Ci a ion: Hesamian, G.; Johannssen,
A.; Chukh o a, N. A Th ee-S age
Nonpa ame ic Ke nel-Based Time
Se ies Model Based on Fuzzy Da a.
Ma hema ics 2023,11, 2800. h ps://
doi.o g/10.3390/ma h11132800
Academic Edi o : Sal a o e Sessa
Recei ed: 27 May 2023
Re ised: 13 June 2023
Accep ed: 19 June 2023
Published: 21 June 2023
Copy igh : © 2023 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/).
ma hema ics
A icle
A Th ee-S age Nonpa ame ic Ke nel-Based Time Se ies Model
Based on Fuzzy Da a
Gholam eza Hesamian 1, A ne Johannssen 2,* and Na aliya Chukh o a 3
1Depa men o S a is ics, Payame Noo Uni e si y, Teh an 19395-3697, I an; [email p o ec ed]
2Facul y o Business Adminis a ion, Uni e si y o Hambu g, 20146 Hambu g, Ge many
3Ha enCi y Uni e si y o Hambu g, 20457 Hambu g, Ge many; na aliya.chukh [email p o ec ed]
*Co espondence: a ne.johannssen@uni-hambu g.de
Abs ac :
In his pape , a nonlinea ime se ies model is de eloped o he case when he unde lying
ime se ies da a a e epo ed by
LR
uzzy numbe s. To his end, we p esen a h ee-s age nonpa a-
me ic ke nel-based es ima ion p ocedu e o he cen e as well as he le and igh sp eads o he
unknown nonlinea uzzy smoo h unc ion. In each s age, he nonpa ame ic Nada aya–Wa son
es ima o is used o e alua e he cen e and he sp eads o he uzzy smoo h unc ion. A hyb id
algo i hm is p oposed o es ima e he unknown op imal bandwid hs and au o eg essi e o de simul-
aneously. Va ious goodness-o - i measu es a e u ilized o pe o mance assessmen o he uzzy
nonlinea ke nel-based ime se ies model and o compa a i e analysis. The p ac ical applicabili y
and supe io i y o he no el app oach in compa ison wi h u he uzzy ime se ies models a e
demons a ed ia a simula ion s udy and some eal-li e applica ions.
Keywo ds:
uzzy eg ession; uzzy ime se ies model; nonpa ame ic ime se ies analysis; ime
se ies analysis
MSC: 03E72; 37M10; 62A86
1. In oduc ion
The ield o ime se ies analysis comp ises me hods used o analyze he cha ac e is ics
o a esponse a iable wi h espec o ime. I akes in o conside a ion he ac ha obse -
a ions made o e ime may ha e an in e nal s uc u e (such as au oco ela ions, ends,
seasonal and/o cyclic a ia ions) ha should be accoun ed o . The main aims o ime
se ies analysis a e as ollows:
•
T end analysis: o iden i y he unde lying pa e n o end in he da a o e ime, such
as an upwa d o downwa d end.
•
Seasonali y analysis: o iden i y i he da a exhibi a epea ing pa e n o e a se pe iod,
such as daily, weekly, o yea ly.
•Fo ecas ing: o o ecas u u e alues using his o ical da a.
•
Anomaly de ec ion: o iden i y any unusual o unexpec ed obse a ions in he da a ha
de ia e om he no mal pa e n.
•
Model selec ion: o choose an app op ia e model o ep esen he unde lying ela ion-
ships be ween a iables in he da a.
•
Noise educ ion: o emo e any unwan ed a iabili y o andom luc ua ions om he da a
o imp o e he accu acy o p edic ions and make he unde lying pa e ns mo e clea .
These aims can in o m decision make s, p o ide insigh in o he unde lying pa e ns
and ela ionships in he da a, and suppo he de elopmen o da a-d i en s a egies in
a ious ields such as economics, enginee ing, inance, and mo e (see, e.g., [1–8]).
Common ime se ies models ely on exac obse a ions and ensu e c isp p edic ions.
Howe e , due o a ious unce ain y ac o s, i is some imes p e e able o make p edic ions
Ma hema ics 2023,11, 2800. h ps://doi.o g/10.3390/ma h11132800 h ps://www.mdpi.com/jou nal/ma hema ics
Ma hema ics 2023,11, 2800 2 o 17
using imp ecise alues. Fo ins ance, we usually obse e imp ecise obse a ions in ca bon
emissions, social bene i s and oil ese es, among o he s [
9
]. T adi ional s a is ical ime
se ies models ail o add ess p edic ion p oblems based on ambiguous o ague in o ma ion
ep esen ed by uzzy da a. This sho coming can be o e come by ime se ies models ha
use echniques o uzzy s a is ics. In gene al, uzzy s a is ics is a b anch o s a is ics
ha deals wi h unce ain y and imp ecision, e.g., in he da a. I includes, o ins ance,
he ields o uzzy es ima ion, uzzy eg ession, uzzy clus e ing, and uzzy hypo hesis
es ing [10–12].
Fuzzy ime se ies models we e o iginally in oduced in 1993 [
13
], and since hen hey
ha e eplaced con en ional (c isp) ime se ies app oaches when obse a ions a e unce ain.
When conside ing uzzy ime se ies models, he p edic ion o u u e alues equi es h ee
p incipal s eps. In s ep 1, he exac da a a e epo ed. In s ep 2, h ough he iden i ica ion o
uzzy logical ela ions [
14
,
15
], he p edic ions a e ans o med in o uzzy quan i ies. Finally,
s ep 3 p o ides a de uzzi ica ion app oach [
16
–
22
] o ans o m he uzzy alues in o c isp
ones. The echniques used o iden i y uzzy logical ela ions in s ep 2 p ima ily in ol e
uzzy logical ela ion g oups and ma ices [
13
,
23
–
34
], so compu ing me hods [
35
–
44
],
and s a is ical app oaches in in e ac ion wi h uzzy logic [
21
,
24
,
45
–
47
]. S ep 2 is an essen ial
pa o he p edic i e powe o he p esen ed model. Fuzzy ime se ies models ha ely on
imp ecise obse a ions ha e a ac ed subs an ial a en ion in ecen yea s, mainly due o
hei high applicabili y o eal-li e p oblems.
In ac , a lo o esea che s ha e ocused on ime se ies models using imp ecise obse a-
ions. The so compu ing echniques employed in his amewo k a e mos ly combina ions
o a i icial neu al ne wo ks, e olu iona y algo i hms, uzzy and ough se s. These ap-
p oaches a e widely used o c isp o uzzy o ecas s based on c isp pas obse a ions such
as elec ici y load, s ock index p ices and empe a u e ( o a e iew o hese echniques, we
e e o [
48
–
56
]). In addi ion, a ious me hods combine echniques o ime se ies and uzzy
eg ession analysis [
57
]. Fo some ecen ad ances in uzzy eg ession analysis, see [
58
–
63
].
The eliabili y o o ecas ing me hods gene ally equi es exac obse a ions in he
sample. Bu he e is o en only ague in o ma ion ha is gi en in e ms o imp ecise
quan i ies. Mo eo e , he e a e a ious eal-wo ld p oblems ela ed o biological, economic,
en i onmen al, medical and sociological da a whe e we ace inaccu a e ins ead o accu-
a e da a. In many eal-li e applica ions, e.g., mon hly Co
2
emission, annual sea su ace
empe a u e o he wa e le el o a lake, con en ional obse a ions a e o en epo ed as
mean alues. In such cases, he da a ob ained a e no su icien in o ma i e since some
in o ma ion con ained in he ange o he da a is neglec ed. To o e come his sho coming,
one al e na i e would be o epo such kind o da a as in e al alued (compa able o
con en ional con idence in e als). Howe e , a po en ial sho coming o in e al- alued
da a is he ac ha all alues wi hin he in e al ha e he same impo ance. To a oid his
issue o in e al- alued da a, he epo ed da a can al e na i ely be ep esen ed wi h help
o uzzy numbe s [
64
]. These uzzy quan i ies can be modeled ia expe s opinion, o as
simple al e na i e, hey can be cons uc ed ia a me hod p oposed by Buckley [
65
]. In his
app oach, con en ional con idence in e als a e employed o cons uc uzzy numbe s
a ound he con en ional mean alues.
In addi ion o he abo emen ioned me hods, he e a e also uzzy ime se ies models
ha ely on uzzy da a, bu compa a i ely ew o e all. In his ega d, Hesamian and
Akba i [
66
] i s sugges ed a uzzy semi-pa ame ic ime se ies model (
FSPTSM
) based on
uzzy da a, non- uzzy coe icien s, and uzzy smoo h unc ions. Secondly, Za ei e al. [
67
]
used a speci ic a ian o he
FSPTSM
[
66
] o iangula uzzy da a and di e en dis ance
measu es o uzzy da a. And hi dly, Hesamian e al. [
68
] in oduced a o wa d addi i e
ime se ies model (FATSM) o uzzy obse a ions.
In his pape we de elop a uzzy nonpa ame ic ime se ies model (
FNPTSM
) o
uzzy obse a ions ha is inspi ed by nonpa ame ic eg ession models and ke nel smoo h-
ing me hods [
57
]. As an ini ial idea, no e ha in nonpa ame ic eg ession analysis, he
Nada aya-Wa son es ima o [
69
,
70
] is ai ly common. Now, le us conside he issue o pa-
Ma hema ics 2023,11, 2800 3 o 17
ame e es ima ion in he nonlinea eg ession model
x = (x −1
,
x −2
,
. . .
,
x −p) + e
wi h
:Rp→R
. Based on his gene al model, a simple nonpa ame ic way o es ima ing he
unc ion is o employ he ke nel-based Nada aya–Wa son es ima o
b
(x ) =
T∗
∑
j=p+1
wh( ,j)xj(1)
wi h
wh( ,j) =
∑p
i=1Kx −i−xj−i
h
∑T∗
j=p+1∑p
i=1Kx −i−xj−i
h,
whe e
K
is a ke nel unc ion and
h>
0 he bandwid h pa ame e . No e ha he es ima o
(1)
is
a weigh ed a e age o
x1
,
x2
,
. . .
,
xT
using he weigh s
wh(
,
j)
. As o de e mining he op imal
bandwid h h, he Gene alized C oss Valida ion (GCV) c i e ion
bh=a g min
h>0GCV(h) = a g min
h>0
1
T∗−p
T∗
∑
=p+1
x −∑T∗
j=p+1wh( ,j)xj
1− (Wh)
T∗−p

2
can be u ilized, whe e
(Wh)
is he ace o he ma ix
Wh= [wh(
,
j)]
. I is a ma e o
ac ha he es ima ed alues o
a e ensu ed o be wi hin he ange o he esponse
a iable. This bene icial p ope y is one o he easons why we apply he Nada aya–Wa son
ke nel-based es ima o o ou uzzy ime se ies model. By u ilizing his idea, he p oposed
FNPTSM
p o ides an es ima ion p ocedu e o he unknown (nonlinea ) ela ionship be-
ween he uzzy obse a ions in h ee s ages. The ad an age o his me hodology is ha i
conside ably dec eases he complexi y in he es ima ion p ocedu e. While he o he uzzy
ime se ies models [
66
–
68
] a e based on es ima ing he unknown componen s o he model
by uni ying he cen e s and sp eads o uzzy da a and hei co esponding p edic ed alues,
ou p oposed me hod p o ides a smoo h es ima ion p ocedu e acco ding o h ee sepa a e
s ages. In he amewo k o a simula ion s udy and wo eal-da a examples, he e iciency
and app op ia eness o he
FNPTSM
is assessed in compa ison wi h p e ious ime se ies
models o uzzy da a by u ilizing ou app o ed goodness-o - i c i e ia.
The pape is o ganized as ollows. Fi s , we ecall some necessa y concep s ela ed o
uzzy numbe s in Sec ion 2. In Sec ion 3, he h ee-s age nonpa ame ic ke nel-based ime
se ies model using uzzy da a is p esen ed. In Sec ion 4, a ious applica ion examples a e
gi en. Concluding ema ks a e p o ided in Sec ion 5.
2. Fuzzy Numbe s
In his sec ion, we in oduce basic de ini ions o uzzy numbe s ha a e needed o
de elop ou p oposed me hod.
A uzzy se
e
A
is a mapping on
X
ha assigns a speci ic deg ee o membe ship
0
≤µe
A(x)≤
1 o each
x∈X
. In addi ion, a uzzy numbe (
FN
)
e
A
is a con ex no -
malized uzzy se on he eal line
R
wi h an uppe semi-con inuous membe ship unc ion
o bounded suppo [
71
]. In many eal applica ions, ague da a
a
can be epo ed as
e
A
:
“abou
a
”. Such uzzy da a can o en be ep esen ed ia a special case o
FN
s, so called
LR
-
FN
s, which spli
µe
A
in o wo cu es: a pa on he le and a pa on he igh o he
modal alue. So, when conside ing eal-li e applica ions in uzzy en i onmen s,
LR
-
FN
s
play an impo an ole. The membe ship unc ion o an
LR
-
FN µe
A(x) = (a
;
la
,
a)LR
can be
de ined by:
Ma hema ics 2023,11, 2800 4 o 17
µe
A(x) = 






La−x
lai x≤a
Rx−a
ai x>a
(2)
In
(2)
,
L
and
R
a e con inuous and s ic ly dec easing unc ions om
[
0, 1
]
o
[
0, 1
]
sa is ying
L(
0
) = R(
0
) =
1 and
L(
1
) = R(
1
) =
0. In addi ion,
a∈R
ep esen s he
modal alue, while
la>
0 and
a>
0 a e he le sp eads and igh sp eads o
e
A
, espec i ely.
The se o all
LR
-
FN
s is ep esen ed by
FLR(R)
. A special case o an
LR
-
FN
is he so-called
iangula uzzy numbe (TFN), whose membe ship unc ion has he ollowing o m:
µe
A(x) = 






x−(a−la)
laa−la≤x≤a
a+ a−x
aa<x≤a+ a
0 o he wise
The e a e a ious ope a ions ha can be de ined be ween wo
LR
-
FN
s, i.e., be ween
e
A= (a
;
la
,
a)LR
and
e
B= (b
;
lb
,
b)LR
. Fo ins ance, as we need bo h ope a ions in his
pape , we de ine Addi ion and Scala mul iplica ion o e
Aand e
Bin he ollowing [72]:
• Addi ion: e
A⊕e
B= (a+b;la+lb, a+ b)LR
• Scala mul iplica ion:
λ⊗e
A=(λa;λla,λ a)LR i λ>0
(λa;−λ a,−λla)RL i λ<0
Mo eo e , he e a e nume ous concep s used o de ine dis ances be ween wo
LR
-
FN
s
e
A= (a
;
la
,
a)LR
and
e
B= (b
;
lb
,
b)LR
[
71
]. He e, we u ilize he squa ed e o dis ance measu e
D
o pe o mance e alua ion o he
FNPTSM
in compa ison wi h o he models. I is
de ined as
D(e
A,e
B) = (((a−b)2+c1(la−lb)2+c2( a− b)2)/3)0.5
wi h c1=R1
0L−1(α)dαand c2=R1
0R−1(α)dα[73].
3. Nonpa ame ic Ke nel-Based Time Se ies Model o Fuzzy Da a
In his sec ion, he
FNPTSM
is de eloped along wi h he sugges ed pa ame e es ima-
ion me hod.
3.1. The Model
Fi s , we ecall he de ini ion o uzzy ime se ies da a.
De ini ion 1.
Le
exT={ex1
,
ex2
,
. . .
,
exT
be a se o
FN
s o size
T
. Then,
exT
is called uzzy ime se ies
da a i {ex1,ex2, . . . , exTis he ague concep o o dina y ime se ies da a {x1,x2, . . . , xT}[68,74].
As discussed in he In oduc ion, he e a e many si ua ions whe e i is p e e able o
epo exac da a xby an FN e
xas “abou x”. Then, e
xis he espec i e ague concep o x.
De ini ion 2.
Le
exT={e
x1
,
e
x2
,
. . .
,
e
xT
be uzzy ime se ies da a. The
FNPTSM
o uzzy ime
se ies da a exTis hen de ined by
e
x =e
(e
x −1,e
x −2, . . . , e
x −p)⊕ee , (3)
whe e
1. e
x = (x ;lx , x )LR,
2. e
(e
x −1,e
x −2, . . . , e
x −p) = ( (x −1,x −2, . . . , x −p);
l (lx −1,lx −2,...,lx −p), ( x −1, x −2,..., x −p))LR,
Ma hema ics 2023,11, 2800 5 o 17
3. ee = (e ;le , e )LR’s a e uzzy e o s, whe e e ∈Rand le , e ∈R+.
Rema k 1.
No e ha
(3)
p o ides an
FN
in he o m
e
x∗
= (x∗
;
lx∗
,
x∗
)LR
wi h
x∗
= (x −1
,
x −2
,
. . .
,
x −p) + e
,
lx∗
=l (lx −1,...,lx −p)+le
and
x∗
= ( x −1,..., x −p)+ e
wi h
=
1, 2,
. . .
,
T
.
Acco ding o De ini ion 1, as
{x1
,
x2
,
. . .
,
xT}
is o dina y ime se ies da a,
ex∗
T={e
x∗
1
,
e
x∗
2
,
. . .
,
e
x∗
T
is also a ague concep o o dina y ime se ies da a
{x∗
1
,
x∗
2
,
. . .
,
x∗
T
. Thus, he p oposed uzzy ime
se ies model (3)gene a es new uzzy ime se ies da a.
3.2. Th ee-S age Es ima ion Me hod o he Nonlinea Fuzzy Smoo h Func ion
Below, we sugges a h ee-s age me hod o es ima e he unknown uzzy smoo h unc-
ion
e
in
(3)
. Fo his pu pose, he uzzy p edic ions a e ob ained based on a wi hin-sample
o ecas
xT∗=x1
,
x2
,
. . .
,
xT∗>
wi h
T∗<T
. F om
(3)
, one can ge h ee o dina y non-
linea ime se ies models as (1)
x = (x −1
,
x −2
,
. . .
,
x −p) + e
, (2)
lx =l (lx −1,...,lx −p)+le
,
and (3)
x = ( x −1,..., x −p)+ e
o
=
1, 2,
. . .
,
T∗
. The e o e, o es ima e he uzzy smoo h
unc ion a
ex= (x
;
lx
,
x)T
wi h
x= (x1
,
x2
,
. . .
,
xp)>
,
lx= (lx1
,
lx2
,
. . .
,
lxp)>
and
x= ( x1
,
x2, . . . , xp)>, we ollow he h ee-s age p ocedu e below:
•S age
(1): Conside he nonlinea eg ession model
lx =l (x −1,x −2,...,x −p)+le
.
Based on he ime se ies da a
lx = (lx −1
,
. . .
,
lx −p)>
, we employ he weigh ed
Nada aya–Wa son es ima o o es ima e
l
o a wi hin-sample o ecas
T∗≤T
a
lx= (lx1, . . . , lxp)>)as
lb
(lx )=
T∗
∑
j=p+1
whl( ,j)lxj,
whe e
whl( ,j) =
∑p
i=1Klx −i−lxj−i
hl
∑T∗
j=p+1∑p
i=1Klx −i−lxj−i
hl(4)
wi h ke nel unc ion
K(
.
)
and bandwid h pa ame e
hl>
0. The op imal alue o
hl
can be es ima ed by implemen ing he GCV c i e ion,
bhl=a g min
hl>0GCV(h) = a g min
hl>0
1
T∗−p
T∗
∑
=p+1

lx −∑T∗
j=p+1whl( ,j)lxj
1− (Whl)
T∗−p



2
, (5)
whe e
(Whl)
is he ace o he ma ix
Whl= [whl(
,
j)]
wi h
whl(
,
j)
as de ined in
(4)
.
•S age
(2): Conside he nonlinea eg ession model
x = (x −1,x −2,...,x −p)+ e
. Based
on he wi hin-sample ime se ies o ecas da a
x = ( x −1
,
. . .
,
x −p)>
,
=
1, 2,
. . .
,
T∗
,
he weigh ed Nada aya–Wa son es ima ion o
a
x= ( x1
,
. . .
,
xp)>)
can be es ab-
lished ia
b
( x )=
T∗
∑
j=p+1
wh ( ,j) xj,
whe e
wh ( ,j) =
∑p
i=1K x −i− xj−i
h 
∑T∗
j=p+1∑p
i=1K x −i− xj−i
h (6)

Ma hema ics 2023,11, 2800 6 o 17
and
h >
0 is a bandwid h pa ame e . The op imal alue o
h
can be es ima ed using
he GCV c i e ion,
bh =a g min
h >0GCV(h) = a g min
h >0
1
T∗−p
T∗
∑
=p+1
 x −∑T∗
j=p+1wh ( ,j) xj
1− (Wh )
T∗−p

2
, (7)
whe e (Wh )is he ace o he ma ix Wh = [wh ( ,j)] wi h wh ( ,j)as de ined in (6).
•S age
(3): Conside he nonlinea eg ession model
x = (x −1
,
x −2
,
. . .
,
x −p) + e
.
Based on he wi hin-sample ime se ies o ecas da a
(x = (x −1
,
x −2
,
. . .
,
x −p)>)
,
=1, 2, . . . , T∗, a nonpa ame ic es ima o can be achie ed as
b
(x ) =
T∗
∑
j=p+1
wh( ,j)xj,
whe e
wh( ,j) =
∑p
i=1Kx −i−xj−i
h
∑T∗
j=p+1∑p
i=1Kx −i−xj−i
h(8)
and bandwid h pa ame e
h>
0. Simila o he p e ious s ages, he op imal alue o
his es ima ed wi h he help o he GCV c i e ion,
bh=a g min
h>0GCV(h) = a g min
h>0
1
T∗−p
T∗
∑
=p+1
x −∑T∗
j=p+1wh( ,j)xj
1− (Wh)
T∗−p

2
, (9)
whe e (Wh)is he ace o he ma ix Wh= [wh( ,j)] wi h wh( ,j), as de ined in (8).
The e o e, he o ecas
e
xT∗+k
wi h ime lag
k∈N
can be achie ed by an
LR
-
FN
ia
eb
xT∗+k= (b
xT∗+k;lb
xT∗+k, b
xT∗+k)LR wi h
b
xT∗+k=
T∗+k−1
∑
j=p+1
∑p
i=1Kx −i−xj−i
bh
∑T∗+k−1
j=p+1∑p
i=1Kx −i−xj−i
bh·xj,
lb
xT∗+k=
T∗+k−1
∑
j=p+1
∑p
i=1Klx −i−lxj−i
bhl
∑T∗+k−1
j=p+1∑p
i=1Klx −i−lxj−i
bhl·lxj,
b
xT∗+k=
T∗+k−1
∑
j=p+1
∑p
i=1K x −i− xj−i
bh 
∑T∗+k
j=p+1∑p
i=1K x −i− xj−i
bh · xj.
Acco ding o S ages (2) and (3), i can be seen ha he sp eads o he uzzy p edic ion
e
xT∗+ka e always non-nega i e.
Rema k 2.
Since he p oposed ime se ies model elies on uzzy da a, le us ecall he p e ious
ime se ies models based on uzzy da a [
66
–
68
]. Fi s , Hesamian and Akba i [
66
] p oposed a uzzy
semi-pa ame ic au o eg essi e in eg a ed mo ing a e age (ARIMA) model as ollows:
e
xi=
p
M
l=1
(θl⊗e
xi−l⊕e
( i)⊕eei),i=p+1, . . . , T.
Ma hema ics 2023,11, 2800 7 o 17
The pa ame e s o he model a e es ima ed by employing a hyb id me hod including a non-
pa ame ic ke nel-based me hod and leas absolu e de ia ions. Fo a second ime se ies model based
on uzzy da a, Za ei e al. [
67
] applied he me hod [
66
] o es ima e he model pa ame e s and he
uzzy smoo h unc ion based on a speci ic dis ance, ke nel and iangula uzzy numbe s. Finally,
Hesamian e al. [68] p oposed he uzzy nonlinea ime se ies model
e
x =e
(e
x −1,e
x −2, . . . , e
x −p)⊕ee , =1, 2, . . . , T,
whe e
e
(e
x −1,e
x −2, . . . , e
x −p) =
p
M
l=1
l(e
x −l).
As o he es ima ion o he unknown uzzy smoo h unc ions
e
l
, hey applied a o wa d addi i e
nonpa ame ic echnique.
Rema k 3.
We ha e ex ended some common pe o mance measu es used o compa e he p edic i e
accu acy o di e en ime se ies models ha we implemen in Sec ion 4. Fo his pu pose, a ime se ies
model is i s es ima ed based on a wi hin-sample uzzy ime se ies da ase o size
T∗<T
and hen he
pe o mance o he model is e alua ed ia he emaining uzzy ime se ies da ase o size T −T∗.
1. Mean Fo ecas E o :
MFE =∑T
=T∗+1D2(eb
x ,e
x )
T−T∗
2. Mean Absolu e Scaled E o :
MASE =∑T
=T∗+1q
T−T∗
wi h
q =D(eb
x ,e
x )
1
T−T∗∑T
=T∗+1D2(e
x ,e
x −1)
3. Basis o he Index o Ag eemen :
BIA =1−∑T
=T∗+1D2(e
x ,eb
x )
∑T
=T∗+1(D(e
x ,ex) + D(ex,eb
x ))2
wi h
ex=∑T
=T∗+1e
x
T−T∗
4. Mean Simila i y Measu e:
MSM =1
T−T∗
T
∑
=T∗+1Rmin{eb
x (x),e
x (x)}dx
Rmax{eb
x (x),e
x (x)}dx
Le
A
and
B
be wo uzzy ime se ies models. As
MSM :FLR(R)×FLR(R)→[
0, 1
]
is a
simila i y measu e, alues o
MSM
abo e 0.5 show a good deg ee o simila i y be ween he uzzy
esponses and hei uzzy p edic ions. I we obse e
MSMB<MSMA
, hen model
A
ou pe o ms
model
B
. Fu he , i
MFEA<MFEB
,
MASEA<MASEB
o
BIAA<BIAB
, hen model
A
ac s
be e in e ms o p edic ion accu acy compa ed o model B.
Rema k 4.
While he p oposed es ima ion p ocedu e does no depend on he shape unc ions
L
and
R
co esponding o uzzy da a, he pe o mance measu es
MFE
,
MASE
and
MSM
depend on
hese shape unc ions. The e o e, he selec ed ype o he shape unc ions
L
and
R
may a ec he
p edic ion c i e ia. Fo ins ance, assume ha he da a ha e epo ed by
e
x = (x
,
lx
,
x )LR
wi h
L(x) =
1
−x
and
R(x) = √1−x
. Tha is,
c1=1
2
and
c2=2
3
. The e o e, he dis ance be ween
Ma hema ics 2023,11, 2800 8 o 17
e
x
and i s p edic ion is
D2(e
x
,
eb
x ) = (x −b
x )2+1
2(lx −lb
x )2+2
3( x − b
x )2
. This implies ha
he
MFE
c i e ion is mo e sensi i e o igh sp eads han o le sp eads in his case. Conside ing
L(x) = R(x) =
1
−x
, i can be seen ha
D2(e
x
,
eb
x )
would be equally dependen om he le
and igh sp eads. Howe e , when we compa e he pe o mance o uzzy ime se ies models, i is
easonable ha he shape unc ions
L
and
R
a e assumed o be he same o all he conside ed models.
Thus, ollowing his app oach, he pe o mance c i e ia a e no sensi i e o he selec ion o
L
and
R
since c1and c2 emain ixed o each model.
3.3. Selec ion o Au o eg essi e O de and Op imal Bandwid hs
When implemen ing he
FNPTSM (3)
, i is necessa y o selec he op imal bandwid hs
h
,
hl
and
h
, o choose he ke nel unc ion and o de e mine he au o eg essi e o de
p
.
The p ocedu e used o selec he au o eg essi e o de and he op imal bandwid hs is
p oposed as ollows:
(1)
Le p=1.
(2)
(2.1) Compu e bhp
lbased on (5).
(2.2) Compu e bhp
based on (7).
(2.3) Compu e bhpbased on (9).
(3)
Le p=p+1 and e u n o (2) un il
b
p=a g min
pRMSEp,
whe e
RMSEp=
u
u
∑T∗
i=p+1D2(eb
xi,e
xi)
T∗−p.
Then, b
p,bhp,bhp
land bhp
a e he op imal alues.
4. Nume ical Examples
In his sec ion, he e ec i eness o he
FNPTSM
is in es iga ed conside ing a simula-
ion s udy and applica ion examples ha ely on uzzy da a. Recall ha he e a e h ee o he
ime se ies models ha a e based on uzzy da a (see Rema k 2), i.e., he models in oduced
by Hesamian and Akba i [
66
], Za ei e al. [
67
] and Hesamian e al. [
68
]. Howe e , as he
me hod o Za ei e al. [
67
] is based on Hesamian and Akba i’s me hod [
66
] (wi h a di e en
dis ance measu e), we omi his echnique in he compa isons below. Thus, we compa e ou
p oposed me hod wi h he models sugges ed by Hesamian and Akba i (
FSPTSM
) [
66
] and
Hesamian e al. (
FATSM
) [
68
] ia h ee di e en ke nel unc ions (Gaussian, Epanechniko ,
and iweigh ).
Example 1.
In his example, 10 uzzy da ase s, each o size 300, a e gene a ed by he ollowing
FNPTSM:
e
x = (e
x −1,e
x −2,e
x −3)⊕ee , =4, 5, . . . , 300,
whe e
1.
(e
x1,e
x2,e
x3) = x1−cos(x2)−expx3
1+|x3|; cos2 0.9
3
∏
j=1
lxj!, exp 0.002
3
∏
j=1
xj!!LR
2. e
xj= (xj
;
lxj
,
xj)LR
,
j=
1, 2, 3 a e he ini ial alues wi h
xj∼N(
0, 1
)
, and
lxj
and
xj
a e
andom a iables ollowing U(0, 0.2)and U(0, 0.9), espec i ely,
3. ee = (e
;
le
,
e )LR
wi h
e ∼N(
0, 4
)
,
le
and
e
a e andom a iables ollowing
U(
0, 0.4
)
and U(0, 0.5), espec i ely, and
4. L(x) = 1−x2and R(x) = 1−x.
Ma hema ics 2023,11, 2800 9 o 17
The ke nels Gaussian, Epanechniko , and iweigh a e applied o p edic
e
x
. Based on he
10 sample uzzy da ase s (each o size 300), he mean alues o he goodness-o - i measu es and hei
co esponding bandwid h mean alues a e summa ized in Table 1. Consul ing he esul s o he
FNPTSM
, i is e iden ha he bes esul s among a ious ke nels a e ob ained ia he Gaussian
ke nel (lowes alues o
MFE
,
MASE
and la ges alues o
BIA
,
MSM
). In addi ion, he esul s o
he
FSPTSM
and
FATSM
can also be ound in Table 1. Compa ing hese esul s wi h he esul s o
he
FNPTSM
, i is ob ious ha he
FNPTSM
p o ides mo e accu a e p edic ions compa ed o bo h
o he me hods o all h ee ke nels, as all he conside ed goodness-o - i measu es show be e esul s
o he
FNPTSM
. Tha is, we obse e he lowes alues o
MFE
,
MASE
and he la ges alues o
BIA, MSM o he FNPTSM.
Table 1.
The mean pe o mance measu es o he
FNPTSM
,
FSPTSM
and
FATSM
co esponding o
some speci ic ke nels in Example 1.
Me hod Ke nel Resul s Goodness-o -Fi
C i e ia
FNPTSM
Gaussian
MFE =1.0452
bh=0.45 MASE =1.6089
b
hl=0.04 BIA =0.9996
b
h =0.22 MSM =0.4167
Epanechniko
MFE =1.1728
bh=0.66 MASE =1.6478
b
hl=0.05 BIA =0.9992
b
h =0.39 MSM =0.3953
iweigh
MFE =1.2482
bh=1.89 MASE =1.6339
b
hl=0.08 BIA =0.9991
b
h =0.54 MSM =0.3721
FSPTSM
Gaussian
hop =0.07 MFE =8.9728
b
θ1=0.5575 MASE =4.2652
b
θ2=−0.0956 BIA =0.9536
b
θ3=−0.1247 MSM =0.2207
Epanechniko
hop =0.02 MFE =5.2530
b
θ1=−0.6424 MASE =4.5383
b
θ2=−0.5168 BIA =0.9603
b
θ3=−0.4258 MSM =0.2920
iweigh
hop =0.13 MFE =11.5231
b
θ1=0.3757 MASE =3.8643
b
θ2=−0.1576 BIA =0.9738
b
θ3=−0.2568 MSM =0.2133
Ma hema ics 2023,11, 2800 16 o 17
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