A flexible soft nonlinear quantile-based regression model

Hesamian, Gholam eza; Johannssen, A ne; Chukh o a, Na aliya
A icle — Published Ve sion
A lexible so nonlinea quan ile-based eg ession model
Fuzzy Op imiza ion and Decision Making
P o ided in Coope a ion wi h:
Sp inge Na u e
Sugges ed Ci a ion: Hesamian, Gholam eza; Johannssen, A ne; Chukh o a, Na aliya (2025) : A lexible
so nonlinea quan ile-based eg ession model, Fuzzy Op imiza ion and Decision Making, ISSN
1573-2908, Sp inge US, New Yo k, NY, Vol. 24, Iss. 1, pp. 129-153,
h ps://doi.o g/10.1007/s10700-025-09441-5
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Fuzzy Op imiza ion and Decision Making (2025) 24:129–153
h ps://doi.o g/10.1007/s10700-025-09441-5
A lexible so nonlinea quan ile‑based eg ession model
Gholam ezaHesamian1· A neJohannssen2· Na aliyaChukh o a3
Accep ed: 25 Janua y 2025 / Published online: 6 Ma ch 2025
© The Au ho (s) 2025
Abs ac
The e a e se e al models o so eg ession analysis in he li e a u e, bu ela i ely
ew a e based on quan iles, and hese models a e limi ed o he linea case. As quan-
ile-based eg ession models o e a se ies o bene i s (like obus ness and handling
o asymme ic dis ibu ions) bu ha e no been conside ed in he nonlinea case, we
p esen he i s so nonlinea quan ile-based eg ession model in his pape . Con-
side ing nonlinea i y ins ead o limi ing o linea i y in he modeling b ings nume -
ous ad an ages such as a highe lexibili y, mo e accu a e p edic ions, a be e model
i and an imp o ed explainabili y/in e p e abili y o he model. In pa icula , we
embed uzzy quan iles in o nonlinea eg ession analysis wi h c isp p edic o a i-
ables and uzzy esponses. We p opose a new me hod o pa ame e es ima ion by
implemen ing a h ee-s age echnique on he basis o he cen e and he sp eads. In
he amewo k o his p ocedu e, we u ilize ke nel- i ing, a leas quan ile loss unc-
ion, leas absolu e e o s, and gene alized c oss- alida ion c i e ia o es ima e he
model pa ame e s. We pe o m comp ehensi e compa a i e analysis wi h o he so
nonlinea eg ession models ha ha e demons a ed supe io i y in p e ious s udies.
The esul s e eal ha he p oposed nonlinea quan ile-based eg ession echnique
leads o be e ou comes compa ed o he compe i o s.
Keywo ds C oss- alida ion· Explainabili y· Fuzzy quan iles· Fuzzy eg ession·
Ke nel- i ing· Leas absolu e e o s· Robus ness
* A ne Johannssen
[email p o ec ed]
Gholam eza Hesamian
[email p o ec ed]
Na aliya Chukh o a
na aliya.chukh o [email p o ec ed]
1 Depa men o S a is ics, Payame Noo Uni e si y, Teh an19395-3697, I an
2 Facul y o Business S udies, Ha z Uni e si y o Applied Sciences, 38855We nige ode,
Ge many
3 Depa men o Heal h Economics, Facul y o Medicine Mannheim, Uni e si y o Heidelbe g,
68167Mannheim, Ge many
130
G.Hesamian e al.
1 In oduc ion
Reg ession analysis can be employed o p edic ou comes, in es iga e ela ionships,
build models, and es hypo heses, making hese models a e sa ile and widely-used
s a is ical echnique. In many si ua ions, howe e , he unde lying da a is imp ecise,
which causes se ious p oblems in common eg ession analysis as da a is assumed o
be c isp. In eal-li e applica ions, da a is o en imp ecise due o e o s (e.g., meas-
u emen and sampling e o s), missing obse a ions, unce ain y, ambigui y o a i-
abili y ha educe he accu acy and eliabili y o he da a. The e o e, i is c ucial
o conside hese aspec s when analyzing he da a o ensu e he alidi y and eli-
abili y o he conclusions. Fo his eason, so / uzzy eg ession analysis has been
in oduced o handle imp ecise da a and ela ed issues. This ield is he mos ac i e
esea ch a ea wi hin uzzy s a is ics and he e a e nume ous p oposals o add ess
uzziness in he amewo k o eg ession models. The mos impo an ields a e pos-
sibilis ic eg ession, leas squa es / leas absolu es eg ession and eg ession models
based on machine lea ning echniques, see he sys ema ic e iew on his opic p o-
ided by Chukh o a and Johannssen (2019).
Among a ious eg ession echniques, quan ile-based eg ession models can be
seen as gene aliza ion o leas squa es es ima ion o condi ional mean models o
p edic condi ional quan ile unc ions. Quan ile-based eg ession models o e a i-
ous bene i s o e common mean-based eg ession models, such as:
• Quan ile-based eg ession models a e less sensi i e o ou lie s in he da a,
because hey es ima e he condi ional dis ibu ion o he esponse a iable a he
han jus he condi ional mean.
• By es ima ing he condi ional dis ibu ion o he esponse a iable, quan ile-
based eg ession models can cap u e changes in he ela ionship be ween he
a iables ac oss di e en pa s o he dis ibu ion, which is why hey allow o a
mo e lexible analysis o he ela ionship be ween he a iables.
• The coe icien s es ima ed by quan ile-based eg ession models ha e a clea
in e p e a ion, as hey ep esen he e ec o he p edic o a iable on he co -
esponding quan ile o he esponse a iable.
• When he dis ibu ion o he esponse a iable is asymme ic, quan ile-based
eg ession models can p o ide a mo e accu a e and nuanced analysis o he ela-
ionship be ween he a iables.
Recen ly, quan ile eg ession models ha e ound hei way in o he li e a u e o
uzzy eg ession analysis. Fi s , A e i (2019) in oduced a quan ile linea eg es-
sion model based on uzzy esponses and uzzy p edic o s by ex ending he con-
en ional quan ile-based loss unc ion o es ima e he uzzy coe icien s. Second,
Hesamian and Akba i (2019) p oposed a uzzy semi-pa ame ic quan ile-based
linea eg ession model, also by conside ing uzzy esponses and uzzy p e-
dic o s. They used bo h he common semi-pa ame ic me hod and he quan ile
eg ession model o he es ima ion o he unknown model pa ame e s. Thi d,
Chachi and Chaji (2021) p esen ed a pa ame e es ima ion me hod o a uzzy
131
A lexible so nonlinea quan ile-based eg ession model
linea eg ession model wi h uzzy esponses and c isp p edic o s h ough ma h-
ema ical p og amming based on a weigh ed agg ega ion o o de ed esiduals.
Howe e , he abo e-men ioned app oaches canno include nonlinea i ies in he
modeling, as hey a e limi ed o he in lexible linea case. In many si ua ions, we
do no know much abou he unde lying na u e o he ela ionship wi hin he se
{yi,xi}
, whe e
xi
=(x
i1
,x
i2
,…,x
ip
)
⊤
. Fo such cases, i is assumed ha he e is a
nonlinea ela ionship be ween
yi
and
xi
, i.e.,
yi= (xi)+𝜖i
,
i=1, 2, …,n
. The e
a e se e al popula pa ame ic echniques such as polynomials, B-splines, Gauss-
ian unc ions, wa ele bases o adial basis unc ions o app oxima e . In he con-
ex o uzzy nonlinea eg ession models wi h c isp p edic o and uzzy esponse
a iables, se e al uzzy pa ame ic (Asadolahi e al., 2021; D’U so & Gas aldi,
2002; De Hie o e al., 2016; Fe a o e al., 2010; Jiang e al., 2013; Hesamian &
Akba i, 2020a) and nonpa ame ic (Cheng & Lee, 1999; Hong & Hwang, 2003;
Wang e al., 2007) nonlinea eg ession models ha e been p oposed.
As he known uzzy quan ile-based eg ession models a e limi ed o he linea
case, we p opose a lexible uzzy nonlinea quan ile-based eg ession model in
his pape . When conduc ing nonlinea eg ession analysis, he app oach de el-
oped by Balasunda am and Meena (2019) is one o he mos e ec i e echniques.
The main p ope y o his app oach is ha i p o ides a simple and e ec i e es i-
ma o o he eg ession unc ion ha is always wi hin he ange o he esponse
a iable. In his pape , we choose his way o pa ame e es ima ion due o i s
inhe en simplici y compa ed o mo e sophis ica ed nonlinea echniques. In pa -
icula , he p oposed uzzy nonlinea quan ile-based eg ession model o e s se -
e al ad an ages and is based on he ollowing inno a i e me hodology:
• I is a combined me hod ha uni es he bene i s o quan ile-based eg ession
and uzzy nonlinea eg ession analysis. Thus, i can handle asymme ies in
he unde lying dis ibu ion, is mo e obus and mo e lexible, leads in gene al
o a highe model i , has a highe p edic i e pe o mance, and i p o ides an
imp o ed explainabili y and hence a be e in e p e abili y o he esul s.
• A ke nel-based nonlinea eg ession model is conside ed and he concep o
uzzy quan ile unc ions o an LR uzzy andom a iable is employed. In his
way, a new nonlinea eg ession echnique based on uzzy quan iles wi h c isp
p edic o s and LR uzzy esponses is de eloped.
• A h ee-s age p ocedu e in ol ing a nonlinea quan ile- and ke nel-based
eg ession model and wo nonlinea ke nel-based eg ession models is sug-
ges ed o pa ame e es ima ion. Wi hin his h ee-s age es ima ion p ocedu e,
some hyb id algo i hms a e implemen ed using gene alized c oss- alida ion,
leas absolu e e o s as well as quan ile loss unc ion.
The pape is s uc u ed as ollows. In Sec .2, necessa y basics on uzzy numbe s
a e b ie ly gi en and he concep o a uzzy quan ile is in oduced. In Sec .3,
he new me hodology is p esen ed o de elop he uzzy nonlinea quan ile-based
eg ession model. A e wa d, in Sec . 4 an algo i hm is discussed o es ima e
he model pa ame e s. Sec ion 5 p esen s ou applica ion s udies whe e he
132
G.Hesamian e al.
e ec i eness o he new eg ession model is compa ed wi h a ious es ablished
uzzy eg ession app oaches. Finally, he pape concludes in Sec .6.
2 Fuzzy numbe s and uzzy quan ile unc ion
In his sec ion some basics a e b ie ly discussed ha we need o de elop he nonlin-
ea quan ile-based eg ession model wi h c isp p edic o s and uzzy esponses.
2.1 Essen ials on uzzy numbe s
A uzzy se

A
is cha ac e ized by a membe ship unc ion
𝜇
A(x)
ha assigns membe -
ship deg ees be ween ze o and one o he elemen s
x∈𝕏
, i.e., i holds
0≤𝜇

A(x)≤1
.
A uzzy numbe (FN)

A
is a uzzy se , which is con ex and no malized on
ℝ
, and
𝜇
A(x)
is an uppe semi-con inuous unc ion. Wi hin he class o FNs so called Le -
Righ FNs (LR-FNs)

A=(aL
,a,aU)
LR
,
aL<a<aU
, wi h
a e mos p ominen as hey p o ide a p omising possibili y o model unce ain y in
a simple and p ac ical way. In (2.1), he con inuous unc ions L(.) and R(.) cha -
ac e ize he le and igh sp eads o he LR-FN and i holds
L(0)=R(0)=1
,
L(1)=R(1)=0
. Le
FLR(ℝ)
deno e he se o all LR-FNs, and LL-FNs s ands o
symme ic LR-FNs. The handling o LR-FNs is especially easy when conside ing a
so called iangula uzzy numbe (TFN) cha ac e ized by
Le

A=(a
L
,a,a
U
)LR
and

B=(b
L
,b,b
U
)LR
be wo LR-FNs, hen he ollowing ope -
a ions a e de ined:
• Addi ion:
�
A⊕�
B=(a
L
+b
L
,a+b,a
U
+b
U
)LR
.
• Scala mul iplica ion:
The squa ed dis ance be ween wo LR-FNs

A
and

B
is de ined by
(2.1)
𝜇
�
A(x)=
⎧
⎪
⎨
⎪
⎩
L
�
a−x
a−aL
�
i aL≤x≤a
,
R
�
x−a
aU−a
�
i a<x<aU
(2.2)
𝜇

A(x)=
⎧
⎪
⎨
⎪
⎩
x−aL
a−aLi aL≤x≤a,
aU−x
a+aUi a≤x≤aU,
0 i x−[aL,aU]
.
(2.3)
𝜆⊗
�
A=
{
(𝜆a
L
,𝜆a;𝜆a
U
)LR i 𝜆>
0,
(𝜆aU,𝜆a;𝜆aL)
RL
i 𝜆<
0.

133
A lexible so nonlinea quan ile-based eg ession model
wi h
c1
=
∫1
0
L−1(𝛼)d
𝛼
and
c2
=
∫1
0
R−1(𝛼)d𝛼.
The e a e se e al app oaches o ank uzzy quan i ies based on a eal- alued c i-
e ion o a p e e ence deg ee. He e, a simple anking c i e ion is sugges ed o com-
pa e wo LR-FNs.
De ini ion 2.1 Le

A=
(aL
,
a
,a
U
)
LR
and

B=
(
b
L
,b,b
U
)
LR
, hen i holds

A⪰

B
i
aL≥bL
,a≥b
and
aU≥bU
.
Fu he , he pa ial o de “
⪰
” mee s he p ope ies below o h ee LR-FNs

A
,

B
and

C
:
(1)

A⪰

A
.
(2) I

A⪰

B
and

B⪰

A
hen

A=

B
.
(3) I

A⪰

B
and

B⪰

C
hen

A⪰

C
.
Finally, he e a e di e en ways in exp essing an LR-FN, e.g.,

A=(a
L
,a,a
U
)LR
is
equi alen o

A=(a;la, a)LR
wi h
la=a−aL
and
a=aU−a
.
2.2 Fuzzy quan ile unc ion
Fi s , we de ine LR uzzy andom a iables (LR-FRVs).
De ini ion 2.2 Le
(Ω,A,P)
be a common p obabili y space. Then he uzzy-
alued mapping

Y∶Ω
→F
LR(
ℝ
)
is e e ed o as LR-FRV i he mappings
YL
,Y,YU∶Ω
→
ℝ
a e andom a iables (RVs) wi h

Y=(Y
L
,Y,Y
U
)LR
and
P(YL<Y<YU)=1
. LR-FRVs

Y1
and

Y2
a e i.i.d. i
(Y1,Y2)
,
(
Y
L
1
,Y
L
2)
and
(
Y
U
1
,Y
U
2)
a e i.i.d. RVs. In addi ion,

Y1,…,

Yn
is an LR uzzy andom sample i all he

Yi
’s a e
i.i.d. LR-FRVs. An obse ed LR uzzy andom sample is deno ed by
y1,…,yn
.
Two ypes o LR-FRVs ha a e commonly employed a e (1)

Y
= ((1−aY
1
)Y,Y,(1+aY
2
)Y)
LR
o a posi i e RV Y wi h
aY
1
,a
Y
2
∈(0, 1
]
and (2)

Y
=(Y−bY
1
,Y,Y+bY
2
)
LR
, whe e ze o is pa o he suppo o Y and
bY
1
,b
Y
2
>
0
.
In his pape we employ he no ion o a uzzy cumula i e dis ibu ion unc ion
(c.d. .) inspi ed by Hesamian and Chachi (2015):
De ini ion 2.3 Le
YL
, Y and
YU
be common RVs wi h c.d. .
FYL
,
FY
, and
FYU
,
espec i ely, whe e
P(YL<Y<YU)=1
. The uzzy c.d. . o

Y
a
y=(yL
,y,yU)LR
is
hen de ined in he way
D
2(

A,

B)=(
|
a−b
|2
+c1
|
(a−a
L
)−(b−b
L
)
|2
+c2
|
(a
U
−a)−(b
U
−b)
|2
)
3,
(2.5)

F

Y
(y)=(F
Y
U(yL),F
Y
(y),F
Y
L(yU))
LR,
134
G.Hesamian e al.
whe e
FY
U(y
L
)=P(Y
U≤
y
L)
,
FY(y)=P(Y≤y)
and
FY
L(y
U
)=P(Y
L≤
y
U)
.
De ini ion 2.4 Le
YL
, Y and
YU
be common RVs wi h c.d. .
FYL
,
FY
, and
FYU
,
espec i ely, whe e
P(YL<Y<YU)=1
. The uzzy quan ile unc ion o

Y
a uzzy
quan ile le el
𝜏 =(
𝜏
L,
𝜏
,
𝜏
U)LR ∈
F
LR(0, 1)
is de ined by
wi h
Acco ding o he de ini ion o “
⪰
” (see De ini ion 2.1), we can s a e some p ope -
ies o he uzzy quan ile o an LR-FRV:
Lemma 2.1 Le

Y
=(YL
,Y,YU)
LR
hen i holds:
(1)

Q
Y
(

F

Y
(y)) =
y
o any
y
=(y
L
,y,y
U
)
LR
(2)

F
Y
(

Q

Y
(𝜏 )) ⪰ 𝜏 o any
𝜏
=(𝜏
L
,𝜏,𝜏
U
)
L
(3)
�
QI(Y)⊕
�
b
(𝜏)=
�
b⊕I(QY(𝜏
))
o any RV Y and le el
𝜏∈(0, 1)
wi h

b
=(bL
,b,bU)
LR
P oo Acco ding o De ini ions 2.3 and 2.4, we ha e
which p o es (1). To p o e (2), i s , no e ha
In addi ion, i holds
Acco ding o De ini ion 2.1, i can be concluded ha

F
Y
(

Q

Y
(𝜏 )) ⪰ 𝜏 . Mo eo e , o
p o e (3), acco ding o a i hme ic ope a ions o LR-FNs, we ind ha
◻
(2.6)

Q

Y
(𝜏 )=(F−1
Y
U(𝜏L),F−1
Y
(𝜏),F−1
Y
L(𝜏U))
LR,
(2.7)
F
−1
YU(𝜏
L
)=in {y∶FYU(y)≥𝜏
L
}
,
F−1
Y(𝜏)=in {y∶FY(y)≥𝜏},
F
−1
Y
L(𝜏U)=in {y∶F
Y
L(y)≥𝜏U}
.
(2.8)

Q

Y
(

F
Y
(y))=(F−1
YU
(F
Y
U(yL)),F−1
Y
(F
Y
(y)),F−1
YL
(F
Y
L(yU)))
LR
=(yL,y,yU)
LR
=y
,
(2.9)

F

Y
(

Q
Y
(

))=(F
Y
U(F−1
Y
U(𝜏L)),F
Y
(F−1
Y
(𝜏)),F
Y
L(F−1
Y
L(𝜏U)))
LR.
(2.10)
FY
U(F
−1
Y
U(𝜏
L
)) ≥𝜏
L
,F
Y
(F
−1
Y
(𝜏)) ≥𝜏,F
Y
L(F
−1
Y
L(𝜏
U
)) ≥𝜏
U.
(2.11)
�
Q
Y⊕�
b(𝜏)=(F−1
Y+bL(𝜏),F−1
Y+b(𝜏),F−1
Y+bU(𝜏))LR
=(bL+QY(𝜏),QY(𝜏),bU+QY(𝜏))
LR
=
�
b
⊕
I(QY(
𝜏
)).
135
A lexible so nonlinea quan ile-based eg ession model
Following Lemma 2.1, i is ob ious ha he uzzy quan ile ex ends he con en-
ional p ope ies o quan ile unc ions. This is ue because we used he anking c i-
e ion
⪰
in his s udy.
Example 2.1 Conside an FRV

Y=(Y
L
,Y,Y
U
)LR
, whe e
YL∼exp(
𝜆
L)
,
Y∼exp(𝜆)
,
YU∼exp(
𝜆
U)
wi h
𝜆L<𝜆<𝜆
U
. Fi s no e ha
The e o e, acco ding o De ini ion 2.4, we ha e
wi h
Example 2.2 Le
�
Y=�𝜇 ⊕ I(𝜖)
whe e
𝜇 =(𝜇L
,𝜇,𝜇U)LR
and
𝜖∼N(0, 𝜎2)
. Acco ding
o Lemma 2.1 (3), i holds
�
Q�
Y
(𝜏)=�𝜇 ⊕ I(Z
1−𝜏
𝜎)=(𝜇L+Z
1−𝜏
𝜎,𝜇+Z
1−𝜏
𝜎,𝜇U
+
Z1−𝜏
𝜎
)LR
. He e,
Z1−𝜏
is he
(1−𝜏)
-quan ile o he s anda d no mal dis ibu ion.
3 The uzzy nonlinea quan ile‑based eg ession model
Le
{(
x
i
,

y
i
)}
i=1,…,n
be a uzzy da a se , whe e x
i
=(x
i1
,…,x
im
)
⊤
( ixed alues) and
yi
∈F
LR
(ℝ
)
a e obse ed alues o he FRVs. Inspi ed by Balasunda am and Meena
(2019), we conside he ollowing uzzy (ke nel-based) nonlinea eg ession model
whe e
1.
Kh(xi,A)=(Kh(xi,x1),…,Kh(xi,xn))
wi h bandwid h h and ke nel unc ion
Kh(., .)
,
2.

Yi
=(YL
i
,Y
i
,YU
i
)
LR
,
3.

w=(
w
L
,
w
,
w
U)LR
wi h
wL
=(w
L
1
,…,w
L
n
)
⊤
,
w=(w1,…,wn)⊤
and
wU
=(w
U
1
,…,w
U
n
)
⊤
.
4.
𝜖 i=𝜇 +I(𝜖i)
indica es an LR uzzy e o e m, whe e
𝜖1,…,𝜖n
a e i.i.d. no mally
dis ibu ed RVs wi h ze o mean and a iance
𝜎2
, and i holds
𝜇 =(𝜇L
,𝜇,𝜇U)LR
.
Acco ding o De ini ion 2.4, he p oposed model (3.1) can be w i en in he o m
(2.12)
FY
(y)=1−e−𝜆y,F
Y
L(y)=1−e−𝜆
L
y,F
Y
U(y)=1−e−𝜆
U
y
.
(2.13)

Q

Y
(𝜏 )=(F
−
1
Y
U(𝜏L),F
−
1
Y
(𝜏),F
−
1
Y
L(𝜏U))
LR,
(2.14)
F−1
YU(𝜏L)=in {y∶FYU(y)≥𝜏L}=
−1
𝜆Ulog(1−𝜏L),
F−1
Y(𝜏)=in {y∶FY(y)≥𝜏}=−1
𝜆log(1−𝜏),
F
−1
YL(𝜏U)=in {y∶FYL(y)≥𝜏U}=−1
𝜆
Llog(1−𝜏U)
.
(3.1)
�
Yi=(
K
h(
x
i,A)⊗

w
)⊕ �𝜖 i,i=1, …,n,
136
G.Hesamian e al.
Now, he uzzy quan ile o an FRV in oduced in Sec .2.2 is u ilized o ob aining
he uzzy quan ile o he model (3.2). Following Lemma 2.1, he uzzy quan ile o

Yi
can be e alua ed a he exac quan ile le el
𝜏
in he way
whe e
�
b
(𝜏)=�𝜇 ⊕ I(Z
1−𝜏
𝜎)=(𝜇L+Z
1−𝜏
𝜎,𝜇+Z
1−𝜏
𝜎,𝜇U+Z
1−𝜏
𝜎)
LR
. Since
K(xi,A)
a e posi i e quan i ies, he igh -hand side o (3.3) is an LR-FN. Based on
he second no a ion o LR-FNs (see Sec .2.1, las pa ag aph), he uzzy quan ile o

Yi
can be ew i en as
whe e
w=(w1,…,wn)⊤
,
l
w=(l
w1
,…,l
wn
)
⊤
,
w=(
w1
,…,
wn
)
⊤
.
Conside ing he equali y o wo LR-FNs in (3.4), h ee sepa a e (nonlinea )
eg ession models can be s a ed in he ollowing way:
(L)
l
QY
i
=K
hl
(x
i
,A)l
w
+lb
,
i=1, …,n
,
(C)
QYi(𝜏)=Kh(xi,A)w+b(𝜏)
,
i=1, …,n
,
(R)
QY
i
=K
h
(x
i
,A)
w
+ b
,
i=1, …,n
.
The pa ame e s

w
=(
�
w
1
,…,
�
w
n
)
⊤
,
𝜏
, h and

b
(𝜏)=(b(𝜏);l
b
,
b
)
LR
can be es ima ed by
conside ing h ee o dina y nonlinea eg ession models L, C, R. To achie e his, we
decompose he aining pa e ns
{(x
i
,y
i
=(y
i
;l
y
i,
y
i)
LR
)}
in o h ee da a se s
(l
y
i,x
i
)
,
(yi,xi)
, and
(
y
i,x
i
)
,
i=1, …,n
. Thus, he cen e and he sp eads o he uzzy coe -
icien s
w
=(w
1
,…,w
n
)
⊤
,
𝜏
, h and

b(𝜏)
can be sepa a ely es ima ed in each s ep. All
eg ession coe icien s should be simul aneously es ima ed in each s ep. No e ha
L and R p o ide wo nonlinea eg ession models, while C is a nonlinea quan ile-
based eg ession model.
4 Pa ame e es ima ion
In o de o es ima e he model pa ame e s, he ollowing s eps can be implemen ed
o he nonlinea eg ession models L, C, R de ined in Sec .3.
4.1 Es ima ing hepa ame e s o model C
As o pa ame e es ima ion ela ed o he eg ession model
QYi(𝜏)=Kh(xi,A)w+b(𝜏)
(model C) using he da a
(yi,xi)
,
i=1, …,n
, h ee a ge unc ions a e equi ed o
(3.2)
�
Yi
=
(
(K
h
(x
i
,A)⊗
w)⊕ �𝜇
)
⊕I(𝜖
i
)
.
(3.3)
�
Q
�
Yi
(𝜏)=
(
(Kh(xi,A)⊗
w)⊕ �𝜇
)
⊕I(Z1−𝜏𝜎)) = (K(xi,A)⊗
w)⊕
�
b(𝜏)
,
(3.4)

Q

Y
i
(𝜏)=(Kh(xi,A)w+b(𝜏);K(xi,A)lw+lb,K(xi,A) w+ b)LR
,
143
A lexible so nonlinea quan ile-based eg ession model
lowe alue o
RMSE
o he in oduced model in his pape . Thus, he p edic i e
pe o mance o he newly p oposed uzzy nonlinea quan ile-based eg ession model
is be e compa ed o he o he models, and he e o e i is supe io o all he compe i-
o s o he in es iga ed da a se in his example.
Example 5.2 This example is based on a uzzy da a se ha is ela ed o he de elop-
men o a sophis ica ed so wa e sys em o un a i ual mall, whe e he p og am-
me s wo ked in pa allel wi h a eam o se en be a es e s (D’U so & Gas aldi, 2002;
Hesamian e al., 2024). Du ing he 30-mon hs so wa e de elopmen , he eam o
be a es e s ha e w i en comp ehensi e mon hly epo s. In any o hese epo s, he
eam p o ided an o e all a e age sco e in he o m o a symme ic TFN. In he ol-
lowing, we use he uzzy uni a ia e nonlinea eg ession model
o he abo e da a se , whe e
K(xi,A)=(K(xi,x1),…,K(xi,xn))
and
The op imal alues o h,
hl
,
𝜏
,

b
(
𝜏
)=(
b
(
𝜏
);
l
b)T
, and

w
can be de e mined by means
o he ollowing wo s ages:
(L)
l
QY
i
=K
hl
(x
i
,A)l
w
+lb
,
i=1, …, 30
,
(5.5)
�
Yi
=(K
h
(x
i
,A)⊗
w)⊕ �𝜖
i
,i=1, …
, 30,
(5.6)
K
h(xi,xj)=
√
1
2𝜋
e−0.5 (xi−xj)2
h
.
Table 1 Values o

yi
=(y
i
;l
yi
)
LL
,
S
(y
i
,

y
i)
and D2(y
i
,

y
i)
o he
p oposed me hod in Example
5.1
No

yi
=(y
i
;l
yi
)
LL
S
(
yi
,


yi)
D
2(
yi
,


yi)
1
(12.30;1.845)LL
1 0
2
(20.90;3.135)LL
1 0
3
(39;5.85)LL
1 0
4
(47.90;0.185)LL
1 0
5
(5.60;0.84)LL
1 0
6
(25.90;3.885)LL
1 0
7
(37.30;5.595)LL
1 0
8
(21.90;3.285)LL
1 0
9
(18.10;2.715)LL
1 0
10
(21;3.15)LL
1 0
11
(34.90;5.23)LL
1 0.00005
12
(57.20;8.58)LL
1 0
13
(0.69;1.0389)LL
0.527 1.74
14
(25.90;3.885)LL
1 0
15
(54.90;8.235)LL
1 0

144
G.Hesamian e al.
(C)
Q
Y
i(𝜏)=K
h
(x
i
,A)w+b(𝜏)
,
i=1, …, 30
.
Tha is, (1) he op imal alues o
hl
,
l
wh
l
and
lb
can be e alua ed ia s eps L1–L2
based on he obse ed alues
(l
y
i,x
i
)
,
i=1, …, 30
, and (2) he unknown model
componen s h,
wh
and
b(𝜏)
can be es ima ed based on s eps C1–C4 by using he
obse ed da a se
(yi,xi)
,
i=1, …, 30
. The alues o


y
i
=(y
i
;l
yi
)
LL
,
D
2(y
i
,

y
i)
, and
S
(y
i
,

y
i)
a e gi en in Table3. In his example, we compa e ou me hod wi h he
uzzy eg ession models by De Hie o e al. (2016), D’U so and Gas aldi (2002),
Table 2 Compa a i e analysis in Example 5.1
Me hod Model componen s
MSM
RMSE
D’U so and Gas aldi (2002)

y
(x)=(

(x);l
(x)
)
LL

(
x
)=−120 +38.76
x1
−2.415
x2
+3.26
x3
0.63 5.36
l

(x)=10.5 +6.25x
1
+1.71x
2
−4.46x3
Kula and Apaydin (2008)


𝛽0
= (−120.36;24.26)
LL
,


𝛽1
=(32.95;7.21)
LL


𝛽2
= (−4.01;1.22)
LL
,


𝛽3
=(2.62;2.64)
LL
0.618 5.75
Wang e al. (2007)

h=8
0.639 5.239
Hesamian and Akba i (2020b)

h=2
,

k=8
0.702 4.55
Hong and Hwang (2003)

w0
= (−145.32;18.21)
LL
,


w1
=(29.51;9.29)
LL

w2
= (−3.21;1.44)
LL
,

w3
=(3.41;3.24)
LL
0.65 5.15
Asadolahi e al. (2021)

w0=(
11.80;1.30
)LL
,
w1= (−
1.04;0.0006
)LL
w2=(2.61;0.33)LL
,

w3
=(3.74;2.60)
LL

w4
=(6.43;0)
LL
,


w5
=(0.08;0)
LL

w6=(
3.79;0
)LL
,

w7=(
4.46;0.20
)LL
0.734 3.59

w8
=(6.21;0.0001)
LL
,


w9
=(1.21;0)
LL

w10
=(2.54;0)
LL
,

w11
=(6.39;0.95)
LL

w12 =(
6.37;3.36
)LL
,

w13 = (−
1.14;0
)LL

w14
=(2.39;0.69)
LL
,

w15
=(7.32;0.52)
LL
P oposed model


b=(
26.68;0.10
)LL,

w1=(
20.17;0.12
)LL,

w2
= (−23.79;0.784)
LL,


w3
=(19.24;0.54)
LL

w4
=(27.42;1.63)
LL,

w5
= (−25.67;0.26)
LL,

w6= (−
2.91;0.50
)LL,

w7=(
11.49;0.17
)LL
0.927 1.899

w8
= (−39.73;0.99)
LL,


w9
=(7.57;0.44)
LL,

w10
= (−6.89;0.52)
LL,


w11
=(3.84;0.43)
LL

w12 =(
49.72;4.37
)LL,

w13 = (−
40.40;0.13
)LL,

w14
=(12.51;0.72)
LL
,


w15
=(26.48;4.73)
LL
𝜏 =0.7
,

h=4
,

hl
=
1
145
A lexible so nonlinea quan ile-based eg ession model
Kula and Apaydin (2008), Chachi e al. (2016), Wang e al. (2007), Fe a o e al.
(2010), Hesamian and Akba i (2020a, 2020b), Hong and Hwang (2003), and Asa-
dolahi e al. (2021). In Table4 he esul s o ou compa a i e analysis including
es ima ed model pa ame e s and pe o mance measu es can be seen. Analyzing
he esul s in Table4, we ind he highes alue o MSM and he lowes alue o
RMSE o ou p oposed eg ession model (
MSM =0.82
,
RMSE =1.46
). Thus,
he new model p o ides a highe deg ee o simila i y on he one hand and a lowe
p edic ion e o on he o he hand in compa ison o he uzzy eg ession models
unde conside a ion, con i ming he supe io i y o he nonlinea quan ile-based
eg ession model o e he o he app oaches.
Table 3 Values o

yi
=(y
i
;l
yi
)
LL
,
S
(y
i
,

y
i)
and D2(y
i
,

y
i)
o he
p esen ed model in Example 5.2
No

yi
=(y
i
;l
yi
)
LL
S
(
yi
,


yi)
D
2(
yi
,


yi)
1
(6.0293;1.99721)T
0.798339 0.0851244
2
(5;3.4)T
1 0
3
(8;5.3)T
1 0
4
(13;2.4)T
0.0140845 9.48
5
(19;5.331)T
0.0257109 38.8636
6
(19;5.6)T
1 0
7
(20;4.3)T
1 0
8
(25.3461;3.01)T
0.245816 5.53222
9
(25;2.2)T
1 0
10
(26.9619;4.35)T
0.982134 0.00228494
11
(26;5.1)T
1 0
12
(23.8419;3.0245)T
0.54447 1.8018
13
(5.11;1.2843)T
0.813722 0.0234222
14
(8;2.1)T
1 0
15
(10;5.02393)T
0.570272 2.10891
16
(10;4.1)T
1 0
17
(13;4.498)T
0.597673 1.5616
18
(13;3.5)T
1 0
19
(20;6.4)T
1 0
20
(28;7.7)T
1 0
21
(29;5.2)T
1 0
22
(30.0157;3.85790)T
0.89707 0.0653973
23
(35;5.2)T
1 0
24
(44;6.3)T
1 0
25
(44.3;6.82091)T
0.81555 0.506267
26
(43.0991;5.15)T
0.962597 0.0106541
27
(47;4.2)T
1 0
28
(48;5.3)T
1 0
29
(49.0162;5.13680)T
0.693062 1.88994
30
(48.36;5.3896)T
0.795188 0.413663
146
G.Hesamian e al.
Table 4 Compa a i e analysis in Example 5.2
Me hod Model componen s MSM RMSE
De Hie o e al. (2016)

y
(
x
)=(

(
x
);
s
(
x
))
T

(
x
)=−14.419 +13.516
x
−1.724
x
2−0.081
x
3−0.001
x4
0.39 9.87
s(x)=0.581 +2.29x−0.26x2+0.01x3−0.0001x4
D’U so and Gas aldi (2002)

y
(x)=(

(x);l
(x)
)
T

(x)=
5.31
+
2.52
x−
0.18
x
2
+
0.005
x3
0.31 12.10
l

(x)=5.96 −0.03x
Kula and Apaydin (2008)


𝛽0
=(3.18;1.12)
T
,


𝛽1
=(1.5;0.4)
T


𝛽2=(
0.19;0.13
)T
0.31 12.27
Chachi e al. (2016)


𝛽0
=(2.45;0.52)
T
,


𝛽1
=(0.66;0.17)
T


𝛽2=(
0.000001;0
)T
0.28 13.73
Wang e al. (2007)

h=0.08
0.41 10.49
Fe a o e al. (2010)

y
(x)=(y(x);l
y(x)
)
T
y(x)=−18.21 +14.74x−1.66x2+0.072x3−0.001x4
l

y
(
x
)=−0.17 +2.29x−0.17x
2
+0.004x
3
−1.96 ×10
−5
x
4
0.43 10.23
Hesamian and Akba i
(2020a)


𝛽0
=(0.019;1.56)
T
,


𝛽1
=(1.68;0.232)
T


𝛽2=(
0.307;0
)T
,


b1= (−
1.024;0
)T


b2
=(1.85;0)
T
,


b3
= (−1.198;0.00001)
T
0.42 9.82

K=3
,
1=5, 2=12, 3=16
,

𝜆=1
Hesamian and Akba i
(2020b)
h=5
,
k=1
0.40 10.208
Hong and Hwang (2003)

w0= (−
14.41;0.58
)T
,

w1=(
13.51;2.29
)T

w2
= (−1.72;0.26)
T
,

w3
=(0.08;0.01)
T
0.46 8.24
Asadolahi e al. (2021)

w0
=(10.72;0.93)
T
,

w1
= (−2.72;0)
T
w2= (−
2.42;0
)T
,

w3= (−
1.59;0.132
)T

w4
= (−0.89;0.882)
T
,

w5
= (−0.1;1.35)
T

w6
=(0.245;1.480)
T
,


w7
=(2.64;1.32)
T

w8=(
3.55;1
)T
,

w9=(
4.19;0.665
)9T

w10
=(4.83;0.402)
T
,

w11
=(3.06;0.242)
T

w12
=(1.07;0.170)
T
,

w13
=(0.33;0.148, 0)
T

w14 = (−
2.84;0.158
)T
,

w15 = (−
2.94;0.211
)T
0.55 7.47

w16
= (−3.30;0.333)
T
,

w17
= (−1.78;0.529)
T

w18
=(0.94;0.764)
T
,

w19
=(1.46;0.954)
T

w20 =(
2.11;0.991
)T
,

w21 =(
2.67;0.807
)T

w22
=(3.89;0.437)
T
,

w23
=(2.23;0.001)
T

w24
=(2.11;0)
T
,

w25
=(2.82;0)
T
147
A lexible so nonlinea quan ile-based eg ession model
Example 5.3 In his hi d example we conside a eal da a se wi h 21 obse a ions
ha aims o analyze he ela ionship be ween he a mosphe ic concen a ion o ca -
bon monoxide (
y
) and h ee me eo ological p edic o a iables in Rome: humidi y
(
x1
), ain (
x2
), wind speed (
x3
) (Hesamian e al., 2024). The esponse a iable has
been epo ed in he o m o LR-FNs wi h
L(x)=exp(−x)
,
R(
x
)=1∕(1+
x
2)
, while
he p edic o a iables ha e c isp alues. Wang e al. (2007) employed he ollowing
uzzy mul iple eg ession model o cap u e he ela ion be ween
y
and
x1,x2,x3
:
He e, we in es iga e he abo e ela ionship by implemen ing he ollowing model:
The esul s o he compa a i e analysis a e summa ized in Table 5. We conside
13 compe i o s o a ious kinds in o de o ob ain a ep esen a i e and comp ehen-
si e compa ison: Choi and Yoon (2010), Zeng e al. (2017), Tahe i and Kelkinnama
(5.7)
�
yi=�w0⊕
3
⨁
j=1
(�wij ⊗xij)⊕ �𝜀 i,i=1, …
, 21
(5.8)
�
Yi
=(K
h
(x
i
,A)⊗
w)⊕ �𝜖
i
,i=1, …
, 21
Table 4 (con inued)
Me hod Model componen s MSM RMSE

w26 =(
3.56;0
)T
,

w27 =(
5.51;0.178
)T

w28
=(7.25;0.686)
T
,

w29
=(8.92;1.136)
T

w30
=(9.51;1.418)
T
P oposed model

b
(𝜏 )=(17.280;0, 0)
T
,w
1
= (−11.3173;0.58, 7.01)
T,
w2= (−
7.1110;1.33, 0
)T
,

w3= (−
6.5733;0.71, 4.48
)T,

w4
= (−4.0607;5.14, 4.45)
T
,


w5
= (−7.4598;5.45, 2.73)
T,

w6
=(3.8772;0, 4.07)
T
,

w
7
= (−4.9458;3.87, 0.91)
T

w8=(
4.7932;6.21, 2.07
)T
,

w9= (−
2.8439;0, 2.36
)T,

w10
=(10.0855;2.42, 5.71)
T
,


w11
= (−8.9716;5.00, 0)
T

w12
=(21.2993;0.01, 4.66)
T
,

w
13
= (−28.1973;1.18, 0)
T,

w14 =(
5.1703;0.91, 0
)T
,

w15 = (−
10.9986;5.58, 4.07
)T,

w16
= (−3.8492;0, 0)
T
,


w17
= (−3.7330;3.02, 4.12)
T,
0.82 1.46

w18
= (−5.7769;0, 5.86)
T
,

w
19
= (−1.5642;4.25, 0)
T

w20 =(
6.6130;0.93, 6.11
)T
,

w21 =(
1.9484;6.53, 3.15
)T,

w22
=(4.8021;0, 4.22)
T
,


w23
=(3.3954;0.63, 0)
T

w24
=(14.2488;0, 2.17)
T
,

w
25
=(12.2567;4.26, 3.72)
T,

w26 =(
6.5624;1.97, 0
)T
,

w27 =(
15.0507;0.26, 7.96
)T,

w28
=(11.9609;1.17, 0)
T
,


w29
=(11.2484;3.44, 1.57)
T

w30
=(21.4031;2.51, 2.50)
T
,𝜏 =0.7, 
h=
h
l
=
0.7
148
G.Hesamian e al.
Table 5 Compa a i e analysis in Example 5.3
Me hod Model componen s MSM RMSE
Choi and Yoon (2010)


w0
=(0.769;0.361, 0.905)
T
,

w1
=(0.048;0.012, 0.002)
T
w2= (−
0.644;1.101, 0.775
)T
,

w3
= (−0.988;0.520, 0.232)
T
0.54 11.10
Zeng e al. (2017)

w0=(
1.53;0.147, 1.50
)T
,

w1=(
0.037;0.024, 0.002
)T

w2
=(0.753;0.189, 0.234)
T
,

w3= (−
1.025;0.009, 0.005
)T
0.58 9.025
Tahe i and Kelkinnama
(2012)


w0
=(1.34;1.12, 0.87)
T
,


w1
=(0.142;0.023, 0)
T
w2= (−
2.750;0, 0
)T
,

w3= (−
1.038;0, 0
)T
0.50 11.69
Kula and Apaydin (2008)

w0=(
0.535;0.361, 0.014
)T
,

w1=(
0.048;0.018, 0.014
)T
w2= (−0.835;0.398, 0)T
,

w3
= (−0.904;0, 0.020)
T
0.56 9.87
Choi and Buckley (2008)

w0=(
2.99;0.101, 0.020
)T
,

w1=(
0.029;0.023, 0.020
)T
w2= (−
1.278;0, 0
)T
,

w3= (−
1.14;0, 0
)T
0.55 10.36
Wang e al. (2007)

h=5.25
0.58 9.03
Jiang e al. (2013)


w0
=(0.73;0.33, 0.90)
T
,


w1
=(0.69;0, 0)
T
w2
= (−0.49;0.83, 0.51)
T
,

w3
= (−1.03, 0.70, 0.079)
T
,

h=0.5
0.51 12.93
A alay e al. (2015)


w0
=(0.39;0.60, 0.41)
T
,


w1
=(0.106;0, 0.008)
T
w2= (−
0.59;0.52, 0.005
)T
,

w3= (−
1.049;0, 0.016
)T
0.60 8.63

h=
0.5,

k1=

k2=1
Icen and Demi han (2016)

w0
=(1.37;0.172, 0.012)
T
,

w1
=(0.037;0, 0.012)
T
w2= (−0.75;0.143, 0.193)T
,

w3
= (−1.02;0, 0.011)
T
0.57 9.53
Hesamian and Akba i (2020b)

h=1
,

k=0.5
0.53 10.73
A e i (2019)

w0=(
0.341;1.034, 0.821
)T
,

w1=(
0.043;0.121, 0.142
)T

w2= (−
0.721;0.052, 0.072
)T
,

w3
= (−0.008;0.010, 0.012)
T
,
𝜏 =0.5
0.56 8.54
D’U so and Gas aldi (2002)


y(x)=( (x);l
(x)
,
(x)
)
T
(x)=0.420152 +0.038745x1−0.82130129x2−1.02135975x3
l
(
x
)
=0.05 +0.0286x
1
−0.200x
2
−0.05219x3
(x)=−0.384175858 +0.0329x
1
−0.0225264x
2
−0.322588x3
0.502 11.01
Asadolahi e al. (2021)

w0=(
3.605;0, 0
)T
,

w1= (−
4.010;0, 0
)T
w2
= (−
2.1610;0, 0
)
T
,

w3
=(0.9986;0, 0)
T


w4
=(3.2599;1.60, 1.55)
T
,

w5
=(0.9369;1.67, 1.80)
T

w6= (−
0.7129;0.165, 0
)T
,

w7= (−
1.4941;0.71, 0
)T


w8
=(3.9898;0.90, 0.74)
T
,

w9
=(2.2226;0.49, 1.08)
9T


w10
=(0.6054;0, 0.07)
T
,


w11
= (−0.3640;0.42, 0.65)
T
0.64 7.23

w12 =(
1.1437;1.53, 2.17
)T
,

w13 =(
0.3731;1.60, 0.14
)T

149
A lexible so nonlinea quan ile-based eg ession model
(2012), Kula and Apaydin (2008), Choi and Buckley (2008), Wang e al. (2007),
Jiang e al. (2013), A alay e al. (2015), Icen and Demi han (2016), Hesamian and
Akba i (2020b), A e i (2019), D’U so and Gas aldi (2002), and Asadolahi e  al.
(2021). In his example we obse e he same clea pic u e as in he p e ious exam-
ples: he highes alue o MSM and he lowes alue o RMSE can be ound o ou
newly p oposed uzzy eg ession model. Tha is, conside ing he deg ee o simila -
i y as well as he p edic i e pe o mance o all he compa ed models, we ha e a
clea bes me hod, and his is he uzzy nonlinea quan ile-based eg ession model
wi h c isp p edic o and uzzy esponse a iables.
Example 5.4 In his example, we analyze a la ge da a se o Shanghai housing p ice
da a (Hesamian e al., 2024). The uzzy dependen a iable ep esen s he accep -
able pu chase p ice, exp essed by TFNs
y
i
=(y
i
;l
y
i,
y
i)
T . Addi ionally, he e a e six
c isp explana o y a iables: housing size (
x1
), mo gage in e es a e (
x2
), eal es a e
ax (
x3
), down paymen a io (
x4
), annual household income (
x5
), amily popula ion
(
x6
). Acco ding o he p oposed me hod, he uzzy p edic ed alues can be de e -
mined ia

yi
=(y
i
;l
yi
,
yi
)
T
,
i=1, 2, …, 147
, whe e
Table 5 (con inued)
Me hod Model componen s MSM RMSE


w14
= (−2.3193;0.43, 0.41)
T
,

w15
= (−0.9893;0, 0)
T


w16
= (−0.5327;0, 0)
T
,

w17
= (−1.9571;0, 0)
T


w18
= (−1.2844;1.50, 0.93)
T
,

w19
= (−1.2687;0, 0)
T

w20 = (−
0.9986;0, 0
)T
,

w21 =(
0.8715;0, 0
)T
P oposed model


b
(
𝜏
)=(2.6461;0, 0)T,

w
1= (−1.68;0, 0)T,

w
2=(0.0769;0.53, 0.30)T,

w3=(
0.7219;0.53, 0.76
)T
,

w4=(
3.2956;0, 1.39
)T,

w5=(
1.0787;1.42, 1.42
)T
,

w6= (−
1.1128;1.80, 0
)T,


w7
= (−0.1529;0.79, 0.24)
T
,

w
8
=(0.6294;0.21, 1.39)
T,


w9
=(0.7415;1.41, 0.98)
T
,

w
10
=(1.3161;0.97, 1.50)
T,
0.87 4.67

w11 =(
0.0930;1.35, 0.98
)T
,

w12 =(
1.1248;1.62, 2.05
)T,


w13
= (−0.6108;2.42, 0.84)
T
,

w
14
= (−1.5312;0, 0)
T,

w15 =(
0.9069;0, 0.26
)T
,

w16 = (−
3.0121;0, 0
)T,


w17
= (−0.3858;0.35, 0.57)
T
,

w
18
= (−0.4612;1.24, 0.68)
T,


w19
= (−2.0747;0.18, 0)
T
,

w
20
=(0.3412;0, 0)
T,

w21 =(
1.4886;1.49, 0.06
)T
𝜏 =
0.7,

h=
hl=
h =0.7
150
G.Hesamian e al.
(L)
l
y
i=K

hl(x
i
,A)l

w
+l
b
,
(C)

y
i
=K

h
(x
i
,A)
w+

b(𝜏
)
,
(R)
y
i=K

h
(x
i
,A)

w
+ 
b
wi h
xi
=(x
i1
,x
i2
,…,x
i6
)
⊤
. The pa ame e s
𝜏
,

hl
,

h
,

h
,
l
w
=
(
l
�w1
,l
�w2
,…,l
�w147
)
⊤
,

w
=
(�
w
1
,
�
w
2
,…,
�
w
147
)
⊤
,

w
=
(
�w1
,
�w2
,…,
�w147
)
⊤
,
l
b
,

b(𝜏 )
and

b
can be es ima ed ia
Table 6 Es ima ed coe icien s and pe o mance measu es o he conside ed uzzy eg ession models in
Example 5.4
Me hod Model componen s
MSM
RMSE
Zeng e al. (2017)

w0
= (−570.59;5.236, 7.20)
T
,

w1
=(8.243;1.773, 0.982)
T

w2= (−
7.238;0.3058, 0.8429
)T
,

w3= (−
52.663;3.528, 3.762
)T
0.61 30.55

w4
= (−0.127;0.432, 0.472)
T
,

w5
=(11.856;1.89, 1.3621)
T

w6
= (−3.275;0.0893, 0.657)
T
Choi and Buckley
(2008)

w0= (−
496.6;127.25, 13048
)T
,

w1=(
7.26;1.25, 0.968
)T

w2
= (−4.256;0.625, 1.025)
T
,

w3
= (−45.628;17.236, 11.25)
T
0.47 35.94

w4
= (−0.471;1.528, 0.568)
T
,

w5
=(7.25;0.528, 1.856)
T

w6= (−
2.369;1.745, 0.658
)T
Asadolahi e al.
(2021)
S ep 1:

C=30
,
𝛾 =4
,
𝜖 =3
,
𝜇 =0.01
S ep 2:

C=20
,
𝛾 =1
,
𝜖 =2
,
𝜇 =0.02
0.68 47.26
S ep 3:

C=30
,
𝛾 =2
,
𝜖 =1
,
𝜇 =0.01
Khamma e al.
(2021)

w0
= ((−570.63;0;0)
T
,

w1
=(7.61;0.47;0)
T

w2
= (−5.78;0;0)
T
,

w3
= (−54.47;0;0.04)
T

w4= ((−
0.7;0;0
)T
,

w5=(
14.05;3.60;3.16
)T
0.71 37.94

w6
= (−2.10;0;0)
T
,

h=26.4
D’U so and Gas -
aldi (2002)

y
(x)=( (x);l
(x)
,
(x)
)
T
(x)=−572 +8.467x1−6.925x2−50.227x3
−0.4328x4+9.862x5−3.106x6
l (x)=−134 +4.58x1−1.725x2+36.025x3
0.61 30.74
−2.324x4
+
1.253x5
−
22.35x6
(x)=−90.3 +1.115x1+1.28x2+11.925x3
+0.12x4+2.325x5+2.358x6
Wang e al. (2007)

h=5
0.54 32.24
Cheng and Lee
(1999)

h=3.5
0.50 34.76
P oposed model
𝜏 =
0.55,


b(𝜏 )=(−
571;0, 0
)T
,

h=
4.4,

hl=
2.1,

h =2.2
0.84 24.67
151
A lexible so nonlinea quan ile-based eg ession model
h ee dis inc op imiza ion algo i hms L, C and R p oposed in Sec .4. The esul s
can be ound in Table 6. As he e a e nume ous es ima ed pa ame e s
{
l
w
,

w, 
w}
,
hey a e no included in his able. Conside ing he pe o mance measu es we ge
MSM =0.84
and
RMSE =24.67
. We compa e he esul s o ou model wi h he
espec i e esul s o se e al o he models including (Zeng e  al., 2017; Choi &
Buckley, 2008; Asadolahi e al., 2021; Khamma e al., 2021; D’U so & Gas aldi,
2002; Wang e al., 2007), and Cheng and Lee (1999). The esul s o all he models
a e p esen ed in Table6. Upon in e p e ing he alues, i is e iden ha he p oposed
model achie es be e esul s. Consis en wi h he p e ious examples, hese indings
ea i m he supe io i y o he p oposed quan ile-based model o e a ious uzzy
eg ession models o he conside ed da a se .
6 Conclusions
The p oposed me hod in his pape is he i s quan ile-based so eg ession model
ha can embed nonlinea i ies in he modeling while he p e ious app oaches in he
li e a u e a e es ic ed o he in lexible linea case. The in oduced model is based
on quan iles o uzzy andom a iables and embeds hem in o uzzy nonlinea
eg ession modeling. To his aim, he quan ile unc ion o a uzzy andom a iable
was employed and i s p ope ies we e in es iga ed in ela ion o LR uzzy numbe s.
To es ima e he eg ession coe icien s, h ee o dina y nonlinea eg ession models
we e ob ained based on he cen e , le , and igh sp eads o he in oduced quan-
ile eg ession model. In each s age, some o he model pa ame e s a e es ima ed
by implemen ing hyb id op imiza ion algo i hms, ke nel-based i ing, quan ile loss
unc ion, and gene alized c oss- alida ion c i e ia. We ha e conduc ed comp ehen-
si e compa a i e analysis by pe o ming h ee case s udies based on eal da a se s
and by including nume ous compe i o s, i.e., uzzy eg ession models o a ious
kinds ha ha e been p o en o be supe io in p e ious s udies. In he amewo k o
hese compa isons i has been shown ha bo h he deg ee o simila i y and he p e-
dic i e powe o he p oposed model is be e compa ed o all he compe i o s.
Beyond he inno a i e me hodology, he p oposed model has a ious applicabil-
i y and managemen implica ions, which we b ie ly discuss in he ollowing. The e
a e mainly ou applicabili y implica ions, namely he obus ness o da a asymme-
y and ou lie s (i.e., ideal o inancial isk analysis, housing p ice p edic ions and
medical s udies), lexibili y in eal-wo ld da a modeling (well-sui ed o complex
sys ems wi h unce ain y, e.g., in en i onmen al s udies and supply chains), b oad
domain applicabili y (like in ag icul u e, clima e s udies and heal h esea ch), and
imp o ed p edic i e pe o mance (i.e., eliable o high-s akes decision-making).
As o managemen implica ions, we would like o highligh i e, namely enhanced
decision-making (iden i ies isks and oppo uni ies ac oss da a quan iles, suppo ing
s a egic choices), ope a ional e iciency (e icien o big da a, aiding indus ies like
e ail and logis ics), esilien planning (accoun s o unce ain y, helping leade s in
ola ile con ex s like ma ke o ecas ing), policy implica ions (aids a ge ed in e -
en ions and esou ce alloca ion o policymake s), and c oss-disciplina y syne gy
152
G.Hesamian e al.
(encou ages collabo a ion be ween da a scien is s and decision-make s o be e
ou comes).
Finally, as he in oduced quan ile-based eg ession model is ocused on LR uzzy
numbe s, i would be a p omising u u e pa h o ans e he p oposed me hod o
o he ypes o uzzy numbe s. Ex ensions o he model o o he ypes o uzzy quan-
i ies such as Py hago ean uzzy se s, Z- uzzy clouds, and hesi an uzzy linguis ic
e m se s a e o he po en ial opics o u u e s udies.
Acknowledgemen s The au ho s would like o hank he edi o and bo h anonymous e iewe s o hei
aluable eedback and sugges ions, which we e impo an and help ul o imp o e he pape .
Funding Open Access unding enabled and o ganized by P ojek DEAL.
Open Access This a icle is licensed unde a C ea i e Commons A ibu ion 4.0 In e na ional License,
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