Diagnosing Problems of Distribution-Free Multivariate Control Chart

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1 Advaced Materals Research Ole: ISSN: , Vols , 6-66 do:.48/ 4 ras ech Publcatos, Swtzerlad Dagosg Problems of Dstrbuto-Free Multvarate Cotrol Chart Wel Sh, a ad Xuem Z,b, a Uversty of echology ad Educato, a, P. R. Cha a shwel43@63.com.c, b z_xuem@alyu.com.c Keywords: Dstrbuto-Free; Multvarate Statstcal Process Cotrol; Fault Dagoss; LASSO Abstract. I order to solve the roblem of oly have a few hstorcal data that ca be used multvarate rocess motorg, a ew dstrbuto-free multvarate cotrol chart has bee roosed. Ad the cotrol chart structure the cotrol lmts are determed o-le wth the future observatos ad the hstorcal data. herefore, the roosed cotrol chart has very mortat alcato ractce. However, the research does t study the roblem of the fault dagoss after the cotrol chart alarms. So we use LASSO-based dagostc framewor to detfy whe a detected shft has occurred ad to solate the shfted comoets. Itroducto I moder qualty cotrol, t s very commo to motor several qualty characterstcs a rocess smultaeously []. hs motorg ad dagostc about multvarate rocess observatos s ofte called multvarate statstcal rocess cotrol (referred to as MSPC). Because the basc tas of MSPC cludes: Frstly, determg whether statstcal rocess has chaged, that s, whether a sgal occurred cotrol chart. Secod, detfyg the locato of shft occurred ad solatg the comoets of shfted. I the recet research, Che, Z & Zou reseted a dstrbuto-free multvarate cotrol chart []. herefore, the urose of ths aer based o the ther research s comletg the secod tas, amely the fault dagoss or fault detfcato of the dstrbuto-free multvarate cotrol chart. I the recet research, for the dagosg roblems MSPC, Zou, Jag & sug have rased a ufed LASSO-based dagostc framewor [3]. he ma urose of the dagoss s to determe the ma chaged arameters, uder the stuato that other MSPC methods have used to be detected ad have estmated the chage-ot correctly. I ths case, the chage-ot arttos the observatos to two subsets wth dfferet arameter values. Hece the fault solato as two samle selecto roblem to combe Bayesa formato crtero (referred to as BIC)wth a ealzed techque to facltate the fault tracg rocess ad suggest a ractcal LASSO-based dagostc rocedure [4,5]. Wth the method above we ca fd the arameters that are resosble for the chage. herefore, the cotet of ths aer s that usg based o LASSO dagostc framewor to comlete the ost-sgal dagostc aalyss, whch s based o dstrbuto-free multvarate cotrol chart, so that ths cotrol chart has a better alcato ractce. he rest cotet of ths aer s as followed: Secto descrbes dstrbuto-free multvarate cotrol chart. Secto 3 rovdes LASSO-based dagostc method. Secto 4 develos the ost-sgal dagostc aalyss of dstrbuto-free multvarate cotrol chart, whch s the focus of ths aer. Secto 5 gves the coclusos of the aer. Dstrbuto-free Multvarate Cotrol Chart Suose that there are m deedet ad detcally dstrbuted hstorcal observatos, x m,... + x R,where are some teger.ad suose the th future observato x = ( x,..., x ) s collected from the follow multvarate locato chage-ot model All rghts reserved. No art of cotets of ths aer may be reroduced or trasmtted ay form or by ay meas wthout the wrtte ermsso of ras ech Publcatos, (#69868, Pesylvaa State Uversty, Uversty Par, USA-8/9/6,:54:8)

2 Advaced Materals Research Vols x F ( x, µ ), = m +,...,,,..., τ F ( x, µ ), = τ +,... () whereτ s the shft ot, F ad F are the -cotrol ad out-of-cotrol dstrbuto fuctos. I ractcal alcatos, they may be the same or may be dfferet, but for ther locato arameters µ = ( µ,... µ ) ad µ = ( µ,... µ ), we ofte assume that they are uequal. Cosdered the motorg roblem () s close to the locato of the two samle arameter hyothess test of the roblem. herefore, we use oarametrc two-samle test methods to cosder + ad { x,..., x} as F ( x ; µ ) ad (, ) ths roblem. Suose { x m,..., x τ } τ + are two deedet samles, ad dstrbuted F x µ. he cosder the ull hyothess test, H : µ µ = versus H : µ µ. Because the ull hyothess µ = µ s equvalet to the µ = µ,where =,.... So, t s straghtforward to cosder the Wlcoxo ra-sum test for each comoet, = = τ +, () τ r ( τ )( + ) ( + )( τ ) Where r s the ra of x amog the samle. hrough the above dscusso to costruct a EWMA cotrol chart ad use ths cotrol chart to motor the model (). Let X, = ( x,..., x ),ad X ( X,,... X, ) =. Choose a smoothg arameter λ ad a wdow sze ω.at each ot, costruct a chartg statstcs ω, λ = ω, λ, where = ( ω, λ) = ( = ω+ r ( m + + ) ( + m ω )( m + + ), (3) ω Smlarly, r s the ra of x amog the samle X +. Suose τ s the locato of chage ot, m, the ( ω, λ) would become a large value for > τ, ad corresodgly would become a large value, ad the caused a alarm. I order to costruct a comlete cotrol chart after the costructo of the statstc, aother mortat tas s to determe the cotrol lmts. hs ew method roose to determe the cotrol lmts C( α ) by solvg the followg equatos, ( ω λ > C α F) = α ( ) Pr,, Pr ω, λ > C ( α) ( ω, λ) < C α, <, F = α, >, where α s the re-secfed false alarm rate, F = τ τ I( x ) t, ad = F ( t ) ( m = + ) I ( x t ). So ths cotrol chart costtuted by the ad C( α ), termed as = m + dstrbuto-free multvarate EWMA chart(referred to as DFEWMA). he LASSO-based EBIC Dagoss Method I ths secto, we wll troduce the method we use ths aer, amely the LASSO-based EBIC dagoss method [3]. I may alcatos, all varables shft at the same tme s very rare, ad the umber of smultaeously chaged varables s relatvely small. herefore, we suose µ = µ + δ, (4)

3 64 New echologes for Egeerg Research ad Desg Idustry whereδ = ( δ,..., δ ). We geerally suose that the maorty comoets are zero, whch s called sarse features [6]. So the fault dagoss s essetally smlar to the model or varable selecto roblem, that s, oe wshes to choose those values that devate sgfcatly from δ. Because of Schwarz s Bayesa formato crtero teds to better detfy the true sarse model, so we wll cosder the dagostc ad wth BIC [4]. However, whe the model sace s large but the samle sze s moderate, the ordary BIC s somewhat lberal for model selecto. herefore, we wll cosder tae advatage of the exteded famly of BIC (referred to as EBIC) [7]. Accordg to a heurstc dervato rovded by Zou & Qu [3], the defto of EBIC s EBICs = f ( δ s ) + l + l s, (5) + Where f ( δ s ) = ( µ µ δ s ) ( Σ + Σ) ( µ µ δ s ), ad s s a caddate model, whch cotas the corresodg dex of chaged arameters. I ractcal alcatos, whe the dmeso s large, usg the (5) to calculate all the values of the EBIC s mossble. herefore, ths method combed some ealzed techques wth the EBIC. Next, we wll cosder the ealzed loss fucto, ( δ ; ) ( δ ) ( δ ) PL = f + g, (6) = Where (,..., ) g s the ealty fucto. By usg the adatve LASSO [8], the ealzed loss fucto coverted to APL = are the ealty arameters, ( ; ) f = δ = δ + δ, (7) herefore, for a gve, model δ ca be obtaed by mmzg ( ; ) s = { : δ }. By substtutg δ to (5) we wll get APL δ. hus we get a caddate l EBIC = f δ + + l. (8) + Where reresets the umber of o-zero values r δ. Let = δ, where r > s some gve costat, e.g..5 or, ad δ = µ µ, the the ealzed loss fucto (7) coverted to ( ; ) = PL α = δ α Γ δ α + α. (9) Where r r r α = δ δ, = dag( δ,..., δ ),ad Γ = Ω + Ω. herefore, wth the hel of LARS algorthm[9], the dagostc results ca be easly obtaed. hs dagostc method s dvded to three stes: ().Fd corresodg estmates of µ ad covarace matrx Σ, where =, determe f ( δ ). ().Use the LARS algorthm to solve (9), the obta ALASSO solutos. (3).Substtute these solutos to (8), ad fd a δ m dagostc corresodg result s s = { : δ m }. m. hus we so that EBIC value s the smallest. he the

4 Advaced Materals Research Vols he Post-Sgal Dagoss of DFEWMA Chart Che, Z & Zou (4) roosed the method of dstrbuto-free multvarate cotrol chart, but ther study dd ot clude the ost-sgal dagoss of the DFEWMA chart [], so ths art of aer wll study the fault dagoss of DFEWMA based o ther research. Frstly, we suose that the cotrol chart sgals at th observato the we wll have m hstorcal IC observatos ad ew observatos. Now, we assume that a shft occurs after the τ th samle ( τ < ), the gve a estmate of shft locato v, we ca calculate the oarametrc test statstc ( v, arg max τ < (,.herefore, we ca get the estmator of the shft locato,τ, s τ = v. () After the shft locato s estmated, the chage-otτ wll dvded the observatos to two X τ m + X τ + samles, ad. herefore, we ca use the above-metoed LASSO-based EBIC dagoss rocedure [3], by dagostc aalyss of chage-ot to determe whch comoets have chaged. Suose that the mea vector are µ, the covarace matrx are Σ, where =,. he f δ x x δ x x δ = ( ) Σ + Σ ( ), () Where x ad Σ are the estmates of the mea vector ad covarace matrx for the th samle. herefore, we ca use the above-metoed basc stes to dagose the locatoτ of the shft, ad obta dagostc results s = { : δ }. Next we show the results of our research through data smulato. Here we use a multvarate ormal dstrbuto, the mea vector µ s set to zero vector ad the covarace matrx s ( σ ) Σ =,where σ = ad.5 σ =. Now we are gog to cosder the followg case: x =, 3 + δ s used as a out of cotrol model whe x m =, =., α =. 5 > τ, where δ s shft vector. I our smulato, λ ad let chaged locato at τ = 55. So we frst get a DFEWMA cotrol chart show Fgure. 6 DFEWMA sgal 统统量 Samle Idex Fgure.DFEWMA chart. he blac dot reresets the ( ω,, the red dot reresets the C ( α ), ad the blue sgal ot s = v Fgure. he values of ( v, v <. for

5 66 New echologes for Egeerg Research ad Desg Idustry After the cotrol chart sgals at = 6, the we go to comlete the ost-sgal dagoss. Frst, we v, λ reaches the maxmum use () to fd the chage-otτ. From Fgure we ca see, 57. whe v = 55. So usg () we ca get a very accurate estmate of the chage-ot τ, that s τ = 55. Now, we have = 8 ad = 6. he, we use the LASSO-based EBIC dagoss method to detfy the chaged arameters. Our results are showed able. hese values of EBIC dcates the shft may have occurred the four comoets. herefore, we ca get the dagostc result of DFEWMA cotrol chart s s = {, 7, 9,}. able. Dagostc results of the LASSO-based EBIC dagoss method about DFEWMA chart EBIC δ Cocluso hs aer based o the study of dstrbuto-free multvarate cotrol chart, usg the LASSO-based EBIC dagoss method to comlete the fault dagostc aalyss of DFEWMA cotrol chart. Whe the DFEWMA cotrol chart sgals, we through the dagostc aalyss ca accurately determe the locato of the shft, ad detfy the ma elemets of the shft. I ractcal alcatos, ths wll be able to hel busess maagers ad egeers to detfy ad elmate the root causes of fault qucly ad accurately, so that ths cotrol chart has better alcato. Acowledgemet hs research was suorted by the NNSF of Cha Grats 36. Refereces [] Woodall, W. H. ad Motgomery, D. Some Curret Drectos the heory ad Alcato of Statstcal Process Motorg, Joural of Qualty echology, to aear. (3) [] Na Che,Xuem Z ad Chaglag Zou.A Dstrbuto-free Multvarate Cotrol Chart(4) [3] Zou, C., Jag, W., ad sug, F. A Lasso-Based Dagostc Framewor for Multvarate Statstcal Process Cotrol, echometrcs, 53, () [4] Schwarz, G. Estmatg the Dmeso of a Model, he Aals of Statstcs, 6, (978) [5] bshra, R. J. Regresso Shrage ad Selecto va the LASSO, Joural of the Royal Statstcal Socety: Seres B, 58, (996) [6] Zou, C., ad Qu, P. Multvarate Statstcal Process Cotrol Usg LASSO, Joural of the Amerca Statstcal Assocato, 4, (9) [7] Che, J., ad Che, Z. Exteded Bayesa Iformato Crtero for Model Selecto wth Large Model Saces, Bometra, 95, (8) [8] Zou, H. he Adatve Lasso ad Its Oracle Proertes, Joural of the Amerca Statstcal Assocato,, (6) [9] Efro, B., Haste,., Johstoe, I., ad bshra, R. Least Agle Regresso, he Aals of Statstcs, 3, (4)

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