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. Nikolopoulos, and E. Siemsen,
‘‘Judgmen al selec ion o o ecas ing models,’’ J. Ope . Manage., ol. 60,
no. 1, pp. 34–46, May 2018.
[2] J. H. F iedman, ‘‘An o e iew o p edic i e lea ning and unc ion app ox-
ima ion,’’ in F om S a is ics o Neu al Ne wo ks: Theo y and Pa e n
Recogni ion Applica ions (NATO ASI Se ies), ol. 136, V. Che kassky,
J. H. F iedman, and H. Wechsle , Eds. Be lin, Ge many: Sp inge , 1994,
doi: 10.1007/978-3-642-79119-2_1.
[3] A. Boisbunon, ‘‘Model selec ion: A decision- heo e ic app oach,’’
M.S. hesis, Dep . IT Lab., In . P ocess. Sys ., Uni e si é de Rouen, Rouen,
F ance, 2013.
[4] J. Rissanen, ‘‘Modeling by sho es da a desc ip ion,’’ Au oma ica, ol. 14,
no. 5, pp. 465–471, Sep. 1978.
[5] P. D. G ünwald, The Minimum Desc ip ion Leng h P inciple. Camb idge,
MA, USA: MIT P ess, 2007.
[6] H. Akaike, ‘‘A new look a he s a is ical model iden i ica ion,’’ IEEE
T ans. Au om. Con ol, ol. AC-19, no. 6, pp. 716–723, Dec. 1974.
[7] M. A. Villegas, D. J. Ped egal, and J. R. T ape o, ‘‘A suppo ec o
machine o model selec ion in demand o ecas ing applica ions,’’ Compu .
Ind. Eng., ol. 121, pp. 1–7, Jul. 2018.
[8] A. A. Ghobba and C. H. F iend, ‘‘E alua ion o o ecas ing me hods o
in e mi en pa s demand in he ield o a ia ion: A p edic i e model,’’
Compu . Ope . Res., ol. 30, no. 14, pp. 2097–2114, Dec. 2003.
19406 VOLUME 13, 2025
P. Silha y e al.: MDL and Mul i-C i e ia Decision Analysis in P edic i e Modeling
[9] C.-O. Oh and B. J. Mo zuch, ‘‘E alua ing ime-se ies models o o ecas
he demand o ou ism in Singapo e: Compa ing wi hin-sample and
pos sample esul s,’’ J. T a el Res., ol. 43, no. 4, pp. 404–413, May 2005.
[10] J. W. Taylo and P. E. McSha y, ‘‘Sho - e m load o ecas ing me hods:
An e alua ion based on Eu opean da a,’’ IEEE T ans. Powe Sys ., ol. 22,
no. 4, pp. 2213–2219, No . 2007.
[11] W. Han, X. Wang, F. Pe opoulos, and J. Wang, ‘‘B ain imaging and
o ecas ing: Insigh s om judgmen al model selec ion,’’ Omega-In .
J. Manage. Sci., ol. 87, pp. 1–9, Sep. 2019.
[12] A. Ma dani, A. Jusoh, K. M. D. No , Z. Khali ah, N. Zakwan, and
A. Valipou , ‘‘Mul iple c i e ia decision-making echniques and hei
applica ions—A e iew o he li e a u e om 2000 o 2014,’’ Econ.
Res.-Ekonomska Is až, ol. 28, no. 1, pp. 516–571, Jan. 2015, doi:
10.1080/1331677x.2015.1075139.
[13] I. N. Du bach and T. J. S ewa , ‘‘Modeling unce ain y in mul i-c i e ia
decision analysis,’’ Eu . J. Ope . Res., ol. 223, no. 1, pp. 1–14, No . 2012,
doi: 10.1016/j.ejo .2012.04.038.
[14] Y. Badulescu, A.-P. Hame i, and N. Cheikh ouhou, ‘‘E alua ing demand
o ecas ing models using mul i-c i e ia decision-making app oach,’’ J. Ad .
Manage. Res., ol. 18, no. 5, pp. 661–683, Oc . 2021, doi: 10.1108/jam -
05-2020-0080.
[15] A. Bansal, B. Kuma , and R. Ga g, ‘‘Mul i-c i e ia decision making
app oach o he selec ion o so wa e e o es ima ion model,’’ Manage.
Sci. Le ., ol. 7, no. 6, pp. 285–296, 2017, doi: 10.5267/j.msl.2017.3.003.
[16] L. G. Va gas and J. J. Doughe y, ‘‘The analy ic hie a chy p ocess and
mul ic i e ion decision making,’’ Ame . J. Ma h. Manage. Sci., ol. 2, no. 1,
pp. 59–92, Jan. 1982, doi: 10.1080/01966324.1982.10737086.
[17] T. L. Saa y, ‘‘How o make a decision—The analy ic hie a chy p ocess,’’
Eu . J. Ope . Res., ol. 48, no. 1, pp. 9–26, 1990.
[18] F. Sa o, A. Pe ozziello, and M. Ha man, ‘‘Mul i-objec i e so wa e e o
es ima ion,’’ in P oc. 38 h In . Con . So w. Eng., 2016, pp. 619–630, doi:
10.1145/2884781.2884830.
[19] G. Schwa z, ‘‘Es ima ing he dimension o a model,’’ Ann. S a is ., ol. 6,
no. 2, pp. 461–464, Ma . 1978.
[20] J. D. Ang is and J.-S. Pischke, Mos ly Ha mless Econome ics: An
Empi icis ’s Companion. P ince on, NJ, USA: P ince on Uni . P ess, 2009.
[21] J. Rissanen, S ochas ic Complexi y in S a is ical Inqui y, ol. 15.
Singapo e: Wo ld Scien i ic, 1998.
[22] J. Rissanen, In o ma ion and Complexi y in S a is ical Modeling. Cham,
Swi ze land: Sp inge , 2007.
[23] A. Ba on, J. Rissanen, and B. Yu, ‘‘The minimum desc ip ion leng h
p inciple in coding and modeling,’’ IEEE T ans. In . Theo y, ol. 44, no. 6,
pp. 2743–2760, Jun. 1998.
[24] P. G ünwald and T. Roos, ‘‘Minimum desc ip ion leng h e isi ed,’’ In .
J. Ma h. Ind., ol. 11, no. 1, Dec. 2019, A . no. 1930001.
[25] T. C. M. Lee, ‘‘An in oduc ion o coding heo y and he wo-pa minimum
desc ip ion leng h p inciple,’’ In . S a . Re ., ol. 69, no. 2, pp. 169–183,
Aug. 2001.
[26] C. E. Shannon, ‘‘A ma hema ical heo y o communica ion,’’ Bell Sys .
Tech. J., ol. 27, no. 3, pp. 379–423, Jul. 1948.
[27] D. J. MacKay, In o ma ion Theo y, In e ence and Lea ning Algo i hms.
Camb idge, U.K.: Camb idge Uni . P ess, 2003.
[28] C. D. Giu căneanu, S. A. Raza i, and A. Liski, ‘‘Va iable selec ion in
linea eg ession: Se e al app oaches based on no malized maximum
likelihood,’’ Signal P ocess., ol. 91, no. 8, pp. 1671–1692, Aug. 2011.
[29] J. Rissanen, ‘‘MDL denoising,’’ IEEE T ans. In . Theo y, ol. 46, no. 7,
pp. 2537–2543, Jul. 2000.
[30] T. L. Saa y, ‘‘Wha is he analy ic hie a chy p ocess?’’ in Ma hema ical
Models o Decision Suppo . Cham, Swi ze land: Sp inge , 1988,
pp. 109–121.
[31] M. Ta ana, M. Sol ani a , and F. J. San os-A eaga, ‘‘Analy ical hie a chy
p ocess: Re olu ion and e olu ion,’’ Ann. Ope . Res., ol. 326, no. 2,
pp. 879–907, Jul. 2023.
[32] R. Silha y, P. Silha y, and Z. P okopo a, ‘‘Analysis and selec ion o
a eg ession model o he use case poin s me hod using a s epwise
app oach,’’ J. Sys . So w., ol. 125, pp. 1–14, Ma . 2017.
[33] Z. P okopo a, R. Silha y, and P. Silha y, ‘‘The e ec s o clus e ing o
so wa e size es ima ion o he use case poin s me hods,’’ in P oc. Compu .
Sci. On-Line Con ., in Ad ances in In elligen Sys ems and Compu ing,
ol. 575, Jan. 2017, pp. 479–490.
[34] P. Silha y, R. Silha y, and Z. P okopo a, ‘‘E alua ion o da a clus e ing o
s epwise linea eg ession on use case poin s es ima ion,’’ in P oc. Compu .
Sci. On-Line Con ., in Ad ances in In elligen Sys ems Compu ing,
ol. 575, 2017, pp. 491–496.
[35] R. Silha y, P. Silha y, and Z. P okopo a, ‘‘E alua ing subse selec ion
me hods o use case poin s es ima ion,’’ In . So w. Technol., ol. 97,
pp. 1–9, May 2018.
[36] D. Ja ale and S. Desai, ‘‘Da ase o people o hei blood glucose le el
wi h hei supe icial body ea u e eadings,’’ IEEE Da apo , Jul. 2021,
doi: 10.21227/c4pp-6347.
[37] Yahoo Finance. S ock Ma ke Da a. Accessed: Jul. 11, 2024. [Online].
A ailable: h ps:// inance.yahoo.com
[38] C. S. K. Dash, A. K. Behe a, S. Dehu i, and A. Ghosh, ‘‘An ou lie s
de ec ion and elimina ion amewo k in classi ica ion ask o da a mining,’’
Decis. Anal. J., ol. 6, Ma . 2023, A . no. 100164.
[39] S. Louis. (2024). G oss Domes ic P oduc . Accessed: Jul. 10, 2024.
[Online]. A ailable: h ps:// ed.s louis ed.o g/se ies/GDP
[40] N. S i as a a, G. E. Hin on, A. K izhe sky, I. Su ske e , and
R. 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