A Three-Stage Nonparametric Kernel-Based Time Series Model Based on Fuzzy Data

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 28. Cheng, S.H.; Chen, S.M.; Jian, W.S. Fuzzy ime se ies o ecas ing based on uzzy logical ela ionships and simila i y measu es. In . Sci. 2016,327, 272–287. [C ossRe ] 29. Sadaei, H.J.; Enaya i a , R.; Abdullah, A.H.; Gani, A. 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Disclaime /Publishe ’s No e: The s a emen s, opinions and da a con ained in all publica ions a e solely hose o he indi idual au ho (s) and con ibu o (s) and no o MDPI and/o he edi o (s). MDPI and/o he edi o (s) disclaim esponsibili y o any inju y o people o p ope y esul ing om any ideas, me hods, ins uc ions o p oduc s e e ed o in he con en .