Metallurgical Copper Recovery Prediction Using Conditional Quantile Regression Based on a Copula Model

Ci a ion: He nández, H.; Díaz-Vie a,
M.A.; Albe di, E.; Oya bide-Zubillaga,
A.; Go i, A. Me allu gical Coppe
Reco e y P edic ion Using Condi ional
Quan ile Reg ession Based on a Copula
Model. Mine als 2024,14, 691.
h ps://doi.o g/10.3390/
min14070691
Academic Edi o s: K zysz o
Sk zypkowski, René Gómez,
Fha uwani Sengani, De ek B. Apel,
Faham Tahmasebinia and
Jianhang Chen
Recei ed: 20 May 2024
Re ised: 24 June 2024
Accep ed: 25 June 2024
Published: 1 July 2024
Copy igh : © 2024 by he au ho s.
Licensee MDPI, Basel, Swi ze land.
This a icle is an open access a icle
dis ibu ed unde he e ms and
condi ions o he C ea i e Commons
A ibu ion (CC BY) license (h ps://
c ea i ecommons.o g/licenses/by/
4.0/).
mine als
A icle
Me allu gical Coppe Reco e y P edic ion Using Condi ional
Quan ile Reg ession Based on a Copula Model
Hebe He nández 1,* , Ma ín Albe o Díaz-Vie a 2, Elisabe e Albe di 3, Ai o Oya bide-Zubillaga 4
and Ai o Go i 4
1Facul ad de Ingenie ía y A qui ec u a, Uni e sidad Cen al de Chile, San iago 8370178, Chile
2Ins i u o Mexicano del Pe óleo, Eje Cen al Láza o Cá denas No. 152, Ciudad de México 07730, Mexico;
[email p o ec ed]
3Depa men o Applied Ma hema ics, Uni e si y o he Basque Coun y UPV/EHU, 48013 Bilbao, Spain;
[email p o ec ed]
4Depa men o Mechanics, Design and O ganiza ion, Uni e si y o Deus o, 48007 Bilbao, Spain;
ai o [email p o ec ed] (A.O.-Z.); ai o [email p o ec ed] (A.G.)
*Co espondence: hebe [email p o ec ed]
Abs ac : This a icle p oposes a no el me hodology o es ima ing me allu gical coppe eco e y, a
c i ical ea u e in mining p ojec e alua ions. The complexi y o modeling his nonaddi i e a iable
using geos a is ical me hods due o low sampling densi y, s ong he e o opic ela ionships wi h
o he measu emen s, and nonlinea i y is highligh ed. As an al e na i e, a copula-based condi ional
quan ile eg ession me hod is p oposed, which does no ely on linea i y o addi i i y assump ions
and can i any s a is ical dis ibu ion. The p oposed me hodology was e alua ed using geochemical
log da a and me allu gical es ing om a simula ed block model o a po phy y coppe deposi . A
highly he e o opic sample was p epa ed o coppe eco e y, sampled a 10% wi h espec o o he
a iables. A copula-based nonpa ame ic dependence model was cons uc ed om he sample da a
using a ke nel smoo hing me hod, ollowed by he applica ion o a condi ional quan ile eg ession
o he es ima ion o coppe eco e y wi h chalcoci e con en as seconda y a iable, which u ned ou
o be he mos ela ed. The accu acy o he me hod was e alua ed using he emaining 90% o he
da a no included in he model. The new me hodology was compa ed o cok iging placed unde he
same condi ions, using pe o mance me ics RMSE, MAE, MAPE, and R
2
. The esul s show ha he
p oposed me hodology ep oduces he spa ial a iabili y o he seconda y a iable wi hou he need
o a a iog am model and imp o es all e alua ion me ics compa ed o he geos a is ical me hod.
Keywo ds: me allu gical coppe eco e y; copula model; condi ional quan ile eg ession; ke nel
smoo hing; colloca ed cok iging
1. In oduc ion
Me allu gical eco e y, in he con ex o mine al mining, e e s o he pe cen age o
aluable me al ex ac ed om he o e du ing he p ocessing o bene icia ion s age [
1
,
2
].
I is a c ucial me ic in he mining indus y as i indica es he e iciency o he ex ac ion
p ocess and ul ima ely a ec s he p o i abili y o he mining ope a ion.
F om a mine al p ocessing pe spec i e, con empo a y ea men s o coppe include
he lo a ion o sul ide o es, leaching o oxide o es, and a hyb id app oach ha in eg a es
lo a ion and magne ic sepa a ion o ce ain mixed o es [
3
]. Madeno a and Madani
(2021) [
4
] de ine me allu gical eco e y as a i al geome allu gical a iable o mine plan-
ning, ep esen ing a esponse o he p ocessing plan design and he geological cha ac e is-
ics o he o e.
The p ocess o ex ac ing aluable me als om o e ypically in ol es se e al s ages,
including c ushing, g inding, concen a ion, and e ining. Me allu gical eco e y measu es
Mine als 2024,14, 691. h ps://doi.o g/10.3390/min14070691 h ps://www.mdpi.com/jou nal/mine als
Mine als 2024,14, 691 2 o 21
he e ec i eness o hese p ocesses in sepa a ing and concen a ing he aluable me al om
he o e.
The calcula ion o me allu gical eco e y in ol es compa ing he amoun o me al
eco e ed om he o e o he o al amoun o me al p esen in he o e. This is o en exp essed
as a pe cen age, whe e a highe pe cen age indica es a mo e e icien ex ac ion p ocess.
Fac o s in luencing me allu gical eco e y include he mine alogy and composi ion o
he o e, he e iciency o he p ocessing equipmen and echniques used, and he expe ise
o he pe sonnel in ol ed in he ope a ion. Imp o ing me allu gical eco e y is a key ocus
a ea o mining companies seeking o op imize hei ope a ions and maximize he alue o
hei mine al esou ces.
To maximize p o i , i is c ucial o ha e a eliable es ima ion model o all a iables
in ol ed, bo h p ima y (geological p ope ies, mine al g ades, densi ies, con aminan s,
e c.) and esponse a iables (me allu gical eco e y, bond wo k index, g indabili y index,
p ocessing capaci y, milling pe o mance, e c.) [
5
]. The combina ion o a geological model
and me allu gical da a is cu en ly known as a geome allu gical model [6].
Me allu gical eco e y is a c i ical ea u e in he e alua ion and exploi a ion o mining
p ojec s, as i di ec ly in luences ne economic bene i . This a iable is exp essed as a
pe cen age and ep esen s he yield o mine al p ocessing, in he case o coppe sul ides,
by lo a ion along he mining alue chain [7].
Inco po a ing me allu gical eco e y in o mine planning poses a challenge o esou ce
modele s and mine planne s. In mos p ojec s, he lack o p ope collec ion and analysis
o geome allu gical da a leads o un eliable me allu gical esponse models [
8
]. Samples
wi h his in o ma ion a e o en sca ce, cos ly, and highly he e o opic compa ed o p ima y
a iables [
9
,
10
]. Howe e , hese p ima y a iables a e known o be use ul indica o s o
p edic ing me allu gical esponses [11,12].
In he con ex o coppe me allu gical eco e y, a he e o opic sample e e s o a sample
aken om a loca ion wi hin a mine al deposi ha is geologically dis inc om he p ima y
o e body being a ge ed o ex ac ion.
Fo example, in a coppe mining ope a ion, he p ima y o e body may consis o a
speci ic geological o ma ion o ein whe e coppe mine als a e concen a ed. A he e o opic
sample, in his case, could be aken om a nea by a ea whe e coppe mine aliza ion
occu s bu in a di e en geological se ing. This could include samples om adjacen ock
o ma ions, seconda y eins, o a eas wi h di e en mine alogical cha ac e is ics.
Analyzing he e o opic samples is impo an in coppe me allu gical eco e y because
hey can p o ide insigh s in o he a iabili y o coppe mine aliza ion wi hin a mining a ea.
Unde s anding he dis ibu ion and cha ac e is ics o coppe mine aliza ion in he e o opic
samples can help op imize mining p ocesses, imp o e eco e y a es, and in o m mine
planning and de elopmen s a egies.
Modeling me allu gical eco e y is usually ca ied ou using geos a is ical echniques
and, mo e ecen ly, by machine lea ning me hods. Machine lea ning me hods seek o p e-
dic geome allu gical esponse a iables using assay and mine alogy da a [
13
–
17
]; howe e ,
hey equi e a la ge numbe o a iables and a conside able amoun o da a o consolida e
a obus model [
18
]. On he o he hand, geos a is ics equi es ini ially de ining domains
based on geome allu gical a ibu es [
19
] and hen es ima ing indi ec ly, ha is, by seeking
ma hema ical a angemen s o ced o mee he equi ed assump ions. Techniques like
k iging a e no usually ecommended, as hey can gene a e biased esul s due o he nonad-
di i e na u e o me allu gical eco e y [
20
]. I has been obse ed ha he weigh ed a e age
o wo sample alues is no a good es ima o o he co esponding alue in he blend [
21
,
22
].
Addi ionally, cok iging and i s a ian s ha e di icul ies being applicable, mainly due o
he subjec i i y in modeling a iog ams wi h li le in o ma ion and nonlinea o complex
dependency ela ionships wi h mine al g ades and o he geochemical a iables [23].
Gi en he p oblems and limi a ions men ioned ea lie , he applica ion o copula-based
me hods eme ges as a p omising al e na i e in he mining indus y [
24
]. As is usual in
s a is ics and geos a is ics, wo app oaches ha e been de eloped: one o es ima ion and
Mine als 2024,14, 691 3 o 21
he o he o simula ion. In pa icula , se e al geos a is ical me hods based on copulas ha e
been published ha ha e been success ul in Ea h sciences applica ions [
25
–
29
]. Bu abo e
all, he e a e mul iple de elopmen s in he sphe e o inance using an es ima ion app oach
known as quan ile eg ession based on copulas [30–32].
A copula is a mul i a ia e cumula i e dis ibu ion unc ion o which he ma ginal
p obabili y dis ibu ion o each a iable is uni o m on he in e al [0, 1]. They a e unc ions
ha desc ibe he unde lying dependence be ween andom a iables. Skla ’s heo em [
33
]
s a es ha any mul i a ia e join dis ibu ion can be exp essed in e ms o uni a ia e
ma ginal dis ibu ion unc ions and a copula ha desc ibes he dependency s uc u e
be ween he a iables [34,35].
This esea ch e alua es he pe o mance o he condi ional quan ile eg ession me hod
(CQRM) compa ed o he classical geos a is ical colloca ed cok iging me hod (CCM) o mod-
eling coppe me allu gical eco e y based on a p edominan ly measu ed
geological a ibu e.
The s uc u e o he pape is as ollows. In he Sec ion 2, he p oblem s a emen
is es ablished. The Sec ion 3p esen s he colloca ed cok iging and condi ional quan ile
eg ession me hods, and he gene al me hodology o hei applica ion. The Sec ion 4gi es
a desc ip ion o he da ase used in he compa a i e s udy. The Sec ion 5shows he esul s
o he applica ion o bo h me hods o he case s udy. In he Sec ion 6, he compa ison o he
pe o mance o he wo me hods is discussed, and, inally, in he Sec ion 7, he conclusions
and u u e wo k a e gi en.
2. P oblem S a emen
In a basic mining exploi a ion uni o calcula e mining p o i , Equa ion (1), me allu -
gical eco e y, o e g ade, and me al p ice a e c i ical a iables, all subjec o signi ican
unce ain y, as can be in e ed om i s own de ini ion [36,37]:
P o i =RCu
c ·PCu
c·GCu ·TCu −CCu
c(1)
whe e
PCu
c
is he p ice o he coppe concen a e,
GCu
is he coppe g ade,
TCu
is he mine al
onnage, and
CCu
c
a e he cos s associa ed wi h he p oduc ion, p ocessing, and sale o he
coppe concen a e.
Due o he low sampling densi y, hei s ong he e o opic ela ionship wi h o he
mine al deposi measu emen s, and he nonlinea i y in hese ela ionships, geos a is ical
modeling appea s o be a complex al e na i e.
Fo mally, me allu gical eco e y can be de ined o a coppe mine by he ollowing
exp ession [8]:
RCu
c =mCu
c
mCu =xCu
c c1
xcu (2)
whe e
RCu
c
is he coppe eco e y,
mCu
c
is he mass o coppe in he concen a e (i.e.,
he amoun o eco e ed coppe ), and
mCu
is he ini ial mass o coppe in he eed.
xCu
c
is he coppe g ade in he concen a e,
xcu
is he eed g ade, and
c
is he ac ion o
mass eco e ed.
Small a ia ions in me allu gical eco e y es ima ion will impac he p o i alua ion,
which de ines he ma e ial’s des ina ions in s a egic planning, whe he as p ocessed o e in
he plan , was e ock ex ac ed o deposi ion in dumps, o low-g ade o e o s ockpiling.
Gi en he impo ance o his a iable o downs eam p ocesses such as block model
op imiza ion, mine design, and mine li e planning, i is essen ial o seek he bes p ac ices
o i s p edic ion.
3. Me hodology
In his pape , a compa ison o he condi ional quan ile eg ession me hod (CQRM)
is made wi h espec o he adi ional colloca ed cok iging me hod (CCM) in e ms o
accu acy and pe o mance. To apply bo h me hods, he e a e a se ies o s eps ha a e
common, as is shown in he gene al me hodological wo k low o Figu e 1.
Mine als 2024,14, 691 4 o 21
The i s s ep, consis ing o he explo a o y analysis o he da a, is s anda d o any
s a is ical p ocedu e and consis s o a summa y o i s s a is ics and he p obabili y dis ibu-
ion g aphics wi h i s boxplo s and his og ams. Also, an e alua ion o he impac o he
p esence o ou lie s in he da a sample on he s a is ics is also ca ied ou .
In he a iable selec ion s ep in bo h cases, he a iable pai ha has he g ea es
dependence is sough , bu he di e ence is ha o CCM i mus be a linea dependence
ha should be es ima ed wi h he Pea son co ela ion coe icien , while o CQRM i is
ecommendable o use a mo e obus measu e o dependence by using Spea man o Kendall
ank co ela ion coe icien s.
In he modeling pa , di e en models a e buil . Fo CCM i is a spa ial co ela ion
model wi h he a iog am o he p ima y a iable, while o CQRM i is he dependence
model ha consis s o he es ima ion and i ing o he ma ginal dis ibu ions o each
a iable as well as he copula.
Figu e 1. Gene al me hodological wo k low.
The alida ion o he models is ca ied ou o CCM wi h he usual c oss- alida ion
me hod, i.e., a lea e one ou me hod [
38
], while CQRM is alida ed pe o ming an es i-
ma ion wi h he same da a used o build i . The quali y o each me hod is e alua ed in
e ms o pe o mance me ics, such as oo mean squa ed e o (RMSE), mean absolu e
e o (MAE), mean absolu e pe cen age e o (MAPE), and de e mina ion coe icien (R2).
Finally, he p edic ion o coppe me allu gical eco e y condi ioned by a seconda y
a ibu e is pe o med a each poin whe e he e a e no sample alues.
3.1. Colloca ed Cok iging Me hod
Colloca ed cok iging is a geos a is ical me hod used o spa ial in e pola ion o es i-
ma ion o a a ge a iable a unsampled loca ions based on a ailable da a om sampled
loca ions [
38
]. This me hod is pa icula ly use ul when dealing wi h mul iple co ela ed
a iables o when auxilia y in o ma ion is a ailable. In colloca ed cok iging, he ela ion-
ship be ween he a ge a iable and auxilia y a iables is modeled using he concep
o spa ial dependence, which assumes ha nea by loca ions end o ha e simila alues.
The me hod es ima es he a ge a iable a an unsampled loca ion by combining in o ma-
ion om bo h he a ge a iable and auxilia y a iables a nea by sampled loca ions (see
Appendix B).
The usual s eps in ol ed in he applica ion o colloca ed cok iging me hod a e
he ollowing:
•
Da a collec ion: Collec da a on he a ge a iable and auxilia y a iables om
sampled loca ions wi hin he s udy a ea.
•
Spa ial co ela ion analysis: Assess he spa ial co ela ion o dependence be ween he
a ge a iable and auxilia y a iables using a iog ams o co a iance unc ions.
•
Modeling: Model he spa ial dependence s uc u e be ween he a ge a iable and
auxilia y a iables using geos a is ical echniques such as o dina y k iging.
•
Valida ion: Valida e he accu acy o he p edic ions using c oss- alida ion o compa i-
son wi h independen da a, i a ailable.
•
P edic ion: Es ima e he alue o he a ge a iable a unsampled loca ions by com-
bining in o ma ion om nea by sampled loca ions and auxilia y a iables, aking in o
accoun hei spa ial co ela ion.
Mine als 2024,14, 691 5 o 21
Colloca ed cok iging o e s ad an ages o e adi ional k iging me hods by inco po-
a ing addi ional in o ma ion om auxilia y a iables, which can imp o e he accu acy o
p edic ions, especially in a eas wi h limi ed o spa se da a co e age o he a ge a iable.
3.2. Condi ional Quan ile Reg ession Me hod
Condi ional quan ile eg ession based on copulas is a s a is ical me hod used o
modeling he ela ionship be ween a iables, pa icula ly when dealing wi h non-no mal
o skewed dis ibu ions, and when he e a e complex dependencies among a iables. Cop-
ulas a e ma hema ical unc ions ha desc ibe he dependence s uc u e be ween andom
a iables, independen o hei ma ginal dis ibu ions. They cap u e he join dis ibu ion
o a iables wi hou making assump ions abou hei indi idual dis ibu ions. Copulas
allow he modeling o bo h linea and nonlinea dependencies, making hem use ul o
cap u ing complex ela ionships be ween a iables.
Condi ional quan ile eg ession ex ends he concep o linea eg ession by es ima ing
condi ional quan iles o he esponse a iable gi en he alues o p edic o a iables. Un-
like o dina y leas squa es eg ession, which models he condi ional mean o he esponse
a iable, quan ile eg ession models di e en quan iles (e.g., median, uppe /lowe quan-
iles), p o iding a mo e comp ehensi e unde s anding o he condi ional dis ibu ion o he
esponse a iable. In quan ile eg ession based on copulas, copulas a e used o model he
dependence s uc u e be ween a iables, while quan ile eg ession is applied o es ima e
he condi ional quan iles o he esponse a iable gi en he alues o p edic o a iables.
This app oach allows he cap u ing o he join dis ibu ion and condi ional dis ibu ions
o a iables simul aneously, aking in o accoun complex dependencies among hem.
Condi ional quan ile eg ession based on copulas o e s se e al ad an ages:
• Flexibili y: I can model nonlinea dependencies and accoun o he e oscedas ici y.
• Robus ness: I is obus o ou lie s and non-no mali y in he da a.
•
In e p e abili y: I p o ides insigh s in o how di e en quan iles o he esponse
a iable a e a ec ed by changes in he p edic o a iables.
•
Tail beha io : I can cap u e ex eme e en s o ail beha io in he condi ional dis i-
bu ions o a iables.
This me hod is widely used in inance, economics, en i onmen al science, and o he
ields whe e unde s anding he ela ionship be ween a iables ac oss di e en quan iles
is essen ial. I can be used o isk managemen , o ecas ing, and decision making unde
unce ain y. O e all, quan ile eg ession based on copulas is a powe ul ool o model-
ing complex dependencies and unde s anding he condi ional dis ibu ion o a iables,
especially in si ua ions whe e adi ional eg ession me hods may no be adequa e.
In his a icle, a nonpa ame ic copula by he ke nel smoo hing me hod [
39
] is used,
gi en i s good adap abili y o any ype o dis ibu ion and compu a ional e iciency. In pa -
icula , ke nel smoo hing, a ype o weigh ed mo ing a e age me hod, is applied o
p obabili y densi y es ima ion, ha is, o es ima e he p obabili y densi y unc ion o a
andom a iable based on ke nels as weigh s, whe e he e m ke nel in his con ex means
a window unc ion.
A mo e de ailed explana ion o copula heo y and he quan ile eg ession me hod can
be ound in Appendix A. In pa icula , addi ional explana ions abou he ke nel smoo hing
me hod can be ound in Appendix A.2.
4. Da ase Desc ip ion
Wi h he pu pose o ca ying ou he compa ison be ween he colloca ed cok iging
and condi ional quan ile eg ession me hods wi h mining da a, a da ase ex ac ed om
a syn he ic po phy y coppe deposi was selec ed, which was published in [
19
], and is
openly a ailable o academic use. F om he h ee-dimensional block model, he le el 2080
is chosen o con igu e a 2D space and we use i as expe imen al da a o his applica ion.
This da ase consis s o 3479 (71 × 49) cells o 20 × 20 me e s, each spa ially geo e e enced
by hei X (eas ing) and Y (no hing) coo dina es. Each cell con ains geochemical eco ds

Mine als 2024,14, 691 6 o 21
o 1—clays, 2—chalcoci e, 3—bo ni e, 4—chalcopy i e, 5— ennan i e, 6—molybdeni e,
7—py i e, 8—coppe (Cu), 9—molybdenum (Mo), and 10—a senic (As), as well as he
esul s o me allu gical es s o 11—coppe eco e y and 12—bond wo k index.
This exe cise consis s o ex ac ing, in a andom and spa ially uni o m manne , a
sample equi alen o 10% o he o al a ailable in o ma ion o he me allu gical eco e y,
cons uc ing a s ongly he e o opic case wi h espec o he o he a iables. Figu e 2shows
he coppe eco e y maps o he comple e da ase on he le side and he co esponding
map o he 10% sampling on he igh side, espec i ely.
Figu e 2. Coppe eco e y o he 100% (in he le ) and 10% (in he igh ) sample maps, espec i ely.
The compa ison o he CCM and CQRM me hods is ca ied ou by applying hem
o he es o he da a co esponding o 90% o he da ase , allowing a di ec compa ison
be ween he eal in o ma ion and he esul s ob ained by he wo me hods unde he same
he e o opic sampling condi ions and alida ion me ics.
A compa a i e s a is ical summa y be ween he ull (100%) da ase and he 10% sample
o coppe eco e y is shown in Table 1and Figu es 3and 4, espec i ely. I can be seen
ha he 10% subse o da a is s a is ically equi alen o he o al da ase in e ms o he
s a is ical alues and p obabili y dis ibu ion. He eina e , he subse co esponding o 10%
o he da ase will be e e ed o as “10% sample”.
Figu e 3. Coppe eco e y his og am and boxplo o he ull (100%) da ase .
Mine als 2024,14, 691 7 o 21
Figu e 4. Coppe eco e y his og am and boxplo o 10% sample da ase .
Table 1. Coppe eco e y s a is ics summa y o he o al and 10% sample, espec i ely.
S a is ics To al Sample 10% Sample
Size 3479 347
Minimum 68.7907 75.7572
1s qua ile 84.2624 84.8486
Median 87.2162 87.503
Mean 86.5376 86.8988
3 d qua ile 89.5262 89.6185
Maximum 93.2732 93.0413
Range 24.4825 17.2841
In e qua ile ange 5.2638 4.7699
Va iance 13.8434 12.3059
S anda d de ia ion 3.7207 3.508
Skewness −0.8964 −0.8003
Ku osis 0.6558 0.2731
5. Case S udy Applica ion
5.1. Explo a o y Da a Analysis
A s a is ical summa y o he geochemical a ibu es o he 10% sample is shown in
Table 2.
Table 2. S a is ics summa y o a 10% sample o all geochemical a ibu es, whe e he numbe ing
co esponds o: 1—clays, 2—chalcoci e, 3—bo ni e, 4—chalcopy i e, 5— ennan i e, 6—molybdeni e,
7—py i e, 8—coppe (Cu), 9—molybdenum (Mo), and 10—a senic (As), 11—coppe eco e y and
12—bond wo k index.
S a is ics 1 2 3 4 5 6 7 8 9 10 11 12
Size 347 347 347 347 347 347 347 347 347 347 347 347
Minimum 0.7294 0.0045 0.0047 0.205 0.0046 0.0047 0.0053 0.0766 0.0028 0.0009 75.7572 11.39
1s qua ile 2.1378 0.009 0.0674 0.5892 0.005 0.0089 0.6665 0.3199 0.0054 0.001 84.8486 12.5312
Median 3.2251 0.0457 0.1308 0.789 0.0067 0.0127 1.6151 0.4129 0.0076 0.0014 87.503 12.8145
Mean 4.08 0.0804 0.1733 0.9003 0.0128 0.0185 1.8393 0.4569 0.0111 0.0026 86.8988 12.9392
3 d qua ile 5.0217 0.1261 0.2246 1.0561 0.0123 0.0218 2.7769 0.5312 0.0131 0.0025 89.6185 13.1052
Maximum 20.7767 0.7682 0.9131 3.6683 0.133 0.125 7.1561 1.6023 0.075 0.027 93.0413 22.2781
Range 20.0473 0.7637 0.9084 3.4633 0.1285 0.1203 7.1509 1.5257 0.0722 0.0261 17.2841 10.8881
In e qua ile ange 2.884 0.1172 0.1571 0.4669 0.0072 0.0129 2.1103 0.2113 0.0077 0.0015 4.7699 0.5739
Va iance 8.0645 0.0101 0.0255 0.2463 0.0003 0.0002 1.911 0.0461 0.0001 0 12.3059 0.9168
S anda d de ia ion 2.8398 0.1003 0.1597 0.4963 0.0174 0.0153 1.3824 0.2146 0.0092 0.0035 3.508 0.9575
Skewness 1.7563 2.9569 1.7926 2.3118 4.4297 2.4914 0.664 1.7473 2.4914 4.4297 −0.8003 5.058
Ku osis 4.3005 14.3695 3.6934 7.8782 23.0514 9.0065 −0.1015 4.9835 9.0065 23.0514 0.2731 36.8668
Mine als 2024,14, 691 8 o 21
5.2. Va iable Selec ion
A bi a ia e analysis is ca ied ou o selec he geochemical a ibu e ha has a g ea e
dependence ela ionship wi h espec o coppe eco e y. Fo his pu pose, hea maps o
he Pea son, Kendall, and Spea man co ela ion coe icien s a e ob ained (see he la e in
Figu e 5).
Figu e 5. Spea man co ela ion hea map o all geochemical a ibu es o a 10% sample.
The geochemical a ibu e ha shows he s onges co ela ion wi h coppe eco e y is
chalcoci e, wi h co ela ion coe icien s o
−
0.75,
−
0.84, and
−
0.63 acco ding o Pea son,
Spea man, and Kendall, espec i ely (see Table 3). Howe e , i is impo an o no e ha
he dependency ela ionship be ween he a iables, as shown in Figu e 6, p esen s a
clea ly nonlinea beha io . The a iable chalcoci e, which is mos ly sampled, is selec ed
o pe o m he compa ison be ween he colloca ed cok iging and condi ional quan ile
eg ession me hods.
Figu e 6. Coppe eco e y s. chalcoci e sca e plo o 10% sample da ase .
Mine als 2024,14, 691 9 o 21
Table 3. Coppe eco e y and chalcoci e co ela ion coe icien s.
P ima y Va iable Seconda y Va iable Pea son Kendall Spea man
Coppe eco e y chalcoci e −0.75 −0.63 −0.84
5.3. Modeling
As was men ioned in Sec ion 3, he modeling s ep is di e en o each me hod. CCM
co esponds o a spa ial co ela ion modeling s ep, while CQRM o a dependence modeling s ep,
as shown below.
5.3.1. Spa ial Co ela ion Modeling
The spa ial co ela ion model o CCM consis s o es ima ing he sample a iog am
and op imally adjus ing a a iog am model o he p ima y a iable, which in his case is
coppe eco e y.
The chalcoci e spa ial dis ibu ion o he comple e da ase on he le side and he
co esponding map o he 10% sampling on he igh side a e shown in Figu e 7, while
Figu e 8shows he coppe eco e y empi ical a iog am calcula ed by Ma he on es ima o
and a sphe ical i ed model wi h an e ec i e ange o 141.12 m, a sill o 12.20, and no nugge
e ec . This spa ial co ela ion model is used in conjunc ion wi h Pea son’s co ela ion
coe icien in he colloca ed cok iging applica ion o chalcoci e comple e da ase (le side
o Figu e 7).
Figu e 7. Chalcoci e o he 100% (in he le ) and 10% (in he igh ) sample maps, espec i ely.
Figu e 8. Coppe eco e y a iog am model. The blue do s a e he empi ical a iog am and he
con inuous g een line is he i ed a iog am model.
Mine als 2024,14, 691 16 o 21
The app oach p oposed in his a icle consis s o he applica ion o a no el condi ional
quan ile eg ession me hod o me allu gical coppe eco e y condi ioned on a seconda y
a iable ha is widely sampled and exhibi s he maximum possible dependence. The pa -
icula i y o his app oach lies in he op imal es ima ion o he codependency model o he
a iables using a ke nel smoo hing me hod o he ma ginals as well as o he copula.
The p esen ed applica ion is based on a syn he ic bu ealis ic case o a po phy y
coppe deposi , whe e he median eg ession o coppe eco e y is es ima ed a all known
chalcoci e loca ions, used as an explana o y a iable. The compa ison o he esul s ob ained
by CQRM e sus CCM shows ha , bo h in e ms o p ecision and e iciency, as well as
ep oduc ion o spa ial a iabili y, he me hod p oposed in his a icle is supe io when he
dependencies a e nonlinea .
Al hough his is a speci ic applica ion o a case s udy, he me hodology can be ex ended
o o he scena ios, such as an explo a ion d illing campaign, whe e he la ges scale o
in o ma ion comes om d illing in e al logs, wi h me allu gical eco e y measu ed in
ewe quan i ies and no necessa ily in he same loca ions.
CQRM is an inno a i e, p ac ical, e icien , and e sa ile app oach. I a oids s ong as-
sump ions abou he da a and heo e ical cons ain s in i s implemen a ion, demons a ing
a clea capaci y o adap o di e en le els o in o ma ion and scales, making i applicable
o a wide a ie y o scena ios.
As u u e wo k, he implemen a ion o condi ional quan ile eg ession me hod based
on copula model wi h mul i a ia e dependencies is p oposed. This model ex ension is
expec ed o u he imp o e he accu acy o p edic ions while main aining p ac icali y,
e sa ili y, and compu a ional e iciency.
As can be deduc ed om he Funding sec ion o his manusc ip , his esea ch has
aised he in e es o o he sec o s, such as he s eel indus y, who wishes o maximize he
e iciency o all i s p ocesses in e ms o bo h indus ial and social impac .
Au ho Con ibu ions: Concep ualiza ion, H.H. and M.A.D.-V.; me hodology, M.A.D.-V.; so wa e,
M.A.D.-V.; alida ion, H.H.; o mal analysis, M.A.D.-V.; w i ing—o iginal d a p epa a ion, H.H.
and M.A.D.-V.; w i ing— e iew and edi ing, E.A., A.G., and A.O.-Z.; supe ision, E.A.; und-
ing acquisi ion, A.G. and A.O.-Z. All au ho s ha e ead and ag eed o he published e sion o
he manusc ip .
Funding: Wo k unded by p ojec SILENCE—Eu opean Commission—Resea ch P og am o he
Resea ch Funds o Coal and S eel—P j. No.: 101112516.
Da a A ailabili y S a emen : Da a used in his a icle a e aken om he publica ion [
19
] and a e
publicly a ailable.
Con lic s o In e es : The au ho s decla e no con lic s o in e es .
Abb e ia ions
The ollowing abb e ia ions a e used in his manusc ip :
CCM Colloca ed cok iging me hod
CQRM Condi ional quan ile eg ession me hod based on copulas
MAE Mean absolu e e o
MAPE Mean absolu e pe cen age e o
PDF P obabili y densi y unc ion
R2De e mina ion coe icien
RMSE Roo mean squa ed e o
Appendix A. Condi ional Quan ile Reg ession Me hod Based on Copulas
Skla in 1959 [
33
] es ablished a heo em indica ing a unc ional ela ionship be ween
he join p obabili y dis ibu ion unc ion o a andom ec o and i s uni a ia e ma ginal
dis ibu ion unc ions. Fo ins ance, in he case o wo a iables, i
(X
,
Y)
is a andom
ec o wi h a join p obabili y dis ibu ion
HXY(x
,
y) = P(X≤x
,
Y≤y)
, hen he ma ginal

Mine als 2024,14, 691 17 o 21
dis ibu ion unc ions o
X
and
Y
a e
FX(x) = P(X≤x) = HXY(x
,
∞)
and
GY(y) = P(Y≤
y) = HXY(∞
,
y)
, espec i ely. Howe e , when ma ginalizing
HXY
, some in o ma ion is
los , as he ma ginal dis ibu ions
FX
and
GY
gene ally do no su ice o ully de e mine
HXY
. This is because he ma ginal dis ibu ions only desc ibe he indi idual p obabili y
beha io o he andom a iables hey ep esen . Skla ’s heo em demons a es ha he e
exis s a unc ion CXY:[0, 1]27−→ [0, 1]such ha
HXY(x,y) = CXY(FX(x),GY(y)) (A1)
whe e
CXY
is he copula unc ion associa ed wi h he bi a ia e andom ec o
(X
,
Y)
and
desc ibes i s dependence ela ionship,
H
is he bi a ia e p obabili y dis ibu ion unc ion,
and Fand Ga e he uni a ia e (ma ginal) p obabili y dis ibu ions.
Copula unc ions a e a aluable ool o cons uc ing join p obabili y models wi h
g ea e lexibili y. They allow us o independen ly selec uni a ia e models o he andom
a iables o in e es and choose a copula unc ion ha bes ep esen s hei dependence,
ei he pa ame ically o nonpa ame ically. Fo example, in a mul i a ia e no mal model, all
ma ginal dis ibu ions mus be no mal, wi h no ail dependence and ini e second momen s
o well-de ined co ela ions. The mul i a ia e no mal model is a speci ic ins ance whe e he
unde lying copula is Gaussian and all uni a ia e ma ginals ollow a no mal dis ibu ion.
When
FX
and
GY
a e con inuous, elemen a y p obabili y heo y ells us ha
U=FX(X)
and
V=GY(Y)
a e con inuous uni o m andom a iables on
(
0, 1
)
. The unde lying copula
C
o he andom ec o
(U
,
V)
is he same copula co esponding o
(X
,
Y)
. Acco d-
ing o Skla ’s heo em, he join p obabili y dis ibu ion unc ion o
(U
,
V)
is gi en by
HUV(u
,
) = C(FU(u)
,
GV( )) = C(u
,
)
. The e o e, i
FX
and
GY
a e known bu
HXY
is unknown, and we ha e an obse ed andom sample
{(x1
,
y1)
,
. . .
,
(xn
,
yn)}
o
(X
,
Y)
,
he se
{(uk
,
k) = (FX(xk)
,
GY(yk)) :k=
1,
. . .
,
n}
will be an obse ed andom sample o
(U
,
V)
wi h he same unde lying copula
C
as
(X
,
Y)
. Since
C=FUV
, we can use he alues
(uk, k)(known as copula obse a ions) o es ima e Cas a join empi ical dis ibu ion:
ˆ
C(u, ) = 1
n
n
∑
k=1
I{uk≤u, k≤ }(A2)
S ic ly, he es ima e
ˆ
C
is no a copula since i is discon inuous and copulas a e always
con inuous. I
FX
,
GY
, and
HXY
a e all unknown, which is he mos common case,
FX
and
GYa e es ima ed by hei empi ical uni a ia e dis ibu ion unc ions:
ˆ
FX(x) = 1
n
n
∑
k=1
I{xk≤x},ˆ
GY(y) = 1
n
n
∑
k=1
I{yk≤y}(A3)
whe e I ep esen s an indica o unc ion equal o 1 when i s a gumen is ue and 0 o he wise.
We will e e o he se o pai s
(ˆ
uk,ˆ
k) = ( ˆ
FX(xk),ˆ
GY(yk)) :k=1, . . . , n
as pseudo
obse a ions o he copula. I can be e i ied di ec ly ha
ˆ
FX(xk) = 1
n ank(xk)
and
ˆ
GY(yk) = 1
n ank(yk)
. In his case, he concep o empi ical copula, see [
34
], is de ined
as he ollowing unc ion Cn:I2
n7−→ [0, 1], whe e In=ni
n:i=0, . . . , no, gi en by
Cni
n,j
n=1
n
n
∑
k=1
I{ ank(xk)≤i, ank(yk)≤j)}(A4)
Again,
Cn
is no a copula bu is an es ima e o he unde lying copula in he mesh
I2
n
ha
can be ex ended o a copula in
[
0, 1
]2
o , o example, Be ns ein polynomials, as p oposed
Mine als 2024,14, 691 18 o 21
and s udied in [
43
], leading o wha is known as a nonpa ame ic es ima e o he Be ns ein
copula ˜
C:[0, 1]27−→ [0, 1]gi en by
˜
C(u, ) =
n
∑
i=0
n
∑
j=0
Cni
n,j
nn
iui(1−u)n−in
j j(1− )n−j(A5)
Appendix A.1. Th ee App oaches o Building a Copula-Based Dependency Model
The e a e h ee app oaches o building a copula-based dependency model: pa ame ic,
nonpa ame ic, and semipa ame ic.
The pa ame ic app oach consis s o being able o i a known copula model o he em-
pi ical copula, such as he F ank, Gumbel, o Clay on copulas, which belong o he amily o
A chimedean copulas [
34
,
35
], as well as being able o i he empi ical ma ginal p obabili y
dis ibu ions o known dis ibu ion unc ions such as no mal o Gaussian, logno mal,
gamma, Weibull, e c. In his way, a join p obabili y dis ibu ion model is ob ained.
The nonpa ame ic app oach consis s o nume ically app oxima ing he empi ical cop-
ula and i s ma ginals, usually by means o some polynomial exp ession. In his app oach,
Be ns ein polynomials [
28
,
43
] and splines [
44
], as well as he ke nel smoo hing me hod,
ha e been used [35], whe e ke nel smoo hing is a ype o weigh ed mo ing a e age.
While a semipa ame ic app oach is a combina ion o he wo p e ious app oaches,
his allows wo op ions, ha is, a model could be i ed o he empi ical copula and app ox-
ima e he ma ginals wi h a polynomial, o he empi ical copula could be app oxima ed by
a polynomial exp ession and adjus he ma ginals wi h a known dis ibu ion model [45].
Appendix A.2. Ke nel Densi y Es ima ion wi h Ke nel Smoo hing Me hod
Conside
(x1
,
x2
,
. . .
,
xn)
as independen and iden ically dis ibu ed samples d awn
om a uni a ia e dis ibu ion cha ac e ized by an unknown densi y unc ion
a any
a bi a y poin
x
. Ou ocus lies in app oxima ing he shape o his unc ion
. I s ke nel
densi y es ima o is exp essed as ollows:
b
h(x) = 1
n
n
∑
i=1
Kh(x−xi) = 1
nh
n
∑
i=1
Kx−xi
h, (A6)
In his con ex ,
K
deno es he ke nel, a unc ion ha yields only non-nega i e alues,
while
h>
0 ep esen s a smoo hing pa ame e e e ed o as he bandwid h. A ke nel
labeled wi h he subsc ip
h
is e med he scaled ke nel, de ined by
Kh(x) = 1
hK(x
h)
.
In essence, one aims o selec
h
o be as small as he da a pe mi s in ui i ely; ne e heless,
he e in a iably exis s a ade-o be ween he es ima o ’s bias and i s a iance.
Appendix A.3. Copula Simula ion Algo i hm
As summa ized in [
45
], in o de o simula e he epe i ions o he andom ec o
(X
,
Y)
wi h he dependence s uc u e in e ed om he obse ed da a
(x1,y1), . . . , (xn,yn)
, we
ha e he ollowing algo i hm:
(i)
Gene a e wo independen and con inuous andom a iables
u
and
uni o mly
dis ibu ed in (0, 1).
(ii)
Se =c−1
u( ), whe e cu( ) = ∂˜
C(u, )
∂u.
(iii)
The desi ed pai is
(x
,
y) = ( ˜
Qn(u)
,
˜
Rn( ))
, whe e
˜
Qn
and
˜
Rn
a e he empi ical
quan ile unc ions o Xand Y, espec i ely.
Appendix A.4. Condi ional Quan ile Reg ession Algo i hm
Fo a alue
x
in he ange o he andom a iable
X
and a gi en 0
<α<
1, le
y=φα(x)
be he solu ion o he equa ion
P(Y≤y|X=x) = α
. Then he g aph o
Mine als 2024,14, 691 19 o 21
y=φα(x)
is he quan ile eg ession cu e
α
o
Y
condi ional on
X=x
. In [
34
] i is
p o en ha
P(Y≤y|X=x) = cu( )|u=FX(x), =GY(y)(A7)
This esul leads o he ollowing algo i hm o ob ain quan ile eg ession cu e
α
o
Y
condi ional on X=x:
(i)
Se cu( ) = α.
(ii)
Sol e he eg ession cu e o :
=gα(u). (A8)
(iii)
Replace uby ˜
Q−1
n(x)and by ˜
R−1
n(y).
(i )
Sol e he eg ession cu e o y:
y=φα(x). (A9)
Appendix B. Colloca ed Cok iging Me hod wi h Ma ko Model
The k iging me hod is well known as he bes unbiased linea spa ial es ima o o a
single andom unc ion and he cok iging me hod is i s gene aliza ion o wo o mo e
andom unc ions. This equi es calcula ing he p ima y a iog am (e.g., coppe eco e y),
he seconda y a iog am (e.g., geochemical a ibu e), and he c oss- a iog am [
38
,
46
].
In pa icula , he c oss- a iog am akes in o accoun he spa ial dependence be ween he
wo andom unc ions and is de ined as ollows:
γ12(h) = 1
2N(h)
N
∑
α=1
(Z1() −Z1(xα+h))(Z2(xα)−Z2(xα+h)) (A10)
whe e
γ12
is he semi a iance be ween he andom unc ions
Z1
(coppe eco e y) and
Z2
(geochemical a ibu e),
h
is he lag dis ance,
xα
is a spa ial loca ion, and
N
is he numbe
o lag dis ances.
The igo ous applica ion o he o dina y cok iging me hod is equen ly e y di icul
and complica ed since i also equi es ha he se o a iog ams comply wi h he linea
co egionaliza ion model. This model is e y es ic i e since i equi es ha all a iog ams
i he same model wi h a common ange.
A mo e e icien and p ac ical al e na i e is he colloca ed cok iging me hod ha was
in oduced by Almeida and Jou nel (1994) [
47
]. I is a a ian o cok iging me hod o
spa ial in e pola ion ha le e ages bo h p ima y and seconda y da a. Bu i simpli ies he
compu a ion by using only he seconda y a iable’s alue ha is colloca ed (a he same
loca ion) as he p ima y a iable. This me hod is pa icula ly use ul when seconda y da a
a e mo e densely sampled han p ima y da a.
Howe e , his app oach equi es a condi ional independence be ween he p ima y
a iable and he seconda y a iable, gi en hei colloca ed alues. This is known as a
Ma ko assump ion o condi ional independence, which simpli ies he modeling p ocess.
The e a e wo ypes o Ma ko models: Ma ko models 1 and 2, espec i ely. He e, Ma ko
model 1 (MM1) is used as i is he simples and mos s aigh o wa d op ion.
In Ma ko model 1, he ollowing condi ional independence assump ion is made:
E(Z2(u)|Z1(u) = z1,Z1(u′) = z1(u′)) = E(Z2(u)|Z1(u) = z1)(A11)
This implies, unde he assump ions o MM1, a simpli ica ion in he s a is ical ela-
ionship be ween he p ima y and seconda y a iables. Thus, he c oss-co elog am model
can be w i en as
ρ12(h) = ρ12(0)ρ1(h)(A12)
whe e he co elog am ρis exp essed by
ρ(h) = 1−γ(h)(A13)
Mine als 2024,14, 691 20 o 21
Assuming ha
γ(h)
is he a iog am o da a wi h a s anda d no mal dis ibu ion,
he co elog am is simply he a iog am in e ed and shi ed upwa d by one. While
he a iog am measu es spa ial co a iance, he co elog am measu es spa ial co ela ion.
The e o e,
ρ12(
0
)
, he co elog am a a lag dis ance o ze o, is equi alen o he co ela ion
coe icien be ween a iables 1 and 2.
In summa y, o apply he colloca ed cok iging me hod wi h Ma ko model 1 (MM1),
all ha is needed is he p ima y a iable a iog am and he linea co ela ion coe icien
be ween he p ima y and seconda y a iables. The a iog am o seconda y a iable is
no necessa y.
No e ha you MUST o pe o m a no mal sco e ans o ma ion o he p ima y and
seconda y da a p io o in oking MM1.
Re e ences
1.
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Min. Sci. 2022,67, 699–713.
2.
Gholami, A.; Asga i, K.; Khoshdas , H.; Hassanzadeh, A. A hyb id geome allu gical s udy using coupled His o ical Da a (HD)
and Deep Lea ning (DL) echniques on a coppe o e mine. Physicochem. P obl. Mine . P ocess. 2022,58, 147841. [C ossRe ]
3.
Oumesaoud, H.; Faouzi, R.; Aboulhassan, M.A.; Naji, K.; Benzakou , I.; Faqi , H.; Oukh ib, R.; Elboughdi i, N. I on Ox-
ide–coppe Mine al Associa ions in Supe gene Zones: Insigh s in o Flo a ion Challenges and Op imiza ion Using Response
Su ace Me hodology. ACS Omega 2024,9, 24438–24452. [C ossRe ]
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