses. Though some au ho s in oduced he da a sou ce, i is ha d o epea hei expe imen wi hou knowing he da a ex ac ion de ails and which speci ic c i e ia he in es iga ion in ol ed. Hund eds o con lic c i e ia a e challenging o DIY use s. Las bu no leas , as he s a emen s a es, almos no one buys he s ock o op lis s om a sea ch engine a gi en p ices. The s ock selec ion echniques p oduce dis inc solu ions when gi en a di e en si ua ion/en i onmen . The DIY in es men selec ion in ol es he use ’s expec a ions (pa ame e s o en i onmen se ing). This wo k ocuses on he p ice p edic ion and s ock selec ion p ob- lem, p o iding a comp ehensi e use -in e ac i e amewo k, de ailed da a ex ac ing, da a cons uc ion, analysis, and selec ion, ensu ing one o in e es can ep oduce he ou pu , b inging con enience o academic pu pose s udies. The p oposed amewo k is p oposed only o academic s udy pu poses, which does no accommoda e ac o s such as en i onmen al change o policy elemen s. Hence, we do no sugges in es o s apply i in eal in es men . We ha e no ound a simila amewo k wi h a use -in e ac i e in e ace o he s ock se- lec ion p oblem. Such a amewo k bene i s he quan i a i e s udy o complex s ock selec ion p oblems, p o iding a pla o m o compa ison and gene aliza ion. This amewo k con ains wo p ima y expe imen al In o ma ion Fusion 102 (2024) 102018 4 V. Snášel e al. Table 1 A summa y o a icles ha ela e o po olio managemen . (Ti : Ti le; RLA: egula ized linea algo i hms; EM: ee-based ensemble models; FFNs: eed- o wa d neu al ne wo ks and LSTM: long sho - e m memo y ne wo k, ecu en ne wo ks; RF: Random Fo es ; PCA: a p incipal componen analysis; c i e ia (𝑥): a bi a y numbe o c i e ia [11–17].) Pape name Me hods No. c i e ia/T ading da a/sou ce Yea Selec ion Fo ecas ing Type 2019 No Yes Analysis ML: AdaBoos ; ee; SVM; NN: MLP 194 c i e ia; 1994 o 2016; Assembled by IHS ma ke Ti : Machine lea ning o s ock selec ion [11] 2021 No Yes Analysis RLA(Lasso, Ridge), RF, FFN, LSTM, AdaBoos , XgBoos 150 c i e ia; 2009 o 2021; S&P Global, Uniges ion.Ti : Nex gene a ion o machine lea ning based s ock selec ion models [12] 2020 No Yes Clus e PCA, Bol zmann machine, au oencode , clus e analysis 12 c i e ia; 2006 o 2013; (h p://s ock.eas money.com/Ti : E ec o dimensionali y educ ion on s ock selec ion wi h clus e analysis (...) [13] 2019 Yes Yes Selec ion Ex eme lea ning machine; E olu ion op imiza ion 2(main) c i e ia; 2006 o 2016; h ps://www.wind.com.cn/Ti : A no el hyb id s ock selec ion me hod wi h s ock p edic ion [10] 2021 Yes Yes Po olio G adien boos ing decision ee-Rank me hod 20 c i e ia; 2010 o 2020; h ps://www.wind.com.cn/Ti : Resea ch on sho e m s ock selec ion s a egy based on machine lea ning [14] 2019 No Yes po olio Mean– a iance po olio op imiza ion, Ma kowi z Unde ined; 2014 o 2018; www.coinma ke cap.comTi : Po olio managemen wi h c yp ocu encies: The ole o es ima ion isk [15] 2021 No Yes Op imiza ion Hedging a io and mean– a iance app oach Unclea ; 1007 ading; Chicago Me can ile exchangeTi : Po olio managemen and dependence s uc u e... [16] 2022 No Yes P edic ion Hidden Ma ko model C i e ia (𝑥); 2012 o 2022; Pakis an s ock exchangeTi : S ock selec ion h ough hidden Ma ko model: .. [17] Ou Yes Yes Selec ion CAPM model; XGBoos wi h 11(𝑥)+6 c i e ia; 2019 o 2022; Focus: cus omized DIY s ock selec ion amewo k wi h use -in e ac i e in e ace cus omized loss un, MCDM; {YahooFinance, In es ing}.com, Alpha an age e alua ions, p ice p edic ion and MCDM selec ion. Hence we e i y he p oposed amewo k om hese wo aspec s showing he high p edic ion accu acy and p ac ical selec ion esul s. 1.4. Main goal and con ibu ion The de eloped model aims o selec a subse o desi ed s ocks wi h es ablished epu a ions and ends. Hence a usion-based model om da a-le el and decision-making suppo aspec s is necessa y. The da a usion on he mul i-sou ce aw da a is o ob ain he consensus analysis, hus d opping he subs anda d candida es. A cus omized loss unc ion is de eloped o moni o he aining p ocess o he XGBReg esso [18] model o enhancing he s ock p ice o ecas ing accu acy. Then he p edic ed da a will be used o an indi idual-based analysis. A e ha , an MCDM decision-le el usion is applied o he p ocessed da a o ob ain a consis en s ock decision o in es men . The main con ibu ions o his wo k include he ollowing: •We gene alize a usion-based model ha pe o ms he s ock se- lec ion o DIY in es men . This model s a s wi h da a usion on he mul i-sou ce da a and ex ac s a consensus analysis. Then he model pe o ms decision-le el usion o in eg a e he inal decisions ha ex end he s eng h o a model and o e come he weakness o a single-pe o med model. •The p oposed model is scalable o de eloping, es ing, and ex- pe imen ing. An in e es ed pa y can plug in echniques o any one o h ee sec ions, da a usion, indi idual-based analysis, and g oup decision-making sepa a ely, bene i ing academic s udy. •Besides, we p opose a cus omized loss objec i e unc ion ha combines dynamic ime wa ping and s anda d loss e alua ion o moni o he aining p ocess in a s ock p ice p edic ion model, aiming o imp o e he da a p edic ion accu acy. •Ul ima ely, we illus a e he model in a eal-li e s ock selec ion p oblem. And we jus i y he e ec i eness and p ac icali y o he p oposed model h ough a de ailed abla ion s udy. The pape is o ganized as ollows: Sec ion 2p o ides he p e- limina y and ela ed wo ks o he echniques/app oach used in his wo k. Then he p oposed loss unc ion o he ime-se ies o ecas ing is p esen ed in Sec ion 3. Nex , in Sec ion 4, we p esen and illus a e he p oposed usion-based s ock selec ion model wi h a eal-li e s ock selec ion applica ion. A las , we gi e he analysis o he esul s and conclude he pape . 2. P elimina y and ela ed wo ks This sec ion conce ns he p elimina y and ela ed wo ks we in es- iga ed and adop ed in he p oposed a chi ec u e. These echniques include Ex eme G adien Boos ing (XGBoos ) [19], Dynamic Time Wa ping (DTW) [20], and b ie ime-se ies o ecas ing. 2.1. Mul i-sou ce usion echniques Some esea che s [21–23] ocused on he s ock selec ion h ough he MCDM me hod, such as he hyb id MCDM app oach combining Analy ic Hie a chy P ocess [24] me hod wi h mode n po olio he- o y [25], MCDM hyb id app oach [26] including Complex P opo ional Assessmen [27], simple addi i e weigh ing [28], and TOPSIS com- bined wi h Spea man’s ank co ela ion coe icien [29]. Al hough he men ioned li e a u e [30–32] p oposed compa a i e me hods o he s ock selec ion and alked abou wo po en ial ques ions ha canno be neglec ed: (1) Is he one sou ce o da a eliable, o should one acqui e mul i-sou ce da a o he analysis? (2) Which usion le el is mo e sui able o he s ock selec ion p oblem? How o design a sui able usion-based model o he co esponding s ock selec ion p oblem. Mul i-sou ce da a usion (MDF) [33,34] me hods ha e a ac ed ex ensi e in e es and e i ied o ge mo e accu a e esul s compa ed wi h he app oach on he applica ion o using indi idual sou ces sepa a ely. Fusion echniques a e also u ilized in an ensemble way as a hyb id model, such as he Model ha combines di e en le els o usion. To he bes knowledge o us, MDF me hods ha e no been ex- ensi ely s udied and applied o he s ock selec ion p oblem. Anki [9] p o ided he la es e iew on he usion-based s ock ma ke p edic ion me hods. In his e iew, he au ho conside ed nine c i e ia o analyze hei su ey. Th ee c i e ia we e ela ed o he p edic ion/ o ecas ing (s ock p ice, s ock end, isk/ e u n) aspec s; Only one c i e ion was abou po olio managemen ; The o he i e c i e ia we e Financial ma ke concep ualiza ion, In o ma ion usion, Fea u e usion, Model usion, and o he s ock applica ion. The applica ions o In o ma ion usion, Fea u e usion, and Model usion echniques in his e iew mainly ocused on he p edic ion/ o ecas ing p oblems [35]. S ock ecommenda ion and managemen p oblems ecei ed li le a en ion. In o ma ion Fusion 102 (2024) 102018 5 V. Snášel e al. 2.2. S ock p ice da a p edic ion As men ioned abo e, s ock p ice p edic ion se es as a subsec ion p oblem o s ock in es men (selec ion), which e lec s he pa e n o inancial ac i i ies and p edic s hei de elopmen and changes. Ne e heless, i is conside ed one o he mos challenging asks o accomplish in inancial o ecas ing due o he complica ed na u e o he s ock ma ke . Resea che s [36,37] in es iga ed s ock p ice p edic ion p oblems by a ious deep neu al ne wo ks echniques, such as Con olu ion ne - wo k [38], Long Sho -Te m Memo y (LSTM) [39], Recu en ne - wo k [40], e c. Neu al ne wo ks a e cons uc ed by laye s o nodes, like he human b ain composed o neu ons, and enable i ual ma- chine lea ning by emula ing human beha io . The lea ning p ocess i e a es and imp o es by he backp opaga ion based on he lea ning loss. Tha loss calcula ion compa es he g ound u h ( he ac ual s ock p ice alue) o he p edic ion esul s (neu al ne wo k ou pu ). Loss measu emen s commonly include squa ed-log e o (SLE), mean ab- solu e e o (MAE), oo -mean-squa e e o , e c. Howe e , a neu al ne wo k-based model equi es much aining da a o pe o m well and p oduce sa is ac o y accu acy. The his o y s ock p ice o h ee yea s only con ains less han a housand i em da a. Mo eo e , some lea ning model equi es expensi e GPUs o educe lea ning ime cos . Fu he mo e, some esea che s p e e ed ligh machine lea ning me hods o add essing s ock p ice p edic ion, such as he hyb id GA- XGBoos algo i hm used in [41], ime-se ies o ecas ing Au o eg essi e Mo ing A e age (ARMA) and SARIMA in [42,43]; embedded p incipal componen analysis, disc e e wa ele ans o m [44] o XGBoos ; he hyb id me hod based on XGBoos and Long Sho -Te m Memo y [45] e c. XGBoos e icien ly implemen s a ee-based g adien -boos ing al- go i hm o supe ised classi ica ion and eg ession (XGBReg esso ) p oblems. I became a popula p edic i e model, as we ha e seen in ime-se ies da a applica ions, wi h cha ac e is ics o as and e icien and is a s eady compe i ion winne , such as hose on Kaggle [46]. Howe e , hese wo ks men ioned abo e and mos o he wo ks [47,48] measu ed he p edic ion loss and alida ed he ou pu accu acy based on he s anda d loss unc ion, which did no ake he ime a iance ac o in o accoun . The s ock p ice ending is a ime-se ies da a. Howe e , he dis o ion o wo ime-se ies da a migh appea du ing he p edic ion p ocess. Tha is o say; he p edic ed alue may no one- o-one ma ch he obse ed sequences by he exac ime index. Hence, weigh ing he p edic ion di e ence o he obse a ion om one- o-one index poin s o one- o-one a ied ime index is necessa y and p ac ical. 2.3. XGBoos eg esso and dynamic ime wa ping Fo a quick unde s anding o he p oposed model in oduced in Sec ion 4, we p o ide he p elimina y in oduc ion and ope a ion o XGBoos and DTW. XGBoos is an open-sou ce supe ised Machine Lea ning [49] so - wa e lib a y ha in eg a es and op imizes decision ees [50] and g a- dien boos ing [51] echniques. XGBoos i e a i ely co ec s he e o s d i en h ough he cu en ensemble model and eaches a inal solu ion by conside ing all p edic ion esul s o a single ou pu . One o he ben- e i s o adop ing XGBoos is ha i is an ex ensible lib a y p o iding an in e ace o he cus omized objec i e unc ion and co esponding me ic o moni o ing pe o mance, sou ce om h ps://xgboos . ead hedocs. io/ whe e wi h hash ag cus omized-objec i e- unc ion. As men ioned abo e, ime-se ies analysis in ol es measu ing he simila i y be ween wo sequences, which may a y in speed. Any da a ype can be ans o med in o a linea sequence and hen be analyzed by DTW [52,53]. In gene al, when calcula ing he ma ch [54,55] be ween wo gi en da a, he e a e ce ain cons ain s and ules as ollows: •One- o-many (one) mapping. No consecu i e skip-o e ; e e y poin mus be ma ched wi h one o mo e poin s om he o he sequence. •Bounda y condi ions. The alignmen pa h s a s a he i s poin and ends a he las poin , gua an eeing no pa ially ma ched o one sequence. And a good ma ch is unlikely o wande oo a om he diagonal alignmen . •Slop cons ain , mono onici y, and con inui y. The ma ched se- quence does no go back in ‘ ime’ and does no jump o a ‘ ime’ poin (no skip any ‘ ime’ poin ), which gua an ees he ea u es o da a a e no epea ed and does no omi impo an ea u es in he sequence. Excep o he es ic ions abo e, i also equi es a local cons ain ha gua an ees he ma ched ime indices 𝑖and 𝑗(whe e 𝑖, 𝑗, ∈Z), om wo da a, ha e a maximum limi whe e |𝑖−𝑗|≤𝑤,𝑤is a window pa ame e (o window size) and 𝑤∈Z.Fig. 5 p esen s an example o a wa ping pa h pai ed o ime se ies (𝑎) and (𝑏). The wa ping pa h s a s wi h a ma ix size o 10 ×10 g id plo , as he leng h o da a (𝑎) and (𝑏) is 10, hen each s ep ollows (𝑎𝑖, 𝑏𝑗) o mula ion whe e 𝑖, 𝑗 ∈ [0,…,9] and |𝑖−𝑗|≤𝜔, indica ing ha he wa ping index di e ence be ween wo sequences is less han a window size wi h a alue o ‘ h ee’ in Fig. 5( igh ), as he col- o ed diagonal mo ing g id is h ee in a ow o he wa ping ma- ix. Fig. 5(le ) shows he wa ping pa h esul is pai ed sequences {[(𝑎0, 𝑏1),(𝑎1, 𝑏1),(𝑎2, 𝑏2),(𝑎3, 𝑏3),(𝑎4, 𝑏4),(𝑎5, 𝑏5),(𝑎5, 𝑏6),(𝑎6, 𝑏7),(𝑎7, 𝑏8), (𝑎8, 𝑏8)}. To ob ain an op imal wa ping pa h, one could es e e y pos- sible pa h be ween wo gi en wo sequences. Howe e , his exhaus i e me hod is compu a ional complexi y. DTW applies ea ly abandoning and p uning in some as DTW p ocess [56,57]. In summa y, aking ad an age o he echniques men ioned abo e, we add ess he s ock ecommenda ion p oblem and p opose a usion- based model ha in eg a es he da a- usion and decision-le el usion p ocess on mul i-sou ce aw da a, enhancing he eliabili y and con- ciseness o he inpu . Mo eo e , we p opose a cus omized loss unc ion ha no only measu es he s anda d loss on he di e ence be ween p edic ion and he g ound u h bu also conside s he di e ence a ies in ime. 3. P oposed s ock p ice da a p edic ion In his sec ion, we in oduce he p oposed s ock p ice p edic ion s a egy. This s a egy is essen ially an XGBReg esso wi h a a ian loss unc ion. We cus omize a new loss unc ion ha uses he DTW cos and he p edic ion e o . We i s b ie ly ecall he p oblem o employing XGBoos wi h a s anda d loss unc ion on s ock p ice p edic ion, hen p esen he s ock p ice p edic ion me hod in de ail. Fo ecas ing sequen ial da a h ough machine lea ning is usually e alua ed by s anda d me ics unc ions, indica ing he di e ence be- ween he p edic ion and obse a ion a e ma ched ins an aneously. Howe e , hese classical me ics may ail o cap u e he ea u e pa e n o beha io o he sequen ial da a. Some esea che s [58] p oposed o design combined me ics o e alua e he p edic ion esul s, while his combined me ic played no ole in moni o ing he aining in machine lea ning. Hence we p opose he cus omized loss unc ion ha uses DTW cos and p edic ion loss (e.g., SLE) o cap u e he ending and luc ua ion in s ock p ice da a. This cus omized objec i e unc ion in luences he o ecas ing p ocess by he alignmen pe o - mance and di ec s he op imiza ion owa ds he desi ed solu ion a he han e alua ing he p edic ion ou pu [58]. One key aspec in de- signing he cus omized objec i e unc ion is he g adien ha decides he op imiza ion di ec ion and plays an essen ial ole du ing model aining. I is wo h men ioning ha he e is an ex a cons ain in ou cus omized loss unc ion ha he p edic ions only compa e wi h he obse a ions (g ound u h) in he cu en and pas ime index and do no compa e wi h u u e ime index poin s. This cons ain is easonable because we can only e e o he pas ime-se ies da a o he compa - ison and ha e no knowledge o he u u e da a. Hence, he wa ping In o ma ion Fusion 102 (2024) 102018 6 V. Snášel e al. Fig. 5. Illus a ion o he ma ched pa h o index pai s o da a ime se ies (a) and ime se ies(b) o leng h 10; (a) ={0, 1, 2, 3, 2, 1, 1, 2, 1, 1} and (b) ={1, 1, 2, 3, 2, 1, 2, 1, 1, 3}; Le : simila ends in c ocodile and pink colo s, whe e (1,1), (2,2), ..., (7, 8) a e he ma ched ime-indices pai s om wo ime-se ies da a. Righ : The wa ping pa h sa is ies one- o-one mapping, bounda y limi , mono onici y, and con inui y condi ions, and he ma ch window is h ee. ules a e modi ied o mee his cons ain . Fo example, i we deno e 𝑝𝑖is he p edic ion alue, and le 𝑜𝑗be he obse a ion, hen he ex a ule is se 0≤𝑖−𝑗≤𝑤, whe e 𝑖, 𝑗 ∈Zdeno es he ime index, and 𝑤is a use -de ined window pa ame e , whe e 𝑤∈Z. Mo e speci ically, we emo e he condi ion o 0≤𝑗−𝑖≤𝑤. Algo i hm 1: 𝐹𝑆𝐿𝐸 o XGBoos (Re e o h ps://xgboos . ead hedocs.io/en/s able/ u o ials/cus om_me ic_obj.h ml) INPUT: p ed , a ge ; OUTPUT: ,; /* Squa ed Log E o objec i e. A simpli ied e sion o he SLE */ 𝑓𝑙𝑜𝑠𝑠(𝑝𝑟𝑒𝑑𝑡, 𝑡𝑎𝑟𝑔𝑒𝑡) = 1 2[log(𝑝𝑟𝑒𝑑𝑡 + 1) − log(𝑡𝑎𝑟𝑔𝑒𝑡 + 1)] ; /* loss unc ion o XGBoos */ 1: p ed [p ed <-1] ←-1 + 1e-6; 2: ←g adien (p ed , a ge ) ; 3: ←hessian(p ed , a ge ); /* g adien , hessian → e e o Algo i hm 3 */ Algo i hm 2: 𝐹𝐷𝑇 𝑊 +𝑆𝐿𝐸 o he p oposed model INPUT: p ed , a ge ; OUTPUT: ′,′; /* Cus omized objec i e unc ion, whe e we only need o p o ide g adien , hessian */ 1: p ed [p ed <-1] ←-1 + 1e-6; 2: dis ←d w_cal(p ed , a ge ) ; /* d w_cal → e e o Algo i hm 4 */ 3: ′←g adien (p ed , a ge ) + 𝛼×dis ; 4: ′←hessian(p ed , a ge ) + 𝛼×dis ; Algo i hm 3: g adien (p ed , a ge ) ; hessian(p ed , a ge ) INPUT: p ed , a ge ; OUTPUT: g adien (p ed , a ge ),hessian(p ed , a ge ) ; g adien (p ed , a ge )←𝑙𝑜𝑔1𝑝(𝑝𝑟𝑒𝑑𝑡) − 𝑙𝑜𝑔1𝑝(𝑡𝑎𝑟𝑔𝑒𝑡) 𝑝𝑟𝑒𝑑𝑡 + 1 ; hessian(p ed , a ge )←(−𝑙𝑜𝑔1𝑝(𝑝𝑟𝑒𝑑𝑡) + 𝑙𝑜𝑔1𝑝(𝑡𝑎𝑟𝑔𝑒𝑡)+1 (𝑝𝑟𝑒𝑑𝑡 + 1)2; /* whe e 𝑙𝑜𝑝1𝑝(𝑥)is a unc ion o 𝑥de ined as 𝑙𝑜𝑝1𝑝(𝑥) = 𝑙𝑜𝑔(1 + 𝑥)*/ Algo i hm 4: d w_cal(p ed , a ge ) INPUT: p ed , a ge , ; OUTPUT: dis ; o k, (s, d), 𝑡 2do /* calcula e wa ping cos om ime 𝑠 o ime 𝑑 wi h window sliding size o 𝑡 2, whe e 𝑡is a pa ame e ela ed o ime s amp. */ dis ←d w(p ed [k:k+ ], a ge [k:k+ ], w); /* 𝑤is a use -de ined window pa ame e ela ed o DTW */ dis (k:k+ 𝑡 2)←dis; /* sliding assign dis ance cos o each ime s amp */ /* d w is a unc ion calcula ing he wa ping cos , e e o [58] */ The ole o he objec i e unc ion is o p o ide he i s and second- o de g adien in o ma ion, labeled as g adien and hessian. The cos o DTW is a alue measu ing a ‘window’ leng h o da a co esponding o se e al ime indices. In con as , he g adien in o ma ion is assessed a each ime index poin . Fo his eason, we conside spli ing he DTW cos equally and hen passing i o each ela ed index ha con ibu es o he g adien . He e, DTW plays a sub le ole in e alua ing he ending o p edic ion. Hence we a ange a lowe weigh o he DTW cos han he SLE loss. Fo example, he cus omized loss unc ion is he weigh ed accumula ion esul o DTW cos and SLE loss, whe e we scale down he DTW cos wi h a scale pa ame e 𝛼(𝛼 < 1)and hen add his cos o he SLE loss. Ou expe imen s use 𝛼= 0.1(Algo i hm 2). This cus omized loss unc ion is inspi ed by he obse a ion [59] and he Regis e Hook in e ace (The hook will be called e e y ime a g adien conce ning he objec is compu ed), ha as he p oblem is complex and he g adien may no di ec ly mo e owa ds o op imal solu ion, i is possible and necessa y o modi y he g adien in o ma ion as he use s’ need. The implemen a ion o he p oposed cus omized objec i e unc ion deno ed as 𝐹𝐷𝑇 𝑊 +𝑆𝐿𝐸 is shown in Algo i hm 2, along wi h he e e - ence objec i e unc ion as 𝐹𝑆𝐿𝐸 in Algo i hm 1. Algo i hm 3p o ides he de ail calcula ion o g adien and hessian o SLE, and he 𝑑𝑡𝑤_𝑐𝑎𝑙 unc ion men ioned in Algo i hm 2is shown in Algo i hm 4. In summa y, o use XGBoos o ime se ies o ecas ing, we made wo changes du ing aining: (1) added he walk- o wa d alida ion [60] scheme ha makes su e he aining p ocess is on he pas da a. In o ma ion Fusion 102 (2024) 102018 7 V. Snášel e al. Fig. 6. The p oposed amewo k o s ock selec ion wi h expe imen al da a. (2) adjus ed he DTW by applying an addi ional cons ain ha a oided compa ing p edic ions and he obse a ions in he u u e ime index. The cus omized loss unc ion measu es he op imiza ion pe o mance and di ec s he op imiza ion di ec ion. A sui able loss unc ion can di ec he model aining o a be e s a us wi h lowe e o . 4. Fusion-based model o s ock selec ion This sec ion will in oduce he p oposed usion-based model, which analyzes s ock- ela ed da a and p o ides p e e ed in es men op ions. This p oposed a chi ec u e comp ises he ollowing s eps: da a- usion p ocessing and decision- usion suppo . Da a- usion p ocessing aims o e ine he aw da a and ope a es planned obsolescence o educe he numbe o s ock candida es. The decision- usion p ocess pe o ms MCDM o ob ain he inal in es men op ions. 4.1. P oposal This model is owa ds capi al-low in es o s wi h no desi e o ake high isks o long- e m p ospec s bu a he o sa e y and small luc ua ions in es men s. Hence he al e na i e s ocks il e ou hose s ocks wi h la ge luc ua ions based on calcula ed me ics. We se he undamen al ma ke benchma k based on he S anda d and Poo ’s 500 (S&P 500) [61] o es ima ing he isk-based esul s. S&P 500 s alks he pe o mance o 500 la ge companies lis ed on s ock exchanges and commonly wo ks as equi y indices de e mining he o e all economy’s pe o mance. Ou goal is o inalize a se o Tech s ocks o in es ing. The iewpoin o his p oposed usion-based model is he implemen a- ion and applica ion o he succession o p ocedu es ha selec s a subse o op imal s ock candida es om a se o s ocks in he Tech sec o based on he analysis o use -p e e ed c i e ia. Those chosen candida e s ocks would be eliable in es men op ions. To be e illus a e he model, we apply i o a p ac ical s ock in es men applica ion and p esen he esolu ion p ocedu e in de ail wi h Tech s ock da a. A pic o ial p ocess o he da a usion is shown in Fig. 6.Fig. 6 shows a de ailed p ocess o he p oposed amewo k om he Inpu (bold in black) wi h s a ic da a and ime-se ies p ice da a o he Final ou pu (in yellow). This amewo k comp ises h ee sec ions, 1. Da a P epa a ion; as ime-se ies p ice da a ex ac ion and s a ic da a ex ac ion om he da abase (shown in pink and ligh -yellow shadowed a eas), 2. Indi idual-based analysis; s ock il e ing based on me ic compu a ion (shown in a squa ed a ea wi h blue no a ion), 3. G oup-based analysis; p ocess based on MCDM, as ollows: 1. Da a P epa a ion •Da a selec ion (In es men ho izon): Based on he expec a- ions and goals ho oughly desc ibed below, we manually conclude upon 34 Tech s ocks. •Da a loading: Fo hese 34 selec ed s ocks we download his- o ical ime-se ies p ice da a o a speci ic pe iod om h ee di e en da a sou ces (YahooFinance.com, In es ing.com, Al- pha Van age), and download 11- ea u e s a ic da a, whe e he undamen al analysis includes 6 ea u es, he pe o - mance analysis owns 3 ea u es and he emaining 2 ea- u es is o he echnical analysis. 2. Indi idual-based Analysis In his sec ion, he his o ical p ice da a o all 34 s ocks sou ce om YahooFinance.com a e used and analyzed indi idually. •Risk-based analysis: Fo each o hese 34 s ocks, he well- known linea eg ession inance model CAPM is applied o calcula e se en isk-adjus ed me ics. Then we d op s ocks i any me ic o his s ock is unde he p e-de e mined h eshold. A e his sc eening, we conclude 18 s ocks ha mee all he h eshold condi ions. •Pe o mance analysis: Fo each o hese 18 emaining s ocks, h ee ea u es o pe o mance analysis a e measu ed, (YTD, 1-Yea ,3-Yea ). •Technical analysis: Subsequen ly, we calcula e 2 echnical ea u es weekly, mon hly. This echnical analysis consis s o wo s eps, ime-se ies da a o ecas ing and he inal ecom- menda ion con e sion. Fi s ly, we p edic he ime-se ies p ice da a by XGBReg esso wi h he p oposed cus omized loss unc ion, hen calcula e he week and mon h co e- sponding esul s. Nex , disc e ized he calcula ed alues in o a i e-scale ecommenda ion [0,1,2,3,4], which inan- cially co esponds o he echnical indica o s S ong Sell, Sell,Neu al,Buy and S ong Buy. In o ma ion Fusion 102 (2024) 102018 8 V. Snášel e al. 3. G oup-based Analysis This sec ion analyzes he s a ic da a wi h 11- ea u es o all e- maining 18 s ocks in a g oup compa ison iew om wo DMs. The i s DM da a is om In es ing.com. The use cons uc ed he second DM da a wi h 11 ea u es; he 6 undamen al analysis ea u es a e de i ed om a da abase. The es 5 ea u es a e hose calcula ed in he p e ious Indi idual-based analysis sec ion. •G oup-decision making: We apply he g oup decision- making p ocess o u he d opping s ocks ha a e no mee expec a ions. Speci ically, we calcula e (𝑡𝑜𝑡𝑎𝑙𝑠𝑐𝑜𝑟𝑒, 𝑎𝑣𝑔𝑠𝑐𝑜𝑟𝑒, 𝑠𝑡𝑑𝑠𝑐𝑜𝑟𝑒, 𝑐𝑜𝑛𝑠𝑒𝑛𝑠𝑢𝑠)me ics based on wo DMs da a and compa ed hem wi h he h esholds se by he use . We conclude upon 13 s ocks ha mee all he h eshold condi ions. (No e: h esholds depend on da a; a use could de e mine i by h eshold- elax me hod.) •Mul i-c i e ia decision making (MCDM): Finally, h ee di - e en MCDM algo i hms a e applied, namely TOPSIS, ELECTRE and PROMETHEE. Each one eaches a sligh ly di e en subse o s ocks. The inal consis en pool includes i e s ocks o e lapping he h ee subse s, indica ing he ecommended s ocks by he p oposed model. 4.2. Da a p epa a ion The da a p epa a ion includes wo majo s eps, s ock selec ion based on he in es men ho izon and he implemen a ion o da a loading o ob ain he s ock da a ha we will wo k on la e . 4.2.1. S ock selec ion: In es men ho izon This is he i s da a selec ion p ocess we apply based on he in es men ho izon. Mo e han 3300 companies a e aded publicly on he NASDAQ exchange. In 2022, he 51.13% (o app oxima ely 1689) we e high Tech s ocks. In his sec ion, we inalize ou i s se o ules (me ics e alua ions) o il e ou he undesi ed s ocks and conclude upon a decen se o s ocks. Selec ed ea u es o calcula ed ea u es accompany each s ock. Fo his usion-based model analysis, 11 ea u es (c i e ia) we e selec ed o analyze s ocks. All hese c i e ia lay unde sepa a e umb ellas o analysis, e.g., undamen al analysis c i e ia, pe o mance analysis c i e ia, and echnical analysis c i e ia. The selec ion o da a ea u es (c i e ia) is based on he da a able head whe e he able was downloaded om in es ing.com o iles named ‘ undamen al.cs ’, ‘ echnical.cs ’ and ‘pe o mance.cs ’. Fo a de ailed da a ex ac ion p ocess, please e e o he ollowing subsec ion. A. Fundamen al analysis c i e ia Fo a s ock, he undamen al analysis c i e ia [62] ypically con ain inspec ing many ac o s associa ed wi h s ock p ices, e.g., (1) he o e all pe o mance in a domain ha he company pa icipa es in, (2) he domes ic poli ical en i onmen , (3) ele an ade ag eemen s and ex e nal poli ics, (4) a company’s inancial s a emen s, (5) a company’s p ess eleases, e c. The e o e, we selec he ollowing co esponding c i e ia o comple e ou analysis. •Ma ke Capi aliza ion (Ma ke cap) e e s o he o al alue o all a company’s sha es o s ock. –No e: The mos signi ican ad an age o adding la ge-cap s ocks o an in es men po olio is hei s abili y and well- es ablished epu a ion wi h consume s. –Goal:𝑚𝑎𝑟𝑘𝑒𝑡_𝑐𝑎𝑝 ∈ (6𝐵, 10𝑇) •P ice–ea nings a io (P/E Ra io) ells in es o s how much a com- pany is wo h, and i con eys how much in es o s will pay pe sha e o 1 (uni ) o ea nings. –No e: This c i e ion de e mines i he expec ed g ow h wa - an s he p emium ha someone paid. –Goal:𝑝𝑒 ∈ (5,35) •Re enue (G oss Sales) depic s he o al income gene a ed by selling goods o se ices ela ed o he company’s p ima y ope a ions. –No e: This c i e ion is essen ial o ex ac ing a o al sco e o a s ock. –Goal: None •Ea nings pe sha e (EPS) indica es how much money a company makes o each sha e o i s s ock and is a widely used me ic o es ima ing he co po a e alue –No e: We aim a a s eady sou ce o income, and his c i e ion implies he oom a company has o inc easing i s cu en di idend. –Goal:𝑒𝑝𝑠 > 2.5 •3 Mon h A e age Volume (A e age Vol (3 m)) is he daily a e age o he cumula i e ading olume du ing he las h ee mon hs –No e: The sha e p ice p obably anks i he in es o decides o sell. This scena io is no ideal o indi idual in es o s. –Goal:𝑣𝑜𝑙 > 500000 B. Pe o mance analysis ea u es Fea u es ega ding he annualized yea - o-da e benchma k pe o - mance a e selec ed. These h ee c i e ia selec ed a e as ollows: •YTD, 1-Yea , 3-Yea Re u ns Yea o da e (YTD) e e s o he pe iod beginning he i s day o he cu en calenda yea o iscal yea up o he cu en da e. –No e: We deal wi h la ge-cap Tech s ocks, so we expec hem o ou pe o m he ma ke in a s anda d-based in es men . –Goal:YTD >25% C. Technical analysis ea u es The p ima y cha ac e is ic o echnical analysis in ol es employing models and ading ules depending on p ice and olume ans o - ma ions, e.g., mo ing a e ages, ela i e s eng h index, eg essions, business cycles, and in e -ma ke and in a-ma ke p ice co ela ions. In his usion-based analysis, we a e in e es ed in s ock p ice mo emen . Mo e speci ically, he wo c i e ia selec ed a e as ollows: •Weekly, Mon hly (Technical indica o s) a e heu is ic o pa e n- based signals p oduced by he p ice, olume, and/o open in e es o a secu i y o con ac used by ade s who ollow echnical analysis. –No e: This p ojec aims o selec a bunch o Tech s ocks ha a e p e e able o in es ing based on a se o ules. Focusing on ‘S ong Buy’ and ‘Buy’ s ocks is essen ial. Technical indi- ca o s scale anges om 0 o 4, co esponding om ‘S ong Sell’ o ‘S ong Buy’ i e le els. –Goal:𝑤𝑒𝑒𝑘𝑙𝑦 ∈ {3,4},𝑚𝑜𝑛𝑡ℎ𝑙𝑦 ∈ {3,4} All he expec a ions and goals o all he c i e ia/ ea u es/me ics can be ound unde Table 2. Unde his i s se o ules om he au ho s’ p e e ence, we conclude on 34 candida e companies and make he in es men decision on hei s ocks; please e e o Table 3 o companies index and ull name. The da a is om public in es ing websi es, co espondingly, Tech s ocks aded in NASDAQ. 4.2.2. Da a loading Fo he sake o comple eness, we used h ee di e en APIs o load he eques ed da a o he selec ed s ocks. We used h ee di e en Py hon w appe lib a ies ha es ablished well-s uc u ed unc ions o communica e wi h he espec i e APIs. All his in o ma ion can be In o ma ion Fusion 102 (2024) 102018 9 V. Snášel e al. Table 2 C i e ia/ ea u e scope, expec a ions, and goals on he s ocks; (exp): expec a ion, M: millions, T: illion. Fea u e class Me ic/Fea u e Mo emen (exp) Goal (exp) Range Fundamen al analysis Ma ke cap High ↑∈ (6𝐵, 10𝑇)– P/E a io High ↑∈ (5,35) – Re enues High ↑– – EPS High ↑>2.5– A e age Vol. (3 m) Medium ↕>5𝑀– Be a Low ↓∈ (0.4,1.6) – Pe o mance analysis YTD e u n High ↑>25% – 1-Yea e u n High ↑>25% – 3-Yea e u n High ↑>25% – Technical analysis Weekly pe o mance High ↑∈ {3,4} {1, 2, 3, 4} Mon hly pe o mance High ↑∈ {3,4} {1, 2, 3, 4} Table 3 The de ails in o ma ion ac onym and co esponding company. ID Name Ac onym ID Name Ac onym 0 Qualcomm Inco po a ed QCOM 17 In ui Inc INTU 1 Tesla Inc TSLA 18 Sapiens In e na ional Co po a ion NV SPNS 2 NVIDIA Co po a ion NVDA 19 B uke Co po a ion BRKR 3 Tai on Componen s Inco po a ed TAIT 20 In e Digi al Inc IDCC 4 In el Co po a ion INTC 21 MIND CTI L d MNDO 5 AudioCodes L d AUDC 22 Ebix Inc EBIX 6 Uni e sal Display OLED 23 Wayside Technology G oup Inc WSTG 7 Cisco Sys ems Inc CSCO 24 CDK Global Holdings LLC CDK 8 Mic oso Co po a ion MSFT 25 Wes e n Digi al Co po a ion WDC 9 Apple Inc AAPL 26 Ac i ision Blizza d Inc ATVI 10 As oNo a Inc ALOT 27 MKS Ins umen s Inc MKSI 11 Ga min L d GRMN 28 Te adyne Inc TER 12 Elbi Sys ems L d ESLT 29 Analog De ices Inc ADI 13 Texas Ins umen s Inco po a ed TXN 30 Magic MGIC 14 Equinix Inc EQIX 31 Logi ech In e na ional SA LOGI 15 CDW Co p CDW 32 Asia Paci ic Wi e & Cable Co p L d APWC 16 B oadcom Inc AVGO 33 Skywo ks Solu ions Inc SWKS Table 4 APIs used o ex ac he da a. API Py hon module w appe Link YahooFinance API y inance yahoo inance.com Alpha Van age API alpha an age alpha an age.com In es ing.com API in es py in es ing.com ound in Table 4. Fo each o hese 34 s ocks, we load all he a ailable ea u es. A ailabili y di e s o e e y API.3 We ex ac wo di e en ypes o da a o each s ock. The i s ype is his o ical ime se ies da a and conce ns he unique ea u e o adjus ed closed p ice o he indi idual s ock o a speci ic pe iod (mul iple samples pe s ock). The second ype is s a ic da a and conce ns he 11 ea u es desc ibed in he p e ious sec ion (1 piece o sample pe s ock). Fo he speci ic p ojec , we a e in e es ed in a long- e m analysis, so he pe iod examined is a 3 Yea pe iod (be ween 01-04-2019 and 01- 04-2022), which is ansla ed in o 759 di e en imes amps. The da a ames ha e he ollowing shapes: We e ie e he abo e-men ioned da a om each API sepa a ely, while some ea u es a e conside ed p emium and canno be accessed. The a ailabili y can be inspec ed in Table 5. As a esul , he downloaded s a ic da a om In es ing.com a e shown in Table 6. The his o ical ime-se ies p ice da a o 34 s ocks om YahooFinance.com and Alpha an age.com a e shown in Tables 7 and 8, espec i ely. We no iced ha he adjus ed close p ices om he wo ables men ioned abo e a e sligh ly di e ence. Fo a ai and eliable compa ison o he sou ce da a, Fig. 7 shows he eco ded his o ical p ice da a in a plo o m, whe e he ed, blue, and g een lines ep esen 3The echnical indica o s a e conside ed p emium ea u es o AlphaVan- age and canno be accessed o ee. da a om Alpha an age.com,YahooFinance.com and In es ing.com, e- spec i ely. Fig. 7 shows he ime-se ies da a on s ocks AAPL and NVDA a e ma ched. I can be seen ha MSFT s ock da a om h ee sou ces a y much bu wi h a ma ched inc easing end. 4.3. Indi idual-based analysis In his sec ion, he his o ical p ice ime-se ies da a o all 34 s ocks a e used and analyzed indi idually. This p ocess aims o d op undesi able s ocks om candida es based on indi idual pe o mance. 4.3.1. Risk-based analysis This sec ion YahooFinance API4 emo es undesi able s ocks based on hei pe o mance on isk-e alua ed me ics calcula ed on he his o ical p ice da a. The isk- ela ed me ics a e calcula ed based on Capi al Asse P icing Model (CAPM) model [63] using annualized loga i hmic e u ns s ock p ice da a a he han aw s ock p ice da a. A. Annualized e u n We calcula e he daily loga i hmic e u n be ween imes amp 𝑡and 𝑡− 1, whe e 𝑡∈Z. This e u n is hen annualized by mul iplying he esul wi h he numbe o ading days 𝑡𝑟_𝑑𝑎𝑦𝑠 ∼ 253.Annual pe cen age a e (APR) ep esen s he annual a e cha ged o ea ning o bo owing money and is used o calcula e in e es o in es men . ⎧ ⎪ ⎨ ⎪ ⎩ 𝑙𝑜𝑔_𝑟𝑒𝑡𝑡=𝑙𝑜𝑔(𝑝𝑟𝑖𝑐𝑒𝑡) − 𝑙𝑜𝑔(𝑝𝑟𝑖𝑐𝑒𝑡−1) 𝑟𝑓= isk- ee a e o annualized YTD o he 3-mon h T-bill 𝑎𝑝𝑟𝑡=𝑙𝑜𝑔_𝑟𝑒𝑡𝑡× 253 − 𝑟𝑓 (1) 4We explici ly showed API’s us wo hiness compa ed o he es wo APIs. In o ma ion Fusion 102 (2024) 102018 16 V. Snášel e al. Table 18 The p edic ion on Weekly and Mon hly c i e ia alue by he p oposed cus omized loss unc ion o each s ock. Ticke s MSFT TXN AAPL AVGO CDW INTU NVDA BRKR TER Weekly −4.674 −1.036 −4.947 −0.224 0.152 43.581 10.886 2.039 0.901 Mon hly −1.957 0.248 1.626 0.471 −0.176 55.140 6.531 1.349 1.189 Ticke s QCOM EQIX SPNS MGIC ESLT LOGI TSLA AUDC ATVI Weekly −0.868 −4.857 0.064 0.419 −51.675 1.555 −9.692 0.300 −0.473 Mon hly 1.781 −2.191 0.024 0.339 −37.214 2.157 −11.716 0.206 −0.236 Table 19 Recommenda ion con e sion esul om p edic ed p ice alue o Weekly and Mon hly c i e ia. Symbol 𝛼 𝛽 𝑅2Sha pe T eyno F_ es 1 Yea 2 Yea 3 Yea YTD Weekly Mon hly MSFT 15.37 1.15 72.22 1.04 0.30 1968.15 25.24 43.93 38.95 −7.81 2 1 TXN 2.23 1.15 63.85 0.58 0.15 1337.02 − 5.31 40.67 21.95 −2.75 2 2 AAPL 26.14 1.17 63.27 1.28 0.43 1304.25 39.27 71.29 55.28 −1.71 2 2 AVGO 7.49 1.32 61.96 0.70 0.20 1232.76 32.03 72.87 31.76 −5.14 2 2 CDW 2.24 1.21 61.38 0.56 0.15 1203.06 4.69 43.07 23.07 −12.92 2 2 QCOM 14.05 1.28 44.11 0.74 0.27 597.54 6.61 52.35 39.82 −19.29 2 2 EQIX 6.29 0.80 36.63 0.59 0.19 437.51 12.26 14.95 20.64 −9.40 2 1 TSLA 76.35 1.41 22.59 1.42 1.14 220.90 56.95 235.58 165.68 2.63 2 1 AUDC 7.17 0.92 19.19 0.43 0.20 179.77 − 8.27 6.65 23.88 −26.11 2 2 ATVI 8.37 0.64 17.01 0.50 0.24 155.11 −17.02 19.23 20.16 21.48 2 2 INTU 1.76 1.24 61.09 0.55 0.14 1188.59 20.95 50.00 23.00 −24.68 4 4 NVDA 35.63 1.55 52.17 1.18 0.49 825.60 91.10 109.88 80.64 −9.16 3 2 Table 20 Weigh s o each c i e ion, he weigh s a e p e-de ined by a use based on some in es iga ion wi h g ea e alues o isk me ics. Ve o Th eshold Ma ke Cap 1% P/E Ra io 2.5% Re enue 2.5% A e age Vol 1.5% EPS 10% Be a 25% YTD 15% 1 Yea 6.5% 3 Yea 1% Weekly 25% Mon hly 10% S ep 3: We calcula e sco e o each s ock, and he sco e is de ined as he summa ion o all ea u es’ alues, conside ed as he inal anking o he s ocks o he po olio. Fo example, he sco e o he 𝑖 h s ock is calcula ed as: 𝑠𝑐𝑜𝑟𝑒𝑖= 11 ∑ 𝑗=1 𝑥′ 𝑖𝑗 S ep 4: Then each s ock we calcula e i s o al sco e, he a e age sco e, he s anda d de ia ion and he consensus sco e o da a o wo gi en DMs. 𝑡𝑜𝑡𝑎𝑙𝑖= 2 ∑ 𝑘=1 𝑠𝑐𝑜𝑟𝑒𝑖𝐷𝑀𝑘 𝑎𝑣𝑔𝑖=𝑡𝑜𝑡𝑎𝑙𝑖 2 𝑠𝑡𝑑𝑖=√∑𝐾 𝑘=1(𝑠𝑐𝑜𝑟𝑒𝑖𝐷𝑀𝑘−𝑎𝑣𝑔𝑖)2 2 𝑐𝑜𝑛𝑠𝑒𝑛𝑠𝑢𝑠𝑖= 1 − 𝑠𝑡𝑑𝑖 𝑠𝑡𝑑𝑡ℎ (4) The 𝑠𝑡𝑑𝑡ℎ is he h eshold o he sco es’ s anda d de ia ion. 𝐷𝑀𝐾 is he 𝑘 h decision make , and we cons uc ed wo DMs in ou case. S ep 5: A e ha ing accu a ely de ined all he g oup decision- making me ics and he desi ed h esholds (as shown in he Table 22), we d op all s ocks ha ha e a leas one me ic alue ou o he espec ed h esholds. Thus, we conclude on 13 candida e companies and make he in es men decision on hei s ocks, while he 5s ocks wi h icke s INTU, ESLT, LOGI, TSLA, AUDC, ATVI a e d opped. The a e age Consensus Sco e o 𝐷𝑀1,𝐷𝑀2on hese emaining 13 s ocks is 60.45%. The g oup decision-making analysis esul s a e shown in Table 23. 4.4.2. MCDM applica ion In his sec ion, we apply 3di e en MCDM me hods, namely TOP- SIS, ELECTRE and PROMETHEE, on he emaining 13 s ocks. We aim o each a inal consis en pool o desi able s ocks om a DIY in es men pe spec i e. Fo each o hese emaining 13 s ocks, we use he 11- ea u e s a ic da a p o ided by In es ing.com. All ea u es a e hose downloaded di ec ly om he da a sou ce. The basic en i onmen con igu a ion and he expec ed numbe o s ock in o ma ion a e shown in Table 24. Be o e applying all MCDM me hods, he s a ic da a a e no malized o ease he compa ison among a ious c i e ia, as A. De ine he ini ial ma ix and he weigh o c i e ia. The no malized da a a e he inpu o all 3MCDM me hods, which p ocess hem and each 3di e en subse s o s ocks as shown in sec ions B. TOPSIS me hod,C. ELECTREEE me hod and D. PROMETHEE. Finally, we conclude on a inal consis en pool o s ocks, including hose p esen in all 3subse s as shown in E. Conclusion. A. De ine he ini ial ma ix and he weigh o c i e ia The ini ial da a has 13 ows co esponding o 13 esidual s ocks and 11 columns co esponding o 11 c i e ia/ ea u es da a sou ces om In es ing.com. The anno a ions o he MCDM p ocess a e shown in Table 25. We no malize he ini ial c i e ia da a by weigh s 𝑊 o ease he a ious scale p oblems. The weigh ec o 𝑊di e s om he one used in he G oup Decision Making sec ion, whe e we assign highe weigh s o Pe o mance and Technical c i e ia o o e come insecu i y and each good consensus sco es. In his no maliza ion p ocess wi h 𝑊, we emphasize Fundamen al c i e ia as his class ea u e e lec s he o e all pe o mance o a company o some deg ee. The weigh s 𝑊a e shown in Table 26. Fo he no malized decision ma ix  𝑋= { 𝑥𝑖𝑗 }, The no maliza ion alue 𝑥𝑖𝑗 o 𝑗 h ea u e o 𝑖 h s ock 𝑥𝑖𝑗 is calcula ed by Sec ion 4.4.1, whe e in his case, 𝑁= 13 deno ing he emaining s ocks. B. TOPSIS me hod applica ion The p ima y egula ion o he TOPSIS me hod is ha he op i- mal al e na i e should be close o he posi i e ideal solu ion. In he mean ime, his al e na i e should be dis an om he nega i e i ual In o ma ion Fusion 102 (2024) 102018 17 V. Snášel e al. Table 21 No malized esul s on s a ic da a. No malized esul o s ock In esing.com Ticke Ma ke cap Re enue A e age olume EPS P/E Ra io Be a YTD 1 Yea 3 Yea s Weekly Mon hly DM1 MSFT 59.31 17.24 12.73 21.10 1.12 0.74 −0.70 2.87 8.32 1.31 2.86 126.89 TXN 4.30 1.71 2.13 18.54 0.76 0.78 −0.30 −0.54 3.39 1.31 2.86 34.93 AAPL 72.60 35.26 31.61 13.51 0.99 0.96 −0.16 4.33 13.51 1.31 2.86 176.80 AVGO 6.54 2.66 0.88 39.40 1.23 0.84 −0.51 3.30 5.57 1.31 2.86 64.07 CDW 0.61 1.94 0.31 15.78 0.87 0.90 −1.16 0.51 4.12 0.65 2.86 27.40 INTU 3.50 1.06 0.66 17.60 2.10 0.90 −2.18 2.33 4.19 0.33 2.14 32.64 NVDA 17.11 2.51 17.86 8.64 2.43 1.16 −0.81 9.70 25.22 1.31 2.86 87.99 No malized esul o s ock YahooFinance.com Symbol Ma ke cap Re enue A e age olume EPS P/E Ra io Be a YTD 1 Yea 3 Yea s Weekly Mon hly DM2 MSFT 59.31 17.24 12.73 21.10 1.12 0.96 −0.69 2.87 8.32 0.87 1.19 125.01 TXN 4.30 1.71 2.13 18.54 0.76 0.97 −0.24 −0.54 3.39 0.87 2.37 34.25 AAPL 72.60 35.26 31.61 13.51 0.99 0.98 −0.15 4.33 13.51 0.87 2.37 175.91 AVGO 6.54 2.66 0.88 39.40 1.23 1.11 −0.46 3.30 5.57 0.87 2.37 63.46 CDW 0.61 1.94 0.31 15.78 0.87 1.02 −1.15 0.51 4.12 0.87 2.37 27.26 INTU 3.50 1.06 0.66 17.60 2.10 1.04 −2.19 2.33 4.19 1.74 4.75 36.78 NVDA 17.11 2.51 17.86 8.64 2.43 1.30 −0.81 9.70 25.22 1.30 2.37 87.64 BRKR 0.25 0.23 0.28 4.06 1.22 0.95 −2.00 0.01 3.58 0.87 2.37 11.81 Table 22 G oup decision making me ics h esholds. Sco e Th eshold Condi ion To al sco e 0 𝑡𝑜𝑡𝑎𝑙𝑖≥0 A e age sco e – – S anda d de ia ion 1.5 𝑠𝑡𝑑𝑖≤1.5 Consensus sco e 15 𝑐𝑜𝑛𝑠𝑒𝑛𝑠𝑢𝑠𝑖≥15 solu ion (ideal in an opposing di ec ion) candida e. The e o e, he TOPSIS me hod s a s calcula ing he posi i e/nega i e ideal solu ion, and hen calcula es each candida e’s dis ance sco e (closeness sco e) o hese wo ideal solu ions. Las ly, he s ocks/al e na i es a e anked based on he closeness sco e. A ela i e closeness sco e is always be ween 0 and 1, and an op ion is bes when i is close o 1. The solu ion o he p oblem o he bes al e na i e is he one wi h he highes closeness sco e. We se he h eshold alue as 0.25, 𝑅𝑗≥0.25, o closeness measu emen o selec ing he po en ial al e na i e, o which each selec ed s ock should ha e closeness alues la ge han his h eshold. We p o ide he in e media e calcula ion s eps o TOPSIS on expe i- men al da a in Appendix. The in e media e pic o ial esul is shown in Table 27, whe e he posi i e ideal solu ion, nega i e i ual solu ion, and closeness deno ed as D_plus,D_minus and D_closeness, espec i ely. And he inal selec ed s ocks a e shown in Table 28. The e o e, he candida es ha a e desi able o in es a e: 1. AAPL →Apple Inc 2. MSFT →Mic oso Co po a ion 3. NVDA →NVIDIA Co po a ion 4. AVGO →B oadcom Inc 5. QCOM →Qualcomm Inco po a ed C. ELECTRE I wi h & wi hou e o me hod applica ion ELECTRE me hod c ea es a anking o al e na i es desc ibed on some c i e ia [68] by a comp ehensi e e alua ion app oach. The ELEC- TRE I me hod is he i s me hod o he ELECTRE amily and conce ns he pai wise compa ison o he al e na i es. The me hod’s goal is no necessa ily o ind he op imal al e na i e bu a he a subse o al e na i es, h ough a se ies o successi e checks, ha will be supe io o he es . The s eps a e ho oughly desc ibed in Appendix. The inal selec ion esul s a e shown in Tables 29 and 30 o me hods ELECTRE I wi h and wi hou e o, espec i ely. Bo h me hods eached he same pool o selec ed s ocks (al e na i es), alida ing ha he esul s a e consis en . The candida es ha a e desi able o in es in a e: 1. AAPL →Apple Inc 2. MSFT →Mic oso Co po a ion 3. NVDA →NVIDIA Co po a ion 4. AVGO →B oadcom Inc 5. EQIX →Equinix Inc D. PROMETTHEE me hod applica ion PROMETTHEE [69] me hod o de s he al e na i es based on he c i e ia pe o mance and assigns weigh s wi h espec o he c i e ia. PROMETTHEE I aims o ge a pa ial anking o al e na i es, and PROMETTHEE II ob ains a ull anking o all al e na i es. The numbe o s ocks/al e na i es is 13, and he c i e ia numbe is 11. The p ocess o p oducing he ne ou anking low on al e na i es by he PROMETTHEE II me hod is p o ided in Appendix. A e ha , ank all he al e na i es based on he ne ou anking low alue 𝑓(𝐴𝑘), co esponding o he ank lis o desi able s ocks. The esul s show he desi ed s ocks wo hy o in es men a e in Table 30 and as ollows: 1. NVDA →NVIDIA Co po a ion 2. AAPL →Apple Inc 3. MSFT →Mic oso Co po a ion 4. AVGO →B oadcom Inc 5. EQIX →Equinix Inc 6. TXN →Texas Ins umen s Inco po a ed 7. CDW →CDW Co p E. Conclusion The p ocess o MCDM is o each uni o m esul s in which each can- dida e’s pe o mance is gua an eed and con i med. The inal s ocks a e he c ossed s ocks anked in he op esul s lis by TOPSIS, ELECTRE, and PROMETTHEE. Based on he analysis and pe o mance abo e, we conclude ha NVDA, AAPL, MSFT, AVGO, and EQIX, a e he bes in es men op ions. Co espondingly hei ull names a e ‘NVIDIA Co po a ion’, ‘Apple Inc’, ‘Mic oso Co po a ion’, ‘B oadcom Inc’, and ‘Equinix Inc’. •N idia Co po a ion is a global leade in a i icial in elligence ha dwa e and so wa e. NVIDIA designs applica ion p og am- ming in e aces o da a science, high-pe o mance compu ing, and sys em-on-chip uni s o he mobile compu ing and au o- mo i e ma ke . Especially g aphics p ocessing uni s de eloped by NVIDIA a e becoming mo e popula in c ea i e p oduc ion and a i icial in elligence, which makes NVIDIA s ocks a s ong selec ion. In o ma ion Fusion 102 (2024) 102018 18 V. Snášel e al. Table 23 G oup decision making esul s. Ticke DM1 DM2 To al sco e A e age S . de S anda d consensus CDW 27.40 27.26 54.66 27.33 0.10 95.02 NVDA 87.99 87.64 175.63 87.81 0.25 87.44 AVGO 64.07 63.46 127.52 63.76 0.43 78.58 TXN 34.93 34.25 69.18 34.59 0.48 75.99 ATVI 23.67 22.86 46.54 23.27 0.57 71.41 AAPL 176.80 175.91 352.71 176.36 0.63 68.31 QCOM 41.65 42.85 84.49 42.25 0.85 57.61 TER 23.30 24.72 48.01 24.01 1.00 49.84 SPNS 4.87 6.37 11.24 5.62 1.07 46.67 MGIC 10.26 11.79 22.04 11.02 1.08 45.88 BRKR 10.07 11.81 21.89 10.94 1.23 38.57 EQIX 27.98 26.19 54.18 27.09 1.26 36.83 MSFT 126.89 125.01 251.90 125.95 1.33 33.71 Table 24 MCDM se ings o selec ing he desi ed s ocks. Me hod # S ocks # Selec ed Th eshold TOPSIS 13 5 Closeness =0.25 ELECTRE I (wi h e o) 13 5 Ag eemen /disag eemen (ini ial) 1.0/0.0 ELECTRE I (wi hou e o) 13 5 – PROMETTHEE 13 7 P e e ence unc ion: TypeI, TypeV Final-decision 5 Common s ocks ha selec ed by all me hods Table 25 Anno a ion used in MCDM sec ion on he esidual da a. Symbols Anno a ion 𝑛The numbe o esidual al e na i e s ocks, 𝑛= 13 𝑚The numbe o c i e ia, whe e 𝑚= 11 𝑖, 𝑗, 𝑘 The co esponding index symbol. 𝑊The weigh s conce ning 11 c i e ia. 𝑊= [𝑤1,…, 𝑤11]. 𝑋The ini ial da a 𝑋= {𝑥𝑖𝑗 }, whe e 𝑖= 1,2,…, 𝑛;𝑗= 1,2,…, 𝑚.  𝑋The no malized da a ma ix on 𝑋, 𝑋= { 𝑥𝑖𝑗 } A The symbol o al e na i e, whe e 𝐴𝑖 ep esen s he 𝑖 h al e na i e (s ock) K The ke nel size used in he ELECTREE me hod. I is a a iable, which is ini ialized as 𝐾0=𝑛= 13 D(𝐴𝑖, 𝐴𝑗) A ela ion be ween al e na i e 𝐴𝑖and 𝐴𝑗 C(𝐴𝑖, 𝐴𝑗) A ela ion be ween al e na i e 𝐴𝑖and 𝐴𝑗 P(𝐴𝑖, 𝐴𝑗) A ela ion be ween al e na i e 𝐴𝑖and 𝐴𝑗 Table 26 Use -de ined weigh s o da a no maliza ion. Those weigh s emphasize undamen al c i e ia/ ea u es. Ve o Th eshold Ma ke cap 2.5% P/E a io 10% Re enue 15% A e age Vol 2.5% EPS 15% Be a 12.5% YTD 10% 1 Yea 10% 3 Yea 2.5% Weekly 12.5% Mon hly 7.5% •Apple Inc. owns he mos p ominen ma ke capi aliza ion in he wo ld, which p o ides and designs consume elec onics, so - wa e, and online se ices ha in ol e ou daily li e. I s p oduc s ha e always become bes selle s and one o he mass-p oduced mic ocompu e s. •Mic oso Co po a ion p oduces compu e so wa e, consume elec- onics, and pe sonal compu e s, and i s bes -known so wa e p oduc s include Windows ope a ing sys ems, Mic oso O ice sui e, and Edge web b owse s. Those p oduc s a e ypical con- sump ion in ou mode n daily li e. •B oadcom Inc. is a designe , de elope , manu ac u e , and global semiconduc o and in as uc u e p oduc supplie o da a cen- e s, ne wo king, so wa e, b oadband, wi eless, s o age, and in- dus ial ma ke s. •Equinix, Inc. specializes in In e ne connec ion and da a cen e s. These i e companies ea n hei epu a ions h ough hei well- designed p oduc s and se ices, and hese i e s ocks a e nomina ed as he p e e ed s ocks o in es men by he p oposed usion-based model. 5. De ailed abla ion s udies In his sec ion, we will compa e he selec ion esul s when gi en a single echnique o he p oposed usion-based model. The abla ion s udies include wo pa s; we emo e he da a- usion p ocess and decision-le el usion om ou usion-based model sepa a ely, namely he model wi h only he da a- usion p ocess and he model wi h only decision usion. 5.1. Model wi h da a- usion We name his model wi h only he da a usion p ocess as Vmodel-1, and i s a s wi h da a selec ion and da a loading, hen ends wi h he indi idual-based analysis. Vmodel-1 d ops he s ocks ha canno each he use ’s expec a ions. The da a selec ion and loading o Vmodel-1 ollow he same p oce- du e as ou p oposed model. Fi s , he e a e 34 s ocks selec ed. Then we analyze he s ock indi idually using isk- ela ed me ics and each a s anda d consensus sco e. The isk- ela ed e alua ion esul o Vmodel-1 is shown in Ta- ble 10. A e ha , Vmodel-1 pe o ms he s anda d consensus analysis ollowing ou p oposed model, and he esul is shown in Table 31 ending up wi h 13 s ocks. Though hese 13 s ocks sa is ied he expec ed c i e ia, we can see ha he o al sco e, a e age o s anda d consensus e alua ion a ies much om he Table 31, i.e., he e alua ion o al sco e di e om alue 11.24 o 352.71, he e alua ion A e age sco e a e in a ange be ween 5.62 o 176.36. The S anda d consensus alues o 13 s ocks also a y widely. Those alues imply a p elimina y selec ion esul ha needs o be u he handou om con lic c i e ia e alua ion. Ne e heless, he use s o DIY in es o s ha e li le chance o accep ing hese con using esul s wi h a high di e ence sco e gap on he S anda d consensus sco e o To al sco e. This esul equi es u he p ocess on he esul s. In o ma ion Fusion 102 (2024) 102018 19 V. Snášel e al. Table 27 Calcula ed me ics esul s by TOPSIS o all 13 esidual s ocks. The da a a e he co esponding ea u es/c i e ia om In es ing.com. Ticke Ma ke cap Re enue A g. olume EPS P/E a io Be a YTD 1 Yea 3 Yea s Weekly Mon hly Dplus Dminus Closeness CDW 0.26 0.49 0.05 10.49 0.95 2.15 −2.22 0.43 4.81 1.48 4.31 44.35 10.63 0.19 NVDA 7.15 0.63 3.06 5.74 2.66 2.78 −1.54 8.09 29.40 2.96 4.31 32.58 29.32 0.47 AVGO 2.73 0.67 0.15 26.19 1.34 2.02 −0.97 2.75 6.49 2.96 4.31 38.02 26.53 0.41 TXN 1.79 0.43 0.36 12.32 0.83 1.86 −0.57 −0.45 3.96 2.96 4.31 43.13 12.98 0.23 ATVI 0.67 0.21 0.71 5.13 0.87 1.06 3.61 −1.33 4.27 2.96 4.31 46.49 10.21 0.18 AAPL 30.33 8.89 5.41 8.98 1.09 2.31 −0.31 3.61 15.76 2.96 4.31 23.09 36.01 0.61 QCOM 1.77 0.85 0.65 13.03 0.66 2.35 −3.30 0.58 9.30 0.00 3.23 40.17 13.81 0.26 TER 0.20 0.09 0.12 8.24 0.80 2.90 −4.85 −0.84 10.75 0.74 2.16 43.15 10.27 0.19 SPNS 0.02 0.01 0.00 1.27 1.12 2.68 −4.17 −1.71 4.09 0.00 2.16 49.74 2.00 0.04 MGIC 0.01 0.01 0.00 0.90 1.08 2.62 −2.74 1.00 6.48 0.00 2.16 48.09 4.61 0.09 BRKR 0.10 0.06 0.05 2.70 1.33 2.41 −3.79 0.01 4.17 0.00 2.16 48.54 3.18 0.06 EQIX 0.74 0.16 0.03 8.27 5.02 0.90 −1.65 1.09 3.99 2.96 4.31 45.04 10.27 0.19 MSFT 24.78 4.35 2.18 14.03 1.23 1.76 −1.35 2.40 9.70 2.96 4.31 25.90 29.77 0.53 Table 28 The inal selec ion esul s by TOPSIS. Ticke Ma ke cap Re enue A e age olume EPS P/E a io Be a YTD 1 Yea 3 Yea s Weekly Mon hly AAPL 2.84E+12 3.78E+11 9.33E+07 6.02 28.99 1.18 −1.84 41.72 259.33 4 4 MSFT 2.32E+12 1.85E+11 3.76E+07 9.40 32.80 0.90 −8.00 27.67 159.60 4 4 NVDA 6.69E+11 2.69E+10 5.27E+07 3.85 70.96 1.42 −9.18 93.40 483.87 4 4 AVGO 2.56E+11 2.85E+10 2.58E+06 17.55 35.87 1.03 −5.78 31.73 106.89 4 4 QCOM 1.66E+11 3.60E+10 1.12E+07 8.73 17.51 1.20 −19.62 6.68 153.04 0 3 Table 29 Resul s o selec ed desi ed s ocks by ELECTRE I wi h and wi hou a e o. Wi hou e o esul Ticke Ma ke cap Re enue A e age olume EPS P/E a io Be a YTD 1 Yea 3 Yea s Weekly Mon hly NVDA 6.69E+11 2.69E+10 5.27E+07 3.85 70.96 1.42 −9.18 93.40 483.87 4 4 AVGO 2.56E+11 2.85E+10 2.58E+06 17.55 35.87 1.03 −5.78 31.73 106.89 4 4 AAPL 2.84E+12 3.78E+11 9.33E+07 6.02 28.99 1.18 −1.84 41.72 259.33 4 4 EQIX 6.92E+10 6.64E+09 4.67E+05 5.54 133.97 0.46 −9.79 12.54 65.71 4 4 MSFT 2.32E+12 1.85E+11 3.76E+07 9.40 32.80 0.90 −8.00 27.67 159.60 4 4 Wi h e o esul Ticke Ma ke cap Re enue A e age olume EPS P/E a io Be a YTD 1 Yea 3 Yea s Weekly Mon hly NVDA 6.69E+11 2.69E+10 5.27E+07 3.85 70.96 1.42 −9.18 93.40 483.87 4 4 AVGO 2.56E+11 2.85E+10 2.58E+06 17.55 35.87 1.03 −5.78 31.73 106.89 4 4 AAPL 2.84E+12 3.78E+11 9.33E+07 6.02 28.99 1.18 −1.84 41.72 259.33 4 4 EQIX 6.92E+10 6.64E+09 4.67E+05 5.54 133.97 0.46 −9.79 12.54 65.71 4 4 MSFT 2.32E+12 1.85E+11 3.76E+07 9.40 32.80 0.90 −8.00 27.67 159.60 4 4 Table 30 Resul s o selec ed desi ed s ocks by PROMETTHEE. Ticke Ma ke cap Re enue A e age olume (3 m) EPS P/E a io Be a YTD 1 Yea 3 Yea s Weekly Mon hly NVDA 6.69E+11 2.69E+10 5.27E+07 3.85 70.96 1.42 −9.18 93.40 483.87 4 4 AAPL 2.84E+12 3.78E+11 9.33E+07 6.02 28.99 1.18 −1.84 41.72 259.33 4 4 MSFT 2.32E+12 1.85E+11 3.76E+07 9.40 32.80 0.90 −8.00 27.67 159.60 4 4 AVGO 2.56E+11 2.85E+10 2.58E+06 17.55 35.87 1.03 −5.78 31.73 106.89 4 4 EQIX 6.92E+10 6.64E+09 4.67E+05 5.54 133.97 0.46 −9.79 12.54 65.71 4 4 TXN 1.68E+11 1.83E+10 6.27E+06 8.26 22.20 0.95 −3.39 −5.20 65.11 4 4 CDW 2.40E+10 2.08E+10 9.12E+05 7.03 25.43 1.10 −13.18 4.93 79.11 2 4 5.2. Model wi h MCDM selec ion We name his model wi h only he decision-le el usion p ocess as Vmodel-2, and i s a s wi h he da a selec ion and ends wi h g oup decision-making based on MCDM analysis. Vmodel-2 selec s he op- anked s ocks as a esul . The da a is selec ed om he In es ing.com, and he candida es a e he same 34 s ocks, ollowing ou p oposed model. Then Vmodel-2 pe o ms he MCDM me hod on all 34 s ocks. Table 32 shows he esul s by he TOPSIS me hod ending up wi h h ee selec ed s ock op ions, TSLA, AAPL, MSFT. Then Table 33 shows he selec ion TSLA, AAPL by ELECTRE wi h/wi hou e o me hod as hese wo condi ions p oduce he same esul s. PROMETTHEE p oduces se en selec ion s ocks (as in Table 34) as op ions based on he same h eshold we se in he p oposed usion models. Hence he inal selec ion esul is TSLA, AAPL by he MCDM me hod. Though s ock TSLA sa is ies he goals se on he i s ound o isk- ela ed expec a ion, hey each an undesi able low S anda d Con- sensus alue (Table 35), which implies high inconsis en pe o mance be ween wo DMs (In es ing.com and YahooFinance). Besides, he ESLT selec ed by PROMETTHEE ge s a nega i e s anda d consensus alue. In summa y, single-p ocessed models such as Vmodel-1 and Vmodel-2 canno mee he use s’ expec a ions when wo king alone, and Vmodel- 2 only e alua es he cu en s a is ical da a and does no conside he hidden ends and isks. Fu he mo e, he Vmodel-1 wi h only he da a usion p ocess hands in he selec ed 13 s ocks, which is way oo la ge o a use wi h a DIY in es men goal. Tha selec ion esul s in 13 s ocks s ill makes use s con used. Finally, he p oposed usion-based model conside s he use s’ expec a ions and can make a p ac ical and easonable s ock selec ion. In o ma ion Fusion 102 (2024) 102018 20 V. Snášel e al. Table 31 Abla ion s udy esul s o selec ed s ocks by Vmodel-1 o da a usion p ocess. Ticke DM1 DM2 To al sco e A e age S . de S anda d consensus CDW 27.40 27.26 54.66 27.33 0.10 95.02 NVDA 87.99 87.64 175.63 87.81 0.25 87.44 AVGO 64.07 63.46 127.52 63.76 0.43 78.58 TXN 34.93 34.25 69.18 34.59 0.48 75.99 ATVI 23.67 22.86 46.54 23.27 0.57 71.41 AAPL 176.80 175.91 352.71 176.36 0.63 68.31 QCOM 41.65 42.85 84.49 42.25 0.85 57.61 TER 23.30 24.72 48.01 24.01 1.00 49.84 SPNS 4.87 6.37 11.24 5.62 1.07 46.67 MGIC 10.26 11.79 22.04 11.02 1.08 45.88 BRKR 10.07 11.81 21.89 10.94 1.23 38.57 EQIX 27.98 26.19 54.18 27.09 1.26 36.83 MSFT 126.89 125.01 251.90 125.95 1.33 33.71 To al =13 6. Conclusion This pape ocuses on he s ock selec ion p oblem, which has no ye been ex ensi ely s udied and published. The di icul ies o his p oblem lie in wo signi ican aspec s: Dynamic and ola ile om public policies; and con lic ing c i e ia o pe o mance anking. Hence, we p oposed a usion-based model o add essing he s ock selec ion p oblem, which includes da a usion and decision-le el usion sub-p ocedu es. Da a usion p ocesses he mul i-sou ce aw da a and acqui es a consensus analysis o candida e s ocks. In addi ion, we cus omized ou loss unc- ion o ob ain a mo e eliable da a o ecas ing module and combined i wi h XGBReg esso o s ock p icing p edic ions. As a esul , his p oposed loss unc ion achie es lowe e o compa ed o he baseline alue o he s anda d loss unc ion. Meanwhile, he p oposed me hod educes he p edic ion e o and he wa ping pa h cos by 6.3 pe cen and 5.6 pe cen on a e age, espec i ely, compa ed o he s anda d loss me hod in a bes case wi h he expe imen al eal-si ua ion da a, indica ing he e ec i eness o he p oposed me hod. Mo eo e , MCDM echniques a e he decision-making suppo o ade o he con lic c i e ia. A las , we expe imen ed on an ac ual li e s ock om In es ing.com and YahooFinance da a aded in NASDAQ by he p oposed model. The esul s show ha he selec ed s ocks mee he goal and he expec a ions om he iew o he model plan and p e e ence. Fu he mo e, o jus i y he p ac ical applica ion o he p oposed usion-based model, we p esen he abla ion s udies o e i y he necessi y o he da a-le el usion and he decision-le el usion p o- cesses. In summa y, ou mul i-sou ce usion-based model can p o ide a op lis o he desi able s ocks o in es men , which o e comes he si ua ion o o e -numbe candida e selec ion. This Vmodel-1 canno ank hose ague candida es (13 ou o 34 s ocks), which makes i ha d and un iendly o use s. The p oposed model is be e han he di ec MCDM me hods o selec ing unde use expec a ions. Vmodel-2 does no conside he basic goal/expec a ion like low- isk and s able equi emen , which is unsui able o a DIY in es men . In o he wo ds, he da a usion and decision le el usion oge he make a comple e and cus omized model o a use o DIY in es men . No e ha he p oposed usion-based model is designed o s udy pu poses o he low-capi al and conse a i e DIY in es o . Despi e being op imis ic abou he economy’s long- e m p ospec s, he model pe o ms expec a ions o in es men s ha p o ide sa e y o p incipal and mode a e-income and igno es he s ocks wi h highe luc ua ions. The weigh s o c i e ia used in his model ollow he designe ’s expec- a ions, in luencing he inal decision esul s. The p oposed amewo k is a s udy ool. The s ock ma ke is isky and in luenced no by he en i onmen bu by policy. In es men needs o be cau ious. 7. Limi a ion and u u e wo ks •The p oposed amewo k only conside s objec i e in o ma ion and da a. Some imes, a s ock app ecia ion ela es o he com- pany’s s uc u al adjus men o whe he he company can be lis ed, which is subjec i e in o ma ion. •This amewo k employs a limi ed numbe o c i e ia o s ock selec ion, namely ep esen a i e c i e ia om pe o mance and echnical analysis da a. Changes in ampli udes o c i e ion alues a e also indica o s wo hy o analysis, e.g., di idends (paying ou a po ion o p o i s o sha eholde s). •The p oposed amewo k ocuses on p ice p edic ion and s ock selec ion wi hou po olio op imiza ion, a necessa y sec ion in po olio managemen . Po olio op imiza ion is op imizing he esou ce alloca ion, o en equi ing in e ac ion wi h use s. •The o e all amewo k is o s udy pu poses and p o ides com- ple e use con ol. Howe e , o o e an e icien se ice wi h p o i , we should e e o he se ice by s ock ad iso s, such as obo-ad iso y and Mo ley Fool. Decla a ion o compe ing in e es The au ho s decla e ha hey ha e no known compe ing inan- cial in e es s o pe sonal ela ionships ha could ha e appea ed o in luence he wo k epo ed in his pape . Da a a ailabili y Public da a, download link is p o ided in he pape . Acknowledgmen s The au ho s g a e ully acknowledge inancial suppo ANID Fonde- cy 1231122, PIA/PUENTE AFB220003, Chile, DST/ INT/ Czech/ P- 12/ 2019, eg.no. LTAIN19176 by he Czech Republic Minis y o Educa ion, You h and Spo s in he p ojec ‘‘Me aheu is ics F ame- wo k o Mul iobjec i e Combina o ial Op imiza ion P oblems (META MO-COP)’’. Appendix In Sec ion 4.4.2, we p o ide a simple in oduc ion o MCDM appli- ca ions; he e, we p o ide he de ailed ope a ion s eps o PROMETTHEE me hods on he expe imen al da a. MCDM applica ion The ini ial da a has 13 ows co esponding o 13 esidual s ocks and 11 columns co esponding o 11 c i e ia/ ea u es da a sou ces om In es ing.com. Th ee MCDM applica ions s a wi h he cons uc ion o he ini ial ma ix, no malized da a ma ix, and he de e mina ion o he weigh o c i e ia, which is pe o med in Pa -A in Sec ion 4.4.2. Then we apply TOPSIS, ELECTRE, and PROMETTHEE indi idually. The example applica ion is on PROMETTHEE as ollows: PROMETTHEE applica ion S ep A.1: PROMETTHEE s a s wi h no malized ini ial da a, done in Sec ion A (Abo e) wi h a use ’s de ined weigh s 𝑊. S ep A.2: Calcula e he pai wise e alua i e di e ences be ween al e na i es co esponding o each c i e ion. Le di (𝐴𝑘,𝑗 , 𝐴𝓁,𝑗 ) be he di e ence be ween al e na i e 𝐴𝑘and 𝐴𝓁 on c i e ia 𝑗. Then he di e ence be ween all pai s o al e na i es will be a ma ix da a (le us call i di ma ix) wi h a ow size o 𝑛× (𝑛− 1) and column size o 𝑚. Mo e speci ic, in 𝑛× (𝑛− 1) ows, he i s 𝑛− 1 ows co esponds o he di e ence be ween 𝐴1 o {𝐴2,…, 𝐴𝑛}, he second 𝑛− 1 ows co esponds o he di e ence be ween 𝐴2 o {𝐴1, 𝐴3,…, 𝐴𝑛}. All he way o he las 𝑛 h (𝑛− 1) ows co esponds o In o ma ion Fusion 102 (2024) 102018 21 V. Snášel e al. Table 32 Abla ion s udy o he inal selec ion by TOPSIS and in e media e esul (closeness). Ticke Ma ke cap Re enue A g. olume EPS P/E a io Be a YTD 1 Yea 3 Yea s Weekly Mon hly Dplus Dminus Closeness AAPL 2.8E+12 3.8E+11 9.3E+07 6.0 29.0 1.2 −1.8 41.7 259.3 4 4 1.6E+03 2.9E+12 1.0 MSFT 2.3E+12 1.8E+11 3.8E+07 9.4 32.8 0.9 −8.0 27.7 159.6 4 4 5.5E+11 2.3E+12 0.8 TSLA 1.1E+12 5.4E+10 2.7E+07 4.9 219.0 2.1 2.6 63.9 1796.8 4 4 1.8E+12 1.1E+12 0.4 Table 33 Abla ion s udy esul by ELECTRE I wi h/wi hou e o, wo expe imen s p oduce he same esul s. Ticke Ma ke cap Re enue A e age olume EPS P/E a io Be a YTD 1 Yea 3 Yea s Weekly Mon hly TSLA 1.12E+12 5.38E+10 2.67E+07 4.92 218.95 2.08 2.63 63.90 1796.80 4 4 AAPL 2.84E+12 3.78E+11 9.33E+07 6.02 28.99 1.18 −1.84 41.72 259.33 4 4 Table 34 Abla ion s udy esul by PROMETTHEE in abla ion s udy. Ticke Ma ke cap Re enue A e age olume EPS P/E a io Be a YTD 1 Yea 3 Yea s Weekly Mon hly TSLA 1.12E+12 5.38E+10 2.67E+07 4.92 218.95 2.08 2.63 63.90 1796.80 4 4 AAPL 2.84E+12 3.78E+11 9.33E+07 6.02 28.99 1.18 −1.84 41.72 259.33 4 4 NVDA 6.69E+11 2.69E+10 5.27E+07 3.85 70.96 1.42 −9.18 93.40 483.87 4 4 MSFT 2.32E+12 1.85E+11 3.76E+07 9.40 32.80 0.90 −8.00 27.67 159.60 4 4 AVGO 2.56E+11 2.85E+10 2.58E+06 17.55 35.87 1.03 −5.78 31.73 106.89 4 4 TXN 1.68E+11 1.83E+10 6.27E+06 8.26 22.20 0.95 −3.39 −5.20 65.11 4 4 ESLT 9.74E+09 5.16E+09 5.18E+04 7.55 29.18 0.79 26.33 52.76 63.33 4 4 Table 35 The expec a ion iew on he selec ed s ocks in abla ion s udy. Expec a ion DM1 DM2 To al sco e A e age S . de S anda d consensus >0<2>15 TSLA 167.60 164.97 332.57 166.28 1.86 7.13 ESLT 34.60 30.36 64.96 32.48 3.00 −49.95 he di e ence be ween 𝐴𝑛 o {𝐴1, 𝐴2,…, 𝐴𝑛−1}. A he same ime, each ow has 𝑚elemen s co esponding o he di e ence in speci ic c i e ia. The e alua ion di e ence calcula ion is de ined as: 𝑑𝑖𝑓𝑓(𝐴𝑘,𝑗 , 𝐴𝓁,𝑗 ) = 𝑥𝑘𝑗 −𝑥𝓁𝑗 whe e, anno a ions please e e o Table 25. S ep A.3: De ine he p e e ence unc ion and ans o m he e alua- ion di e ence di da a o unc ion alue. We use TypeV and TypeI, wo p e e ence unc ions. Then each elemen in di da a will be ans o med by p e e ence unc ion o a alue, which cons uc s a new ma ix da a 𝑃o he same size o ma ix da a di . Two p e e ence unc ions a e de ined: 𝑇 𝑦𝑝𝑒𝑉 (𝑑𝑖𝑓𝑓 , 𝑞, 𝑝) = 𝑃𝑗(𝐴𝑘,𝑗 , 𝐴𝓁,𝑗 ) =⎧ ⎪ ⎨ ⎪ ⎩ 0,i 𝑑𝑖𝑓𝑓(𝐴𝑘,𝑗 , 𝐴𝓁,𝑗 )≤𝑞 1,eli 𝑑𝑖𝑓𝑓(𝐴𝑘,𝑗 , 𝐴𝓁,𝑗 )> 𝑝 𝑑𝑖𝑓 𝑓(𝑥)−𝑞 𝑝−𝑞,else {𝑥→(𝐴𝑘,𝑗 , 𝐴𝓁,𝑗 )} and 𝑇 𝑦𝑝𝑒𝐼(𝑑𝑖𝑓𝑓, 𝑞, 𝑝) = 𝑃𝑗(𝐴𝑘,𝑗 , 𝐴𝓁,𝑗 ) = {1,i 𝑑𝑖𝑓𝑓 (𝐴𝑘,𝑗 , 𝐴𝓁,𝑗 )>0 0,else whe e, in TypeV and TypeI,di (x) is he e alua ions di e ences be- ween wo s ocks on speci ic c i e ia, such as di (𝐴𝑘,𝑗 , 𝐴𝓁,𝑗 ); and (𝑞, 𝑝) is de ined in Table 36. S ep A.4: Calcula e he agg ega ed p e e ence. The agg ega ed p e - e ence is he weigh ed ow summa ion o ma ix da a 𝑃, leading o a size o 𝑛× (𝑛− 1) alues. The agg ega ed unc ions a e de ined as: 𝜋(𝐴𝑘, 𝐴𝓁) = 𝑚 ∑ 𝑗=1 𝑤𝑗×𝑃𝑗(𝐴𝑘,𝑗 , 𝐴𝓁,𝑗 ) S ep A.5: Cons uc an al e na i e o an al e na i e pai wise ma ix. The al e na i e o al e na i e pai wise ma ix 𝐹is a squa e ma ix o size 𝑛×𝑛wi h 𝑛 ows ep esen ing al e na i es and 𝑛columns o Table 36 Th eshold and ype in o ma ion used in PROMETTHEE. (𝑞, 𝑝)Type Ma ke cap (109,1011) TypeV P/E a io (15,35) TypeV Re enue (108,1010) TypeV A e age Vol (107,108) TypeV EPS (1,6) TypeV Be a (0,0.6) TypeV YTD (5,25) TypeV 1 Yea (5,45) TypeV 3 Yea (10,75) TypeV Weekly (0,1) TypeI Mon hly (0,1) TypeI al e na i es, whe e each elemen o his ma ix is gi en by he alue o 𝜋(𝐴𝑘, 𝐴𝓁). Then we assign 𝜋(𝐴𝑘, 𝐴𝑘)a ‘0’ alue. S ep A.6: De e mine he lea ing and en e ing ou anking lows. The lea ing/en e ing low is a calcula ed alue co esponding o each al e na i e; hey a e de ined as ollows: 𝑓(𝐴𝑘)+=1 𝑛− 1 𝑛 ∑ 𝑖=1 𝜋(𝐴𝑘, 𝐴𝑖), 𝑖 ≠𝑘, 𝑙𝑒𝑎𝑣𝑖𝑛𝑔𝑓𝑙𝑜𝑤 𝑓(𝐴𝑘)−=1 𝑛− 1 𝑛 ∑ 𝑖=1 𝜋(𝐴𝑖, 𝐴𝑘), 𝑖 ≠𝑘, 𝑒𝑛𝑡𝑒𝑟𝑖𝑛𝑔𝑓𝑙𝑜𝑤 whe e he symbols please e e o Table 25. 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