Control Charts for Joint Monitoring of Mean and Variance: An Overview

Transcription

1 Vol. 10, No. 1, pp , 013 ICAQM 013 Control Charts for Jont Montorng of Mean and Varance: An Overvew A. K. McCracken and S. Chakrabort Department of Informaton Systems, Statstcs and Management Scence Unversty of Alabama, Tuscaloosa, AL 35487, USA (Receved November 011, accepted August 01) Abstract: In the control chart lterature, a number of one- and two-chart schemes has been developed to smultaneously montor the mean and varance parameters of normally dstrbuted processes. These jont montorng schemes are useful for stuatons n whch specal causes can result n a change n both the mean and the varance, and they allow practtoners to avod the nflated false alarm rate whch results from smply usng two ndependent control charts (one each for mean and varance) wthout adjustng for multple testng. We present an overvew of ths lterature coverng some of the one- and two-chart schemes, ncludng those that are approprate n parameters known (standards known) and unknown (standards unknown) stuatons. We also dscuss some of the jont montorng schemes for multvarate processes, autocorrelated data, and ndvdual observatons. In addton, notng that normalty s often an elusve assumpton, we dscuss some avalable nonparametrc schemes for jontly montorng locaton and scale. We end wth a concluson and some recommendatons for areas of further research. Keywords: Nonparametrc control charts, normal dstrbuton, one- and two-chart schemes, parameter estmaton, parametrc control charts. 1. Introducton ontrol charts are wdely used n montorng or survellance of processes n a varety of C ndustres. These graphcal dsplays are desgned to allow a practtoner to determne whether a process s n-control (IC) or out-of-control (OOC) by takng samples at specfed samplng ntervals and plottng values of some statstcs on a graphcal nterface whch ncludes decson lnes called control lmts. The vast majorty of control charts are desgned to montor a sngle process parameter, such as the mean or the varance, but t s often desrable to montor the mean and the varance smultaneously, snce both may shft at the same tme and snce a change n the varance can affect the control lmts of the mean chart. For some processes, specal causes can result n a smultaneous change n both the mean and the varance. For example, n crcut manufacturng, an mproperly fxed stencl can result n a shft n both the mean and varance of the thckness of the solder paste prnted onto crcut boards (Gan et al. [17]).In such cases, smultaneous montorng of both parameters s a logcal approach to process control. As expressed by Gan et al. [17], when specal causes exst and cause both the mean and varance to shft smultaneously, then t s more reasonable to combne the mean and varance nformaton on one scheme and look at ther behavor jontly. In many cases, practtoners are nterested n the general queston of whether the process contnues to produce results whch have the same, specfed dstrbuton. The dstrbuton s most commonly assumed to be the normal dstrbuton, whch s completely characterzed by

2 18 McCracken and Chakrabort ts mean and varance. Thus, montorng both mean and varance allows one to determne whether each collected sample appears to be from the specfed normal dstrbuton or come from a normal dstrbuton whch s dfferent from the IC normal dstrbuton n some way. Gan [15] noted that the mean and varance charts are vrtually always used together. The common practce of usng a mean chart together wth a varance (or range or standard devaton) chart s one type of jont montorng scheme. However, as noted by Gan [16], dong so s bascally lookng at a bvarate problem usng two unvarate procedures. Furthermore, schemes consstng of two ndependent charts can be affected by the classcal multple testng problem, and f adjustments are not made to these charts control lmts to account for ths fact, the false alarm rate (FAR) s nflated, snce the process s deemed to be OOC whenever a sgnal occurs on ether chart. For example, f each chart s set at a nomnal FAR of and the charts operate ndependently, the overall FAR (the probablty of a false alarm on at least one chart) s 1 ( ) , a 100% ncrease from the nomnal Ths FAR nflaton can run the effcacy of the resultng montorng procedure. Therefore, practtoners usng a two-chart scheme should select control lmts for each chart such that the overall FAR s a specfed value. In the earler example, f an overall FAR of 0.01 s desred, each chart should be calbrated at an FAR of As noted earler, the most typcal assumpton n statstcal process control (SPC) has been that the process output follows a normal dstrbuton. Under ths model assumpton, jont montorng of processes nvolves two parameters, the mean (locaton) and the varance (scale) and typcally uses an effcent statstc for montorng each parameter. A jont montorng scheme lnks these statstcs or the correspondng control charts n some way. These schemes can be broadly classfed as one- or two-chart control schemes, respectvely. Ideally, a montorng scheme should be smple to use, easy to understand, and quck to mplement n order to maxmze ts usefulness (Chao and Cheng [3]). A jont montorng scheme should also clearly ndcate the parameter or parameters whch are OOC. We dscuss these varous schemes and the assocated detals below. Though many processes do produce outputs whch follow a normal dstrbuton, there are also many whch do not. As noted by Yang et al. [51], many servce processes produce non-normally dstrbuted outputs. For example, a varable of nterest such as the tme to a certan event that s beng montored may follow a rght-skewed exponental dstrbuton; another varable may have an underlyng dstrbuton that s symmetrc but heaver taled, such as the logstc dstrbuton. A number of authors have recommended that practtoners should avod usng charts based on the normal dstrbuton for processes whch are non-normal, snce these charts may be qute neffectve or perform rather erratcally for other dstrbutons, ncludng hghly skewed or heaver taled processes. Addtonally, these exstng chartng procedures assume the exstence and ndependence of the sample mean and the sample varance. It s not clear how these normal theory charts would perform for dstrbutons whch lack ths trat, such as the skewed gamma dstrbuton, n whch the mean and the varance both depend on the scale and the shape parameters, or the Cauchy dstrbuton, whch s symmetrc and has a fnte medan but has no fnte mean or varance. So far, however, research n the area of parametrc jont montorng has largely overlooked cases n whch processes are known to be non-normal. Ths s an mportant area for further research, snce few non-normal parametrc jont montorng schemes are currently avalable n the lterature. On the other hand, there s not always enough knowledge or nformaton to support the assumpton that the process dstrbuton s of a specfc shape or form (such as normal). In such cases, nonparametrc or dstrbuton-free charts can be useful. However, ths s a

3 Control Charts for Jont Montorng of Mean and Varance: An Overvew 19 relatvely new area of research, and only a handful of nonparametrc jont montorng charts are currently avalable n the lterature. We wll dscuss these procedures later n the revew.. One-Chart Montorng Schemes Of the two major types of jont montorng schemes, one-chart schemes have receved the most attenton n the recent lterature. These schemes are appealng for several reasons. Frst of all, they are smpler than two-chart schemes, n the sense that they allow the practtoner to focus on a sole chart (and, n most cases, a sngle chartng statstc) whch makes the operaton easer, partcularly when the process s IC (whch s far more often than not). It s also relatvely easy to set the control lmts for these charts based on the dstrbuton of the chartng statstc, whch s often a combnaton of two statstcs, one for the mean and one for the varance. Many of these schemes also have dagnostc capablty; that s, they can ndcate whch of the two parameters may have shfted n case the chart sgnals. We frst dscuss some one-chart schemes under the normal dstrbuton when the IC mean and varance are specfed or known (the so-called standards known case)..1. Standards Known, Parametrc One-Chart Schemes There are stuatons n practce where the IC mean and varance of a normally dstrbuted process are known, for example, from external specfcatons or long-term experence. The term standards known or case K s often used n the SPC lterature for these stuatons n whch all relevant process parameters are known. The development, mplementaton, and nterpretaton of control charts are smpler and more straghtforward n case K. Thus, although a wde varety of one-chart jont montorng schemes have been developed over the years, the vast majorty of these charts have been proposed for case K. Among the one-chart jont montorng schemes n case K, there are two major classes: smultaneous control charts, whch use two statstcs (one each for mean and varance) plotted on the same chart, and sngle control charts, whch use a sngle statstc that may actually be a combnaton of two separate statstcs, one each for the mean and the varance (Cheng and Thaga [9]). Sngle control charts can further be broken down nto those wth a two-dmensonal control regon, wheren the chartng statstcs are plotted on a two-dmensonal plane, and those that have tradtonal control lmts (.e. horzontal lne boundares),wheren the chartng statstcs are plotted aganst tme. The three fgures below use smulated data to llustrate the varous categores of one-chart jont montorng schemes. Fgure 1 shows an example of a smultaneous control chart developed by Yeh et al. [5], whle Fgures and 3 show examples of sngle control charts wth tradtonal control lmts (Chen and Cheng s [5] Max chart) and a control regon (Chao and Cheng s [3] sem-crcle chart), respectvely. These charts wll be dscussed n more detal n the sectons that follow. Cheng and Thaga [9] presented a nce overvew of the smultaneous and sngle control charts avalable before 006 and concluded that sngle charts are typcally preferable to smultaneous charts due to ther smplcty and clarty. Ther artcle outlned fve Shewhart-type charts, sx EWMA-type charts, and two CUSUM-type charts, as well as addtonal charts for multvarate and autocorrelated processes. However, major contrbutons have been made to the area of jont montorng n recent years, and therefore ths paper wll predomnately focus on those sngle and smultaneous charts as well as two-chart and other mscellaneous schemes whch have been developed snce 005 or were not covered by Cheng and Thaga [9].

4 0 McCracken and Chakrabort Fgure 1. A smultaneous montorng scheme. Fgure. A sngle chart montorng scheme wth a tradtonal upper control lmt. Fgure 3. A sngle chart montorng scheme wth a control regon.

5 Control Charts for Jont Montorng of Mean and Varance: An Overvew Smultaneous Charts A few montorng schemes exst n the lterature n whch a mean statstc and a varance statstc are plotted wthn the same chart. These smultaneous charts can be thought of as a compromse between sngle charts and two-chart schemes, snce they mantan separate statstcs but only a sngle graphcal nterface. A bref dscusson of these charts s provded by Cheng and Thaga [9], who concluded that sngle charts are more appealng..1.. Sngle Charts wth Tradtonal Control Lmts Sngle charts wth tradtonal control lmts comprse the area of research n jont montorng that has seen the most development over the years. These schemes are based on a sngle chartng statstc whch s usually some combnaton (functon) of the mnmal suffcent statstcs X and S. However, a few charts have been consdered that consst of a sngle chartng statstc whch s not a drect combnaton of a mean statstc and a varance statstc. Instead, these schemes ncorporate the target process mean and varance ( 0 and 0 ) drectly nto the chartng statstc. For example, Domangue and Patch [14] consdered an EWMA chart based on the chartng statstc n( X ) 0 A r r A 1 0 a (1 ), where n s the sample sze and r s a weghtng constant such that 0 r 1. Later, Costa and Rahm [1] consdered an EWMA non-central ch-square (NCS) chart based on the statstc n 0 0 W x j j 1 where 1,,, denotes the sample number, j 1,,, n denotes the observaton number wthn each sample, d f ( X 0 ) 0 and d f ( X 0 ) 0. Costa and Rahm [13] demonstrated that ths NCS chart detects changes n the mean and ncreases n the varance qucker than the X R jont montorng scheme. However they dd not consder decreases n the varance. Furthermore, t s not clear why the dstrbuton of W s a non-central ch-square as clamed by these authors, as the random varable x j 0 0 does not appear to follow a normal dstrbuton. Among the functons to combne the mean and varance statstcs, the maxmum has been qute popular. Chen and Cheng [5] presented the Max chart whch combnes two normalzed statstcs, one for the mean and one for the varance, by takng the maxmum of the absolute values of the two statstcs. The resultng chartng statstc s M max ( U, V ) where U X, 1 n 1 S V H ; n 1, / n H ( wv ; ) s the cumulatve dstrbuton functon (cdf) of the ch-square dstrbuton wth v degrees of freedom, n s the sze of the th sample, and (.) s the cdf of the standard normal dstrbuton. Huang and Chen [1] provded nsght on the economcally optmal choce of sample sze, samplng nterval, and control lmts for Max charts. They also compared the economcally-desgned versons of the Max chart and jont X and S charts,

6 McCracken and Chakrabort and determned that they performed smlarly. Chen et al. [6] proposed a MaxEWMA chart whch smlarly combnes two EWMA statstcs for mean and varance, Y 1 Y 1 U and Z (1 ) Z 1 V,0 1, by takng the maxmum of the absolute values of the two, where U and V are the same statstcs used n the orgnal Max chart and s a specfed smoothng constant. However, Costa and Rahm [1] showed usng smulaton studes that ther NCS chart s preferable to the MaxEWMA chart for detectng ncreases n process varablty, whether or not they are accompaned by a shft n the mean. The MaxEWMA chart has been a popular chart n the lterature. Khoo et al. [4] presented a slghtly dfferent MaxEWMA chart whch utlzes the range R rather than S as the bass for the varance statstc. They stated that ths EWMA X R chart s not ntended to replace the standard MaxEWMA chart and s smply an attempt to construct a sngle chart usng R nstead of S. As noted by Chao and Cheng [4], any approach usng R s bound to be less effectve than the approaches whch combne the mnmal suffcent statstcs. Mahmoud et al. [30] compared the relatve effcency of S and R and concluded that S s strongly preferable for normally dstrbuted data. Thus, t seems unlkely that a practtoner would choose ths chart for process montorng. As a further modfcaton of the MaxEWMA chart, Khoo et al. [5] proposed the Max-DEWMA chart, whch combnes two double EWMA statstcs, W (1 ) W 1 Y and Q (1 ) Q 1 Z, where Y and Z are the EWMA statstcs used n the MaxEWMA chart. Whle ths chart has an addtonal layer of complexty, t outperforms the MaxEWMA chart for small and moderate shfts n mean and/or varance, at least when the same smoothng constant s used for both the constructon of the EWMA statstcs Y and Z and the constructon of the DEWMA statstcs W and Q. As s typcally the case for EWMA-type charts, choosng a small value for ncreases the senstvty of both the Max-DEWMA and MaxEWMA charts to small shfts. Another varaton on the MaxEWMA chart s Memar and Nak s [31] Max EWMAMS chart, whch combnes normalzed versons of the EWMA X statstc, A k A 1 1 k X k, and the EWMA mean-squared devatons statstc, B k X kj 0 1 Bk 1. j 1 n n The resultng chartng statstc s M k Ak 0 1 B k max k, H ; n 0 Ths chart s effectve for most changes n mean and varance but s outperformed by other schemes for decreases n varance. Other ways of combnng a mean statstc and a varance statstc for jont montorng nclude usng the sum of squares or a weghted loss functon or adaptng exstng tests for comparng the dstrbutons of two samples, such as the lkelhood rato approach. A loss functon s an equaton whch ndcates the severty of the dfference between a pont estmate and the true value t s estmatng. Some researchers have proposed charts based on the loss

7 Control Charts for Jont Montorng of Mean and Varance: An Overvew 3 functon n 1 x 0 n 1 n1 n L s x 0 n n n1 n1, but recently, charts based on a general weghted loss functon, WL s x 1 where s an approprate weghng factor between 0 and 1, have been shown to be more effectve (Wu and Tan [48]).Wu and Tan [48] suggested a CUSUM chart, known as the WLC chart, whch s based on ths weghted loss functon and s useful for jontly montorng changes n mean and ncreases (but not decreases) n varance. The statstc used for ths chart s At max0, At 1 WLt ka, where ka s a reference parameter whch s determned usng a desgn algorthm, along wth the weghtng factor and the control lmt. Whle these approaches have been rather ad-hoc, the lkelhood rato-based approach appears to be a promsng dea, rooted n sold statstcal theory, for constructng a sngle chart, snce the jont montorng problem under an assumed parametrc dstrbuton (such as the normal) s analogous to testng the null hypothess that the most recently taken sample comes from a completely specfed populaton (a smple null hypothess) versus all alternatves (a composte alternatve hypothess), repeatedly over tme as more test samples become avalable. The exact (fnte sample) dstrbuton of the lkelhood rato (LR) statstc (also known as the generalzed lkelhood rato (GLR) n ths case of composte alternatves) s often dffcult to obtan, so n hypothess testng, the asymptotc propertes of ths statstc are generally used nstead. However, as noted by Hawkns and Deng [19], n the control chartng settng, practtoners rarely use sample szes large enough to make the asymptotc dstrbuton useful, so the effcacy of ths approach could be a concern. Hawkns and Deng [19] proposed a par of new charts: a generalzed lkelhood rato (GLR) chart whch has the chartng statstc 0 1 n X n S n1 S G nln nlnnn and a Fsher chart based on the statstc W loga1a where A1 s the p-value gven by the X chart for a specfed mean and A s the p-value gven by the S chart for a specfed standard devaton. They then compared the performance of these two charts to that of the Max chart and found that the GLR chart has superor performance for detectng decreases n varance, though another chart performs better for any mean ncrease not accompaned by a varance decrease. Addtonally, they noted that the Max chart and the Fsher chart both exhbt bas, that s, the OOC ARL can be larger than the IC ARL, whle the GLR chart does not share ths problem. A few other authors have also utlzed the lkelhood rato approach to control chartng. Zhang et al. [55] proposed a GLR-based chart for detectng non-sustaned shfts n the mean and/or varance, whch may occur, for example, n healthcare settngs. Zhang et al. [55] developed a lkelhood rato-based EWMA (ELR) chart whch lkewse s able to detect decreases n varance. Convenently, settng up the GLR and ELR charts does not requre specfcaton of an addtonal parameter such as n Domangue and Patch s chart or d n 0

8 4 McCracken and Chakrabort Costa and Rahm s non-central ch-square chart. However, the lmts for the ELR chart are obtaned usng a complcated Markov chan approxmaton, makng t dffcult to mplement n general, though the authors dd provde a table of control lmts for certan sample szes, IC ARLs, and EWMA weght smoothng parameters. Another method whch s sometmes used for detectng an mpendng change s the Shryaev Roberts test. Ths test has been proved to be optmal for detectng a change that occurs at dstant tme horzon when the observatons are..d. and the pre- and post-change dstrbutons are known and s based upon the lkelhood rato between the two specfed dstrbutons Zhang et al. [56]. A Shryaev Roberts chart based upon ths test was proposed by these authors and was shown to perform comparably to the ELR and WLC charts Sngle Charts wth Control Regons Some researchers have developed jont montorng schemes n whch the process data are plotted on a two-dmensonal plane and are consdered IC f they fall wthn some defned control regon. Otherwse, the process s declared OOC. A varety of these sngle control charts has been developed, wth sem-crcular, crcular, and ellptcal control regons. As noted by Chao and Cheng [4], the combned vsual effect of ponts that fall n/out of a certan closed regon s more strkng than ponts just crossng the lnes. Whle vsually appealng, a dsadvantage of such control schemes s that the tme-ordered nature of the data s lost, removng the opportunty for the practtoner to spot tme-related trends. Charts wth sem-crcular control regons have been popular n the lterature. Takahash [47] studed several possbly control regons (rectangular, sectoral and ellptc) and noted that each has advantages n detectng a partcular type of change: rectangular for changes n mean only, sectoral for changes n varance only, and ellptc for changes n both. He developed a very complex chart for whch the control regon s the common area of a rectangle and a sectoral. * Chao and Cheng [3] developed a control chart n whch the ponts X, S are plotted * on the X, S plane, where * n S (1/ n) 1( X X). The control regon for the chart s based on the statstc 0 * T X S. If the sample data are normally dstrbuted, ( n 0 ) T has a ch-square dstrbuton wth n degrees of freedom. Furthermore, the equaton for the statstc T neatly defnes a crcular regon. * However, snce S must be non-negatve, only half of the regon s needed, and ths scheme s known as a semcrcle (SC) chart. Chao and Cheng [4] expanded upon ths concept to construct an SC chart wth mnmum coverage area. Chen et al. [7] utlzed ths T statstc as the bass for an EWMA-SC chart. However, despte the chart s name, the control regon for ths chart s the area under a lne, not a sem-crcle. The T statstc s decomposed nto mean and varance components, 0 n X 0 1 and S 1 n 1, 0 respectvely, whch are used to form the EWMA statstcs, U and V. The sample ponts are then plotted n the U, V -plane (rather than sequentally), and all ponts fallng below a specfed lne are taken to be IC.

9 Control Charts for Jont Montorng of Mean and Varance: An Overvew 5.. Standards Unknown, Parametrc One-Chart Schemes Nearly all of the jont mean/ varance control charts avalable n the lterature are desgned under the assumpton of normalty of the process dstrbuton wth specfed parameters (case K). However, n practce, more often than not, one or more of these parameters are unknown and unspecfed. Ths s referred to as the standards unknown case or case U. Devsng and nterpretng control charts n case U s more nterestng and challengng, because even under the assumpton of a known process dstrbuton, such as the normal, the estmaton of the mean and varance parameters from the data and the use of them to construct the tral control lmts result n statstcal dependency (see for example, Chakrabort et al. []). Ths dependency can affect the performance of the chart n a sgnfcant way (such as resultng n many more false alarms than expected) to such a large extent that the practtoner mght lose fath n the whole endeavor. Thus, the effects of parameter estmaton on the performance of control charts has become an mportant area of research. A lttle background s n order. The development of control charts for statstcal montorng of a process s typcally undertaken n two stages: Phase I, whch nvolves retrospectve examnaton of the process, and Phase II, whch focuses on prospectve montorng. In Phase II, data are collected at regular ntervals, and a chartng statstc of nterest s calculated from the data and placed on a control chart. When a chartng statstc plots above the upper control lmt (UCL) or below the lower control lmt (LCL), a sgnal occurs, and the process s sad to be OOC. In case U, a major objectve of the Phase I study s to obtan a set of IC process data (.e. a reference sample) from whch the parameters needed to determne the control lmts for the Phase II charts can be estmated. The sze and the qualty of ths reference sample can and do greatly mpact the performance of the chart. Ths has been an actve area of research for the last decade or so. Jensen et al. [] presented a comprehensve revew of the lterature on the effects of parameter estmaton on varous types of control charts, notng that t s partcularly problematc when a small Phase I sample s utlzed. Although they dd not explctly menton jont montorng schemes, ther observatons on the mpact of parameter estmaton almost certanly extend to these charts. Further work s necessary to better understand the precse mpact of parameter estmaton on jont montorng schemes. Tradtonally, obtanng the reference sample s accomplshed usng an teratve procedure n Phase I n whch tral lmts are frst constructed from the data (several subgroups or a number of observatons), and then some chartng statstcs are placed on a control chart. It may be noted that the constructon of Phase I control charts has dfferent objectves (see for example the 009 revew artcle by Chakrabort et al. []) from those n Phase II. Phase II control charts are desgned to have a specfc IC ARL, whle Phase I control charts should be constructed based on a specfed IC false alarm probablty (FAP), whch requres consderaton of the jont dstrbuton of the chartng statstcs. In any case, subgroups correspondng to the chartng statstcs whch are located outsde the control lmts are generally consdered OOC and removed from the analyss, and new lmts are estmated from the remanng data. 1 Ths step s repeated untl no more chartng statstcs appear OOC 1 There are two competng schools of thought concernng the handlng of subgroups whch plot OOC. Some practtoners automatcally dscard these subgroups, whle others remove them only f they can be attrbuted to assgnable causes. In our dscusson, we focus prmarly on the former approach; however, problems can arse wth ether method. In the case of the latter approach, the practtoner may fal to dentfy the cause of a partcular subgroup whch plots OOC and therefore not dscard t. If, however, the subgroup was, n fact, OOC, falure to remove t could lkewse result IC lmts whch are too tght or too loose.

10 6 McCracken and Chakrabort so that the remanng data s taken to be IC, at whch pont the fnal control lmts may be constructed for Phase II montorng. However, control charts for jont montorng n Phase I, case U, have been largely absent from the lterature so far. Ths s an mportant problem snce f the Phase I chart used to dentfy ths IC data s ncorrect or neffectve at dentfyng the OOC samples for removal, the resultng parameter estmates may not be very accurate, whch wll affect the performance of the Phase II chart. In fact, several researchers have demonstrated the ptfalls of estmatng the parameters usng data whch contans OOC samples. The resultng control lmts may be too tght or too loose, causng an nflated FAR or a hgher-than-nomnal OOC ARL, respectvely. On ths pont, for example, Mabouduo-Tchao and Hawkns [9] noted, Pluggng n parameter estmates fundamentally changes the run length dstrbuton from those assumed n the known-parameter theory and dmnshes chart performance, even for large calbraton samples. Jensen et al. [] ponted out that the mpact of parameter estmaton depends on the drecton of the estmaton error. Clearly, then, n case U stuatons, t s mportant to have very good Phase I charts; otherwse, the resultng Phase II charts wll suffer. A few case U charts for jont montorng are present n the lterature. A recent example s the smultaneous chart proposed by Yeh et al. [5] whch s comprsed of a par of CUSUM mean and varance statstcs whch have the same scale and are plotted on one control chart wth a sngle set of control lmts. The CUSUM mean and varance statstcs are computed by takng approprate functons of X and S and applyng the probablty ntegral transformaton (PIT) to each, producng statstcs whch have the unform dstrbuton. Because the PIT s used, ths chart can be extended for processes wth known, non-normal dstrbutons as well. A substantal quantty of recent research has demonstrated that n case U, a large quantty of reference data s needed before the Phase II charts actually dsplay ther expected (nomnal) behavor. Jensen et al. [] noted that dependng on the type of control chart beng utlzed, hundreds or even thousands of reference data ponts may be necessary. However, t s not always possble to have such a large clean dataset. In these stuatons an alternatve class of control charts may be useful. Among these are the self startng charts (Hawkns [18]) and the Q-charts (Quesenberry [33]). For the jont montorng problem, L et al. [7] recently proposed a self-startng EWMA lkelhood-rato (SSELR) chart whch has ths advantage. However, ths chart s only approprate for unvarate data. Hawkns and Zamba [0] presented a sngle chart based on a changepont model whch also avods the problem of needng a large reference dataset. In ths model, the changepont s the unknown nstant at whch a specal cause brngs about a shft n one or both of the parameters; as a result, the observatons taken pror to the changepont have a dfferent mean and/ or varance than the observatons taken after t. A GLR test can be used to determne whether the samples before and after a suspected changepont seem to have the same parameters. However, the tme of the true changepont s unknown, and t could occur between any two samples. Thus, the chart proposed by Hawkns and Zamba [0] uses the maxmum of the GLR statstcs resultng from consderng all possble changeponts. They demonstrated that ths chart s effectve when as few as three Phase I data ponts are avalable. However, they acknowledge that ther procedure s nadequate for non-normal dstrbutons. The statstcs can also be placed on separate charts, resultng n a two-chart scheme.

11 Control Charts for Jont Montorng of Mean and Varance: An Overvew 7.3. Dsadvantages of One-Chart Montorng Schemes In general, one-chart schemes are not wthout some weaknesses. As noted by Chao and Cheng [4], one-chart schemes lack a desrable feature present n two-chart schemes; that s, they lose ether the tme-sequental presentaton of the data, n the case of sngle charts wth control regons, or the separate treatment of the mean and varance statstcs, n the case of sngle charts wth tradtonal control lmts. In the former case, the charts do not ndcate the order n whch the data are collected, makng t dffcult to spot tme-related trends. In the latter case, there s a reducton n the nformaton to be gleaned by lookng at the chartng scheme. A sgnal on a sngle mean-varance chart wth tradtonal control lmts cannot be attrbuted to varance or mean wthout dagnostc follow-up. 3 Sngle charts wth control regons, however, do not share ths lmtaton. Another dsadvantage of one-chart jont montorng schemes s that they often utlze very complex chartng statstcs. Addtonally, many sngle charts have some chart-specfc tunng parameters, such as n Domangue and Patch s [14] chart or d n Costa and Rahm s [1] non-central ch-square chart, whch must be specfed and whch greatly affect the chart s performance (Zhang et al. [55]). Choosng approprate values for these parameters can be qute complex. Some one-chart schemes also have the dsadvantage of beng nsenstve to large or small shfts n the parameters or to decreases n varance. Whle ndvdual one-chart schemes have been shown to have performance advantages over two-chart schemes, t s naccurate to clam that one-chart schemes always have superor performance. 3. Two-Chart Montorng Schemes Snce the early days of SPC, there have been jont montorng schemes, used ether mplctly or explctly, consstng of two charts. For normally dstrbuted data, these two-chart montorng schemes are made up of a mean chart (such as the X chart) and a varance chart (such as the S chart or the R chart). They can consst of a par of Shewhart, CUSUM, or EWMA charts, one for the mean and one for the varance, or even a combnaton of one CUSUM and one EWMA chart. The mean charts utlzed n these schemes are typcally two-sded, whle the varance charts may be ether one-sded or two-sded. Fgure 4 shows an example of a two-chart montorng scheme, constructed usng smulated data. The mean of the process appears to go OOC around subgroup 70. A major advantage of two-chart montorng schemes s ther famlarty and the apparent ease of vsual nterpretaton. Such schemes have been used for many years. Furthermore, t seems natural to some practtoners to montor the mean and the varance separately and smultaneously, snce the statstcs X and S are ndependent. However, two-chart schemes have some dsadvantages as well. Cheng and Thaga [9] noted that these schemes utlze more personnel, tme and other resources than do one-chart schemes. In addton, typcal two-chart schemes gve the lluson of clearly showng whether a process s OOC wth respect to mean, varance, or both, snce a sgnal on the mean chart appears to ndcate that mean s OOC, a sgnal on the varance chart appears to ndcate that varance s OOC, and sgnals on both charts seem to ndcate that both are OOC. Contrary to 3 In contrast, a sgnal on the varance chart of a two-chart scheme s mmedately attrbutable to a shft n varance, although the same cannot be sad of a sgnal on the mean chart. Ths fact wll be dscussed n greater detal later n ths paper, but the man pont here s that nether two-chart nor one-chart schemes wth tradtonal control lmts can mmedately ndcate whether a chart sgnal s caused by the mean, the varance, or both, though two-chart schemes do provde slghtly more nformaton.

12 8 McCracken and Chakrabort popular percepton, ths can actually be qute msleadng. As noted by Hawkns and Deng [19], snce the control lmts for an X chart are functons of the IC standard devaton, an ncrease n the varablty may lead to a false sgnal on the X chart, and lkewse, a decrease n varablty may cause the X to fal to sgnal even though a shft n the mean has also occurred. Furthermore, n many two-chart montorng schemes, the mean and varance charts are constructed completely ndependently, so that each has a specfed ARL. Approprate two-chart jont montorng schemes, however, should have control lmts whch have been adjusted so that the overall ARL of the scheme s a specfed value, otherwse the false alarm rate wll be nflated. Fgure 4. A two-chart montorng scheme consstng of two EWMA charts Standards Known, Parametrc Two-Chart Schemes As wth one-chart montorng schemes, the vast majorty of two-chart montorng schemes presented n the lterature are for data from a specfed dstrbuton, such as the normal, for whch the true parameters are specfed. Several authors have consdered the constructon and mprovement of these schemes. Gan [15] dscussed scheme CC, consstng of a two-sded CUSUM mean chart and a two-sded CUSUM varance chart; scheme EE u, consstng of a two-sded EWMA mean chart and a hgh-sded EWMA varance chart; and scheme EE, consstng of a two-sded EWMA mean chart and a two-sded EWMA varance chart, comparng these to Domangue and Patch s [14] omnbus chart. He demonstrated that scheme EE performs smlarly to scheme CC and s slghtly more senstve than CC when there s a small shft n the mean concurrent to a slght decrease n the varance. The other schemes performed poorly n comparson. Reynolds and Stoumbos [40] also nvestgated and compared popular Shewhart, EWMA, and CUSUM schemes, lkewse recommendng a par of EWMA (or CUSUM) charts, though wth a dfferent varance statstc than the one dscussed by Gan [15]. Reynolds and Stoumbos [41] expanded upon ths work, comparng a

13 Control Charts for Jont Montorng of Mean and Varance: An Overvew 9 varety of other two- (and three-) chart combnatons, ncludng some Shewhart-EWMA schemes. They, too, ultmately recommended a par of EWMA (or CUSUM) charts. 3.. Standards Unknown, Parametrc Charts The standards unknown stuaton presents even more complex problems for two-chart montorng schemes than t does for one-chart schemes. Perhaps for ths reason, such charts (for both Phases I and II) have not yet been studed n the lterature. In the Phase I settng, a major problem whch needs to be addressed s how to conduct the teratve, parameter estmaton procedure for a two-chart scheme. 4 Should tral lmts be constructed for the two charts smultaneously and a subgroup dscarded f t plots OOC on ether chart? Alternatvely, should the teratve procedure be conducted frst for the varance chart (snce t s ndependent of the mean parameter) and then for the mean chart usng only the data not already dscarded? If ths second method s utlzed, should the fnal control lmts be recomputed for the varance chart usng only those samples not dscarded durng the estmaton of the process mean? Another dffculty s desgnng the scheme so that t has a specfed FAP. Further research s needed to address all of these ssues and make two-chart jont montorng schemes a vable opton for case U. For the Phase II problem, there s also a current shortage of charts that account for parameter estmaton from a Phase I reference data set. Snce the mpact of parameter estmaton on Phase II charts can be serous and parameters are estmated based on reference data obtaned by the use of approprate Phase I charts, two-chart schemes for both settngs should be an area of further research. 4. Nonparametrc Charts All of the methodologes dscussed above, and ndeed, the vast majorty of avalable jont montorng charts, have focused on normally dstrbuted data. Gven the exstng lterature on the performance of ndvdual normal theory charts for the mean and the varance, t s reasonable to assume that departures from normalty can have a dramatc mpact on the effectveness of at least some such charts. Ths can be remeded by applyng a nonparametrc or a dstrbuton-free control chart that does not requre the assumpton of a specfc form of the underlyng dstrbuton. A recent revew of nonparametrc control charts can be found n Chakrabort et al. [1]; however, the lterature n the area of nonparametrc jont montorng, both one- and two-chart schemes, s currently very lmted and thus presents a great opportunty for research and development. The few nonparametrc jont montorng schemes avalable n the lterature are all one-chart schemes. Zou and Tsung [58] proposed an EWMA control chart based on Zhang s [53] goodness-of-ft test. They showed that ths chart s effectve for detectng changes n locaton, scale, and shape. Mukherjee and Chakrabort [3] adapted the nonparametrc test for locaton-scale by Lepage [6] and constructed a chart based on the Lepage statstc, whch combnes the Wlcoxon rank-sum locaton statstc wth the Ansar-Bradley scale statstc. Ther Shewhart-type chart s called the Lepage-Shewhart (LS) chart, and t has post-sgnal dagnostc capablty for determnng the nature of the shft. Wthn the nonparametrc jont montorng lterature, ther paper appears to be the frst major foray nto the standards unknown case, though more work s currently n progress. Whle there s no doubt that nonparametrc two-chart montorng schemes could be developed, as far as we know, there are none n the current SPC lterature. However, Reynolds 4 The approprate approach may depend, n part, on whether practtoner chooses to dscard all subgroups whch plot OOC or only those for whch assgnable causes can be dentfed.

14 30 McCracken and Chakrabort and Stoumbos [44] presented some schemes consstng of a par of CUSUM charts whch are robust to the normalty assumpton. 5. Mscellaneous Problems n Jont Montorng Much of the recent jont montorng research has focused on specal stuatons, such as multvarate processes, autocorrelated processes, and patterned (rather than sustaned) shfts n parameters. These jont montorng schemes are typcally developed by alterng an exstng scheme. Addtonally, some research efforts have focused on ncorporatng varable sample szes and/or samplng ntervals nto such charts. We dscuss some of these recent developments brefly Multvarate Processes When practtoners wsh to montor multple qualty characterstcs of nterest for a partcular process, a multvarate control chart s needed. For jont montorng of the mean and varance n such cases, a montorng scheme consstng of ether one or two multvarate control charts s necessary. Reynolds and Cho [35] and Reynolds and Stoumbos [43] consdered a varety of two-chart jont montorng schemes for the mean vector and covarance matrx and outlned whch performed best for varous types of shfts. Cheng and Thaga [9] dscussed four one-chart multvarate jont montorng schemes, of whch three are sngle charts whch use the maxmum functon to lnk the mean and varance statstcs together and the fourth s a smultaneous chart. Khoo [3] presented detals of the bvarate verson of the multvarate Max chart. Recently, several artcles have been publshed whch extend other types of one-chart jont montorng schemes for use wth multvarate processes. Zhang et al. [54] proposed an expanson of the ELR chart for ths stuaton. Maboudou-Tchao and Hawkns [9] developed a self-startng chart for jont montorng of mean and varance, n whch the mean and standard devaton estmates are re-calculated each tme a new sample s taken and each sample s transformed so that t follows the multvarate standard normal dstrbuton. Ths s a growng area of actvty as many problems n practce nvolve montorng multvarate dstrbutons. Much further work s needed here for both Phase I and II charts. 5.. Autocorrelaton An mportant assumpton of most control charts, both unvarate and multvarate, s that the samples (or observatons) are ndependent; however, autocorrelaton among observatons or samples occurs n many stuatons. Cheng and Thaga [9] noted that tanks, reactors, and recycle streams are among the places where autocorrelaton can be found. When present, autocorrelaton can nflate the FAR f some adjustment s not made to the chart to account for t (Lu and Reynolds [8]). It s well-known that gnorng autocorrelaton when t exsts can severely affect the performance of a control chart. A few of the approaches to jont montorng of mean and varance of autocorrelated processes were mentoned by Cheng and Thaga [9], and Lu and Reynolds [8] presented an overvew of approaches, dscussng both schemes n whch procedures for parameter estmaton and control lmt selecton are adjusted to account for the autocorrelaton and schemes n whch a tme seres model s utlzed. It was seen that the two-chart approach consstng of an EWMA chart of the observatons and a Shewhart chart of resduals had the best performance for processes wth moderate or low autocorrelaton.

15 Control Charts for Jont Montorng of Mean and Varance: An Overvew Indvdual Observatons Whle most of the charts dscussed above requre sample (subgroup) szes larger than one, n some applcatons, only a sngle tem can be sampled at each pont n tme, due to expense or other dffculty. As noted by Zhang et al. [56], In ndustral practce, samplng may be expensve, tme consumng, and the sample nterval may be relatvely long. These are the sorts of nstances n whch ndvdual observatons, rather than larger samples, are necessary. Wthn jont montorng, there are a few one- and two-chart schemes avalable for these stuatons. Some of them are desgned specfcally for sngle observaton samples, whle others are useful for larger sample szes as well. Reynolds and Stoumbos [38] nvestgated varous two-chart jont montorng schemes for samples of sze n 1 for detectng gradual drfts, rather than sustaned shfts, n the process parameters, such as can occur from the breakdown of machnery over tme. Reynolds and Stoumbus [40] compared the performance of several two-chart jont montorng schemes for samples of sze n 1 and n 1. They determned that for a fxed samplng rate, usng samples of sze n 1 rather than larger, less frequent samples, mproves the detecton and dagnostc performance of the EWMA and CUSUM jont montorng schemes. Wu and Wang [49] developed a one-chart scheme specfcally for sngle observatons, the 1-CUSUM chart, for whch the chartng statstc s based on a lnear combnaton of x and x ) specfcally, 0 ( 0, max 0, At 1 y (1 ) y k A f At 1 0 or At 1=0 and y 0 At mn 0, At 1 y (1 ) y k A f At 1 0 orat 1=0 and y 0 where ka s a reference parameter, 0 0, y x 0. Hawkns and Zamba s [0] changepont model s another one-chart approach for montorng a sequence of ndvdual data ponts. The lkelhood rato-based EWMA chart proposed by Zhang et al. [55], Max EWMAMS chart proposed by Memar and Nak [31], and the Shryaev Roberts chart proposed by Zhang et al. [56] are addtonal one-chart schemes whch work well for n 1 and also for n 1. However, much more work s needed n ths area. A and 5.4. Adaptve Charts Another area of nterest s the development of adaptve control charts n whch some element (sample sze, control lmts, samplng tme nterval, etc.) of the chartng regme can vary dependng on what s observed on the chart so that shft detecton can be faster and more effcent. Adaptve features have been added to both one- and two-chart schemes. Chengalur et al. [10] nvestgated the comparatve performance of fxed and varable samplng nterval (VSI) procedure versons of a one-chart Shewhart scheme and a two-chart Shewhart scheme, showng the VSI procedures to be more effcent n both cases. Reynolds and Stoumbos [39] made such nvestgatons for several two-chart schemes desgned for samples of sze n 1, lkewse demonstratng that the VSI procedures resulted n qucker detecton tmes. Costa and De Magalhães [11] made smlar comparsons for the NCS chart, wth dentcal conclusons. They developed an adaptve verson of the NCS charts that has varable control lmts, sample sze, and samplng nterval. Wu et al. [50] modfed the WLC control chart to ncorporate varable samplng ntervals.

16 3 McCracken and Chakrabort Reynolds and Cho [36] and Reynolds and Km [37] developed adaptve control schemes for multvarate processes whch ncorporate varable samplng ntervals and sequental samplng, an adaptve feature n whch the sample sze at each pont n tme s determned by the most recently collected data. As stated by Reynolds and Km [37], Wth sequental samplng, at each samplng pont observatons are taken n groups of one or more. After each group s taken at a samplng pont, a decson s made to ether take another group at ths samplng pont, wat untl the next samplng pont to sample agan, or sgnal at the current samplng pont Economc Desgn Sometmes, reducng costs s more mportant than the statstcal performance of a control chart. In such cases, t s possble to select parameters for control charts based on economc crtera, such as cost mnmzaton, nstead of statstcal performance. Rahm and Costa [34] nvestgated the jont economc desgn for two-chart Shewhart schemes. Followng ths example, Serel and Moskowtz [45] studed the jont economc (cost-mnmzng) desgn of two-chart EWMA schemes. Addtonally, several researchers have explored the economc desgn of more specalzed jont montorng schemes. Stoumbos and Reynolds [46] nvestgated the jont economc desgn for a two-chart scheme for ndvdual observatons whch ncorporates varable samplng ntervals, and Cheng and Mao [8] studed the jont economc desgn of a one-chart, multvarate jont montorng scheme. 6. Concluson Many control chartng schemes are currently avalable for jontly montorng the mean and the varance of a normal dstrbuton. Both one- and two-chart schemes exst n the lterature and can be useful to practtoners. Though all avalable schemes have advantages and dsadvantages, one-chart schemes appear to be the more promsng opton, partcularly snce there are many unanswered questons regardng the practcal mplementaton of two-chart schemes when parameters are estmated n a Phase I study. There s a growng body of lterature n the area of jont montorng of mean and varance, but much more work s needed. In ths revew, we have hghlghted some areas for further research. These nclude: 1) nonparametrc jont montorng of locaton and scale, ncludng the development and nvestgaton of two-chart nonparametrc schemes and nonparametrc changepont control charts; ) the development of better charts and/or processes for Phase I montorng, especally for two-chart schemes; 3) the adaptaton of case U versons of exstng and new charts as well as extensve comparson and nvestgaton of ther propertes; and 4) the development of locaton-scale jont montorng charts for specfc parametrc non-normal dstrbutons, ncludng the skewed exponental dstrbuton and the symmetrc but heavy-taled dstrbutons such as the double exponental. Acknowledgements The authors would lke to thank Professors Maron R. Reynolds, Jr. and Wllam H. Woodall of Vrgna Tech, Douglas M. Hawkns of the Unversty of Mnnesota, and Smley Cheng of the Unversty of Mantoba for ther comments and encouragement on an earler verson of the paper. Any errors or naccuraces are the responsblty of the authors. The authors also acknowledge Professor Gema Chen, Edtor-n-Chef, QTQM, for hs comments on the paper. Ths research was supported n part under the SARCHI Char at the Unversty of Pretora, South Afrca.

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