Nonlinea Analysis: Modelling and Con ol, Online Fi s , 1–20
h ps://doi.o g/10.15388/namc.2022.27.27493
P ess
Index spaces and s anda d indices in me ic modelling*
Ezgi E do˘
gana, An onia Fe e -Sapenab,
Edua do Jiménez-Fe nándezc, En ique A. Sánchez-Pé ezb
aDepa men o Ma hema ics,
Facul y o A s and Sciences, Ma ma a Uni e si y,
34722, Kadıköy, Is anbul, Tu key
[email p o ec ed]
bIns i u o Uni e si a io de Ma emá ica Pu a y Aplicada,
Uni e si a Poli ècnica de València,
C. de Ve a s/n, 46022 Valencia, Spain
an e sa@up .es; [email p o ec ed].es
cDepa amen o de Teo ía e His o ia Económica,
Facul ad de Ciencias Económicas y Emp esa iales,
Uni e sidad de G anada,
Campus Uni e si a io de La Ca uja, 18071 G anada, Spain
edjim e @ug .es
Recei ed: Sep embe 14, 2021 / Re ised: Ap il 11, 2022 / Published online: May 10, 2022
Abs ac . We analyze he basic s uc u e o ce ain me ic models, which a e cons i u ed by an
index Iac ing on a me ic space (D, d) ep esen ing a ele an p ope y o he elemen s o D.
We call such a s uc u e (D, d, I)an index space and de ine on i no maliza ion and consis ency
cons an s ha measu e o wha ex en Iis compa ible wi h he me ic d. The “bes ” indices a e hose
wi h such cons an s equal o 1(s anda d indices), and we show an app oxima ion me hod o o he
indices using hem. Wi h he help o Lipschi z ex ensions, we show how o apply hese ools: a new
model o he iage p ocess in he eme gency depa men o a hospi al is p esen ed.
Keywo ds: me ic model, index space, s anda d index, Lipschi z ex ension, iage.
1 In oduc ion
Ra ings and indices a e used oday as elemen a y bu undamen al ools ha enable in-
di iduals and ins i u ions o make ele an s a egic decisions. Fo example, in he spe-
ci ic ield o educa ion, indices ha e been widely used in he con ex o highe educa-
ion (ARWU, QS Wo ld Uni e si y Ranking, Webome ics, see, o example, [1]); also,
con ol and moni o ing o he economy o coun ies is mainly based on indices and
*This esea ch was pa ially suppo ed by he g an PID2019-105708RB unded by MCIN/AEI/
10.13039/501100011033 and by “ERDF A way o making Eu ope”. The second, hi d and ou h au ho s
g a e ully acknowledge he suppo o he Cá ed a de T anspa encia y Ges ión de Da os, Uni e si a Poli ècnica
de València, Gene ali a Valenciana.
© 2022 Au ho s. Published by Vilnius Uni e si y P ess
This is an Open Access a icle dis ibu ed unde he e ms o he C ea i e Commons A ibu ion Licence, which
pe mi s un es ic ed use, dis ibu ion, and ep oduc ion in any medium, p o ided he o iginal au ho and sou ce
a e c edi ed.
2E. E do˘gan e al.
ankings (see [4,17,22]). Ou concep ualiza ion o hese models akes he o m o a ma h-
ema ical s uc u e ha makes i possible o con ol he cons uc ion o a gi en index by
ensu ing i s consis ency wi h he dis ance be ween he elemen s o he unde lying me ic
model.
Due o he ma hema ical na u e o he p esen pape , le us men ion i s he main
e e ence ha appea s when we check he scien i ic li e a u e ega ding indices and ank-
ings. Di ec ly ela ed o s a is ical a gumen s, he heo y o ankings is unde s ood o be
he s udy o how, gi en a ma ix o sco es ( o example, gi en by he pai wise compa ison
among he elemen s o a se ), a anking o hese elemen s can be se acco ding o he
gi en ela ions. This happens, o example, when we ha e se e al eams playing agains
each o he in a socce league and we know al eady a pa he esul s. The eade can
ind a comple e explana ion o ela ed me hods in [11] and he e e ences he ein. These
me hods p o ide di e en ways o gi ing o de ings among he elemen s o he se based
on he known da a, bu hey a e no cen e ed in he de ini ion o a simila i y ela ion gi en
by a dis ance among he elemen s. The e o e, al hough he ela ions wi h ou pu pose
can be checked, he s a ing poin o he me hod is exac ly he opposi e one o ou s since
he e is no an unde lying me ic space: o de ed di e ences among elemen s a he han
simila i ies a e used. Indices a e also use ul in image p ocessing in which some ools
a e de ined using speci ic indices in o de o measu e simila i y o pic u es. Fo ins ance,
in [6], i can be ound a comple e explana ion o use o such an ins umen , his ime in
a me ic space amewo k as in he case ha we show in he p esen pape .
Theo e ical app oaches based on amilies o ankings ha ha e al eady been p o en
by use can also be ound. In [5], he compa ison be ween he di e en app oaches is
used o ob ain a gene al idea o wha a anking should be, which allows o gi e he
ounda ions o a common amewo k o analysis a leas in a speci ic a ea. O he ele an
example o he analysis o anking and indices is he pape [22] in which a sys ema ic
me hod o de ining cohe en ankings and a ings is p esen ed. A g ea e o is made o
analyze he co ec ness and applicabili y o he indexes, p oducing a decalogue o ules o
de elop mul idimensional measu es. Essen ially, his decalogue—which can be s uc u ed
as a sequence o consecu i e s eps o be applied— can be summa ized as ollows: a i s
s ep o he cons uc ion o he heo e ical amewo k, a second s ep o da a selec ion and
p ocessing, a hi d s ep ha e e s o ma hema ical (mainly s a is ical) p ocessing, and
a las s ep ha e e s o da a syn hesis and isualiza ion. Le us b ie ly explain how hese
i ems i in o he gene al scheme o ou ma hema ical cons uc ion: (i) In ou pu pose,
he i s s ep ( heo e ical amewo k) is used o de ine a me ic space: he se o en i ies
on which he model wo ks is i s ixed. Then we use a dis ance o quan i y simila i y
among hese elemen s. (ii) An index in i ( he essence o any anking, a ing o indica o
cons uc ion) is no hing mo e han a posi i e unc ion ha is in ended o be consis en wi h
he me ic. Small dis ances in he me ic space ha e o be e lec ed in small di e ences
be ween index alues: Lipschi z unc ions appea . (iii) The analysis o he esul ing
me ic/index s uc u e oge he wi h s a is ics and da a isualiza ion come in o play o
p o ide he inal esul s.
Le us see a conc e e example. We begin by conside ing a gene al class ha we wan
o analyze, ha is, he se D o which he inal index should be applied: he se o all
h ps://www.jou nals. u.l /nonlinea -analysis
Index spaces and s anda d indices in me ic modelling 3
uni e si ies in a coun y. Conside hen some a iables o compa e hem; o example, he
o e all unding o he ins i u ion pe yea o he numbe o s uden s. Using hem, we can
de ine a dis ance dbe ween wo uni e si ies: o example, he no m o he ec o p o ided
by he alues o hese a iables.
Le us now suppose ha we in end o de ine an index Iin his me ic space (a posi i e
eal numbe ), which will allow in he nex s ep o de ine a anking, gi en canonically by
he o de p o ided by he index. We need i o be cohe en wi h he me ic din space since
his s uc u e ep esen s he p ope ies ha we a e in e es ed in aking in o accoun o ou
analysis. A i ial example o such a “cohe en ” index is gi en by he dis ance i sel as
ollows: o all b∈D, we de ine I0by I0(b) := d(a0, b), whe e a0∈Dis a ixed elemen
o D. Indices de ined in his way will play a cen al ole in he pape and will be called
s anda d indices. Fo example, he elemen a0could be he bes uni e si y in a gi en
coun y, and so he index measu es how a any o he uni e si y is om mee ing he
s anda ds o a0. Pu ing he h ee objec s oge he —a se D, a me ic dand an index I—,
we ge wha we call an index space (D, d, I). This has been called a me ic-index model
in [10] in he con ex o he ex ension o indexes de ined o uni e si y ankings.
This s uc u e can be unde s ood as a gene al analy ical ins umen . Fo example,
impu a ion o da a is an impo an p ocess in modeling, especially, in he case o index
cons uc ion. An up- o-da e su ey on he subjec can be ound a [13] (see also [18] and
he e e ences he ein). The p oblem o gi ing conc e e alues o a gi en index when
we do no con ol all he a iables in i s de ini ion becomes a cen al issue in many
cases. The Lipschi z ex ension can be used as a me hod o da a impu a ion when he
s a is ical me hods a e no adequa e. This is why we also s udy he ex ension o he indices
as Lipschi z unc ions: hey allow o ill in missing alues o he indices p ese ing he
simila i y ela ions (dis ances).
The pape is o ganized as ollows. In Sec ion 2, main concep s and de ini ions a e
in oduced. Sec ion 3 is de o ed o he de elopmen o he main ma hema ical esul s
on he indexable spaces and index spaces: na u al opologies, ele an subse s, compac -
ness esul s, app oxima ion and ex ension o indices. In Sec ion 4, we will show he
compa ibili y o he classical o mulas o he ex ension o Lipschi z unc ions wi h he
o iginal me ic s uc u e o he indices, p o iding in his way an analy ical ool o seeing
how a ex ensions can be ep esen ed by means o s anda d indices. O he ex ension
o mulas coming om wha a e called absolu ely minimizing Lipschi z ex ensions ensu e
a be e local beha io han he ones we p opose (see [2, 14]), and could be also used in
u he s udies; howe e , his esea ch plan exceeds he aim o he p esen pape . Finally
(Sec ion 5), we show as an example how o apply ou gene al echnique o c ea ing
a iage au oma ic sys em o es ablish he o de o a endance in an eme gency se ice o
a hospi al based on he p o essional expe ience o some doc o s.
2 Basic de ini ions
We p esen in his sec ion he basic de ini ions and esul s ha a e needed in he pape .
All o hem can be ound in books on gene al opology as [15, 21]. I R+is he se o
Nonlinea Anal. Model. Con ol, 27(Online Fi s ):1–20, 2022
4E. E do˘gan e al.
posi i e eal numbe s (including 0), we de ine a dis ance (o a me ic) in a se Das
a unc ion d:D×D→R+such ha o a, b ∈D, (i) d(a, b)=0i and only i a=b,
(ii) d(a, b) = d(b, a), and (iii) i c∈D, hen d(a, b)⩽d(a, c) + d(c, b).
In [9], he in e es ed eade can ind a lo o examples and pa icula me ics con-
s uc ed o sol ing speci ic p oblems. Each me ic de ines a opology on he space D
in which i is de ined, ha has a basis o neighbo hoods o any elemen x∈D he balls
Bε(x) = {y∈D:d(a, b)< ε},ε > 0. These se s gi e an in ui i e idea o wha a dis ance
is in e ms o who he neighbo s o a gi en elemen aa e: an elemen bbelongs o Bε(x)
i i is as close as ε o a; he lowe he ε, he highe he closeness om b o a.
A unc ion ac ing in a me ic space (D, d)and wi h alues on o he me ic space (R, )
is said o be Lipschi z i ( (a), (b)) ⩽Kd(a, b)holds o a ce ain cons an K > 0and
o e e y a, b ∈D. The Lipschi z cons an o is he in imum o he cons an s Kin he
inequali y. In case we ha e o e e o he unc ion, we will w i e Lip( ) o i , bu o en
we will use simply Kin case his e e ence is no needed. Al hough he de ini ions and
esul s on Lipschi z unc ions can be ound in a lo o wo ks, we mainly use he ecen
exhaus i e book [8].
The main eason o using his amily o unc ions is he use ul ex ension o mulas
ha can be applied o hem. This will allow o apply ou model o he cons uc ion o
speci ic indices and i s gene aliza ion o a b oade class o ins i u ions o educa ion. The
McShane–Whi ney heo em es ablishes ha i Bis a subspace o a me ic space (D, d)
and :B→Ris a Lipschi z unc ion wi h Lipschi z cons an K, he e is an ex ension ˆ
o o Dsuch ha ˆ
is also a Lipschi z unc ion wi h he same Lipschi z cons an K
[8, Thm. 4.1.1].
Two classical ex ensions a e gi en by he so-called McShane and Whi ney o mulas
ha a e ( espec i ely)
M(b) := sup
a∈B (a)−Kd(b, a)and W(b) := in
a∈B (a) + Kd(b, a),
which a e de ined o all b∈Dand equal o i b∈B. We will use as ex ension o mula
o he applica ion o ou s uc u e an in e pola ion o hese wo ex eme si ua ions.
3 Index spaces
The objec o ou model is a me ic space (D, d)and an index I. Such an index is
basically a Lipschi z eal unc ion and in ou model ep esen s a “meaning ul quan i y”—
o he decision make ha is de ining he model— ha allows o cons uc a anking
on Dby o de ing he alues o I(a) o all a∈D. We will call a iple (D, d, I)an
index space. Th oughou he pape , we will assume ha he me ic is bounded, ha is,
supa,b∈Dd(a, b) = d(D, D)<∞.
Unde his assump ion, we will say ha he couple (D, d)is an indexable space.
The con ol o Iusing he me ic dis he main idea o ou model, which assumes ha
Iis in some sense compa ible wi h d, ha is, i ep esen s a quan i y whose p ope ies
a e implici ly ep esen ed by d. This is he eason why we assume ha Iis a Lipschi z
h ps://www.jou nals. u.l /nonlinea -analysis
Index spaces and s anda d indices in me ic modelling 5
unc ion. Also, o ma hema ical easons and in o de o gi e a be e s uc u e o wo k
wi h, we will impose some no maliza ion and cohe ency condi ions o he index Iwi h
ega d o he dis ance d. Basically, we equi e ha he e is some p opo ion among he
index and he dis ance ha can be es ablished in e ms o wo con ol inequali ies. The
i s de ini ion conce ns he con ol on he size o he indices.
De ini ion 1. Le C > 0. An index I: (D, d)→Ris C-bounded i i sa is ies ha
supa∈D|I(a)|⩽C. The cons an B(I) := supa∈D|I(a)|will be called he boundedness
cons an o I. An index will be said o be bounded i i is C-bounded o some C > 0.
Fo u he compa ison be ween se e al indexes, we will impose he ollowing no -
maliza ion p ope y o a gi en cons an Q ha could allow o con ol he ela ion o d
wi h he index I. As we will see, a complemen a y con ol will be gi en by he Lipschi z
inequali y o he index and he me ic.
De ini ion 2. Le Q > 0. Le (D, d)be an indexable space. We will say ha an index
I: (D, d)→Ris Q-no malized i
d(a, b)⩽QI(a)+I(b) o all a, b ∈D.
We will say ha he posi i e numbe N(I)is he no maliza ion cons an o Ii he
in imum o he cons an s Qsa is ying he p ope y abo e is equal o N(I). We will w i e
N(I)<∞ o deno ing ha he e exis s such a no maliza ion cons an o I. No e ha i
Iis Q-no malized, we ha e ha d(D, D)⩽2QB(I).
Ob iously, he de ini ion abo e can be ew i en wi hou he absolu e alue in case
ha he index is posi i e. Howe e , al hough we a e dealing wi h posi i e indices in mos
o he cases, in p inciple, we canno assume ha e e y sui able ex ension o a posi i e
index is posi i e. Since hese ex ensions a e essen ial in ou o malism, we p e e o w i e
he no maliza ion condi ion in his o m. No e also ha , since
I(a)−I(b)⩽
I(a)−I(b)
=I(a)−I(b) o all a, b ∈D,
we ha e ha he new index Abs(I)gi en by Abs(I)(·) := |I(·)|is also Lipschi z i Iis,
and o i s cons an , we ha e ha Lip(Abs(I)) ⩽Lip(I).
Rema k 1. No ice ha he hypo hesis ha N(I)<∞— ha is, Iis no malized o some
K > 0—, imply ha Ican be ze o only a one poin . Indeed, i we ha e ha I(a0) =
I(b0)=0 o a0, b0∈D,d(a0, b0)⩽I(a0)+I(b0)=0, and so d(a0, b0)=0. This will
mean in ou o malism ha he e is only one “op imal” elemen in he space, in he sense
ha he alue o any index is allowed o be 0only in one poin . The ex ension o he esul s
o ha ing se e al op imal elemen s would need o change me ics by pseudome ics.
De ini ion 3. Le K > 0. An index I: (D, d)→R+is K-cohe en i i sa is ies he
Lipschi z inequali y o he cons an K. Tha is,
I(a)−I(b)⩽Kd(a, b) o all a, b ∈D.
Nonlinea Anal. Model. Con ol, 27(Online Fi s ):1–20, 2022

6E. E do˘gan e al.
We will say ha C(I)is he cohe ence cons an o Ii he in imum o all cons an s K
sa is ying his p ope y is equal o C(I), ha is, C(I)is he Lipschi z cons an o I. We
will w i e C(I)<∞ o deno ing ha he e exis s such a cohe ence cons an . No e ha
sup
a,b∈DI(a)−I(b)⩽2 sup
a∈D
I(a)⩽2B(I).
Mo e ob ious ela ion among he jus in oduced cons an s can be seen and will be
use ul in he nex sec ions. Fo example, i in (I) = R∈R+, we ha e ha
B(I)−R= sup
a∈D
I(a)−R⩽sup
a,b∈DI(a)−I(b)⩽C(I)d(D, D),
and so
B(I)⩽C(I)d(D, D) + R,
wha means ha e e y Lipschi z map ( ha is, C(I)<∞) ac ing in an indexable space is
always bounded. The se o indices sa is ying ha in (I)=0will be ele an in he nex
sec ions; so we ha e ha all o hem a e bounded whene e hey a e K-cohe en o any
cons an K < ∞, and
B(I)⩽C(I)·d(D, D).
In case we ha e ha he index has also a ini e no maliza ion cons an N(I), we ge
B(I)⩽C(I)·d(D, D)⩽2C(I)·N(I)·B(I).
3.1 Basic s uc u e and compac ness esul s o index spaces
Le C > 0and conside he space FCo uni o mly bounded indices de ined on an
indexable space (D, d),FC={I:D→R:B(I)⩽C}.
We can de ine wo na u al opologies on i . The i s one is he uni o m opology
gi en by he no m B(·)o he indices as bounded unc ions and wi h basic neighbo hoods
Vε(I0) = {I∈ FC:B(I0−I)< ε},ε > 0,I0∈ FC. The second one is he opology
o he poin wise con e gence wi h basic neighbo hoods Vε,a1,...,an(I0) = {I∈ FC:
|I0(ai)−I(ai)|< ε, i = 1, . . . , n} o I0∈ FC,ε > 0,n∈N, and a1, . . . , an∈D.
P oposi ion 1. The ollowing subspaces o FCa e compac wi h espec o he opology
o poin wise con e gence:
(i) FCand
F0
C:= {I∈ FC:I⩾0}.
(ii) Fo K > 0,
I∈ FC:I⩾0,I(a)−I(b)⩽Kd(a, b), a, b ∈D.
(iii) Fo K > 0,Q > 0and R⩾0,
I∈ FC:I⩾0,I(a)−I(b)⩽Kd(a, b),
R+d(a, b)⩽QI(a) + I(b), a, b ∈D.
h ps://www.jou nals. u.l /nonlinea -analysis
Index spaces and s anda d indices in me ic modelling 7
P oo . (i) Fi s , no e ha he poin wise limi limηIηo a ne o unc ions (Iη)η∈Λin FC
is again a unc ion in FC. Indeed, o each a∈D,limηIη(a)∈[−C, C]; i he unc ion
is in F0
C, we ha e ha limηIη(a)∈[0, C]. A classical a gumen iden i ying each such
a unc ion wi h an elemen o he Ca esian p oduc Πa∈D[−C, C](Πa∈D[0, C] o F0
C)
ha a e compac since hey a e p oduc s o compac spaces wi h he p oduc (Hausdo )
opology, gi es he esul by Tychono ’s heo em.
(ii) Fix a, b ∈Dand no e ha |limηIη(a)−limηIη(b)|⩽Kd(a, b) o a ne o
unc ions (Iη)η∈Λ ha con e ges poin wise since |Iη(a)−Iη(b)|⩽Kd(a, b) o e e y
η∈Λ. So limi s o ne s belonging o he se w i en in (ii) gi es a unc ion also in his
se , wha means ha i is closed. Toge he wi h (i), his gi es he esul .
(iii) Again, o a ixed pai a, b ∈D, we ha e d(a, b)⩽Q(limηIη(a) + limηIη(b)),
and so he se is closed in a compac se .
This basic compac ness esul s o meaning ul subse s o indices opens he doo o
a basic app oxima ion ool ha will be imp o ed in he nex sec ions by conside ing
sequences and pa icula alues o he cons an s appea ing in he subse s in P oposi ion 1.
The na u al app oxima ion ha we can expec ega ding subse s o indices is clea ly
ela ed o he poin wise opology: gi en any ne (Iη)η∈∆, he e is a subne (Iη0)η0∈∆0
and an index I0in he subse such ha limη0Iη0=I0wi h espec o he opology o
poin wise con e gence, ha is, o e e y a∈D,limη0Iη0(a) = I0(a). Mo eo e , i
we know in ad ance ha we ha e a poin wise Cauchy ne (Iη)η∈∆belonging o any o
he subse s in P oposi ion 1, we ha e ha i s poin wise limi I0always de ines an index
belonging o he se , limηIη(a) = I0(a), a ∈D.
3.2 S anda d indices and ep esen a ion o gene al indices
In case we do no ha e a dis inguished index in he model (D, d, I), hose equi emen s
(Q-no maliza ion and K-cohe ence) will gi e us a leas some con ol abou how a
he index Iis o he na u al beha io o he me ic d o he elemen s o D. Howe e ,
he e y s uc u e o he model (D, d)p o ides s anda d indices ha a e associa ed wi h
indi idual poin s a∈Dand which we deno e by Ia. I can be de ined as ollows a e
choosing a e e ence poin a∈D. Tipically, ais unde s ood o be he elemen (s) ha ing
he “minimum alue” o a gi en p ope y, bu i can be chosen o be any elemen o D. We
de ine a ( he) s anda d index (associa ed o a speci ic elemen a∈D) as Ia(b) := d(a, b),
whe e b∈D. Nex p oposi ion explains why an index de ined in his ashion can be
conside ed as “s anda d”.
P oposi ion 2. A s anda d index Iahas bo h he no maliza ion and he cohe ence con-
s an s equal o 1.
P oo . No e i s ha d(b, c)⩽d(b, a)+d(a, c) = Ia(b)+Ia(c), and so he no maliza ion
cons an Q⩽1. On he o he hand, aking b=a, we ge d(a, c) = d(a, a) + d(a, c) =
Ia(a)+Ia(c), and so Q⩾1. The e o e, Q= 1, and Iais 1-no malized. Fo he cohe ence
cons an , i is clea by he iangle inequali y o d ha |Ia(c)−Ia(b)|=|d(a, c)−
Nonlinea Anal. Model. Con ol, 27(Online Fi s ):1–20, 2022
8E. E do˘gan e al.
d(a, b)|⩽d(c, b). So, K⩽1. Mo eo e , o b=a, we ge |Ia(c)−Ia(a)|=d(c, a)−
d(a, a) = d(c, a), and so 1⩽K, and so K= 1.
The ollowing co olla y s a es he ce ain y ha , gi en an indexable space and a poin
ain i , i is always possible o cons uc a s anda d indexed space wi h an index cen e ed
on his poin a.
Co olla y 1. Fo e e y indexable space (D, d)and e e y a∈D, he e is a 1-cohe en
and 1-no malized index Isuch ha I(a) = 0 and I(b)>0 o e e y a6=b∈D.
P oo . Jus ake I:= Iaand apply P oposi ion 2.
A so o con e se o his esul is also ue as we show in P oposi ion 3. I gi es
a cha ac e iza ion o hose indices ha can be iden i ied wi h s anda d indices.
P oposi ion 3. Le I:D→Rbe a 1-cohe en and 1-no malized index, and suppose
ha he e is an elemen a∈Dsuch ha I(a) = 0. Then Icoincides wi h he s anda d
index Ia.
P oo . Fo e e y b∈D, we ha e ha I(b) = I(b)−I(a)⩽|I(b)−I(a)|⩽d(a, b).
Also, we ha e d(a, b)⩽I(a) + I(b) = I(b). Bo h inequali ies oge he imply I(b) =
d(a, b) = Ia(b) o e e y b∈Das desi ed.
In he es o his sec ion, we analyze he s uc u e o he spaces o indices and he
app oxima ion p ope ies ha can be ob ained using he me ic subspace o he s anda d
indices. Fix an indexable space (D, d). We will gi e a desc ip ion o he closu e o his
space wi h espec o he opology o poin wise con e gence. The idea is o show ha ,
unde some equi emen s, e e y index in he space can be app oxima ed by ( ansla ions
o ) s anda d indices. Thus, he na u al ex ension o he class o s anda d indices is gi en
by conside ing i s closu e. Also, he condi ion minb∈DIa(b) = 0 — ha is i ially
sa is ied by s anda d indices— can be adap ed o ge ing a mo e use ul space ha ing
be e opological p ope ies ega ding comple ion. Le us w i e in (I) o in a∈DI(a).
De ini ion 4. Take a sequence (an)in D. Le us say ha i is poin wise Cauchy i o
e e y b∈D, he limi limnd(an, b)o he sequence o eal numbe s (d(an, b)) exis s.
I he sequence (an)is con e gen wi h espec o he me ic d, hen i is i ially
poin wise Cauchy. The es ic ion o he eal line wi h he Euclidean me ic o he in e al
D= (0,1] and he sequence (1/n)⊂Dgi es a i ial example o such a (noncon e gen )
poin wise Cauchy sequence.
The ollowing heo em is he main esul o his sec ion. I is a gene al e sion o
P oposi ion 3 and p o ides a comple e cha ac e iza ion o hose indices ha can be ap-
p oxima ed by a ansla ion o s anda d indices. As he eade may no ice in he de ini ion
o RCbelow, many o he indices we a e in e es ed in — ha is, hose indices ha
occu in he applica ions we p opose, and in gene al, in all p ac ical si ua ions in which
indices appea — sa is y he equi emen o belonging o such a se . In he case ha he
equi emen s o he inequali ies mus be elaxed o conside a gi en index I, one could
use ins ead P oposi ion 4.
h ps://www.jou nals. u.l /nonlinea -analysis
Index spaces and s anda d indices in me ic modelling 9
Theo em 1 [Rep esen a ion heo em o bounded indices]. Conside he se
RC:= I⩾0: I(a)−I(b)⩽d(a, b),2 in (I) + d(a, b)⩽I(a) + I(b),
B(I)⩽C.
Then o e e y I∈ RC, he e is a poin wise Cauchy sequence (an)such ha o e e y
b∈D,I(b) = in (I) + limnd(an, b).
P oo . Conside b∈D. Fix n∈Nand ake an elemen ansuch ha I(an)−1/n ⩽
in (I). Then
in (I) + d(an, b)⩽I(an) + I(b)−in (I)
⩽I(b) + I(an)−I(an)−1
n=I(b) + 1
n.
On he o he hand,
I(b)−I(an)⩽I(b)−I(an)⩽d(an, b),
and so
I(b)⩽d(an, b) + I(an)⩽d(an, b) + in (I) + 1
n⩽I(b) + 2
n.
Consequen ly,
lim
nd(an, b)=I(b)−in (I)∈R
o e e y b, and so (an)is poin wise Cauchy. Also, his gi es he desi ed ep esen a ion
I(b) = in (I) + limnd(an, b),b∈D.
Rema k 2. The con e se o Theo em 1 is also ue. Indeed, we ha e ha i I(a) =
in (I) + limn(an, a) o a ce ain poin wise Cauchy sequence (an)and a, b ∈D,
2 in (I) + d(a, b)⩽in (I) + d(an, a) + in (I) + d(an, b),
and so
2 in (I) + d(a, b)⩽in (I) + lim
nd(an, a) + in (I) + lim
nd(an, b)
=I(a) + I(b).
Thus, we ha e a cha ac e iza ion o hose indices ha can be ep esen ed as a cons an
plus a s anda d index de ined by he me ic d. Fo example, we ha e ha he indices
ha ha e no maliza ion and cohe ence cons an s equal o one (N(I) = C(I)=1)
and in (I)=0a e exac ly he ones wi h a ep esen a ion as I(·) = limnd(an,·) o
a poin wise Cauchy sequence (an). Mo eo e , by P oposi ion 1(iii) we ha e ha he
closu e o his se wi h espec o he opology o poin wise con e gence is compac since
i is included in a (many) se (s) as he one(s) de ined he e.
Nonlinea Anal. Model. Con ol, 27(Online Fi s ):1–20, 2022
16 E. E do˘gan e al.
sensi i e iage is gi en by he decimal numbe s. To see wha i means, conside he las
wo pa ien s wi h he McShane–Whi ney ex ension in he able. The Main Complain s
o hese pa ien s a e he same, and he p edic ed iages a e 1.63 and 1.50. Rounding he
p edic ions, he second o de is gi en o he pa ien s o iage. Bu 1.63 >1.50, and his
shows ha he las pa ien has p io i y. This is signi ican as he las pa ien is elde , and
his is he only main di e ence be ween he pa ien s. The same e ec is also obse ed o
he s anda d index.
Figu e 1 p o ides a ep esen a ion o bo h p edic ed iages oge he wi h he o iginal
iage o he expe s. 55 pa ien s a e ep esen ed in he axis OX. No e ha , as expec ed,
he app oxima ions ob ained by McShane–Whi ney ex ension coincide wi h he o iginal
iage when he pa ien s belong o he es se . The pa ien s in his se a e he i s 11
poin s in he axis OX. This c i e ion is ollowed in all he igu es p esen ed below. To
see a compa ison o he iages, we gi e a ep esen a ion o he g aphics in he same
coo dina e sys em in Fig. 2.
A ep esen a ion o he e o s o bo h p edic ed iages a e p esen ed in Fig. 3, and
compa ison o he e o s is shown in Fig. 4.
As he eade can see, bo h p ospec i e me hods seem o be good enough o p o iding
a use ul au oma ic iage ool. O cou se, i is be e o ge he e alua ion o he iage
numbe by expe s o an small g oup and o ex end i o he es o he cases han make
i necessa y o ge such a decision o he expe s eam a e e y new pa ien en e ing
he hospi al. Only ou a iables a e needed, ha a e he di ec esul o apid physical
measu es and obse a ion. A e he analysis o he igu es, we can s a e h ee main
conclusions:
(a) The se o a iables ha ha e been chosen is good enough o he desc ip ion o
he iage p ocess: expe s should ha e enough in o ma ion wi h hese alues o
doing a good iage: he s anda d index we ha e chosen is gi en by he me ic
wi hou using any expe ’s iage o he ex ension.
(b) The expe ’s iage is also succes ully ex ended by Lipschi z ex apola ion:
McShane–Whi ney ex ension gi es good esul s when compa ed o eal iage
o de ing.
(c) We ha e seen ha he e is a high compa ibili y be ween he dis ance and he
index, so he s anda d index and he McShane–Whi ney ex ension p o ide simila
esul s. The espec i e e o dis ibu ions a e also simila (Figs. 3 and 4). The
coincidence o bo h me hods is a symp om o a co ec ela ionship be ween
dis ance and index, and so he p oposed index space p o ides a s ong model.
The e o e, index spaces, s anda d index app oxima ion and Lipschi z ex ensions on
hem can be used o model iage p ocesses in a hospi al and can p o ide a basis o
a machine lea ning me hod o hospi al managemen . The p oposed example opens he
doo o he cons uc ion o mo e sophis ica ed p ocedu es, whe e he dis ance could
con ain mo e a iables and he aining se could be ex ended.
To inish, le us summa ize he main con en s o he pape . We ha e shown an ele-
men a y s uc u e o modeling physical and social sys ems —index spaces—, which a e
desc ibed as me ic spaces endowed wi h nume ical p ope ies ha can be ep esen ed
h ps://www.jou nals. u.l /nonlinea -analysis

Index spaces and s anda d indices in me ic modelling 17
Figu e 1. Rep esen a ion o he expe s’ o iginal iage and o he iages p edic ed by bo h he McShane–
Whi ney ex ension and he s anda d index.
Figu e 2. Join ep esen a ion o he compa ison o he o iginal iages and he iages p edic ed by ou wo
me hods.
as eal- alued indices. The main p ope ies o he s uc u e o hese models ha e been
p esen ed, as well as some p ope ies o app oxima ion by indices ha beha e well wi h
espec o he me ic —s anda d indices—.
Nonlinea Anal. Model. Con ol, 27(Online Fi s ):1–20, 2022
18 E. E do˘gan e al.
Figu e 3. Rep esen a ion o he e o s commi ed by he McShane–Whi ney ex ension and by he s anda d
index.
Figu e 4. Rep esen a ion o he compa ison o e o s o he p edic ed iages.
I has also been shown how o pe o m Lipschi z ex apola ion on hese index spaces
by showing con ol o mulas on how he ex ension p ese es he good p ope ies o
he o iginal model —no maliza ion and cohe ence cons an s—. These h ee componen s
—index spaces, s anda d indexes and McShane–Whi ney ex ensions— a e hus p oposed
as complemen a y ools o me ic modeling and o esigh , p o iding a comple e analy i-
cal amewo k.
h ps://www.jou nals. u.l /nonlinea -analysis
Index spaces and s anda d indices in me ic modelling 19
Re e ences
1. I. Aguillo, J. Ba -Ilan, M. Le ene, J. O ega, Compa ing uni e si y ankings, Scien ome ics,
85(1):243–256, 2010, h ps://doi.o g/10.1007/s11192-010-01900-z.
2. G. A onsson, M. C andall, P. Juu inen, A ou o he heo y o absolu ely minimizing unc ions,
Bull. Am. Ma h. Soc.,41(4):439–505, 2004, h ps://doi.o g/10.1090/S0273-
0979-04-01035-3.
3. K. Asadi, D. Mis a, M. Li man, Lipschi z con inui y in model-based ein o cemen lea ning,
in J.G. Dy, A. K ause (Eds.), P oceedings o he 35 h In e na ional Con e ence on Machine
Lea ning, ICML 2018, S ockholmsmässan, S ockholm, Sweden, July 10–15, 2018, P oc.
Mach. Lea n. Res., Vol. 80, PMLR, 2018, pp. 264–273, h ps://doi.o g/10.48550/
ARXIV.1804.07193.
4. R. Bandu a, A su ey o composi e indices measu ing coun y pe o mance: 2008 upda e,
UNDP/ODS Wo king Pape , 2008.
5. J. B anko ic, L. Ringel, T. We on, Theo izing uni e si y ankings. a compa a i e esea ch
pe spec i e, P ac ices o Compa ing. Wo king Pape SFB 1288, No. 2, 2019, h ps://
doi.o g/10.4119/unibi/2939561.
6. D. B une , E.R. V scay, Z. Wang, On he ma hema ical p ope ies o he s uc u al simila i y
index, IEEE T ans. Image P ocess.,21(4):1488–1499, 2011, h ps://doi.o g/10.
1109/TIP.2011.2173206.
7. J.M. Calabuig, H. Falciani, E.A. Sánchez-Pé ez, D eaming machine lea ning: Lipschi z
ex ensions o ein o cemen lea ning on inancial ma ke s, Neu ocompu ing,398:172–184,
2020, h ps://doi.o g/10.1016/j.neucom.2020.02.052.
8. ¸S. Cobza¸s, R. Miculescu, A. Nicolae, Lipschi z Func ions, Lec . No es Ma h., Vol. 2241,
Sp inge , Cham, 2019, h ps://doi.o g/10.1007/978-3-030-16489-8.
9. M.M. Deza, E. Deza, Encyclopedia o Dis ances, Sp inge , Be lin, Heidelbe g, 2009, h ps:
//doi.o g/10.1007/978-3-642-00234-2.
10. A. Fe e -Sapena, E. E dogan, E. Jiménez-Fe nández, E.A. Sánchez-Pé ez, F. Pese , Sel -
de ined in o ma ion indices: applica ion o he case o uni e si y ankings, Scien ome ics,
124(3):2443–2456, 2020, h ps://doi.o g/10.1007/s11192-020-03575-6.
11. J. González-Díaz, R. Hend ickx, E. Lohmann, Pai ed compa isons analysis: An axioma ic
app oach o anking me hods, Soc. Choice Wel a e,42(1):139–169, 2014, h ps://doi.
o g/10.1007/s00355-013-0726-2.
12. M.J. Hi n, E.Y. Le G uye , A gene al heo em o exis ence o quasi absolu ely minimal
Lipschi z ex ensions, Ma h. Ann.,359(3):595–628, 2014, h ps://doi.o g/10.1007/
s00208-013-1003-5.
13. M. Huisman, R.W. K ause, Impu a ion o missing ne wo k da a, in R. Alhajj, J. Rokne (Eds.),
Encyclopedia o Social Ne wo k Analysis and Mining, Sp inge , New Yo k, 2017, pp. 1–10,
h ps://doi.o g/10.1007/978-1-4614-7163-9_394-1.
14. P. Juu inen, Absolu ely minimizing lipschi z ex ensions on a me ic space, Ann. Acad. Sci.
Fenn., Ma h.,27(1):57–67, 2002.
15. J.H. Kelley, Gene al Topology, Do e , Mineola, NY, 2017.
Nonlinea Anal. Model. Con ol, 27(Online Fi s ):1–20, 2022
20 E. E do˘gan e al.
16. K. Kyng, A. Rao, S. Sachde a, D.A. Spielman, Algo i hms o Lipschi z lea ning on g aphs,
in P. G ünwald, E. Hazan, S. Kale (Eds.), P oceedings o he 28 h Con e ence on Lea ning
Theo y, 3–6 July 2015, Pa is, F ance, P oc. Mach. Lea n. Res., Vol. 40, PMLR, 2015,
pp. 1190–1223.
17. G. Le anon, E alua ing and compa ing leading and coinciden economic indica o s, Bus.
Econ.,45(1):16–27, 2010, h ps://doi.o g/10.1057/be.2009.29.
18. R.J.A. Li le, D.B. Rubin, S a is ical Analysis wi h Missing Da a, Wiley Se . P obab. S a .,
Vol. 793, John Wiley & Sons, Hoboken, NJ, 2019.
19. S.-H. Moon, J.L. Shim, K.-S. Pa k, C.-S. Pa k, T iage accu acy and causes o mis iage
using he Ko ean iage and acui y scale, h ps:// igsha e.com/a icles/
da ase /T iage_accu acy_and_causes_o _mis iage_using_ he_
Ko ean_T iage_and_Acui y_Scale/9779267/1.
20. S.-H. Moon, J.L. Shim, K.-S. Pa k, C.-S. Pa k, T iage accu acy and causes o mis iage using
he Ko ean iage and acui y scale, PLoS One,14(9):e0216972, 2019, h ps://doi.o g/
10.1371/jou nal.pone.0216972.
21. J. Naga a, Gene alized me ic spaces i, in K. Mo i a, J. Naga a (Eds.), Topics in Gene al
Topology, No h-Holland Ma h. Lib ., Vol. 41, No h-Holland, Ams e dam, 1989, pp. 315–
366.
22. M. Saisana, A. Sal elli, Rankings and a ings: Ins uc ions o use, Hague J. Rule Law,
3(2):247–268, 2011, h ps://doi.o g/10.1017/S1876404511200058.
h ps://www.jou nals. u.l /nonlinea -analysis