Employing machine learning techniques in monitoring autocorrelated profiles

ORIGINAL ARTICLE Employing machine lea ning echniques in moni o ing au oco ela ed p o iles Ali Yeganeh 1 •A ne Johannssen 1 •Na aliya Chukh o a 2 •Saddam Akbe Abbasi 3,4,5 • Fa had Pou panah 6 Recei ed: 31 Oc obe 2022 / Accep ed: 8 Ma ch 2023 / Published online: 29 Ap il 2023 The Au ho (s) 2023 Abs ac In p o ile moni o ing, i is usually assumed ha he obse a ions be ween o wi hin each p o ile a e independen o each o he . Howe e , his assump ion is o en iola ed in manu ac u ing p ac ice, and i is o u mos impo ance o ca e ully conside au oco ela ion e ec s in he unde lying models o p o ile moni o ing. Fo his eason, a ious s a is ical con ol cha s ha e been p oposed o moni o p o iles when be ween- o wi hin-da a is co ela ed in Phase II, in which he main aim is o de elop con ol cha s wi h quicke de ec ion abili y. As a no el app oach, his s udy aims o employ machine lea ning echniques as con ol cha s ins ead o s a is ical app oaches in moni o ing p o iles wi h be ween-p o ile au o- co ela ions. Speci ically, new inpu ea u es based on con en ional s a is ical con ol cha s a is ics and no malized es ima ed pa ame e s a e de ined ha a e capable o adequa ely accoun ing o he be ween-au oco ela ion e ec o p o iles. In addi ion, six machine lea ning echniques a e ex ended and compa ed by means o Mon e Ca lo simula ions. The simula ion esul s indica e ha machine lea ning echniques can ob ain mo e accu a e esul s compa ed wi h s a is ical con ol cha s. Mo eo e , adap i e neu o- uzzy in e ence sys ems ou pe o m o he machine lea ning echniques and he con en ional s a is ical con ol cha s. Keywo ds Adap i e neu o- uzzy in e ence sys em A i icial neu al ne wo k Deep lea ning Long sho - e m memo y  S a is ical p ocess moni o ing Suppo ec o eg ession 1 In oduc ion S a is ical p ocess moni o ing (SPM) is usually employed o indus ial p ocesses o omi assignable causes ha de e io a e he p oduc ou come. SPM is he majo ield o con olling p ocess a ia ions o e en ua e lowe cos s in was e, sc ap, ewo k and claims, be e quali y, and mo e insigh s in o he capabili y o he p ocess. Se en main ools, en ailing sca e diag ams, Pa e o cha s, con ol cha s, his og ams, cause-and-e ec diag ams, check shee s and s a i ica ion, a e u ilized in SPM o implemen inspec ion and moni o ing p ocedu es [1]. Among hem, &A ne Johannssen [email p o ec ed] Ali Yeganeh [email p o ec ed] Na aliya Chukh o a [email p o ec ed] Saddam Akbe Abbasi [email p o ec ed] Fa had Pou panah [email p o ec ed] 1 Facul y o Business Adminis a ion, Uni e si y o Hambu g, 20146 Hambu g, Ge many 2 Ha enCi y Uni e si y, 20457 Hambu g, Ge many 3 Depa men o Ma hema ics, S a is ics and Physics, Qa a Uni e si y, Doha, Qa a 4 Depa men o Ma hema ics, S a is ics and Physics, College o A s and Sciences, Qa a Uni e si y, 2713 Doha, Qa a 5 S a is ical Consul ing Uni , College o A s and Sciences, Qa a Uni e si y, 2713 Doha, Qa a 6 Depa men o Elec ical and Compu e Enginee ing, Cen e o Compu e Vision and Deep Lea ning, Uni e si y o Windso , Windso , Canada 123 Neu al Compu ing and Applica ions (2023) 35:16321–16340 h ps://doi.o g/10.1007/s00521-023-08483-3(0123456789().,- olV)(0123456789().,- olV) con ol cha s a e he mos success ul and e ec i e ools o quali y con ol o manu ac u ing p ocesses [2–4]. To employ a con ol cha o p ocess moni o ing, wo phases en ailing Phase I and II should be ini ially de ined. In Phase I, i is ied o achie e p ope es ima ions o he p ocess pa ame e s, whe eas Phase II moni o ing aims o ind assignable causes in which he p ocess si ua ion changes om In-Con ol (IC) o Ou -o -Con ol (OC) s a e [5–9]. A e age Run Leng h (ARL) and S anda d De ia ion o Run Leng h (SDRL) a e wo common pe o mance indica o s in Phase II. The ARL is he a e age numbe o samples o be ob ained by he p ede ined con ol cha be- o e he cha igge s an OC signal. Thus, a con ol cha wi h la ge (smalle ) alues o he ARL is o be p e e ed when he unde lying p ocess is in IC (OC) s a e [ he mo e common no a ion is ARL 0 (ARL 1 )] [10–12]. In addi ion, he SDRL is de ined in a simila way as a sec- onda y c i e ion in Phase II (i.e., SDRL 0 and SDRL 1 ). The e a e wo common app oaches o moni o a man- u ac u ing p ocesses wi h he help o con ol cha s namely moni o ing quali y cha ac e is ics and p o ile moni o ing [13]. In his pape , we ocus on p o ile moni o ing. He e, he quali y o a p ocess o p oduc is modelled ia a unc ional ela ionship be ween a esponse (dependen ) a iable and one o mo e explana o y (independen ) a i- able(s) [14]. The aim o p o ile moni o ing is o check he s abili y o a p ede ined IC ela ionship (o p o ile) o e ime, and i is essen ial o each a ue OC signal as soon as possible when he IC model shi s o an unknown OC p o ile [15]. Di e en IC models can be employed due o he na u e o he unde lying p oblem, such as ci cula [16], linea [5,17–22], logis ic [23–25], nonlinea [26], nonpa ame ic [27,28], mul ichannel [29], polynomial [30] o quad a ic [31]. Among hem, linea p o iles ha e ecei ed mo e a en ion in he li e a u e [32,33]. The majo i y o p e ious s udies in linea p o ile moni o ing is based on he inde- pendency assump ion ega ding wi hin o be ween p o iles in ela ion o he e o e ms. Howe e , his assump ion is o en iola ed in manu ac u ing p ac ice ha is cha ac e - ized by au oco ela ed p o iles and consequen ly, con en- ional app oaches may lead o inaccu a e ou comes o his ype o p o iles. Au oco ela ed p o iles consis o wi hin- and be ween- co ela ion models in he ela ed li e a u e [34–37]. In he i s g oup, Soleimani, Noo ossana and Ami i [38] de el- oped ou con ol cha s including T 2 and h ee well-known Exponen ially Weigh ed Mo ing A e age (EWMA) cha s conside ing he i s o de Au o eg essi e (AR) model, i.e., AR(1). The esul s showed he supe io i y o EWMA-based app oaches o e T 2 . Au o eg essi e Mo ing A e age (ARMA), Vec o ARMA (VARMA) e c. a e o he mo e complex models ha ha e been de eloped in his ield [39–43]. Due o he highe po en ial o applica ions o he second g oup, his pape ocuses on be ween-p o ile au o- co ela ion. To he bes o he au ho s’ knowledge, he pionee ing wo k is Noo ossana, Ami i and Soleimani [44], in which hey de eloped T 2 , EWMA o esiduals (EWMA/ R) and ans o med indi idual EWMA (EWMA-3) con ol cha s o si ua ions whe e au oco ela ion e ec s exis be ween p o iles. Simila o Soleimani, Noo ossana and Ami i [38], hey concluded ha EWMA-based me hods ou pe o m T 2 . Wang and Lai [45] agg ega ed he indi- idual EWMA s a is ics o a Mul i a ia e EWMA (MEWMA) con ol cha o p o iles wi h be ween-au o- co ela ion, and i has been shown ha MEWMA ou pe - o ms T 2 . Khedma i and Niaki [46] conside ed bo h linea and polynomial p o iles; hey i s u ilized he U s a is ic o emo ing he e ec o au oco ela ion and hen de el- oped a T 2 -based con ol cha . The expe imen al esul s showed ha his me hod pe o ms be e han con en ional T 2 con ol cha s, bu compa isons wi h EWMA a e miss- ing. Koosha and Ami i [47] p oposed a simila T 2 -based con ol cha o moni o ing au oco ela ed logis ic p o iles. Wang and Huang [48] modi ied he es ima ion p ocedu e o he EWMA app oach, and he simula ion esul s demons a ed ha his scheme has a as e de ec ion abili y han ha o con en ional EWMA. F om he li e a u e, i can be in e ed ha he p obabili y o occu ing au oco ela ions in p ac ical applica ions is e y high. The e o e, an ea ly de ec ion o OC condi ions is mo e impo an han simula ion esul s, as a delay in de ec ion may esul in he p oduc ion o noncon o mi ies and addi ional cos s. Howe e , he con en ional con ol cha s such as T 2 , EWMA/R and EWMA-3 a e no able o pe o m well in line wi h his aim as hei pe o mance de e io a e in he occu ence o au oco ela ion in com- pa ison wi h simple si ua ions; o example, i can be e e ed o he esul s o ARL 1 in Noo ossana, Ami i and Soleimani [44] e sus Kim e al. [20]. Hence, p oposing a no el con ol cha wi h a angible abili y in educ ion o he OC signaling ime in au oco - ela ed p o iles is c ucial. To emedy his challenge, in ecen yea s, se e al s udies inco po a ed machine lea ning echniques in he SPM con ex in moni o ing oundness [49], nonlinea [50–52], linea [53,54] and logis ic 16322 Neu al Compu ing and Applica ions (2023) 35:16321–16340 123 [25,55,56] p o iles. As a di e en app oach, Chen e al. [57] employed a deep lea ning echnique, called s acked denoising au oencode s, o moni o au oco ela ed p o iles. Speci ically, his scheme ex ac s a numbe o ea u es om he p ocess using au oencode s, and hen he ex ac ed ea u es a e used o de elop con ol cha s based on T 2 and EWMA. In o he wo ds, he main ask o hei app oach is o selec p ope ea u es om he p ocess, whe eas he di ec usage o machine lea ning echniques as a con ol cha would be mo e p omising. As a as he au ho s know, he e a e no u he a icles whe e machine lea ning echniques a e employed in moni o ing au oco ela ed p o iles. The aim o his pape is o de elop a obus con ol cha based on machine lea ning echniques o alle ia e he abo e-men ioned challenges, i.e., educing he alues o he ARL 1 and SDRL 1 o au oco ela ed linea p o iles ha can esul in ea ly de ec ion o OC si ua ions in Phase II. To achie e his, h ee combina ions o inpu ea u es based on he e ec o he mean o esponses, he mean o e o s, and T 2 s a is ic, each in he cu en and p e ious sample, a e de ined o ed in o he machine lea ning echniques o moni o ing p o iles wi h be ween-au oco ela ion o i s o de , i.e., AR(1). Since each machine lea ning echnique pe o ms di e en ly in ackling a ious p oblems, six machine lea ning echniques anging om shallow o deep s uc u es including adap i e neu o- uzzy in e ence sys em (ANFIS), a i icial neu al ne wo k (ANN) wi h Back- P opaga ion (BP) aining, Con olu ional Neu al Ne wo k (CNN), long sho - e m memo y (LSTM) ne wo k, Radial Basis Func ion (RBF) ne wo k and suppo ec o eg es- sion (SVR), a e employed o ind he mos app op ia e one. To sum up, he main con ibu ions o his pape a e as ollows: •Imp o ing he de ec ion abili y o Phase II con ol cha s o moni o ing linea au oco ela ed p o iles wi h he help o machine lea ning echniques, •De ining di e en combina ions o inpu ea u es based on he e ec o he mean o esponses, he mean o e o s, and T 2 s a is ic, each in he cu en and p e ious sample, o moni o ing he be ween-au oco ela ion e ec o p o iles, •E alua ing he pe o mance o he de ined inpu ea u es and inding he bes combina ion using he p oposed machine lea ning-based con ol cha , and •Iden i ica ion o he mos app op ia e machine lea ning echnique unde he mos sui able inpu combina ion o his p oblem. The es o his a icle is o ganized as ollows. In Sec . 2, de ini ions o au oco ela ed linea p o iles a e discussed. Sec ion 3p esen s he amewo k o he p o- posed app oach. Resul s o simula ion s udies ega ding pe o mance compa isons a e gi en and discussed in Sec . 4. To show he e ec i eness o ou me hod, an illus a i e example is gi en in Sec . 5. Finally, Sec . 6 gi es some conclusions and sugges s u u e esea ch di ec ions. 2 P elimina ies In his sec ion, i s , he gene al ela ions o linea p o iles wi h be ween-au oco ela ion e o e ms a e p esen ed. Then, h ee common con ol cha s, namely T 2 , EWMA/R and EWMA-3, a e b ie ly in oduced. Finally, de ails abou he O dina y Leas Squa es (OLS) es ima ion o he pa ame e s a e discussed. These basics a e necessa y agains he backd op ha (1) ou p oposed me hod employs he T 2 s a is ic as inpu ea u e, and (2) hese con en ional cha s a e used o compa ison pu poses in ou analyses. 2.1 The linea au oco ela ed p o ile in Phase II A common linea p o ile, which is he simples bu he mos undamen al ype o p o iles [19], is de ined as: Yij ¼A0þA1Xiþeij;eij N0; 2  ; i¼1;2;...;n;j¼1;2;...; ð1Þ whe e X i ep esen he explana o y a iable in a linea p o ile and he esponse a iable Y ij is he quali y cha ac- e is ic unde s udy. The pa ame e s o he abo e IC model (in e cep A 0 , slope A 1 and e o a iance 2 ) a e es ima ed om Phase I samples, and i is usually assumed ha sample size nand independen a iable X i a e ixed in each p o ile. When he e is an AR(1) s uc u e be ween he andom e o e ms, (1) becomes: Yij ¼A0þA1Xiþeij;eij ¼/eij1ðÞ þaij;aij N0; 2  ; i¼1;2; :::; n;j¼1;2;...; ð2Þ whe e /is a cons an au oco ela ion coe icien , which is assumed o be known in Phase II. To moni o he abo e IC p o ile, we b ie ly p esen h ee common app oaches in he ollowing subsec ions. Neu al Compu ing and Applica ions (2023) 35:16321–16340 16323 123 2.2 The T 2 con ol cha o moni o ing au oco ela ed p o iles in Phase II By some calcula ions, i can be easily shown ha he es ima ed esponses a e ob ained in he j h gene a ed sam- ple o e ime as ollows [44]: ^ yij ¼/yij1ðÞ þ1/ðÞðA0þA1XiÞ:ð3Þ Thus, he empi ical esiduals can be w i en as: eij ¼^ yij yij:ð4Þ Noo ossana, Ami i and Soleimani [44] used a modi ied o m o he T 2 s a is ics p oposed by Kang and Albin [14] in a simple linea p o ile: 2 j¼ejX 1 e e0 j;ej¼e1j;e2j;...;enj  ;ð5Þ whe e P e is he symme ic n9nma ix 2 I. Since he cha s a is ic ð 2 j) is ensu ed o be la ge han ze o, i is compa ed wi h a p ede ined Uppe Con ol Limi (UCL T ) o each an OC signal. No e ha he Lowe Con ol Limi (LCL T ) is equal o 0. 2.3 The EWMA/R con ol cha o moni o ing au oco ela ed p o iles in Phase II In he EWMA/R con ol cha , wo simul aneous s a is ics moni o he gene a ed p o iles. The i s s a is ic is ela ed o he mean o he esiduals and is de ined as ollows [44,53]: zj¼hejþð1hÞzðj1Þ;ej¼1 nX n i¼1 eij;ð6Þ In (6), i holds z 0 = 0, and his he EWMA cons an ha usually has a alue be ween 0.1 and 0.9 [20]. Following p e ious wo ks [21,53,54,58], his se o 0.2 in his pape . The second s a is ic o he EWMA/R cha is he ange o he empi ical esiduals de ined by [37,48]: j¼maxðeijÞminðeijÞ:ð7Þ The EWMA/R decla es he p ocess as IC i bo h o he ollowing condi ions a e me [14]: L ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h 2hðÞn s zj L ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h 2hðÞn s; d2Ld3 ðÞ j d2þLd3 ðÞ ð8Þ In (8), he alue o Lis assigned o each a p ede ined ARL 0 , while d 2 and d 3 a e wo cons an s ha depend on he sample size (see Mon gome y [1] o mo e de ails). 2.4 The EWMA-3 con ol cha o moni o ing au oco ela ed p o iles in Phase II To sol e he p oblem o dependency be ween he es ima- o s in linea p o iles, Kim, Mahmoud and Woodall [20] sugges ed o ha e ze o mean explana o y a iables, whe e he leas squa es es ima o s o slope and in e cep a e independen andom a iables. By his coding ( ans o - ma ion), he EWMA-3 app oach has he IC model Yij ¼B0þB1Xdi þeij;eij ¼/eij1ðÞ þaij;eij N0; 2  ; i¼1;2;...;n;j¼1;2;...; ð9Þ whe e he coded explana o y a iables (Xdi ¼XiX) lead o he ans o med IC in e cep B 0 =A 0 ?A 1 X,A0and A1 a e de ined as in (1). No e ha he ans o med IC slope is B 1 =A 1 in his app oach. The OLS es ima ion o he pa ame e s ( b B0j;b B1j;b 2j) gene a es h ee sepa a e EWMA- based e o s o in e cep (eIj), slope (eSj) and s anda d de ia ion (eij), as ollows: eIj ¼^ B0jþ;^ B0j1ðÞ 1;ðÞB0; eSj ¼^ B1jþ;^ B1j1ðÞ 1;ðÞB1; eij ¼yij ;yij1ðÞ 1;ðÞB0þB1Xdi; MSEj¼1 nX n i¼1 e2 ij: ð10Þ Based on he OLS es ima ion, he Mean Squa e E o (MSE) o he j h p o ile is conside ed as he es ima o o he e o a iance ( o de ails see Kim e al. [20], Huwang e al. [59] and Yeganeh and Shadman [54]), hus, h ee cha s a is ics can be calcula ed as ollows: EWMAIj ¼heIj þ1hðÞEWMAIj1ðÞ ; EWMASj ¼heSj þ1hðÞEWMASj1ðÞ ; EWMAEj ¼max hMSEj1  þ1hðÞEWMAEj1ðÞ ;0  : ð11Þ The con ol limi s o he h ee sepa a e con ol cha s a e designed as: 16324 Neu al Compu ing and Applica ions (2023) 35:16321–16340 123 UCLI¼LCLI¼LI ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h n2hðÞ s UCLS¼LCLS¼LS ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h n2hðÞ s UCLE¼LEffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hVa MSEj  2hðÞ s ð12Þ I is wo h no ing ha LCLE¼0, and some sugges ions ega ding Va ðMSEjÞcan be ound in Kim, Mahmoud and Woodall [20], Noo ossana e al. [44] and Hosseini a d e al. [53]. EWMA-3 igge s an OC signal when a leas one o he s a is ics exceed he con ol limi s. The p oposed con- s an s en ailing L I ,L S and L E a e usually adjus ed o each a desi ed alue o ARL 0 in such a way ha each o he sepa a e cha s achie es an iden ical indi idual ARL 0 . 2.5 OLS es ima ion o he pa ame e s Since he pa ame e s es ima ed ia OLS a e employed as inpu s o he machine lea ning echniques, some de ails on he OLS es ima o s a e gi en below [20,21,53]. While he in e cep o he o iginal and he ans o med model is es ima ed by b A0j¼Yjb A1jXand b B0j¼Yj, espec i ely, he slope pa ame e in bo h models is es ima ed ia b A1j¼b B1j¼SXYj SXX , whe e SXYj¼Pn i¼1YijðXiXÞand SXY ¼ Pn i¼1ðXiXÞ2(wi h Yj¼1 nPn i¼1Yij and Xj¼1 nPn i¼1Xij). No e ha he de ini ion o MSE j based on (10) is equi alen o he es ima o o he e o a iance. 3 The p oposed con ol cha o moni o ing au oco ela ed p o iles The basic idea in his pape is o use machine lea ning echniques ins ead o s a is ical con ol cha s o moni- o ing p o iles wi h be ween-au oco ela ion. Fo his pu - pose, se e al ea u es a e ex ac ed om he p ocess o embed in o he machine lea ning echniques. Using aining pa e ns and he ob ained con ol limi s help o imp o e decision-making abou he p ocess. To employ a machine lea ning echnique as a con ol cha , ou main s eps a e de ined in he ollowing. In he i s s ep, he s uc u e o he inpu ea u es and ou pu s o a machine lea ning echnique a e de e mined, and hen, a aining da a se based on he inpu ea u es and ou pu s is gene a ed by simula ing IC and OC p o iles in he second s ep. The hi d s ep uses he gene a ed da a se o ain a machine lea ning echnique, and inally, by he de ini ion o a con ol limi , he machine lea ning echnique p o ides in o ma ion abou he p ocess condi ion in he ou h s ep. Figu e 1shows a s ep-by-s ep lowcha ega ding he p oposed me hod in moni o ing au oco ela ed p o iles. Mo eo e , de ails o hese s eps a e p esen ed in he ol- lowing subsec ions. 3.1 De ining he s uc u e o inpu ea u es and ou pu s Ex ac ing p ope inpu ea u es is a key s ep in he implemen a ion o machine lea ning echniques [60]. In he li e a u e, a ious s a egies ha e been employed o ex ac ing ea u es. Fo example, Chen e al. [57] and Se gin and Yan [51] used au oencode s o ob ain Phase I in o ma ion, which is no compa ible o his s udy. Hos- seini a d e al. [53] and Yeganeh and Shadman [54] sug- ges ed o ake he OLS es ima ions o he linea p o ile pa ame e s as inpu ea u es. This inpu s uc u e pe o ms well in p o iles wi hou au oco ela ion, bu i is no able o p o ide eliable esul s when be ween-au oco ela ion o he p o iles is p esen since he e ec o au oco ela ion is no cap u ed by he dis ibu ion o he OLS es ima o s. So, in addi ion o hese inpu s ( b A0j,b A1j,b j), u he app op ia e inpu ea u es a e p oposed in he ollowing, which a e sui able o accoun o be ween-au oco ela ion e ec s o i s o de (i.e., AR(1) au oco ela ion). In pa icula , he p oposed inpu s uc u e add esses he e ec o he mean o esponses, he mean o e o s, and T 2 s a is ic each in he cu en sample (j) and he p e ious sample (j-1). I should be men ioned ha one o he main bene i s o machine lea ning-based algo i hms is hei independency o he basic assump ions abou he p ocess. So, when eplacing p ope es ima ions ela ed o o he au oco ela ion models De ine inpu ea u es Gene a e aining da a se based on he inpu ea u es Selec a machine lea ning echnique Gene a e IC p o iles wi h a ge as 0 Gene a e OC p o iles wi h a ge as 1 T ain he machine lea ning echnique Se CV based on he ou pu o machine lea ning and ARL0 Gene a e andom OC p o iles Compu e p ope inpu s based on he gene a ed p o iles Impo he inpu s o he machine lea ning echnique Compu e ou pu and compa e i wi h CV o decide on he p ocess Fig. 1 The gene al s ep-by-s ep lowcha o he p oposed me hod Neu al Compu ing and Applica ions (2023) 35:16321–16340 16325 123 such as MA, ARMA and ARIMA, he p oposed me hod can be easily applied o o he p ocess condi ions. In machine lea ning, he e a e se e al app oaches o no malizing and scaling o he inpu s, one o which in ol es he pa ame e dis ibu ion o be used. Fo ins ance, Yeganeh and Shadman [54] scaled he pa ame e s o simple linea p o iles, i.e., in e cep , slope and s anda d de ia ion, wi h no mal and chi squa e dis ibu ion ( he ela ions a e no epo ed o b e i y bu he in e es ed eade is e e ed o Eqs. (4) o (9) in Yeganeh and Shad- man [54]). Since he au oco ela ion was no conside ed in Yeganeh and Shadman [54] o OLS es ima o s, we sug- ges o u ilise he be ween-au oco ela ion e ec o he i s o de by means o de ia ions be ween cu en (p e ious) OLS es ima o s and hei espec i e IC alues, i.e., b A0j A0;b A1jA1;b j 0ðb A0ðj1ÞA0;b A1ðj1ÞA1;b ðj1Þ 0Þ. Using his app oach, he means o cu en and p e ious esponses (yj;yj1ðÞ ) a e au oma ically inco po a ed in he inpu s uc u e as hey a e unc ions o he OLS es ima es ^ A0j¼yj^ A1jx;^ A0j1ðÞ ¼ yj1ðÞ ^ A1j1ðÞ xÞ. By conside ing he abo e inpu s, nume ous inpu com- bina ions can be de ined in he p oposed me hod. Ha ing in es iga ed se e al inpu ea u es, he ollowing h ee inpu combina ions I, II and III wi h 8, 4 and 10 inpu s, espec i ely, a e employed o each machine lea ning echnique based on hei abili y o: (1) adequa ely add ess he speci ic e ec s o be ween-au oco ela ion o ype AR(1), and (2) inco po a e he de ia ions be ween OLS es ima es and hei espec i e IC alues o e icien Phase II moni o ing: (I) b A0ðj1ÞA0,b A1ðj1ÞA1,b ðj1Þ 0,b A0jA0, b A1jA1,b j 0,ej,eðj1Þ. (II) b ðj1Þ 0,b j 0, 2 (j-1) , 2 j . (III) b A0ðj1ÞA0,b A1j1ðÞ A1,b ðj1Þ 0,b A0jA0, b A1jA1,b j 0,ej,eðj1Þ, 2 (j-1) , 2 j . These inpu combina ions a e mo i a ed by hei abili y o conside e ec s ega ding au oco ela ion in a ious ways. While inpu combina ion I add esses he aw AR(1) s uc u e o he unde lying model, inpu combina ion II aims o isola e he e ec s o he cu en and p e ious T 2 s a is ics ex ended by an addi ional conside a ion o he cu en and p e ious e o a iances. Finally, inpu com- bina ion III is he union o inpu combina ions I and II, and he e o e, combines bo h main e ec s. In hese h ee inpu combina ions, he ollowing no a ions a e u ilized: •Es ima ed pa ame e s ia OLS in he p e ious sample (b A0ðj1Þ,b A1ðj1Þ,b ðj1Þ). •Es ima ed pa ame e s ia OLS in he cu en sample (b A0j,b A1j,b j). •Mean o e o e ms in he p e ious sample (eðj1Þ). •Mean o e o e ms in he cu en sample (ej). •T 2 s a is ic in he p e ious sample ( 2 (j-1) ). •T 2 s a is ic in he cu en sample ( 2 j ). No e ha he T 2 s a is ic is added as an inpu ea u e and no u he s a is ics om o he con en ional con ol cha s in o de o: (1) inc ease he wo se pe o mance o he T 2 con ol cha , and (2) a oid o e pa ame e iza ion and complexi y wi h ega d o u he common compe i o s. 3.2 Gene a ion o he aining da a se To cons uc he aining da a se , he IC and OC p o iles a e gene a ed by means o simula ions. F om he simula ed p o iles, he inpu s a e cons uc ed based on he p ede ined h ee inpu s uc u es I, II and III in Sec . 3.1. Fo example, he inpu s o he j h gene a ed p o ile consis o 10 ea u es in inpu combina ion III as b A0ðj1ÞA0,b A1j1ðÞ A1, b ðj1Þ 0,b A0jA0,b A1jA1,b j 0,eðj1Þ, ej; 2 ðj1Þand 2 j:Conside ing he sugges ions by Hosseini- a d, Abdollahian and Zeephongsekul [53], equal numbe s o IC and OC p o iles a e gene a ed in a way ha he a ge alues o IC and OC p o iles a e se o 0 and 1, espec i ely. Hence, we conside he size o he aining da a se (numbe o ows) as 6G. Fi s , 3GIC p o iles a e gene a ed and hei inpu ea u es a e eco ded wi h a a ge alue equal o 0. Then, 3GOC p o iles (Gp o iles wi h shi in in e cep , Gp o iles wi h shi in slope, and Gp o iles wi h shi in s anda d de ia ion) a e ob ained in he same way wi h a a ge alue equal o 1. Finally, he aining da a se has 6G ows and 9 (8 ?1), 5 (4 ?1) and 11 (10 ?1) columns o inpu combina ions I, II and III, espec i ely (no e ha he las column ep esen s he a ge alues). Fo be e unde s anding, pseudo code 1 illus a es he p ocess o da a se gene a ion o inpu combina ion III (an anal- ogous p ocedu e also applies o inpu combina ions I and II). 16326 Neu al Compu ing and Applica ions (2023) 35:16321–16340 123 3.3 T aining a machine lea ning echnique A e ob aining a aining da a se , a machine lea ning echnique can be ained on i s basis. In his pape , six common machine lea ning echniques, i.e., ANFIS, ANN, RBF, SVR, CNN and LSTM, wi h he abili y o gene a ing con inuous ou pu s a e in es iga ed. Fo be e unde - s anding, we p o ide a b ie desc ip ion abou he pa am- e e s and adjus men s o each me hod in MATLAB so wa e. Neu al Compu ing and Applica ions (2023) 35:16321–16340 16327 123 •The ANFIS app oach u ilizes uzzy IF–THEN ules o ain he pa ame e s wi h some basic algo i hms such as sub ac i e clus e ing and g id po ioning, in which he idea o ule gene a ion is di e en . The ‘gen is’ unc ion, which is a well-known single-ou pu Sugeno uzzy in e ence sys em, is used o ob ain a g id pa i ion o he aining p ocedu e. •The ANN s uc u e, especially a Mul i-Laye Pe cep- on (MLP), wi h g adien -based op imiza ion is employed he e. An impo an issue in ANNs is ela ed o he adjus men o he numbe o hidden laye s and he neu ons. The unc ion ‘ eed o wa dne ’ u ilises a ully connec ed ne wo k a chi ec u e using he BP Le enbe g–Ma qua d aining algo i hm (‘ ainlm’ op ion). In his s udy, a single hidden laye wi h 10 neu ons is sugges ed o aining. •RBF conside s di e en aining app oaches based on he idea o clus e ing. I has only one hidden laye in a way ha he neu ons wi h a dis inc sp ead ( adius) a e added o i s s uc u e un il he p e-speci ied e o o maximum numbe o neu ons is ob ained. Because he aining p ocedu e is comple ed by he aim o aining e o educ ion, he p obabili y o o e i ing would be gene ally high in his app oach. The unc ion ‘new b’ wi h sp ead ( adius) 1, e o a e 0.05 and maximum neu on size 100 is selec ed o aining pu poses. •As addi ional machine lea ning echniques, SVM and SVR ob ain he pa ame e s based on gene a ing a hype plane in he p oblem space in o de o minimize he gaps be ween he p edic ed and ob ained alues conside ing a ke nel unc ion o mapping he inpu s o he p oblem space. As we aim o each a con inuous ou pu ( eg ession p oblem) in his s udy, he SVR unc ion ‘ i s m’ wi h he gaussian ke nel unc ion is used o aining (mo e de ails abou he classi ica ion and eg ession na u e o machine lea ning-based con ol cha s can be ound in Yeganeh and Shadman [54]). The pa ame e epsilon, which de e mines he dis ance be ween he eal and he es ima ed planes in he space, is an impo an pa ame e o he SVR echnique. As he common ange o epsilon is [0.3–0.5] [61], he alue 0.3 is selec ed in his pape . •The deep leaning echnique CNN is u ilized o in es- iga e i s de ec ion abili y in he SPM ield. Gene ally, a CNN laye mo es some il e s along he inpu e ically and ho izon ally and compu es he do p oduc o he weigh s and he inpu , and hen adds a bias e m o each some no el ea u es om he p ocess. CNNs ha e se e al pa ame e s such as padding and il e size. As he inpu s o his pape a e in ec o o m, he laye s a e c ea ed wi h ‘con olu ion1dLaye ’ unc ion wi h il e size 5. •As a u he deep lea ning echnique, LSTM is ained o e alua e i s pe o mance. Due o he conside a ion o ime dependencies wi h he aim o ime uni s, LSTM can iden i y he ime se ies ela ed pa e ns in an e ec i e way. The mos impo an pa ame e o LSTM is he numbe o hidden neu ons in each uni ha is nea ly like he numbe o hidden laye s in common ANNs. Two LSTM laye s a e de ined by he ‘ls mLaye ’ unc ion wi h 40 hidden neu ons in a way ha he Adam op imize is u ilised o ob ain he bes weigh s. 3.4 Decision on he p ocess condi ion Conside ing he de ini ion o a ge alues, Hosseini a d e al. [53] se he LCL o hei p oposed me hod o 0 and deno ed he UCL as Cu ing Value (CV). The CV is adjus ed by simula ions o each he desi ed alue o ARL 0 . A e adjus men o he CV, he ou pu o he conside ed machine lea ning echnique, e.g., ANN, is compa ed wi h he CV o make a decision on he p ocess [43]. I he ou pu o he ANN in he j h sample (O j ) is la ge han he CV, his indica es an OC condi ion (see Fig. 1 in Hosseini a d e al. [53] o mo e de ails). By employing his app oach in ou p oposed amewo k, we can iden i y he p ocess condi ion when compa ing he ou pu O j o a machine lea ning echnique wi h he espec i e CV. Fo be e unde s anding, pseudo code 2 illus a es he p ocedu e o eaching an OC signal in one i e a ion o simula ions when inpu combina ion III is used. To compu e he ARL and SDRL by means o Mon e Ca lo simula ions, his p ocedu e is i e a ed 10,000 imes. 16328 Neu al Compu ing and Applica ions (2023) 35:16321–16340 123 4 Simula ion s udy To show he e ec i eness o ou p oposed me hod, a comp ehensi e simula ion s udy is conduc ed in his sec- ion. To compa e he six machine lea ning echniques ANFIS, ANNBP, CNN, LSTM, RBF and SVR, he CV o each me hod is se o each ARL 0 equal o 200, as i is he mos common alue in p o ile moni o ing. The nex aim is o ind he bes inpu among he h ee combina ions I, II and III. As such, h ee inpu combina ions unde di e en pa ame e se ings a e used as inpu s o he machine lea ning echniques. Due o he page limi , we only p esen he esul s o h ee inpu combina ions o he shi in in e cep in Table 1unde /= 0.1 and o he esul s can be gi en o he in e es ed eade s upon eques . As can be seen, nea ly all machine lea ning echniques p oduced he bes (i.e., lowes ) alues in e ms o ARL 1 o inpu com- bina ion III. This is due o he ac ha inpu combina ion III is able o combine bo h main e ec s o inpu combi- na ions I and II, namely: (1) app op ia ely add essing he aw AR(1) s uc u e o he unde lying model, and. (2) he e ec s o he cu en and p e ious T 2 s a is ics as well as o cu en and p e ious e o a iances. In o he wo ds, only conside ing (2), i.e., using inpu combina ion II, is no enough o each p ope esul s. On he o he hand, concen a ing on (1), i.e., using inpu combina ion I, enables o ob ain be e pe o mance on a e age han wi h (2). The combina ion o (1) and (2) ia inpu combina ion III clea ly s eng hens he e ec o (1) and leads o supe io esul s. Simila esul s a e ob ained o he es o pa ame e se ings; hus, we only p esen he esul s o inpu combi- na ion III o he es o he expe imen s. In addi ion, due o he i s -p io i y impo ance o de ec ing small shi s in he unde lying p ocess, he ocus is mainly on smalle shi s in he simula ion s udies. As o la ge shi s, he e a e gen- e ally he same pa e ns as o smalle shi s. Fo compa isons o single shi s in in e cep , slope and s anda d de ia ion, he IC model is aken om Noo ossana e al. [44] and Wang and Huang [48], whe e A 0 =3,A 1 =2 and 2 = 1. In addi ion, he explana o y a iables ha e he alues 2, 4, 6, 8 (n= 4), and 0.1, 0.5, 0.9 a e conside ed as ixed alues o /. Fo compa isons o simul aneous shi s, he IC model is ex ac ed om Wang and Lai [45]. In Sec . 4.1, he pe o mance o a ious machine lea ning echniques is compa ed and he me hod wi h he bes pe o mance is selec ed. In Sec . 4.2, he selec ed machine lea ning echnique and con en ional s a is ical con ol cha s in Phase II p o ile moni o ing a e compa ed. Finally, Sec . 4.3 epo s he pe o mance o he bes app oach o simul aneous shi s in p o ile pa ame e s. 4.1 Compa ing di e en machine lea ning echniques based on inpu combina ion III In his subsec ion, h ee indi idual shi s a e conside ed o each pa ame e o compa e he pe o mance o each ech- nique in a way ha he shi ed pa ame e s a e A 0 ?k , A 1 ?g and c . The alues o ARL 1 associa ed wi h di e en machine lea ning echniques a e gi en in Table 2. To ha e a ai compa ison, he alues o he SDRL 1 a e addi ionally p o ided in Table 3 o each o he conside ed shi s. The bold alues ep esen he app oach wi h he bes pe o mance. Neu al Compu ing and Applica ions (2023) 35:16321–16340 16329 123 compe i o s en ailing T 2 , EWMA/R and EWMA-3 a e selec ed ollowing Noo ossana e al. [44]. Table 7shows he esul s o ARL 1 o ANFIS and he compe i o s. No e ha he se ups a e he same as in he p e ious subsec ion, and he esul s o ANFIS and RBF a e ex ac ed om Table 2. As can be seen, ANFIS pe o ms conside ably be e han he o he me hods o shi s ega ding in e cep and slope. The e is a la ge di e ence especially o smalle shi s; o example, ARL 1 is 3.096 o ANFIS gi en k= 0.4, while ARL 1 is 99.7, 21.9 and 19.7 o he con- en ional compe i o s T 2 , EWMA/R and EWMA-3, espec i ely. Howe e , EWMA-3 and RBF ob ain he bes esul s o shi s in he s anda d de ia ion. In addi ion, he s a is ical me hods ou pe o m ANFIS ega ding shi s in he s anda d de ia ion. 4.3 Compa ing he bes machine lea ning echnique wi h con en ional s a is ical app oaches conside ing simul aneous shi s In indus ial p ocesses, simul aneous shi s may occu , so a con ol cha should also be able o de ec such ype o shi s. Wang and Lai [45] conduc ed se e al simula ions abou simul aneous shi s wi h he IC model p oposed by Noo ossana, Ami i and Soleimani [44]. In he ollowing, we compa e ANFIS wi h wo epo ed schemes in Wang and Lai [45], i.e., T 2 and MEWMA. MEWMA is an ad anced e sion o EWMA con ol cha s ha in eg a es he e ec o p e ious samples in one s a is ic, and some esea che s epo ed ha he pe o mance o his app oach in p o ile moni o ing is e y well [17,21,59]. Table 8 shows he esul s in e ms o ARL 1 o simul aneous shi s in in e cep and slope. No e ha we es ic he compa ison o loca ion pa ame e s and do no conside u he simul- aneous shi s, which include shi s in he s anda d de ia- ion, due o he supe io pe o mance o ANFIS ega ding shi s in in e cep and slope. This is also in line wi h he app oach p oposed in Wang and Lai [45]. Acco ding o Table 8, ANFIS ou pe o ms bo h o he me hods. The de ia ions in e ms o ARL 1 a e angible; o example, he alues o ARL 1 a e 135.73, 180.42 and 196.15 (1.91, 34.22 and 49.64) o ANFIS, MEWMA and T 2 , espec i ely, o he smalles (la ges ) shi k= 0.2 and g= 0.025 (k= 1 and g= 0.125). While he e is no dis inc end o he absolu e de ia ions be ween he alues o ARL 1 o ANFIS and each o bo h compe i o s o inc easing shi sizes, he e is gene ally an inc easing beha iou o he co esponding ela i e de ia ions, i.e., ARLMEWMA 1ARLANFIS 1 ARLANFIS 1 and ARLT2 1ARLANFIS 1 ARLANFIS 1 ega ding small shi s (see Table 8). Tha is, he la ge he shi s in slope and/o in in e cep , he la ge he ela i e de ia ions. As o la ge shi s in slope and/o in e cep (k[1 and g[0.125, no abula ed due o lowe ele ance), he al- ues o ARL 1 ega ding ANFIS dec ease o a small ex en , while he alues o ARL 1 ega ding MEWMA and T 2 become close o he espec i e ARL 1 alues o ANFIS, i.e., we obse e a dec easing beha io o he co espond- ing ela i e de ia ions ega ding la ge shi s. To sum up, ANFIS clea ly ou pe o ms bo h me hods in de ec ing simul aneous shi s and i s de ec ion abili y is especially be e o lowe shi sizes. S a is ical con ol cha s usually equi e he ul ilmen o some p incipal assump ions o each he bes pe o mance, while he occu ence o complica ed pa e ns in he man- u ac u ing p ocess may lead o he in alidi y o some o he p esumed assump ions and hus o de e io a ions in hei pe o mance. In con as , machine lea ning echniques encoun e less challenges p o ided ha inpu combina ions and aining p ocedu e a e de ined p ope ly. I could be concluded om he abo e esul s ha he machine lea n- ing-based echniques, and especially ANFIS, pe o m be - e han con en ional s a is ical me hods when moni o ing au oco ela ed p o iles; howe e , some compu a ional e o may be equi ed when implemen ing hese app oa- ches. Due o he exis ence o online da a collec ion sys ems in eal applica ions, big da a s o age and de elopmen o high echnology compu e s, his challenge is becoming easie in a way ha machine lea ning-based sys ems can au oma ically analyze p ocess da a o iden i y OC si ua- ions. To his end, he de ini ion o p ope inpu ea u es, da ase de elopmen , ele an aining adjus men and accep able alse ala m a es a e essen ial asks. These s eps a e usually pe o med as o -line modelling phase while he ope a ion (online) phase e e s o he implemen a ion o he ained model on he online da a o de ec he p ocess Table 9 The IC (black) and OC ( ed) gene a ed esponses in he illus a i e example jy ij 172.253 73.264 72.967 75.188 272.345 73.845 73.610 74.569 372.333 73.363 72.709 74.548 472.193 73.147 73.649 73.942 573.157 74.618 73.373 74.962 673.239 74.219 73.498 75.270 772.352 74.249 73.949 74.897 16336 Neu al Compu ing and Applica ions (2023) 35:16321–16340 123 condi ion [57,60]. By his p ocedu e, he p oposed machine lea ning-based app oach in his pape can imp o e he moni o ing o indus ial p ocesses in e ms o OC de ec ion abili y. 5 Illus a i e example In his sec ion, an illus a i e example o a chemical p o- cess is conduc ed o demons a e a eal applica ion. In ac , his example could be conside ed as a calib a ion sys em in he chemical indus y. Some imes, i is necessa y o con ol a chemical p ocess a om he labo a o y wi h emo e schemes in which some gas senso s a e used as he con- olle . These senso s a e used o moni o such a chemical p ocess o e ime. Al hough i is a bene icial app oach, i needs new calib a ion by changing he senso s’ adjus men s as he a iabili y o gas senso s may a ec he pe o mance o he unde lying calib a ion model [67]. These changes may be caused by di e en chemical ma e ials, p ocess condi ions, and equipmen mo emen s so hei calib a ion should be checked o e ime. The app oach o p o ile moni o ing can be applied o add ess calib a ion issues and o online moni o ing o he p ocess. Fo hese easons, some s udies such as Mahmood e al. [68] and Nadi e al. [36] sugges ed o apply p o ile moni- o ing. Me al oxide (MOX) as a conduc ome ic ype o gas senso s is one o he bes op ions due o i s sensi i i y, ope a ional ease, cos e iciency, apid esponse, and he capabili y o spo ing a high numbe o ola iles. The au ho s supposed MOX as a senso and moni o ed a unc ional ela ionship be ween he esis ance (R) o he senso (i.e., MOX) as he dependen a iable and he concen a ions o ca bon monoxide in he senso as he independen a iable. To moni o his unc ional o mula, hey eco ded he esul s o senso esis ance and di e en concen a ion le els o e ime. Based on he eco ded da a, he explana o y a iables a e ixed a 25, 100, 125, and 150 ppm. To each a be e pe o mance, i is sugges ed o change he p ocess si ua ion wi h some addi i es. These subs ances a e blended o a special p ocess o accele a e he p ocessing abili y o he polyme s, imp o e he cha ac e - is ics such as du abili y, s i ness, and enhance he se ice li e. A wide ange o addi i es such as gas, eed, an i-wea , ood, uel, an ioxidan , plas ic addi i es ha e been ex en- ded ye . Indeed, gas addi i es a e usually added o he gas senso p ocesses o adjus he low o gas du ing he expe imen [67]. Howe e , p e ious wo ks showed ha he ela ion be ween esis ance and ca bon concen a ion migh change in he case o addi i e ma e ials. To add ess hese issues, Nadi e al. [36] in es iga ed si ua ions ela ed o he be o e and a e o adding he addi i e ma e ial in a way ha one addi i e ma e ial was added o he p ocess a e ime 3278; so, he IC model was ex ac ed om he i s 3278 p o iles. Conside ing hese p o iles, Nadi e al. [36] conside ed a simple linea IC model wi h he au oco ela- ion e ec as ollows: Yij ¼71:741 þ0:0176Xiþeij; eij ¼0:565eij1ðÞ þaij;aij N0;0:142ðÞ; i¼1;2;3;4;j¼1;2;... ð13Þ To show he applicabili y o he p oposed me hod in moni o ing he abo e IC model, Nadi e al. [36] u ilized simula ions o OC da a gene a ion (ins ead o using he da a a e he 3278 h p o ile). Following hem, we i s gene a ed i e IC p o iles and hen con inued wi h he OC p o ile gene a ion conside ing a shi in he in e cep un il eaching an OC signal. The magni ude o he OC shi was conside ed as 0.15 (o 0.5 ). Table 9shows he esponse a iables o he gene a ed p o iles ( he black and ed alues a e IC and OC p o iles, espec i ely). To speci y he de ec ion abili y o ANFIS o his da a se , i is ained based on he IC model in Eq. (13) and inpu combina ion III. Conside ing ARL 0 = 200, he CV is se o 0.615. A e adjus men o he CV, he gene a ed da a in Table 9is impo ed o ANFIS and he ou pu o each inpu is compu ed. Table 10 epo s he inpu and ou pu alues Table 10 Inpu and ou pu alues o he i s se en gene a ed p o iles applying ANFIS wi h inpu combina ion III (/= 0.565) jInpu s Oj 10 0 0 0 -0.2 0.001 0.433 4.406 0 -0.08 0.477 2-0.2 0.0012 0.4332 4.406 0.219 -0.001 -0.26 0.963 -0.08 0.14 0.358 30.22 -0.0013 -0.2557 0.963 0.124 -0.004 0.213 3.209 0.14 -0.31 0.230 40.12 -0.004 0.213 3.209 0.076 -0.003 -0.37 1.404 -0.31 -0.12 0.318 50.08 -0.003 -0.3709 1.404 1.194 -0.007 0.304 9.352 -0.12 0.68 -0.357 61.19 -0.007 0.3045 9.352 1.08 -0.005 0.196 1.575 0.68 0.26 0.157 71.08 -0.005 0.196 1.575 0.21 0.002 -0.22 0.971 0.26 0.05 0.849 Neu al Compu ing and Applica ions (2023) 35:16321–16340 16337 123 o he i s se en gene a ed p o iles. Hence, ANFIS only needs wo OC samples o igge a signal. The signal in he 7 h sample appea s because he inal s a is ic exceeds he CV ( ed ho izon al line in Fig. 2), so ANFIS can igge an OC signal (O 7 = 0.849 [0.615 = CV). 6 Conclusions In p o ile moni o ing, he e o e m o en does no ollow a simple s uc u e and is a ec ed by au oco ela ions. Fo his eason, a no el moni o ing scheme o linea au o- co ela ed p o iles wi h be ween-au oco ela ion o i s o de in Phase II o p ocess moni o ing has been p oposed in his pape . Unlike mos o he exis ing me hods ha use common s a is ical con ol cha s, his pape employed a ious machine lea ning echniques, such as ANFIS, ANNBP, CNN, LSTM, RBF and SVR as a con ol cha . To his aim, ou main s eps we e de ined. In he i s s ep, he s uc u e o he inpu ea u es and ou pu s o a machine lea ning echnique we e de e mined, and hen, a aining da a se based on he inpu ea u es and ou pu s was gen- e a ed by simula ing IC and OC p o iles in he second s ep. The hi d s ep u ilized he gene a ed da a se o ain a machine lea ning echnique, and inally, by he de ini ion o a con ol limi , he machine lea ning echnique p o ided in o ma ion abou he p ocess condi ion in he ou h s ep. The s udy conduc ed pu sued h ee main objec i es. Due o he high impo ance o inpu ea u es in machine lea ning, some inpu ea u es, which a e app op ia e o accoun o be ween-au oco ela ion e ec s o i s o de , we e de ined and compa ed o achie e he mos app op ia e inpu combina ion. The esul s indica ed ha inpu com- bina ion III, which is de ined as he union o inpu com- bina ions I and II and combines bo h main e ec s o hese inpu combina ions, is he mos app op ia e one. Fo he second aim, di e en machine lea ning echniques we e compa ed o iden i y he mos adequa e one. Expe imen al s udies showed ha ANNBP, CNN, LSTM and SVR we e mos ly no able o each a sa is ac o y de ec ion abili y in compa ison wi h ANFIS and RBF. Among ANFIS and RBF, ANFIS was p e e able wi h espec o shi s in in e cep and slope, while RBF had he bes pe o mance ega ding shi s in he s anda d de ia ion. This supe io i y was ob ious o low and mode a e au oco ela ion coe i- cien s (i.e., /= 0.1 and 0.5), while i was no possible o iden i y a consis en ly bes me hod o a la ge alue (/ = 0.9). To add ess his issue, we addi ionally implemen ed an o e all pe o mance measu e, called RMI. Following RMI, we ound ha ANFIS u ns ou o be he me hod wi h he bes o e all a e age pe o mance o /= 0.9. The hi d aim o his s udy was o compa e machine lea ning-based echniques wi h s a is ical con ol cha s. This compa ison led o he esul ha he de ec ion abili y o ANFIS ou - pe o med all he compe i o s ega ding shi s in in e cep and slope. Howe e , he de ec ion abili y o ANFIS ega ding shi s in he s anda d de ia ion was in e io compa ed o he selec ed s a is ical con ol cha s. In his ega d, he EWMA-3 con ol cha pe o med be e , and he bes machine lea ning echnique o his pu pose was RBF (wi h a pe o mance ha is ha dly wo se han ha o EWMA-3). Hence, machine lea ning-based con ol cha - s, and ANFIS in he i s place, a e sugges ed o be u ilized in p o iles ha a e cha ac e ized by be ween-sample AR(1) au oco ela ion o conside ably imp o e he de ec ion abili y o he con ol cha . Employing he p oposed no el inpu ea u es wi h o he machine lea ning echniques and o he p o ile ypes such as nonlinea o Gene alized Linea Models (GLMs) in he p esence o au oco ela ions could be a p omising a enue o po en ial u u e esea ch. Also, implemen ing he p o- posed me hod in p o iles ha a e cha ac e ized by wi hin 0.477 0.358 0.230 0.318 -0.357 0.157 0.849 -0.500 -0.300 -0.100 0.100 0.300 0.500 0.700 0.900 1234567 Oj Sample Numbe Fig. 2 The inal cha s a is ics o he i s se en andom gene a ed p o iles in he illus a i e example (/= 0.565) 16338 Neu al Compu ing and Applica ions (2023) 35:16321–16340 123 sample au oco ela ion o in p o iles wi h o he au oco e- la ion pa e ns, such as ARMA o VARMA a e u he sugges ions o po en ial u u e di ec ions. Acknowledgemen s The au ho s would like o hank ou anonymous e iewe s o hei aluable eedback and sugges ions, which we e impo an and help ul o signi ican ly imp o e he pape . Funding Open Access unding enabled and o ganized by P ojek DEAL. Da a a ailabili y s a emen Da a sha ing is no applicable o his a icle as no new da a we e c ea ed o analyzed in his s udy. 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