Minimum description length and multi-criteria decision analysis in predictive modelling

Recei ed 3 Janua y 2025, accep ed 20 Janua y 2025, da e o publica ion 22 Janua y 2025, da e o cu en e sion 30 Janua y 2025. Digi al Objec Iden i ie 10.1109/ACCESS.2025.3532815 Minimum Desc ip ion Leng h and Mul i-C i e ia Decision Analysis in P edic i e Modeling PETR SILHAVY 1, KATEŘINA HLAVÁČKOVÁ-SCHINDLER 2, AND RADEK SILHAVY 1 1Facul y o Applied In o ma ics, Tomas Ba a Uni e si y in Zlín, 760 01 Zlín, Czech Republic 2Da a Mining and Machine Lea ning Resea ch G oup, Facul y o Compu e Science, Uni e si y o Vienna, 1010 Vienna, Aus ia Co esponding au ho s: Radek Silha y ( adek@silha y.cz) and Ka eřina Hla áčko á-Schindle (ka e ina.schindle o [email p o ec ed]) This wo k was suppo ed by Tomas Ba a Uni e si y in Zlín, Facul y o Applied In o ma ics, unde P ojec RO30246061025/2102. ABSTRACT Accu a e model selec ion is essen ial in p edic i e modelling ac oss a ious domains, signi ican ly impac ing decision-making and esou ce alloca ion. Despi e ex ensi e esea ch, he model selec ion p ocess emains challenging. This wo k aims o in eg a e he Minimum Desc ip ion Leng h p inciple wi h he Mul i-C i e ia Decision Analysis o enhance he selec ion o o ecas ing machine lea ning models. The p oposed MDL-MCDA amewo k combines he MDL p inciple, which balances model complexi y and da a i , wi h he MCDA, which inco po a es mul iple e alua ion c i e ia o add ess con lic ing e o measu emen s. Fou da ase s om di e se domains, including so wa e enginee ing (e o es ima ion), heal hca e (glucose le el p edic ion), inance (GDP p edic ion), and s ock ma ke p edic ion, we e used o alida e he amewo k. Va ious eg ession models and eed- o wa d neu al ne wo ks we e e alua ed using c i e ia such as MAE, MAPE, RMSE, and Adjus ed R2. We employed he Analy ic Hie a chy P ocess (AHP) o de e mine he ela i e impo ance o hese c i e ia. We conclude ha he in eg a ion o MDL and MCDA signi ican ly imp o ed model selec ion ac oss all da ase s. The cubic polynomial eg ession model and he mul i-laye pe cep on models ou pe o med o he models in e ms o AHP sco e and MDL c i e ion. Speci ically, he MDL-MCDA app oach p o ided a mo e nuanced e alua ion, ensu ing he selec ed models e ec i ely balanced complexi y and p edic i e accu acy. INDEX TERMS Mul ic i e ia decision analysis, minimum model leng h, machine lea ning, model selec ion p edic ion, MDL-MCDA. I. INTRODUCTION Accu a e model selec ion is essen ial in p edic i e mod- elling ac oss a ious domains. The e icacy o p edic i e models in luences decision-making p ocesses and esou ce alloca ion. Despi e ex ensi e s udies compa ing mul iple p edic i e models, he model selec ion app oach s ill needs o be explo ed. Model selec ion is in insically ied o he objec i es o p edic ion and unde s anding, wi h i s essence cap u ed h ough he o malisa ion o loss and isk, as decla ed by Pe opoulos e al. [1] and by F iedman [2]. The issues o model selec ion lie in na iga ing h ough he complex ela ionship be ween independen a iables and he dependen a iable unde pinned by bo h obse able and unobse able ac o s. The li e a u e iden i ies wo b oad The associa e edi o coo dina ing he e iew o his manusc ip and app o ing i o publica ion was Alba Ama o . ca ego ies o a iables in luencing dependen a iables: explana o y a iables, which a e obse able, and unobse - able a iables, which include ac o s such as measu emen e o s o omi ed independen a iables. This issue has been discussed in a ious p oblem domains. A. HOW MODELS CAN BE SELECTED: MAIN ISSUES Model selec ion is a c ucial challenge wi hin all p edic ion asks, b idging he gap be ween heo e ical cons uc s and p ac ical applica ions. This sec ion desc ibes he co e aspec s o model selec ion, co e ing he undamen al issues, model e alua ion and cons uc ion me hodologies, and he selec ion p ocess. The mul ic i e ia app oach, assump ions unde ly- ing model selec ion s a egies, and he in e play be ween explana o y a iables and unobse able ac o s in luencing he ou come a iable a e essen ial o ou discussion. The e alua ed da ase s a e o en mo e complex because hey 19388 2025 The Au ho s. This wo k is licensed unde a C ea i e Commons A ibu ion 4.0 License. Fo mo e in o ma ion, see h ps://c ea i ecommons.o g/licenses/by/4.0/ VOLUME 13, 2025 P. Silha y e al.: MDL and Mul i-C i e ia Decision Analysis in P edic i e Modeling con ain mo e ea u es, allowing mo e independen a iables o be used. Issues in model selec ion a e mainly ela ed o selec ing a model ha will i da a well, keep a complexi y le el low, and p o ide a easonable, accu a e p edic ion. Fo each p edic ion ask, he ollowing is o be e alua ed [3]: •Complexi y le el – The exis ing me hods o assessing model quali y a e o en based on assump ions o andomness o a iables and may, he e o e, be sensi i e o ex eme alues. On he con a y, some me hods make ew assump ions abou andomness, bu hei inhe en gene ali y may al e hei esul s. This means ha while assump ion-based me hods can be e y accu a e unde ideal condi ions, hey may ail wi h non-ideal da a, whe eas assump ion-ligh me hods a e mo e lexible bu can some imes o e less p ecise insigh s. •Class o models o a speci ic sys em – I is also impo an o use a good se o p edic i e models and selec he mos ele an me hod o model cons uc ion. •E alua ion c i e ia – Selec ing he mos ele an e alua ion c i e ia o p edic ion models is c ucial. A model e alua ion c i e ion, based on i s dis ance o he heo e ical quan i y, assesses he pe o mance in p edic ing a model. Also, c i e ia can o en be in con lic . The p edic ion ask is challenging due o unknown ela ion- ships be ween he a iables in ol ed. A common app oach is o c ea e mul iple models ha ep esen hese ela ionships di e en ly. The ask hen becomes e alua ing and compa ing hese models o selec he bes one, whe e he ‘‘bes ’’ is ask- speci ic. Fou benchma k p oblem domains we e chosen o alida e he p oposed e alua ion me hods: e o es ima ion, p edic ing glucose le els, g oss domes ic p oduc (GDP) p edic ion, and s ock p ice p edic ion. These domains equi e accu a e p edic ions in bo h echnical and inancial ields and he e sa ili y and obus ness o he p oposed MCDA-MDL e alua ion amewo k is demons a ed in ou wo k. The i s p edic ion ask is in so wa e enginee ing, ocusing on p edic ing so wa e de elopmen e o s. This ask is c ucial o p ojec managemen because accu a e es ima es help o plan, budge , and alloca e esou ces. Simple and accu a e p edic i e models a e aluable as hey a e easie o p ojec manage s and s akeholde s o unde s and and use, ensu ing be e p ojec con ol and success. The second p edic ion ask is om he medical and heal h science, explici ly p edic ing glucose le els. Accu a e glucose le el p edic ions a e essen ial o managing diabe es, as hey help o moni o and main ain op imal glucose le els and p e en complica ions. This ask ep esen s a b oade challenge in medical esea ch, whe e accu a e p edic ions a e necessa y o e ec i e pa ien ca e and ea men planning. The hi d p edic ion expe imen is ela ed o he g oss domes ic p oduc p edic ion. Those p edic ions a e essen ial o company inancial planning and li e cos p edic ion. Knowing he g oss domes ic p oduc p edic ion is manda o y o many businesses and public adminis a ion. The ou h domain is ela ed o s ock ma ke p edic ion. This was included as being a ypical ep esen a ion o he ime se ies. Also, his is an impo an ask in economic and inancial analysis. By choosing hese ou signi ican p oblem domains, he s udy aims o show he e sa ili y and obus ness o he p oposed MCDA-MDL e alua ion amewo k o e da a science applica ions. The so wa e enginee ing, medical and inancial analysis/economic asks highligh he need o p ac ical and easy- o-use models in each ield. B. OBJECTIVES OF THE WORK Model selec ion simpli ies he p ocess by educing he numbe o possible models o a limi ed se . Howe e , i emains a challenging p oblem because i equi es de ining wha makes a good model and how o measu e i s quali y. These de ini ions should align wi h he p ima y goal o he s udy. Al hough his seems s aigh o wa d, in p ac ice, he me hods used o c ea e and e alua e models o en need o align be e wi h he s udy’s objec i es. To add ess he challenges o model selec ion in he p esence o con lic ing e o measu emen s, we p opose he in eg a ion o Minimum Desc ip ion Leng h (MDL) and Mul i-C i e ia Decision Analysis (MCDA). The MDL p inciple [4],[5] helps balance he model’s complexi y wi h i s abili y o i he da a. By minimizing he minimal desc ip ion leng h, MDL p o ides a obus way o p e en o e i ing and selec models ha gene alize well o new da a. In eg a ing he MCDA app oach is essen ial when e o measu emen s con lic . MCDA helps o inco po a e an e o sco e, which uses mo e han one e o c i e ion. MCDA sco e can be unde s ood as a goodness-o - i pa o MDL. By combining MDL and MCDA, we can enhance he model selec ion p ocess, ensu ing ha he selec ed model i s he da a well and e ec i ely mee s he ask’s objec i es. This in eg a ed app oach p o ides a s uc u ed amewo k o na iga e he complexi ies o model e alua ion and selec ion. C. RESEARCH QUESTIONS Fo his wo k, he ollowing esea ch ques ions ha e been se : •RQ1: How does he Minimum Desc ip ion Leng h (MDL) and Mul i-C i e ia Decision Analysis (MCDA) in eg a ion a ec p edic i e model selec ion? •RQ2: Wha ad an ages does he MDL-MCDA ha e compa ed o he MDL-RSS1in p edic i e model selec- ion? D. MAIN CONTRIBUTIONS OF THE WORK In his pape , we add ess c i ical challenges in p edic i e modeling and model selec ion by in oducing a no el me hodological amewo k ha in eg a es he s eng hs o he Minimum Desc ip ion Leng h (MDL) p inciple wi h Mul i-C i e ia Decision Analysis (MCDA). While adi ional 1MDL-RSS is he common MDL ha ing he esidual sum o squa es a he goodness-o - i c i e ion. VOLUME 13, 2025 19389 P. Silha y e al.: MDL and Mul i-C i e ia Decision Analysis in P edic i e Modeling MDL elies on Residual Sum o Squa es (RSS) as a measu e o goodness-o - i , his app oach o en alls sho in scena ios whe e mul iple, con lic ing e o c i e ia mus be balanced, pa icula ly in complex, eal-wo ld da ase s. By in eg a ing MDL wi h MCDA, we ex end he scope o model e alua ion beyond a single e o me ic, allowing o a mo e obus and nuanced assessmen ha accoun s o mul iple e alua ion c i e ia. This inno a ion no only enhances he eliabili y o model selec ion bu also add esses c i ical gaps in adi ional MDL app oaches. Below, we de ail he speci ic con ibu ions ha unde sco e he no el y and p ac ical alue o ou p oposed amewo k: •MDL and MCDA In eg a ion: In oduces a no el in e- g a ion o Minimum Desc ip ion Leng h (MDL) wi h Mul i-C i e ia Decision Analysis (MCDA) o imp o e p edic i e model selec ion by esol ing con lic ing e o measu emen s. •Compa a i e Analysis: E alua es MDL-AHP s. MDL-RSS in eg a ions o be e handling o complex da ase s, showing p ac ical bene i s. •Domain Applica ions: Assesses he me hodology ac oss mul iple domains. •Benchma k Da ase s: Valida es he amewo k using da ase s om so wa e enginee ing, medical, and inance domains. •Enhanced Selec ion F amewo k: Demons a es ha MDL-MCDA imp o es model selec ion by balancing complexi y and accu acy. •Impac o MDL-MCDA: MDL-MCDA ou pe o ms adi ional MDL o MCDA in model selec ion ac oss da ase s. •MDL-MCDA s. MDL-RSS: Shows MDL-MCDA selec s models wi h be e gene aliza ion compa ed o MDL-RSS. These con ibu ions ad ance he unde s anding and imple- men a ion o model selec ion me hodologies, o e ing o apply MDL wi h MCDA o a ious domains equi ing p ecise and eliable p edic i e modelling. E. PAPER ORGANIZATION The es o he pape is o ganised as ollows. Sec ion II p o ides a comp ehensi e o e iew o exis ing esea ch and me hodologies ela ed o model selec ion, MDL, MCDA, AHP, and RSS. Sec ion III de ails he me hods used in ou wo k, including da a p epa a ion, model implemen a ion, e alua ion measu es, and he in eg a ion o MDL and MCDA. Sec ion IV p esen s he esul s o he expe imen s using a ious da ase s and p edic i e models. I includes a compa ison o he pe o mance o di e en models based on AHP, MDL wi h AHP, and MDL wi h RSS. Sec ion V discusses he implica ions o he esul s, he e ec i eness o he in eg a ed app oach, and i s applicabili y o di e en p oblem domains. Finally, Sec ion VI summa izes he s udy’s main indings, highligh s he con ibu ions, and sugges s di ec ions o u u e esea ch. II. RELATED WORK A wide ange o iable p edic ion models a e a ailable ac oss di e en indus ies, making i di icul o de e mine he op imal one, especially when aced wi h con lic ing e o measu es. The Minimal Desc ip ion Leng h (MDL) was in oduced o add ess his issue in [4]. MDL is an al e na i e o he Akaike In o ma ion C i e ion (AIC), which was in oduced as a ecognised me hod o au oma ic model selec ion [6]. While he Akaike In o ma ion C i e ion (AIC) is highly e icien in selec ing models wi hin he same class and compa ing non-nes ed models, such as linea and non-linea models, i canno au oma ically choose models om di e en p edic ion model classes, such as exponen ial smoo hing and au o eg essi e models. To add ess his limi a ion, he Bayesian in o ma ion c i e ion (BIC) om Schwa z was in oduced, which, in he same ein as AIC, e alua es he i o he da a wi h a complexi y penal y. Howe e , he BIC imposes a mo e subs an ial penal y o complexi y han he AIC. Ne e heless, his me hod s ill equi es u he de elopmen o assess models wi hin he same class. Villegas e al. [7] sugges employing suppo ec- o machines (SVM) o iden i y he mos sui able p edic ion model om a ange o al e na i es, gi en ha model a iables (such as he deg ee o accu acy and he i ed pa ame e s) may change o e ime. The esea che s disco e ed ha u ilising SVM leads o a g ea e o e all p edic i e accu acy. Ghobba and F iend [8] de ised a p edic i e e o o ecas ing echnique o assessing demand p edic ion models in he ai line manu ac u ing sec o based on hei ac o le els. They employed mean absolu e pe cen age e o (MAPE) as he c i e ion o e alua ion bu did no accoun o hyb id p edic ion models ha inco po a e pe sonal in o ma ion. Oh and Mo zuch [9] assessed eigh demand p edic ion models using six pe o mance measu es ha e alua e bias and o ecas e o , including MAPE, MAE, RMSE, AIC, and BIC. Thei s udy e ealed ha he choice o p edic ion model a ied based on he pe o mance measu es employed. Taylo and McSha y [10] e alua ed six dis inc p edic ion models o es ima e elec ici y demand ac oss en Eu opean coun ies. They used MAPE and MAE as e alua ion measu es and disco e ed ha he ankings gene a ed con lic ing ou comes, excep o he op-pe o ming model, which consis en ly anked i s . Pe opoulos e al. [1] and Han e al. [11] in es iga ed he use o subjec i e expe judgmen in p e- dic ion model selec ion, e ealing ha he chosen models ou pe o med hose selec ed h ough AIC based on e alua ion measu es such as MAE, MAPE and MASE. Fu he mo e, i has been shown ha collec i e judgmen is supe io o a single decision and s a is ical selec ion me hods. Da ydenko and Fildes [6], o ins ance, explo ed he e ec i eness o MAPE and median a e age pe cen age e o (MdAPE) in assessing judgmen al adjus men s o s a is ical p edic ion. They concluded ha elying solely on MAPE o de e mine a model’s pe o mance is insu icien due o inconsis en esul s be ween MAPE and o he e o measu es. The s udy sugges s 19390 VOLUME 13, 2025 P. Silha y e al.: MDL and Mul i-C i e ia Decision Analysis in P edic i e Modeling ha u u e esea ch should de elop an app oach o selec ing he op imal model when e alua ing mul iple e o measu es, pa icula ly in he ace o con lic ing esul s. Mul iple- c i e ia decision analysis (MCDA) is a widely-used app oach o add essing complex p oblems in ol ing mul iple, o en con lic ing, objec i es [12]. Selec ing p edic i e models using MCDA can be pa icula ly use ul when di e en e o measu es, such as mean squa ed e o and mean absolu e e o , p o ide con lic ing guidance on he op imal model. Compa ing AHP and TOPSIS, wo p ominen MCDA me hods, we conside how hey can be applied in his model selec ion con ex . The Analy ic Hie a chy P ocess (AHP) [13] is a s uc u ed echnique o o ganizing and analyzing complex decisions. AHP in ol es decomposing a p oblem in o a hie a chy o goals, objec i es, and al e na i es and hen using pai wise compa isons o de i e p io i ies o he al e na i es. In he model selec ion domain, AHP could be used o es ablish a hie a chy wi h he o e all goal o minimizing p edic ion e o , wi h sub-objec i es o minimizing MSE, MAE, and po en ially o he ele an measu es. Each candida e model would hen be e alua ed agains hese c i e ia, wi h AHP p o iding a composi e sco e o guide he inal model selec ion [14],[15],[16],[17]. The pe cep on, a undamen al building block o neu al ne wo ks, has also been explo ed o e o es ima ion. A s udy by [18] demons a ed he po en ial o pe cep on-based models o cap u e non-linea ela ionships, cha ac e is ic o e o es ima ion p oblems and he po en ial o imp o e adi ional es ima ion echniques. While neu al ne wo k and deep lea ning models ha e shown p omising esul s, hei pe o mance is hea ily dependen on he quali y and cha ac e is ics o he inpu da a. P ope ea u e enginee ing, da a p ep ocessing, and hype pa ame e uning a e c ucial o achie ing eliable and accu a e e o es ima ion using hese ad anced echniques. Hype pa ame e op imiza ion can signi ican ly impac he model’s p edic i e capabili ies and gene alisa ion, such as he numbe o hidden laye s, neu ons, and he lea ning a e. Neu al ne wo ks and deep lea ning models p omise o imp o e so wa e e o es ima ion. Thei abili y o model complex, non-linea ela ionships in da a makes hem well-sui ed o his ask. Con inued ad ancemen s in neu al ne wo k a chi ec u es, aining algo i hms, and hyb id modelling app oaches will likely enhance hei accu acy and applicabili y in so wa e enginee ing. Va ious me hods o e alua ing p edic ion models in di e en domains, p ima ily using e o measu es and in o ma ion c i e ia like AIC and BIC. Howe e , employing AIC and BIC o assess models es ic s he compa ison o models wi hin he same class. Mo eo e , u he esea ch is needed o de e mine an app op ia e app oach o e alua ing mul iclass demand p edic ion models based on se e al in e dependen e o measu es and o selec he bes model based on he simul aneous use o mul iple e o meas- u es [19]. III. METHODOLOGY A. RESEARCH DESIGN This wo k e alua es he in eg a ion o minimum desc ip ion leng h (MDL) and mul i-c i e ia decision analysis (e.g. AHP) in selec ing p edic i e models. To achie e his, a se ies o s eps we e aken du ing expe imen al wo k. This in ol es da a p epa a ion, model implemen a ion, e alua ion, and compa ison. We employ ou da ase s co e ing so wa e enginee ing, medical and inancial p oblem domains. These co e se e al domains and sizes and a e also a combina ion o na u al and syn he ic samples. The conside ed model classes include mul iple linea eg ession (MLR), Ridge eg ession, Lasso eg ession, Elas ic ne eg ession, quad a ic and cubic eg ession (i.e. polynomial eg ession wi h deg ees 2 and 3), and a eed- o wa d neu al ne wo k (FF-NN) wi h a ious con igu a ions, which will be speci ied. The pe o mance o hese models will be e alua ed using c i e ia: Mean Absolu e E o (MAE), Mean Absolu e Pe - cen age E o (MAPE), Roo Mean Squa ed E o (RMSE), Adjus ed R-squa ed (adjR2), P edic ion a 25% (P ed(0.25)), and Weigh ed Quan ile Loss (WQL). Addi ionally, we will apply he Analy ic Hie a chy P ocess (AHP) and Minimum Desc ip ion Leng h (MDL) p inciples o aid in model selec ion. The p ocedu e will in ol e se e al s eps (Figu e 1). Fi s , a da ase is p epa ed and p ocessed by handling missing alues, no malising da a, and spli ing hem in o aining and es se s. Then, he p edic i e models a e ained using he aining da a. A e aining, hey e alua e each model using he speci ied c i e ia on es ing da a. The MDL p inciple will help quan i y he complexi y and goodness-o - i o each model, ocusing on he o al desc ip ion leng h, which includes he model s uc u e, pa ame e s, and da a encoding. MCDA, speci ically AHP, will e alua e models based on mul iple c i e ia. This in ol es making pai wise compa isons o de e mine he ela i e impo ance o each c i e ion in he inal sco e o model selec ion. Finally, he models selec ed using adi ional me hods will be compa ed wi h hose chosen h ough he in eg a ed MDL and MCDA app oach. This compa ison will de e mine i he combined app oach imp o es p edic i e accu acy and model simplici y. The expec ed ou comes o his wo k include iden i ying he impac o in eg a ing MCDA me hods (e.g. AHP) o MDL and compa ing o selec ion using AHP only, o MCDA wi h RSS. Mo eo e , we ob ain insigh s in o how MDL and MCDA can imp o e model selec ion and unde s and he impac o di e en a iables on model pe o mance. B. EVALUATING MEASURES The e alua ion and compa ison o models in ol e a de ailed analysis o a ious models’ pe o mance, ocusing on hei abili y o p edic o explain he dependen a iable accu a ely. Le us conside a sample D= {(xi,yi),i=1, . . . , n},o VOLUME 13, 2025 19391 P. Silha y e al.: MDL and Mul i-C i e ia Decision Analysis in P edic i e Modeling FIGURE 1. Model design lowcha . a iable alues yiand ˆyi, whe e yi ep esen s he ac ual alue and ˆyiis he p edic ed alue. Mean Absolu e Pe cen age E o (MAPE) is a measu e o p edic ion accu acy o a o ecas ing me hod, exp essing he accu acy as a pe cen age. I is de ined as: MAPE =1 n n X i=1 yi− ˆyi yi ×100.(1) Mean Absolu e E o (MAE) and Roo Mean Squa ed E o (RMSE) a e common e alua ion measu es assessing he a e age magni ude o p edic ion e o s. MAE is de ined as: MAE =1 n n X i=1yi− ˆyi(2) and RMSE is de ined as: RMSE = u u 1 n n X i=1 (yi− ˆyi)2.(3) The Median Absolu e Pe cen age E o (MdMAPE) p o- ides a obus measu e by ocusing on he median o he pe cen age e o s, de ined as: MdMAPE =median  yi− ˆyi yi ×100(4) The Adjus ed Coe icien o De e mina ion (Adjus ed R2) is an enhancemen o he egula R2me ic ha adjus s o he numbe o p edic o s in he model. I p o ides a mo e accu a e measu e o goodness o i han R2by conside ing model complexi y. Adjus ed R2is de ined as: Adjus ed R2=1−(1 −R2)(n−1) n−k−1(5) whe e nis he numbe o obse a ions, kis he numbe o p edic o s, and R2is he coe icien o de e mina ion on se D. P ed(0.25) e alua es he p opo ion o p edic ions ha all below a speci ied e o h eshold, such as 25%. I is use ul o assessing he o e all model’s p edic i e accu acy wi hin an accep able e o ange. P ed(0.25) is calcula ed as P ed(0.25) =1 n n X i=1 I yi− ˆyi yi <0.25(6) whe e Iis an indica o unc ion ha equals 1 i he condi ion is ue and 0 o he wise. The Weigh ed Quan ile Loss (WQL) [20] measu es how well a p edic i e model pe o ms ac oss di e en quan iles o he a ge a iable’s dis ibu ion. I is pa icula ly use ul in scena ios whe e i is impo an o unde s and he model’s pe o mance ac oss a ious da a dis ibu ion segmen s. The WQL is gi en by: wQL(τ)=PN i=1Lτ(yi,ˆyi(τ)) PN i=1|yi|(7) whe e τis he quan ile le el (e.g., qua ils), yiis he obse ed alue a he i- h da a poin , and ˆyi(τ) is he p edic ed quan ile alue a he i- h da a poin o he quan ile le el τ. The quan ile loss unc ion, Lτ(yi,ˆyi(τ)), is de ined as Lτ(yi,ˆyi(τ)) =(τ−1{yi<ˆyi(τ)})(yi− ˆyi(τ)), whe e 1{yi< ˆyi(τ)}is an indica o unc ion ha equals 1 i yi<ˆyi(τ) and 0 o he wise. 1) DISCUSSION ON EVALUATION MEASURES Model pe o mance is assessed using a ious c i e ia: Mean Absolu e Pe cen age E o (MAPE), Mean Absolu e E o (MAE), Roo Mean Squa ed E o (RMSE), Median Absolu e Pe cen age E o (MdMAPE), Adjus ed R2, and P edic ions Le el (P ed(0.25)). Each c i e ion o e s a dis inc pe spec i e on model e alua ion, cap u ing di e en ace s o accu acy, obus ness, o complexi y. Howe e , hese measu es may occasionally p oduce con lic ing esul s, necessi a ing ca e ul in e p e a ion. The selec ion o hese c i e ia e lec s hei abili y o balance accu acy, obus ness o ou lie s, and he ade-o be ween model i and complexi y. Accu acy- ocused measu es: •MAPE measu es pe cen age e o s, p o iding an in u- i i e iew o ela i e accu acy o s akeholde s. •MAE a e ages e o magni udes, o e ing a s aigh o - wa d o e all accu acy measu e wi hou ou lie bias. •RMSE highligh s la ge e o s, use ul o signi ican de ia ions bu sensi i e o ou lie s, po en ially con lic - ing wi h MAPE. 19392 VOLUME 13, 2025 P. Silha y e al.: MDL and Mul i-C i e ia Decision Analysis in P edic i e Modeling Robus ness agains ou lie s: •MdMAPE cap u es median pe cen age e o s, ensu ing obus ness o ou lie s and complemen ing RMSE and MAE. Model i and complexi y: •Adjus ed R2measu es a iance explained, accoun ing o p edic o s o balance i and complexi y. •P ed(0.25) measu es p edic ions wi hin 25% e o , p i- o i izing consis en accu acy o e complexi y me ics. Balancing hese c i e ia is c ucial o de eloping obus models. Measu es like MAPE, MAE, RMSE, MdMAPE, and P ed(0.25) ocus on p edic i e accu acy, while complexi y-o ien ed measu es like Adjus ed R2p o ide insigh s in o gene alizabili y. By e alua ing mul iple me ics, a comp ehensi e unde s anding o he model’s s eng hs and weaknesses eme ges. E alua ion measu es can be g ouped based on whe he hey should be minimized, maximized, o ze oed o op imal pe o mance: Minimiza ion C i e ia: •Mean Absolu e Pe cen age E o (MAPE): emphasizes ela i e p edic ion e o s. •Mean Absolu e E o (MAE): cap u es a e age e o magni udes. •Roo Mean Squa ed E o (RMSE): penalizes la ge e o s, highligh ing ex eme de ia ions. •Median Absolu e Pe cen age E o (MdMAPE): o e s obus ness o ou lie s. Maximiza ion C i e ia: •Adjus ed Coe icien o De e mina ion (R2): balances a iance explana ion and model complexi y. •P opo ion o P edic ions Below 25% E o (P ed(0.25)): emphasizes p ac ical p edic i e accu acy. •Weigh ed Quan ile Loss: ensu es balanced pe o mance ac oss quan iles. This di e se se o e alua ion c i e ia ensu es he model is bo h accu a e and gene alizable, mee ing p ac ical needs while a oiding o e i ing o o e emphasis on speci ic e o ypes. C. MINIMUM DESCRIPTION LENGTH The Minimum Desc ip ion Leng h (MDL) p inciple is a o mal me hod o induc i e in e ence ha balances he model complexi y and goodness o i . This p inciple is oo ed in in o ma ion heo y and aims o a oid o e i ing by penalising model complexi y. MDL was in oduced by Rissanen [4] and u he de eloped by Rissanen e al. in e.g. [21],[22],[23], and by G ünwald and Roos in [5] and [24]. MDL is based on he idea ha he bes model o a gi en se o da a is he one ha allows o he sho es o e all desc ip ion o he da a and he model i sel . The o al desc ip ion leng h is he sum o he da a and model encoding leng hs. Ma hema ically, he o al desc ip ion leng h L(D,M) can be exp essed as: L(D,M)=L(M)+L(D|M) (8) whe e L(M) is he leng h o he desc ip ion o he model M and L(D|M) is he leng h o he desc ip ion o he da a Dgi en he model M. MDL p e e s models ha balance simplici y (sho model desc ip ion) and accu acy (sho da a desc ip ion gi en he model) in he sense ha he bes model Dis minimizing Eq. (8). This app oach penalizes mo e complex models unless hey signi ican ly imp o e he da a i . MDL can b ing ad an ages whe e o e i ing is a conce n and model in e p e abili y and simplici y a e alued. MDL helps selec models ha gene alise well o new da a by penalizing model complexi y. MDL ocuses on he o al leng h o encoding bo h he model and he da a, ensu ing ha he model chosen is he one ha bes comp esses he da a. This means MDL inhe en ly balances model i and complexi y by minimizing he in o ma ion equi ed o desc ibe he model and he da a i explains. Unlike AIC and BIC, which a e de i ed om s a is ical conside a ions, MDL di ec ly add esses he issue o o e i ing by penalizing unnecessa ily complex models, hus o en leading o models ha gene alize be e o new da a. This makes MDL a obus c i e ion o selec ing models ha a e no only accu a e bu also pa simonious, enhancing p edic i e pe o mance and in e p e abili y. 1) TWO-PART MDL CODES The minimal desc ip ion leng h as de ined by Eq. (8) is in he li e a u e called a wo-pa MDL. We poin in he beginning, since MDL is a p inciple, he e can be a ious encodings o he models om a class o models and hus he e can be a ious MDL unc ions co esponding o he gene al scheme om Eq. (8). We will now come o a mo e o mal explana ion o he MDL p inciple and i s encodings. Fi s we b ie ly explain he o ginal heo y om Rissanen [4],[22] and his ollowe s [5], [24] as i was de eloped o he case ha a condi ional p obabili y dis ibu ion p(y|x) is known. Secondly we explain MDL when we only know abou he model, om which he da a a e gene a ed, ha i is a membe o a class o unc ional models [25]. 2) ENCODING OF MODELS WITH KNOWN PROBABILITY OF THE DATA GENERATION PROCESS We de ine a model o he p edic ion p oblem as a condi ional p obabili y dis ibu ion p(y|x) o e and inpu space X, i. e. in o he wo ds, Py∈Yp(y|x)=1 (whe e he ou pu space Y can be heo e ically also an in ini e). A model class is a se o models depending on a pa ame e ec o θ θ θ, i.e. M= {pθ θ θ,θ θ θ∈ 2 2 2}. Usually, 2 2 2is a subse o a mul i a ia e Euclidean space. Shannon in [26] p o ed he ollowing undamen al s a emen in in o ma ion heo y, known unde he name Shannon- Hu man code. I a sende and ecei e ag eed in ad ance on a model pand bo h know he inpu xi,i=1, . . . , n hen he e exis s code o ansmi he alues yi,i=1, . . . , n losslessly wi h codeleng h (up o a mos one bi on he whole VOLUME 13, 2025 19393 P. Silha y e al.: MDL and Mul i-C i e ia Decision Analysis in P edic i e Modeling sequence) Lp(y|x)= − n X i=1 log2p(yi|xi) (9) whe e y,xis a sho ened no a ion o se y1, . . . , yn, x1, . . . , xn, espec i ely (which a e om Y,X espec i ely). The one addi ional bi in he Shannon-Hu man code is p esen only once o he whole da a se [27] and wi h la ge da a se s is negligible. Thus i will be omi ed om he encodings. We do no need o know he p ac ical implemen a ion o comp ession algo i hms bu we conside only he heo e ical bi leng h o hei associa ed encodings. We wan o measu e he amoun o in o ma ion con ained in he da a, and how i is ep esen ed by he model. So we will di ec ly wo k wi h codeleng h unc ions. P obabili y dis ibu ion unc ion pcan be unde s ood as he da a gene a ing p ocess and in gene al i is no known bu can be app oxima ed om he da a. To quan i y he complexi y o he compu a ional models o p edic ion (and in gene al o a supe ised lea ning p ob- lem) can be done e.g. by pa ame e coun ing. An in o ma ion- heo e ic way o use he Occam azo p inciple in e ms o he simples model wi h a good gene aliza ion is he minimum desc ip ion leng h (MDL), in oduced by Rissanen [4] and u he de eloped by Rissanen, Ba on, Yu in e.g. [21],[22], [23], and by G ünwald and Roos in [5],[24]. Encodings in which he pa ame e s o a model a e a i s ansmi ed o he ecei e and hen he da a using hese pa ame e s a e encoded, ha e been called wo-pa codes and in oduced by G ünwald [5]. Le Lpa am(θ θ θ) be any encoding scheme o pa ame e s θ θ θ∈ 2 2 2and le θ θ θ∗be any pa ame e . The co esponding wo-pa codeleng h is Lθ θ θ∗(y|x)=Lpa am(θ θ θ∗)+Lpθ θ θ∗(y|x) =Lpa am(θ θ θ∗)− n X i=1 log2pθ θ θ∗(yi|xi).(10) Pn i=1log2pθ θ θ∗(yi|xi) is called he goodness-o - i . The objec i e is o ind θ θ θ∗a he minimum o (10) o e all pa ame e iza ions. 3) ENCODING OF MODELS WITH KNOWN FUNCTIONAL CLASS OF THE DATA GENERATING PROCESS When pis known, i is clea ha he minimum o (10) is equi alen o he maximum likelihood es ima e (MLE). Howe e , wha makes he MDL p inciple so gene ic is ha i can be gene alized o he unc ional cases, i.e. ins ead o p p obabili y, as a gene al unc ion can be conside ed abou which is only known o be a membe o a class o candida e models, see e.g. [25]. I means ha abou he model, om which he da a a e gene a ed, is only known o be a membe l(.|θ θ θl) o a class o models M= { l(.|θ θ θl),θ θ θl∈2 2 2l, θlj ∼πlj(θlj), l=1, . . . , m,j=1, . . . , kl}(11) whe e mis he numbe o models in M,θ θ θl=(θl1, . . . , θlkl) is a kl-dimensional pa ame e ec o associa ed wi h land 2 2 2lis a pa ame e space o θ θ θl.πlj(θlj) is in oduced me ely o simpli y he encoding p ocess as an a i icial de ice o minimize he desc ip ion leng h. I is assumed ha e e y lis known excep o θ θ θl, and ha di e en lmay ha e di e en numbe o pa ame e s kl. Gi en a se o obse ed da a, he goal is o ind he ‘‘ ue’’ l om Mas well as o es ima e he pa ame e θ θ θiassocia ed wi h i . In his sense is (10) eplaced by L(y)=L(ˆ θ θ θl)+L(y|ˆ θ θ θl) (12) whe e L(ˆ θ θ θl), L(y|ˆ θ θ θl) a e code leng hs o encoding l(.|ˆ θ θ θl) and ‘‘ycondi ioned on l(.|ˆ θ θ θl)’’ espec i ely. L(y|ˆ θ θ θl) is called he goodness-o - i . Rissanen in [21] p o ed ha i ˆ θlj is an MLE compu ed om njda a poin s and i nis la ge, hen he p ecision o θlj can be e ec i ely encoded wi h 1 2log2njbi s. Rissanen de i ed a well-known o m when all he pa ame e s θlj a e o be es ima ed by using all da a poin s o size n. Fo subse selec ion in eg ession analysis, based on [22] i is MDL(kl)= − log2 l(y|ˆ θ θ θl)− kl X j=1 log2πlj(ˆ θlj)+kl 2log2n. (13) whe e klis he numbe o he eg esso s. Mo eo e , o n la ge, he choice o πlj(ˆ θlj) is ela i ely unimpo an , as he es ing summands in (13) a e domina ing [22]. So in p ac ice o high n, e m πlj(ˆ θlj) can be omi ed o MDL. 4) MDL WITH MULTI-OBJECTIVE GOODNESS-OF-FIT We p opose o eplace he goodness-o - i measu e, which a e commonly used in he MDL li e a u e, namely MAE,MAPE RMSE e c. by he mul iobjec i e goodness-o - i measu e. We will u ilize his idea o bo h p obabilis ic and unc ional ep esen a ion o models desc ibed abo e. I he p obabili y unc ion pis known: The second pa in Eq. (10) is a goodness-o - i o he model ppθ θ θ∗on da a se D. In his pape , we eplace in he alue Lpθ θ θ∗(y|x):= Pn i=1−log2pθ θ θ∗(yi|xi) by L8θ θ θ∗(y|x)= − log28θ θ θ∗(y|x) (14) whe e 8θ θ θ∗(y|x) is a mul i-objec i e c i e ion, and simila ly, as abo e, he i s pa Lpa am(θ θ θ∗) is he encoding o he selec ed model. The objec i e is o ind θ θ θ∗a he minimum o (14) o e all pa ame e iza ions. I only unc ion lis known, he goodness-o - i in (13) L(y|x):= − log2 l(y|ˆ θ θ θl)= n X i=1 −log2 l(yi|ˆ θ θ θl) (15) will be eplaced analogically by (14). 19394 VOLUME 13, 2025 P. Silha y e al.: MDL and Mul i-C i e ia Decision Analysis in P edic i e Modeling 5) CONSIDERED MACHINE LEARNING METHODS Ou wo k inco po a ed h ee ypes o machine lea ning me h- ods which we call model classes: mul iple linea eg ession (LR), mul iple linea eg ession wi h a penaliza ion e m (penLR), polynomial eg ession (polREG) up o deg ee 3, and eed- o wa d neu al ne wo ks (FF-NN). In his wo k, mul i- laye pe cep ons wi h wo and h ee hidden laye s a e used. In he ollowing subsec ions, we cons uc he MDL desc ip ions o he abo e models o he mul i-objec i e goodness-o - i . 6) MDL FOR LINEAR REGRESSION WITH MULTI-OBJECTIVE GOODNESS-OF-FIT Giu căneanu e al. in [28] cons uc ed se e al in o ma ion- heo e ic c i e ia o he a iable selec ion by mul iple linea eg ession assuming ha he noise ollows a Gaussian dis i- bu ion. We will use hei MDL de i ed om he s ochas ic complexi y [29]. Howe e , we eplace hei goodness-o - i wi h he mul i-objec i e goodness-o - i . We deno e k= |γ γ γ| he numbe o non-ze o alues in he bina y ec o γ γ γ, i. e. he numbe o eg esso s, and we can assume ha k>0. Le β β βγ γ γ∈Rk+1be he ec o o he unknown eg ession coe icien s wi hin he γ γ γ-subse . The ma ix Xγ γ γis gi en by he columns o X ha co espond o he γ γ γ-subse and he eg ession equa ion is y=Xγ γ γβ β βγ γ γ+εγ γ γ,(16) whe e y=(y1, . . . , yn) is he dependen a iable and εγ γ γa e Gaussian dis ibu ed wi h ze o-mean and unknown a iance τγ γ γ. Unde he assump ion ha ma ix Xγ γ γhas ull- ank, he maximum likelihood (ML) es ima es a e ˆ β β βγ γ γ=(X⊤ γ γ γXγ γ γ)−1X⊤ γ γ γy(17) and ˆτγ γ γ= ∥y−Xγ γ γˆ β β βγ γ γ∥2 2/n(18) whe e ˆτγ γ γis a goodness-o - i in Eq. (12) on eg esso om γ γ γ. Pape [28] e alua ed MDL o hese eg essions wi h independen a iables indexed by γ γ γas unc ions depending on ec o yand γ γ γas MDLLR(y, γ γ γ)=n−k 2log2ˆτγ γ γ+k 2log2 ∥Xγ γ γˆ β β βγ γ γ∥2 2 n −log20(n−k 2)−log20(k 2)+n 2log2(nπ) (19) whe e 0deno es he Eule in eg al o he second kind. In ou MDL, we p opose o eplace ˆτγ γ γin (19) by a mul i-objec i e c i e ion 8ˆ β β βγ γ γ(yi|xi), i.e. MDLLR 8(y,γ γ γ)=n−k 2log2( ˆ 8ˆ β β βγ γ γ(y|x) n) +k 2log2 ∥Xγ γ γˆ β β βγ γ γ∥2 2 n −log20(n−k 2)−log20(k 2)+n 2log2(nπ). (20) The objec i e is o ind β β β∗a he minimum o (20) o e all pa ame e iza ions ˆ β β βγ γ γand combina ions o γ γ γ. 7) MDL FOR PENALIZED LINEAR REGRESSION WITH MULTI-OBJECTIVE GOODNESS-OF-FIT We exp ess he encoding o he penaliza ion pa in eg ession as 1 2log2λ o ixed alues o MLE o ˆ β β β. We use he same encoding o he egula iza ion pa ame e o Lasso, Ridge and Elas ic penaliza ion. Howe e , we s ess ha MDL minimiza ion can be used only wi hin he eg ession class wi h he same penaliza ion ype and no wi hin all penal y ypes. Then MDLpenLR(y, γ γ γ) =n−k 2log2ˆτγ γ γ+1 2log2λ +k 2log2 ∥Xγ γ γˆ β β βγ γ γ∥2 2 n−log20(n−k 2)−log20(k 2) +n 2log2(nπ).(21) and MDLpenLR 8(y,γ γ γ) =n−k 2log2( ˆ 8ˆ β β βγ γ γ(y|x) n) +1 2log2λ+k 2log2 ∥Xγ γ γˆ β β βγ γ γ∥2 2 n −log20(n−k 2)−log20(k 2)+n 2log2(nπ).(22) I is well-known ha Lasso, Ridge, and Elas ic ne eg ession can ha e a ious alues o hei egula iza ion pa ame e s. 8) MDL FOR POLYNOMIAL REGRESSION WITH MULTI-OBJECTIVE GOODNESS-OF-FIT Conside now se Mas a se o polynomial eg ession models o deg ee ≤ ′. Deno e ˆ θ θ θ=(ˆa0, . . . , ˆa ) he se o coe icien s in he polynomial o deg ee . Since each ˆas,s= 0, . . . , is a eal numbe es ima ed om nda a poin s, each ˆas equi es 1 2log2nbi s o encode, he same he code o deg ee . Thus L(ˆ θ θ θ)=L(ˆa0, . . . , ˆa )= +1 2log2n+1 2log2n = +2 2log2n.(23) The desc ip ion o goodness-o - i is L(y|ˆ θ θ θ)=n 2log2(RSS n) (24) whe e RSS =Pn i=1(yi−(ˆa0+ ˆa1xi+ · · · + (ˆa x i))2.So he MDL o a polynomial o deg ee ≤ ′is MDLpolREG(y, )= +2 2log2n+n 2log2(RSS n) (25) VOLUME 13, 2025 19395 P. Silha y e al.: MDL and Mul i-C i e ia Decision Analysis in P edic i e Modeling In ou MDL, we p opose o eplace RSS nin (25) by a mul i- objec i e c i e ion 8ˆ θ θ θ(y|x), i.e. MDLpolREG 8(y, )= +2 2log2n+n 2log2( ˆ 8ˆ θ θ θ(y|x) n).(26) 9) MDL FOR A FEED-FORWARD NEURAL NETWORK WITH MULTI-OBJECTIVE GOODNESS-OF-FIT We gene ally conside a eed- o wa d ne wo k (FF-NN) wi h k≥1 hidden laye s, each ha ing hshidden uni s, s= 1, . . . , kand minpu and pou pu uni s. We p opose a simple encoding o such models whe e he model desc ip ion conside s he encodings based on he encoding o he s uc u e o he FF-NN and on he encoding o he lea ning pa . a: THE ENCODING OF THE STRUCTURE The s uc u e will be encoded as numbe o weigh s. In all hidden laye s and in he ou pu laye we conside ed ReLU ac i a ion unc ion, and his is ixed o all FF-NN models. b: THE ENCODING OF THE LEARNING PART We encode he lea ning pa o he FF-NN models so ha we encode he lea ning a e l o he Adam op imize , he ba ch size bs and he numbe o Adam hype pa ame e s Ahyp. Deno e he ec o o all pa ame e s de ining a FF-NN by θ θ θ. We do no encode he alues o θ θ θexplici ely, bu hey a e implici ely gi en by using Adam o hei compu a ion. Then MDLFF−NN (y,θ θ θ) =1 2log2(m×h1) +1 2log2(h1×h2)+ · · · + 1 2log2(hk×p) +1 2log2(l )+1 2log2(bs)+1 2log2(Ahyp) +n 2log2(RSSFF−NN n) (27) whe e RSSFF−NN is he esidual sum o squa es on he ou pu o FF −NN and he alues yand MDLFF−NN 8(y,θ θ θ) =1 2log2(m×h1)+1 2log2(h1×h2) + · · · + 1 2log2(hk×p) +1 2log2(l )+1 2log2(bs)+1 2log2(Ahyp) +n 2log2( ˆ 8ˆ θ θ θ(y|x) n).(28) whe e 8ˆ θ θ θ(y|x) is he mul i-objec i e c i e ion applied on he ou pu o FF −NN and he alues o y. D. ANALYTIC HIERARCHY PROCESS (AHP) The Analy ic Hie a chy P ocess (AHP) is a s uc u ed ech- nique (Mul iple C i e ia Decision Analysis) o o ganizing and analyzing complex decisions [30]. I in ol es b eaking down a p oblem in o a hie a chy o subp oblems ha can be mo e easily comp ehended and e alua ed. The main s eps in AHP a e [31]: •To decompose he decision p oblem in o a hie a chy. •To compa e he elemen s a each hie a chy le el o es ablish p io i ies. •To syn hesize hese compa isons o de e mine weigh s o each elemen . The consis ency a io (CR) [17] is calcula ed o ensu e consis ency in he compa isons: CR =CI RI (29) whe e CI is he consis ency index, and RI is he andom index. The consis ency index (CI) measu es he consis ency o he pai wise compa isons. I is calcula ed as ollows: CI =λmax −n n−1(30) whe e λmax is he la ges eigen alue o he compa ison ma ix, and nis he numbe o i ems being compa ed. The andom index (RI) is he a e age consis ency index o a andomly gene a ed pai wise compa ison ma ix. The alue o he RI depends on he numbe o i ems being compa ed and is used as a benchma k o assess he accep abili y o he calcula ed CI. To implemen AHP o model selec ion, we s a by de in- ing he c i e ia o model e alua ion. Fo example, c i e ia such as Mean Absolu e Pe cen age E o (MAPE), Mean Absolu e E o (MAE), Roo Mean Squa e E o (RMSE), Median Absolu e Pe cen age E o (MdMAPE), Adjus ed R-Squa ed (AdjR2), P edic ion a 0.25 (P ed(0.25)), and Weigh ed Quan ile Loss (wQL) can be used. Each model is e alua ed based on hese c i e ia h ough pai wise com- pa isons o de e mine hei ela i e impo ance. The AHP p ocess helps o syn hesise hese compa isons o assign a weigh o each c i e ion, ul ima ely selec ing he mos sui able p edic ion model based on a comp ehensi e, s uc u ed e alua ion. Each c i e ion has i s own weigh , which is se empi ically o expe imen ally. IV. EXPERIMENTS This chap e ou lines he expe imen s conduc ed using a ious eg ession model classes and wo neu al ne wo ks, namely mul i-laye ed pe cep on model classes o p edic ou comes in he men ioned da ase s. The expe imen s we e di ided in o wo main g oups: Reg ession models and eed- o wa d neu al ne wo ks. Each g oup u ilized speci ic models’ amilies, e alua ed based on hei pe o mance wi h he co esponding da ase s. All expe imen s we e implemen ed using Py hon and lib a ies pandas, numpy, sklea n, enso low and in e ools. A. DATASETS The da ase s employed in his esea ch a e widely acknowl- edged and a e publicly accessible. The his o ical da a u ilized 19396 VOLUME 13, 2025 P. Silha y e al.: MDL and Mul i-C i e ia Decision Analysis in P edic i e Modeling TABLE 10. Reg ession esul s o AHP and MDL on GDP. pe o ms bes wi h he lowes alue o 1066.64, indica ing i has he leas esidual sum o squa es. The Lasso and Elas icNe models using simila p edic o s also demons a e compe i i e pe o mance in his measu e. The Polynomial Reg ession o Deg ee 3 wi h he p edic o s o In e es Ra e, Indus ial P oduc ion, Money Supply, and Pe sonal Income is he bes model when conside ing he AHP Sco e. The Polynomial Reg ession o he Deg ee 2 model s ands ou in e ms o MDLAHP, and he Linea Reg ession model excels in MDLRSS. Thus, i all h ee c i e ia a e aken in o accoun , he Polynomial Reg ession o Deg ee 3 is he mos op imal model due o i s supe io pe o mance in AHP Sco e and compe i i e pe o mance in MDLAHP and MDLRSS. Table 11 p esen s AHP,MDLAHP and MDLRSS o STOCK Da ase . The models es ed include Elas icNe , Lasso, Linea Reg ession, Polynomial Reg ession (o deg ee 2 and 3), and Ridge Reg ession. Each model was e alua ed wi h di e en p edic o s, speci ically Open+High+Low and Da e. The Elas icNe model, when using Open+High+Low as p edic o s, achie ed an AHP Sco e o 0.52, an MDLAHP o −37733.67, and an MDLRSS o 67881.56. When using Da e as he p edic o , he AHP Sco e d opped o 0.04, while MDLAHP and MDLRSS we e −57585.61 and 98004.58, espec i ely. The Lasso model wi h Low as he p edic o had an AHP Sco e o 0.80, MDLAHP o −34271.85, and MDLRSS o 32672.06, whe eas wi h Da e as he p edic o , he AHP Sco e was 0.04, MDLAHP was −57584.76, and MDLRSS was 98005.44. Linea Reg ession using Open+High+Low p edic o s achie ed pe ec AHP Sco es o 1.00 wi h MDLAHP o −32535.83 and MDLRSS o 22562.16, while wi h Da e as he p edic o , he AHP Sco e was 0.04, MDLAHP was −57584.69, and MDLRSS was 98005.51. Polynomial Reg ession models (deg ees 2 and 3) wi h Open+High+Low p edic o s also achie ed pe ec AHP Sco es o 1.00, wi h MDLAHP and MDLRSS alues o −32444.69 and 22851.12 o deg ee 2, and −32230.31 and 22619.67 o deg ee 3. Including he Da e p edic o alongside Open+High+Low in Polynomial Reg ession (deg ee 3) yielded simila esul s. The Ridge Reg ession model wi h Open+High+Low p edic o s had an AHP Sco e o 0.93, MDLAHP o −33062.29, and MDLRSS o 25693.66. Using Da e as he sole p edic o esul ed in an AHP Sco e o 0.04, MDLAHP o −57584.69, and MDLRSS o 98005.51. B. FEED-FORWARD NEURAL NETWORKS-MULTI-LAYER PERCEPTRON Table 12 summa izes he pe o mance o FF-NN models on he UCP da ase , e alua ed h ough he AHP,MDLAHP, and MDLRSS. Two models, FF-NN I and FF-NN II, a e compa ed using di e en combina ions o p edic o s. The pe o mance o FF-NN models on he UCP da ase was compa ed ac oss a ious con igu a ions o p edic o s. The esul s a e summa ised in Table 12. FF-NN I and FF-NN II models we e e alua ed wi h di e en p edic o s. Fo FF-NN I, when using he p edic o s UAW, UUCW, and ECF, he model achie ed an AHP sco e o 0.71, wi h an MDLAHP o −31298.04 and an MDLRSS o 39384.06. Howe e , when using TCF and ECF as p edic o s, he AHP sco e o FF-NN I d opped o 0.60, wi h MDLAHP and MDLRSS alues o −31921.90 and 40464.73, espec i ely. In con as , FF-NN II wi h he p edic o s UAW, UUCW, TCF, and ECF achie ed he highes AHP sco e o 1.00, indica ing a pe ec pe o mance wi h an MDLAHP o −30093.69 and an MDLRSS o 35846.81. When using only TCF as he p edic o , FF-NN II had an AHP sco e o 0.62, and he MDLAHP and MDLRSS we e −31764.88 and 40195.76, espec i ely. Compa ing he models, FF-NN II consis en ly ou pe - o med FF-NN I ac oss all measu es and p edic o se s. This sugges s ha he addi ional complexi y and pa ame e s in FF-NN II p o ide a be e i o he UCP da ase . The combina ion o UAW, UUCW, TCF, and ECF yielded he bes esul s o FF-NN II, achie ing he highes AHP sco e and he lowes MDLRSS . This combina ion cap u es he ele an in o ma ion mo e e ec i ely han he o he es ed se s o p edic o s. The signi ican di e ence in pe o mance measu es be ween he wo models and hei p edic o combina ions Table 13 p esen s he pe o mance VOLUME 13, 2025 19403 P. Silha y e al.: MDL and Mul i-C i e ia Decision Analysis in P edic i e Modeling TABLE 11. AHP,MDLAHP and MDLRSS o eg ession models wi h s ock da ase . TABLE 12. AHP, MDLAHP ,MDLRSS o FF-NN models wi h UCP da ase . o FF-NN models using di e en p edic o s combina ions on he GLP da ase , e alua ed using he AHP sco e, MDLAHP, and MDLRSS. The models compa ed a e FF-NN I and FF- NN II.Fo FF-NN I, when using he p edic o s AGE, DBP, SBP, TE, SPO2, HR, SHI, and DN, he model achie ed an AHP sco e o 0.89, wi h an MDLAHP o −112604.29 and an MDLRSS o 44842.89. Howe e , when using only DN as he p edic o , he AHP sco e o FF-NN I d opped o 0.56, wi h MDLAHP and MDLRSS alues o −117876.50 and 50788.08, espec i ely. In con as , FF-NN II wi h he p edic o s AGE, DBP, SBP, TE, SPO2, HR, and SHI achie ed an AHP sco e o 0.99, indica ing nea -pe ec pe o mance wi h an MDLAHP o −111319.67 and an MDLRSS o 42684.87. FF-NN II had an AHP sco e o 0.50 when using only DN as he p edic o , and he MDLAHP and MDLRSS we e −119184.98 and 50616.86, espec i ely. Addi ionally, when using all p edic o s (AGE, DBP, SBP, TE, SPO2, HR, SHI, and DN), FF-NN II achie ed an AHP sco e o 0.99, wi h an MDLAHP o −111365.58 and an MDLRSS o 42619.76. Compa ing he models, FF-NN II consis en ly ou pe - o med FF-NN I ac oss all measu es and p edic o se s. This sugges s ha he addi ional complexi y and pa ame e s in FF-NN II p o ide a be e i o he GLP da ase . The com- bina ion o AGE, DBP, SBP, TE, SPO2, HR, and SHI yielded he bes esul s o FF-NN II, achie ing he highes AHP sco e and he lowes MDLRSS . This combina ion cap u es he ele an heal h- ela ed in o ma ion mo e e ec i ely han he o he es ed se s o p edic o s. The signi ican di e ence in pe o mance measu es be ween he wo models and hei p edic o combina ions highligh s he impo ance o selec ing app op ia e p edic o s. The p edic o s AGE, DBP, SBP, TE, SPO2, HR, and SHI combined p o ide a obus model capable o accu a ely p edic ing he desi ed heal h ou comes in he GLP da ase . These esul s demons a e he e icacy o using a mo e complex FF-NN model wi h a comp ehensi e se o p edic- o s o supe io pe o mance in he GLP da ase . The nex da ase GDP esuls a e summa ised in Table 14. The measu es e alua ed include he AHP Sco e, MDLAHP, and MDLRSS. The models we e assessed based on di e en se s o p edic o s. Fo FF-NN I, when using he p edic o s In e es Ra e, Unemploymen Ra e, Indus ial P oduc ion, Money Supply, and Pe sonal Income, he model achie ed an AHP sco e o −2.86, wi h an MDLAHP o 12.86 and an MDLRSS o 1255.61. Howe e , when using only In e es Ra e as he p edic o , he AHP sco e o FF-NN I emained a −2.86, wi h MDLAHP and MDLRSS alues o 12.86 and 1255.71, espec i ely. In con as , FF-NN II, wi h he p edic o s o Indus ial P oduc ion and Pe sonal Income, achie ed he highes AHP 19404 VOLUME 13, 2025 P. Silha y e al.: MDL and Mul i-C i e ia Decision Analysis in P edic i e Modeling TABLE 13. AHP, MDLAHP ,MDLRSS o FF-NN models wi h GLP da ase . TABLE 14. AHP, MDLAHP ,MDLRSS o FF-NN models wi h GDP da ase . TABLE 15. AHP, MDLAHP ,MDLRSS o FF-NN models wi h STOCK da ase . TABLE 16. Selec ed FF-NN models pe da ase and pe o mance compa ison. sco e o 1.00, indica ing pe ec pe o mance wi h an MDLAHP o −294.06 and an MDLRSS o 849.81. When using Consume Sen imen as he p edic o , FF-NN II had an AHP sco e o 0.17, and he MDLAHP and MDLRSS we e −415.77 and 1100.61, espec i ely. Compa ing he models, FF-NN II consis en ly ou pe - o med FF-NN I ac oss all measu es and p edic o se s. This sugges s ha he addi ional complexi y and pa ame e s in FF-NN II p o ide a be e i o he GDP da ase . The combina ion o Indus ial P oduc ion and Pe sonal Income yielded he bes esul s o FF-NN II, achie ing he highes AHP sco e and he lowes MDLRSS . This combina ion cap u es he ele an economic in o ma ion mo e e ec i ely han he o he es ed se s o p edic o s. The signi ican di e ence in pe o mance measu es be ween he wo models and hei p edic o combina ions highligh s he impo ance o selec ing app op ia e p edic o s. The Indus ial P oduc ion and Pe sonal Income p edic o s p o ide a obus model capable o accu a ely p edic ing he desi ed economic ou comes in he GDP da ase . These esul s demons a e he e icacy o using a mo e complex FF-NN model wi h a comp ehensi e se o p edic- o s o supe io pe o mance in he GDP da ase . Fo he las da ase (STOCK) he esuls a e in Table 15. Resuls again cons is ing o sco es o he AHP,MDLAHP, and MDLRSS Fo FF-NN I, when using he p edic o s High and Low, he model achie ed an AHP sco e o 0.83, wi h an MDLAHP o −74826.91 and an MDLRSS o −2659.22. Howe e , when using Open, High, and Low as p edic o s, he AHP sco e o FF-NN I d opped o 0.73, wi h MDLAHP and MDLRSS alues o −75867.69 and 7134.96, espec i ely. In con as , FF-NN II wi h he p edic o Low achie ed he highes AHP sco e o 1.00, indica ing pe ec pe o mance wi h an MDLAHP o −73376.47 and an MDLRSS o −11413.85. When using Da e and Open as p edic o s, FF-NN II had an AHP sco e o 0.83, and he MDLAHP and MDLRSS we e −74826.41 and −2457.73, espec i ely. Compa ing he models, FF-NN II consis en ly ou pe - o med FF-NN I ac oss all measu es and p edic o se s. This VOLUME 13, 2025 19405 P. Silha y e al.: MDL and Mul i-C i e ia Decision Analysis in P edic i e Modeling TABLE 17. Selec ed eg ession models pe da ase and pe o mance compa ison. sugges s ha he addi ional complexi y and pa ame e s in FF-NN II p o ide a be e i o he STOCK da ase . The p edic o Low yielded he bes esul s o FF-NN II, achie ing he highes AHP sco e and he lowes MDLRSS . This com- bina ion cap u es he ele an s ock p ice in o ma ion mo e e ec i ely han he o he es ed se s o p edic o s. The sig- ni ican di e ence in pe o mance measu es be ween he wo models and hei p edic o combina ions highligh s he impo ance o selec ing app op ia e p edic o s. The p edic o Low p o ides a obus model ha accu a ely p edic s he desi ed s ock p ice ou comes in he STOCK da ase . These esul s demons a e he e icacy o using a mo e complex FF-NN model wi h a comp ehensi e se o p edic- o s o supe io pe o mance in he STOCK da ase . C. DETAILED DISCUSSION The compa ison be ween eg ession models and Mul i-Laye Pe cep on (FF-NN) models e eals se e al signi ican insigh s ac oss di e en da ase s, as highligh ed in Tables 16 and 17. Fo he GDP da ase , he FF-NN II model achie ed an MDLAHP o −415.77, subs an ially ou pe o ming he Ridge eg ession model wi h In e es _Ra e+Consume _Sen imen p edic o s, which had an MDLAHP o −97.24. Simi- la ly, he FF-NN II model wi h ‘‘Indus ial_P oduc ion, Pe sonal_Income’’ showed supe io pe o mance wi h an MDLAHP o −294.06, compa ed o he Linea Reg ession model using In e es _Ra e+Indus ial_P oduc ion +Pe - sonal_Income, which had an MDLAHP o 10.55. In he GLP da ase , he FF-NN II model wi h DN as he sole p edic o demons a ed an MDLAHP o −119184.98, signi ican ly be e han he Ridge eg ession model wi h AGE, which had an MDLAHP o −57652.15. When mul iple heal h indica o s we e used as p edic o s, FF-NN II again ou - pe o med he Ridge eg ession model, achie ing MDLAHP alues o −111365.58 compa ed o −51369.50, espec i ely. Fo he STOCK da ase , FF-NN I, using Open, High, Low p edic o s, achie ed an MDLAHP o −75867.69, much be e han he Linea Reg ession’s −32535.83. The FF-NN II model using Low alone also pe o med excep ionally well wi h an MDLAHP o −73376.47, su passing he Elas icNe eg ession model using ‘‘Da e,’’ which had an MDLAHP o −57585.61. In he UCP da ase , FF-NN models consis en ly ou pe - o med eg ession models. FF-NN I wi h TCF, ECF achie ed an MDLAHP o −31921.90, be e han Elas icNe wi h he same p edic o s, which had an MDLAHP o −14485.98. The FF-NN II model wi h UAW, UUCW, TCF, ECF showed an MDLAHP o −30093.69, ou pe o ming Elas icNe and any o he eg ession models es ed. This analysis shows ha FF-NN models gene ally ou - pe o m eg ession models ac oss all da ase s in e ms o MDLAHP. This sugges s ha FF-NNs a e mo e capable o cap u ing complex ela ionships wi hin he da a, which sim- ple eg ession models migh miss. Howe e , while FF-NNs p o ide signi ican ad an ages due o hei non-linea i y and dep h, hey also come wi h highe compu a ional cos s and complexi y, which can be a disad an age ega ding in e p e abili y and ease o implemen a ion. Ac oss all da ase s and models, MDLAHP consis en ly p o ides be e pe o mance measu e han MDLRSS. Fo ins ance, in he GDP da ase , he MDLAHP o FF-NN II wi h −415.77, while MDLRSS is 1100.61, showing a con as . This end is obse ed ac oss all da ase s, unde sco ing ha MDL wi h AHP is a be e me hod o model selec ion. I be e cap u es he ade-o s and mul i-c i e ia e alua ions inhe en in complex model selec ion, which MDLRSS may o e simpli y. Howe e , one mus conside ha MDLAHP may also in ol e mo e subjec i e judgmen in de e mining weigh s o di e en c i e ia, which can in oduce bias. REFERENCES [1] F. Pe opoulos, N. Kou en zes, K. 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Salakhu dino , ‘‘D opou : A simple way o p e en neu al ne wo ks om o e i ing,’’ J. Mach. Lea n. Res., ol. 15, no. 1, pp. 1929–1958, Jan. 2014. [41] I. Good ellow, Y. Bengio, and A. Cou ille, Deep Lea ning. Camb idge, MA, USA: MIT P ess, 2016. [42] D. P. Kingma and J. Ba, ‘‘Adam: A me hod o s ochas ic op imiza ion,’’ 2014, a Xi :1412.6980. [43] L. P echel , ‘‘Ea ly s opping-bu when?’’ in Neu al Ne wo ks: T icks o he T ade. Cham, Swi ze land: Sp inge , 1998, pp. 55–69. PETR SILHAVY ecei ed he Ph.D. deg ee in engi- nee ing in o ma ics om he Facul y o Applied In o ma ics, Tomas Ba a Uni e si y in Zlín, Zlín, Czech Republic, in 2009. He is cu en ly an Associa e P o esso wi h he Facul y o Applied In o ma ics, Tomas Ba a Uni e si y in Zlín. He is a Senio Resea ch and an Associa e P o esso o sys em enginee ing and in o ma ics wi h a demons a ed his o y o wo king in esea ch and highe educa ion. He has expe ise as a CTO and a So wa e De elope in da abase p og amming, da abase design, da a managemen , and da a science. His esea ch in e es s include p edic ion and empi ical me hods o so wa e enginee ing. KATEŘINA HLAVÁČKOVÁ-SCHINDLER ecei- ed he M.Sc. deg ee (summa cum laude) in ma hema ics om Cha les Uni e si y, P ague, Czech Republic, he Ph.D. deg ee in compu e science om Czech Academy o Sciences, and he Habili a ion (P i a doz) deg ee om Uni e si y o Vienna, Vienna, Aus ia. She is cu en ly a Senio Scien is wi h he Da a Mining and Machine Lea ning Resea ch G oup, Uni e si y o Vienna. She has mo e han 80 publica ions mos ly on causal in e ence and causal disco e y, machine lea ning, and a i icial neu al ne wo ks. RADEK SILHAVY ecei ed he Ph.D. deg ee in enginee ing in o ma ics om he Facul y o Applied In o ma ics, Tomas Ba a Uni e si y in Zlín, Zlín, Czech Republic, in 2009. He is cu en ly an Associa e P o esso and a Senio Resea che wi h he Facul y o Applied In o - ma ics, Tomas Ba a Uni e si y in Zlín. He is an Associa e P o esso o sys em enginee ing and in o ma ics wi h a demons a ed his o y o wo king in esea ch, highe educa ion, p ojec managemen , and so wa e analysis. His esea ch in e es s include p edic i e analy ics o so wa e enginee ing, empi ical me hods in so wa e enginee - ing, o p edic ion models ocused on cos , size, and e o es ima ions in sys em/so wa e enginee ing. He is also in ol ed in academic publishing as he edi o -in-chie , an edi o , and a e iewe . VOLUME 13, 2025 19407