A Nonparametric Multivariate Control Chart Based on. Data Depth. Abstract

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1 A Nonparameric Mulivariae Conrol Char Based on Daa Deph Amor Messaoud, Claus Weihs and Franz Hering Deparmen of Saisics, Universiy of Dormund, Dormund, Germany Absrac For he design of mos mulivariae conrol chars, i is assumed ha he observaions follow a mulivariae normal disribuion. In pracice, his assumpion is rarely saisfied. In his work, a disribuion-free EWMA conrol char for mulivariae processes is proposed. This char is based on sequenial rank of daa deph measures. 1 Inroducion Reducing variaion in manufacuring is desirable o reduce produc cos and improve produc performance and qualiy. To achieve his objecive saisical process conrol SPC is used. SPC is a se of echniques for monioring a producion process o deermine if i is sable over ime and capable of producing high qualiy producs. One purpose of conrol charing, he feaured ool of SPC, is o disinguish beween wo sources, common and assignable causes, of process variaion. Common or chance causes of variaion canno be economically idenified and correced and considered o be due o he inheren naure of he process. assignable or special causes of variaion are unusual shocks or oher disrupions o he process, he causes of which can and should be removed. A process is said o be in a sae of saisical conrol if i operaes under common causes. Typically conrol chars apply o sysems or processes in which only one qualiy characerisic is measured and esed. However, he rapid growh of daa acquisiion echnology and he use of online compuers for process monioring have led o an increased ineres in he simulaneous 1

2 surveillance of several relaed qualiy characerisics on process variables. These echniques are ofen referred o as mulivariae saisical conrol procedures. For mos of hese procedures, i is assumed ha he underlying disribuion of he process is mulivariae normal. Thus, he saisical properies of commonly employed conrol chars are exac only if his assumpion is saisfied. In pracice, i is well known ha his assumpion rarely holds. Therefore, disribuion-free or nonparameric conrol chars for mulivariae processes are nedded. In his paper, we propose a nonparameric EWMA conrol char for mulivariae processes based on sequenial ranks of daa deph measures. In secion 2, he daa deph noion is inroduced. 2 A nonparameric EWMA for mulivariae processes 2.1 Daa deph Daa deph measures how deep or cenral a given poin X R d is wih respec o w. r.. a probabiliy disribuion F or w. r.. a given daa cloud {Y 1,..., Y m }. There are several measuremens for he deph of he observaions, such as Mahalanobis deph, he simplicial deph, half-space deph, and he majoriy deph of Singh, see Liu e al In his work, he Mahalanobis deph and simplicial deph are considered. 1. The Mahalanobis deph MD F of a given poin X R d w. r.. F is defined o be MD F X = X µ F Σ 1 F X µ F, where µ F and Σ F are he mean vecor and dispersion marix of F, respecively. The sample version of MD F is obained by replacing µ F and Σ F wih heir sample esimaes. In fac, how deep X is w. r.. F is measured by how small is quadraic disance is o he mean. 2. The simplicial deph SD F Liu, 1990 of a given poin X R d w. r.. F is defined o be SD F X = P F {X s[y 1,...,Y d+1 ]}, where s[y 1,...,Y d+1 ] is a d-dimensional simplex whose verices are he random observaions {Y 1,..., Y d+1 } from F. The sample simplicial deph SD Fm X is defined o be SD Fm X = m d I X s[y i1,...,y id+1 ], 1 i 1<...<i d+1 m 2

3 where {Y 1,..., Y m } is a random sample from F, F m denoes he empirical disribuion of {Y 1,..., Y m } and I. is he indicaor funcion. For example, he bivariae SD Fm X relaive o {Y 1,..., Y m } is equal o he proporion of closed riangles wih verices Y i, Y j, Y k ha conain X, 1 i < j < k m. Liu 1990 showed ha if F is absoluely coninuous, hen as m, SD Fm converges uniformly and srongly o SD F X and ha SD F X is affine invarian. A daa deph SD F X induces a cener-ouward ordering of he sample poins if deph values of all poins are compued and compared. If all SD F X s are arranged in an ascending order and X [j] is used o denoe he sample poin associaed wih he jh smalles deph value, hen X [1],..., X [m] are he order saisics of X i s wih X [m] being he mos cenral poin. The smaller he rank of a poin, he more oulying he poin w. r.. he underlying disribuion F Sequenial ranks In his secion, order saisics ha are used in his work are quickly reviewed. Le X, = 1, 2,..., be a sequence of independen random variables from a coninuous disribuion Fx. The sequenial rank R is he rank of X among he mos recen m m > 1 observaions aken from he process X, X 1,..., X m+1. Tha is, R = 1 + i= m+1 IX > X i, 1 where I. is he indicaor funcion. The sandardized sequenial rank R m For all, R m wih mean µ R m R m = 2 m is uniformly disribued on he g poins { 1 m 1, 3 m 1,..., 1 1 } m = 0 and variance σ R m is defined as R m = m2 1 3m 2. For more deails, see Hackl and Ledoler A conrol char based on sequenial rank of daa deph measures Liu 1995 was he firs who used he concep of daa deph o consruc a nonparameric conrol char for monioring mulivariae processes. In his work, we consider an EWMA char based on sequenial ranks of daa deph measures o monior mulivariae processes. The proposed char is a generalizaion on he nonparameric EWMA for individual observaions proposed by Hackl and 3

4 Ledolser I is assumed ha p 1 random vecors X, = 1, 2,..., are observed and moniored over ime. Each vecor X = X 1, X 2,..., X p conains p qualiy characerisic measuremens made on a par from a mulivariae process. The value X j, j = 1,..., p, represens an observaion on he j h qualiy characerisic a ime. For his char, a reference sample is considered as he m mos recen observaions aken from he process X m+1, X m+2,..., X. This sample is be used o decide wheher or no he process is sill in-conrol a ime. The deph of X is calculaed w. r.. his reference sample and he sequenial rank R of D m X among D m X m,..., D m X 1 is compued using equaion 1 The sandardized sequenial rank, defined by equaion 2, are moniored using he exponenially weighed moving average EWMA recursion. Tha is, T = min{b, 1 T 1 + R m }, 3 = 1, 2,..., where 0 < 1 is a smoohing parameer, B is a reflecing boundary and T 0 = u. The process is considered in-conrol as long as T > h, where h < 0 is a lower conrol limi h u B. Noe ha, he lower-sided EWMA is considered because he saisic R m is higher he beer. A reflecing boundary is included o preven he EWMA from drifing o one side indefiniely. I is known ha EWMA schemes can suffer from an ineria problem when here is a process change some ime afer beginning of monioring. Tha is, an EWMA can have wandered away from a cener line in a direcion opposie o ha of a shif ha occurs some ime afer he sar of monioring. In his unhappy circumsance, an EWMA scheme can ake long ime o signal. Hackl and Ledoler 1992 considered a coninuous qualiy crieria. This coninuiy assumpion assures ha ies are impossible. However, in pracice when measuremens or oher numerical observaions are aken, i is ofen ha wo or more observaions are ied. For example, ies may be due o he naure of he phenomenon modelled or rounding of coninuous variables emperaure, blood pressure,... In his work, he simplicial deph is a discree measure and ies may occur. Especially, here always exis a leas d + 1 exreme poins ha share he minimum simplicial deph of d + 1/m, see Soumbos and Reynolds The mos common approach o his problem is o assign o each observaion in a ied se he midrank, ha is, he average of he ranks 4

5 reserved for he observaions in he ied se, see Gibbons and Chakrabori Average run lengh of he in-conrol process As menioned, he parameers of he conrol char are seleced according o a performance of he char. Usually, he performance of conrol chars are evaluaed by he average run lengh ARL. The run lengh is defined as he number of observaions ha are needed o exceed he conrol limi for he firs ime. The ARL should be large when he process is saisically in-conrol in-conrol ARL and small when a shif has occurred ou-of-conrol ARL. In his work, we used he inegral equaion o approximae he in-conrol ARL, see Crowder Le Lu be he ARL of he lower-sided EWMA char given ha T 0 = u, i can be shown ha he inegral equaion for Lu is given by Lu = 1 + LBPr r B 1 u B + L 1 u + r dfr, h where Fr is he cumulaive disribuion of r. We assumed ha ies are no observed. Therefore, R m are uniformly disribued on he m poins {1/m 1, 3/m 1, /m}. For moderae and large m he discree disribuion of R m which leads o Lu = 1 + LBPr r is approximaed by a coninuous uniform disribuion, B 1 u B + L 1 u + r frdr, 4 h where fr is he probabiliy densiy of he uniform disribuion. The soluions o inegral equaion 4 can be obained by replacing he equaion wih a sysem of linear equaions using he collocaion mehod and solving he sysem of equaions. see appendix. As recommended by Calzada and Scariano 2003, he collocaion mehod is used because he coninuous uniform disribuion does no have he enire real line as numerical suppor. In he previous approximaion, we ignored he sligh dependence among successive ranks R m. Therefore, he resul in 4 applies only approximaely, as here are small correlaions among successive ranks. For moderae and large values of m he correlaions are quie small, see Hackl and Ledoler Table 1 shows he lower one sided EWMA ARL s for he same smoohing parameers and conrol limis h as in Hackl and Ledoler 1992 and assuming ha he EWMA sars a 0, ha is T 0 = 0. Table 1 shows a decrease in he ARL wih increasing for fixed conrol limi h. As menioned by Hackl and Ledoler 1992, his is explained by he fac ha σ 2 T increases 5

6 Table 1: ARL s of he one-sided EWMA wih reflecing boundary B = h average run lengh less han 100, + average run lengh greaer han h = 0.1 = 0.2 = 0.3 = 0.4 = wih so ha he probabiliy of crossing he conrol limi h becomes larger. A simulaion sudy is carried in order o validae he ARL approximaion. We generae independen observaions X from a bivariae normal disribuion wih µ= 0, 0 and Σ = Noe ha due o he nonparameric naure of he monioring sraegy, he normaliy is no required and any oher disribuion could be used. The resuls of he simulaion showed ha for m > 100 he approximaion in 4 can be used o selec he parameers of he nonparameric EWMA in order o aain a desired average run lengh in he in-conrol siuaion. 4 Applicaion In his secion, he proposed EWMA conrol char is used o monior a BTA Boring and Trepanning Associaion deep hole drilling process. Deep hole drilling mehods are used for producing holes wih a high lengh-o-diameer raio, good surface finish and sraighness. For drilling holes wih a diameer of 20 mm and above, he BTA Boring and Trepanning Associaion deep hole machining principle is usually employed, for more deails see Theis The process is subjec o dynamic disurbances usually classified as eiher chaer vibraion or spiralling. Chaer leads o excessive wear of he cuing edges of he ool and may also damage 6

7 he boring walls. Spiralling damages he workpiece severely. The defec of form and surface qualiy consiue a significan impairmen of he workpiece. As he deep hole drilling process is ofen used during he las producion phases of expensive workpieces, process reliabiliy is of primary imporance and hence disurbances should be avoided. For his reason, process monioring is necessary o deec dynamic disurbances. In his secion, we will focus on chaer which is dominaed by single frequencies, mosly relaed o he roaional eigenfrequencies of he boring bar. Therefore, we propose o monior he ampliude of he relevan frequencies in order o deec chaer vibraion as early as possible. In pracice, i is necessary o monior several relevan frequencies because he process is subjec o differen kind of chaer i. e., chaer a he beginning of he drilling process, high and low frequency chaer. The EWMA char based on sequenial ranks of daa deph measures is used o monior he ampliudes of frequencies 234 and 703 Hz, which are among he eigenfrequencies of he boring bar, in an experimen wih feed f=0.185 mm, cuing speed v c =90 m/min and amoun of oil V =300 L/min. For more deails, see Weiner a al For he EWMA char, we used B = h. Typical values of are in he range of 0.1 < < 0.3, see Hackl and Ledoler In his work, we used = 0.1, 0.2 and 0.3. The corresponding values for h are respecively 0.314, and The simplicial deph is compued using he FORTRAN algorihm developed by Rousseeuw and Rus Table 2 shows he resuls, for deph 270 mm. The EWMA chars based on MD F produces more ou-of-conrol signals han he EWMA chars based on SD F. This is due o he sensiiviy o he MD F measure o he exreme values. Table 2 shows ha all conrol chars signal a 32 deph 35 mm. In fac, i is known ha approximaely a deph=35 mm he guiding pads of he BTA ool leave he saring bush, which induces a change in he dynamics of he process. From previous experimens, he process has been observed o eiher say sable or sar wih chaer vibraion. A grea number of ou of conrol signals occur a 35 deph 45 mm. Indeed, he new physical sae of he process is represened in he reference sample afer deph 45 mm. All conrol chars signal a deph 110 deph 120 mm and i is known ha deph 110 mm is approximaely he posiion where he ool eners he bore hole compleely. Theis 2004 noed 7

8 Table 2: Ou of conrol signals of he differen conrol chars applied o he ampliude of frequencies 234 Hz and 703 Hz m=100 Hole Deph Observaion EWMA mm number = 0.1 = 0.2 = 0.3 MD F SD F MD F SD F MD F SD F Toal ha his migh lead o changes in he dynamic process because he boring bar is slighly hinner han he ool and herefore he pressures in he hole may change. The imporan ou-of-conrol signals are produced a 250 deph 255 mm. Messaoud e al showed ha a change occurred in he process a deph= mm and hey concluded ha his change may indicae he presence of chaer or ha chaer will sar in a few seconds. Therefore, in his experimen chaer may be avoided if correcive acions are aken afer hese signals. In his experimen, he EWMA char wih =0.3 is he bes, and should be choosen among he hree EWMA chars considered in his work. Indeed, only 14 ou-of-conrol signals are produced and all changes of he physical condiions of he process are deeced. In pracice, a procedure o choose he smoohing parameer is required. 5 Discussion The fuure research should focus on he comparison of he in-conrol and ou-of-conrol performance of he proposed nonparameric EWMA o exising parameric conrol chars for normal and nonnormal daa. This comparison should include he robusly designed parameric mulivariae EWMA MEWMA char. Soumbos and Sullivan 2002 showed ha he MEWMA behaves like disribuion-free conrol chars for an appropriae choice of he smoohing parameer. 8

9 For he process adjusmen, once he EWMA char has produced a signal, a procedure o esimae he shif magniude and o idenify he ime poin a which he shif occurred is required. For example, in our experimen he mos imporan shif occurred a deph mm. The EWMA char, wih = 0.3, based on MD F and SD F deec i afer 2 and 4 samples respecively. In pracice, he minimum of he MD F measure over a shor window, wih a given lengh, before he occurrence of he ou-of-conrol signal may be used o esimae he shif magniude. However, in his case, one limiaion of he SD F is ha once he daa poin is ouside he daa cloud, he SD F measure is equal o d + 1/m. This does no give an informaion abou he shif magniude. For he ou-of-conrol inerpreaion, when he conrol char indicaes an ou-of-conrol condiion, i is imporan o deermine which qualiy characerisic X j, j = 1,..., j = p, or combinaion of X j s, of he mulivariae process caused he process o go ou-of-conrol. For example, for he drilling process, when an ou-of-conrol signal is produced, i is imporan o know which frequency or combinaion of frequencies cause his signal. In fac, in pracice, he idenificaion of he ype of chaer i.e., chaer a he beginning of he drilling process, low-high frequency chaer will usually make i easier for engineers o adjus he process. 6 Conclusion In his work, we proposed o use EWMA conrol chars based on daa deph measures o monior mulivariae processes. These disribuion-free conrol chars are aracive when he mulivariae normal disribuion is no saisfied. A Inegral Equaion Approximaion For more deails on he use of he collocaion mehod used for solving inegral equaion 4, see Calzada and Scariano 2003 pp Firs, he inerval [h, B] is divided ino n subinervals of lengh = B h/n. Equaion 4 can be rewrien as, Lu = 1 + LBPr r B 1 u + 1 B LydF h y 1 u, 5 For he consan collocaion mehod, Ly is approximaed by a consan, say L j, on each subinerval [y j 1, y j ], yielding Lu = 1 + LBPr r B 1 u + 1 n yj y 1 u L j df y j 1 j=1, 6 9

10 Choosing nodes w i in each of he subinervals [y i 1, y i ] and requiring equaion 6 o be exac a hese poins gives he sysem Lw i = 1 + LBPr r B 1 w i + 1 i = 1,..., n. The approximaing linear sysem is A = n yj y 1 wi L j df y j 1 j=1, 1 = AL, 7 wih 1 = [ 1, 1,..., 1] T, L = [LB, L 1, L 2,..., L n ] T, and A is an n + 1 n + 1 marix Pr[r B 1 B ] 1 1 y1 h df y 1 B 1 yi dy y i 1 df y 1 B dy Pr[r B 1 w1 ]. Pr[r B 1 wi ]. A 1, where A 1 is an n n marix wih enries a ij, where 1 yj y a ij = j 1 df 1 yj y j 1 df y 1 wi y 1 wi 1 if i = j if i j i = 1,..., n, j = 1,..., n. The inegrals a ij are calculaed yj y 1 u yj 1 w i yj 1 1 w i df = F F y j 1 he nodes w i are chosen o be he midpoins of he subinervals. If n is chosen o be an odd ineger, hen he ARL0=L0 L n+1/2, which holds if B = h. Acknowledgemens This work has been suppored by he Graduae School of Producion Engineering and Logisics a he universiy of Dormünd and he Collaboraive Research Cenre Reducion of Complexiy in Mulivariae Daa Srucures SFB 475 of he German Research Foundaion DFG. The auhors would like o hank Prof. Regina Liu and Dr. Sven Knoh for heir helpful commens. References CALZADA, M. E. and SCARIANO, S. M. 2003, Reconciling he Inegral Equaion and Markov chain approaches for compuing EWMA Average Run Lengh, Communicaions in Saisics- Simulaion and Compuaion 32, pp

11 GAN, F. F. 1993, Exponenially Weighed Moving Average Conrol Chars Wih Reflecing Boundaries, Journal of Saisical Compuaion and Simulaion, 46, pp GIBBONS, J. D. and CHAKRABORTI, S. 2003, Nonparameric Saisical Inference, 4h edn. New York: Dekker. HACKL, P. and LEDOLTER, J. 1991, A Conrol Char Based On ranks, Journal of Qualiy Technology 18, pp HACKL, P. and LEDOLTER, J. 1992: A New Nonparameric Qualiy Conrol Technique. Communicaions in Saisics-Simulaion and Compuaion 21, pp LIU, Y. R. 1990, On a noion of daa deph based on random simplices, The Annals of Saisics, 18, pp LIU, Y. R. 1995, Conrol Chars for mulivariae Processes, Journal of he American Saisical Associaion, 90, pp LIU, Y. R.; PARELIUS, J. M.; and SINGH, K. 1999, Mulivariae analysis by daa deph: Descripive Saisics, Graphics and Inference, The Annals of Saisics, 27, pp MESSAOUD, A.; THEIS, W.; WEIHS, C.; and HERING, F. 2004, Monioring he BTA Deep Hole Drilling Process Using Residual Conrol Chars, Technical Repor 60/2004 of SFB 475, Universiy of Dormund. ROUSSEEUW, P. and RUTS, I. 1996, AS 307: bivariae locaion deph, Applied Saisics, 45, pp STOUMBOS, Z. G. and REYNOLDS Jr, M. R. 2001, On Shewhar-Type Nonparameric Mulivariae Conrol Chars Based on Daa Deph, Froniers in Saisical Qualiy Conrol 6, pp STOUMBOS, Z. G. and SULLIVAN, J. H. 2002, Robusness o Non-Normaliy of he Mulivariae EWMA Conrol Char, Journal of Qualiy Technology 34, pp THEIS, W. 2004, Modelling Varying Ampliudes, PhD disseraion, Deparmen of Saisics, Universiy of Dormund. URL hp://eldorado.uni-dormund.de:8080/fb5/ls7/forschung/2004/theis WEINERT, K.; WEBBER, O.; HÜSKEN, M.; MEHNEN, J.; and THEIS, W. 2002, Analysis and predicion of dynamic disurbances of he BTA deep hole drilling process, Proceedings of he 3 rd CIRP Inernaional Seminar on Inelligen Compuaion in Manufacuring Engineering. 11