A Change Point Approach for Phase I Analysis in Multistage Processes

Transcription

1 TECH asa v.007/0/3 Prn:5/07/008; 5:3 F:tech07057.tex; Roberta) p. A Change Point Approach for Phase I Anaysis in Mutistage Processes Changiang ZOU Fugee TSUNG 64 LPMC and Department of Statistics Department of Industria Engineering and Schoo of Mathematica Sciences Logistics Management Nankai University Hong Kong University of Science and Technoogy 9 68 Tianjin, China Cear Water Bay, Kowoon, Hong Kong season@ust.uk) 70 Yukun LIU 7 4 LPMC and Department of Statistics 73 5 Schoo of Mathematica Sciences 74 6 Nankai University 75 7 Tianjin, China 76 9 We study the phase I anaysis of a mutistage process where the input of the current process stage may 78 0 be cosey reated to the outputs) of the earier stages). We frame univariate observations from each 79 of the stages in a mutistage process as a singe vector and recognize that the directions in which these 80 vectors can shift are imited when attention is restricted to a singe step shift in the mean of one stage. This aows us to focus detection power on a imited subspace with improved sensitivity. Taking advantage 3 of this particuar characteristic, we propose a change point approach that integrates the cassica binary 8 4 segmentation test with the directiona information based on the state-space mode for testing the stabiity 83 5 of a batch of historica data. We give an accurate approximation for the significance eve of the proposed 84 6 test. Our simuation resuts show that the proposed approach consistenty outperforms existing methods 85 for mutistage processes. 8 KEY WORDS: Binary segmentation; Directiona information; Generaized ikeihood ratio test; Statistica 87 9 process contro. 88. INTRODUCTION Many manufacturing operations invove mutipe operation stages instead of just a singe stage. For instance, in semiconductor manufacturing, Kim and May 999) investigated the via formation procedure to produce mutichip modue dieectric ayers composed of photosensitive benzocycobutene. The via formation process invoves severa sequentia stages, such as spin coating, soft baking, exposing, deveoping, curing, and pasma descumming. In such a case, outputs from operations at downstream stages can be affected by operations at upstream stages. In addition, a product part that is transferred from one stage to the next in a mutistage process may introduce extra variations that do not occur in a generic singe-stage process. Shi 006) has provided an extensive review on the mutistage process contro probem with many industria exampes. Thanks to recent advances in sensor and information technoogies, automatic data acquisition techniques are commony used in increasingy compicated processes with mutipe stages. In addition, a arge amount of data and information reated to quaity measurements in a mutistage process has become avaiabe. Thus engineering and statistica approaches to make use of the mutistage data and information regarding contro and monitoring have become possibe and beneficia in industria practice. One way to use the mutistage information and capture the mutistage reationships e.g., the correation structure between American Statistica Association and 57 seriay dependent stages) is to mode the mutistage process 6 the American Society for Quaity 58 as a inear state-space mode. Exampes have been given by 7 59 DOI 0.98/ Jin and Shi 999), Ding, Shi, and Cegarek 00b), Ding, Cegarek, and Shi 00a), Ding, Jin, Cegarek, and Shi 005), Djurdjanovic and Ni 00), Huang, Zhou, and Shi 00), Zhou, Ding, Chen, and Shi 003a), and Zhou, Huang, and Shi 003b). Even though the state-space modeing approach is popuar and has been particuary infuentia in recent years, investigations of mutistage statistica process contro SPC) based on such a mode, incuding both phase I retrospective anaysis and phase II monitoring and diagnosis, are rare. Xiang and Tsung 006) and Zantek, Gordon, and Robert 006) considered onine monitoring and diagnosis of out-of-contro OC) conditions in the mutistage process. Zhou, Chen, and Shi 004) estimated the process parameters by appying the typica estimation methods of genera mixed inear modes. Xiang and Tsung 006) aternativey derived an EM procedure for maximum ikeihood ML) estimates of the parameters. The estimation methods proposed by Zhou et a. 004) and Xiang and Tsung 006) rey on the assumption that the reference sampe is statisticay stabe. If the coected sampes are indeed not in contro IC) i.e., change points or outiers exist), then the estimates of the parameters wi be adversey affected. Therefore, a phase I anaysis of a mutistage process, aimed at identifying the data from an IC process as accuratey as possibe so that quaity engineers can have a good reference for estabishing contro charts for future operations, is very vauabe and

2 TECH asa v.007/0/3 Prn:5/07/008; 5:3 F:tech07057.tex; Roberta) p. CHANGLIANG ZOU, FUGEE TSUNG, AND YUKUN LIU desirabe. But the phase I anaysis of mutistage processes remains chaenging and has not been thoroughy investigated in the iterature. One technica chaenge is the use of mutistage information in the anaysis. Traditiona methods deveoped for a singe stage cannot fuy use the information from a mutistage operation where the input of the current process stage may be highy reated to the outputs) of the eary stages). Another chaenge is the compexity of phase I anaysis with uncertain mutistage parameters. One task in phase I is to recognize the presence of OC conditions and identify which data if any) are ikey to be from the IC condition. But unike in phase II monitoring, the parameters of the process in phase I anaysis are not known, so that any strategy for phase I anaysis depends on the simutaneous estimation of the parameters. Incuding OC data can ead to biased estimation, which in turn may affect the abiity to detect the presence of the OC conditions) the masking probem). The situation may be more serious for mutistage processes because of the intricacy of the mode and the arge number of parameters. In this artice we integrate the popuar mutivariate change point detection scheme with specific directiona information based on a mutistage state-space mode. Based on this integration, we propose an effective mutivariate change point method to assist in testing and estimating the sustained shift in a fixed mutistage process sampe by using mutistage information. With a focus on the sustained shift case, we demonstrate that the proposed scheme uniformy outperforms existing approaches for mutistage processes. The remainder of the artice is organized as foows. First, in Section we describe the mutistage process mode in detai and review existing mutivariate change point approaches that can be appied to mutistage modes. In Section 3 we consider a mutivariate hypothesis test probem when a set of aternative hypotheses is avaiabe. Then we propose a directiona mutivariate change point method to resove the unique chaenges of mutistage processes. We aso give an appropriate approximation of the threshod for the test statistic. We anayze the test and estimation performance and compare it with the resuts of existing approaches in Section 4. Finay, in Section 5 we summarize the major contributions of the artice and suggest issues for future research. We provide some necessary proofs and derivations in the Appendix MULTISTAGE MODELING AND TRADITIONAL CHANGE POINT APPROACHES. Mutistage Modeing Consider a manufacturing process composed of p stages. Without oss of generaity, the stages are numbered in ascending order, such that if stage k precedes stage, then k <. Atwoeve inear state-space mode generated from a practica appication is usuay used to describe the quaity measurement at the kth stage of an IC process Xiang and Tsung 006), 55 4 y k = C k x k + v k, ) x k = A k x k + w k, for k =,...,p, where x k denotes the unobservabe product quaity characteristic. The first eve of the mode invoves fitting the quaity measurement to the quaity characteristic, with C k used to reate the unobservabe process quaity characteristic, x k, to the observabe quaity measurement, y k. The second eve of the mode invoves modeing the transfer of the quaity characteristic from the previous stage to the present stage, in which A k denotes the transformation coefficient of the quaity characteristic from stage k to stage k. A k and C k for k =,...,p are known constants or matrixes) that are usuay derived or estimated from engineering knowedge see Jin and Shi 999; Ding et a. 00a for detais). The process noise say the common-cause variation and the unmodeed errors) is designated w k, and the measurement error of the product quaity is designated v k. Cameio, Hu, and Cegarek 003) used the foregoing statespace mode to represent variation propagation of a mutistage assemby system with compiant parts. Here x k represents the part deviation at stage k, and y k is the deviation vector containing a measurements at key product characteristic points. A k is the dynamic matrix that represents the deviation change due to part transfer among stages and is determined jointy by reocation, deformation, and sensitivity matrixes. C k is the observation matrix, corresponding to the information on the sensor number and sensor ocations. Such a inear state-space mode structure provides an anaytica engineering too for modeing, anayzing, and diagnosing a mutistage process. Extensive reviews of the state-space mode have been given by Bassevie and Nikiforov 993) and Ding et a. 00a). In this artice we start from a univariate case, where A k C k, x k, and y k in mode ) are one-dimensiona, and foow the mode in the iterature, v k N0,σv ), w k N0,σw k ). The variance depends on the stage index, k, and the initia state, x 0 Na 0,ε ). Usuay in SPC appications, the unknown parameters, a 0, ε, σv, and σ w k, may be estimated through a arge, good reference sampe obtained in the phase I anaysis. Suppose that a mutistage process data set contains m observations in the form y,j, y,j,...,y p,j }, where y k,j is a quaity measurement observed from the jth product operated on the kth stage in the process described by mode ), j =,,...,m. The mode based on the observations then can be written as y k,j = C k x k,j + v k,j, x k,j = A k x k,j + w k,j, where v k,j and w k,j are independent for k =,...,p and j =,,...,m. Here we assume that m > p +. This assumption is not a restrictive one and can be easiy satisfied in practica appications. It can ensure that the estimation of the covariance matrix in the change point approach is nonsinguar, which we discuss ater. In this artice we consider a typica change point probem in which a singe step change occurs at some point, τ,inthe mean of ony one stage, say ζ ; that is, the OC mode can be represented as 59 8 y k,j = C k x k,j + v k,j, x k,j = A k x k,j + w k,j + δi k=ζ,j>τ}, ) 3)

3 TECH asa v.007/0/3 Prn:5/07/008; 5:3 F:tech07057.tex; Roberta) p. 3 A CHANGE POINT APPROACH FOR PHASE I ANALYSIS 3 where k =,...,p, j =,,...,m, and I } is the indicator function. We want to test the no-change nu hypothesis, H 0 : δ = 0, against the aternative, H A : δ 0. We concentrate on this mode assumption, which is common in industria practice, in this artice, and wi investigate various extensions in future research Conventiona Change Point Approaches Conventionay, there are two possibe approaches to deveop testing methods for the mutistage process change point probem. One approach is to conduct a test for the quaity measurements of the product at the fina stage i.e., y p,j ) using cassica univariate techniques, such as the parametric ikeihood ratio test or the Schwarz information criterion Csorgo and Horvath 997; Chen and Gupta 000). Obviousy, using a singe stage in a mutistage process can be miseading and ineffective, because its measure is confounded with the cumuative effects from previous stages. Moreover, the impact of cross-correation and au- tocorreation due to the process dynamics and inertia of seri- j= 0 79 ay dependent stages shoud not be ignored. Xiang and Tsung 006) have discussed the disadvantages of the onine contro 80 scheme based on the observations of the fina stage from the 3 j=+ 8 standpoint of efficient monitoring and diagnosing. Thus we beieve that this method is not appropriate for testing and estimating the change point for the mutistage mode, and we wi not proceed further with it. Another conventiona approach is to obtain the one-step forecast errors OSFEs) i.e., the residuas), based on the process mode and then appy some traditiona methods to them. For mode ), the standardized OSFEs have been given by Xiang and Tsung 006) and are independenty and identicay distributed iid) from the standard norma distribution Durbin and Koopman 00) under IC conditions and with known parameters. But in a phase I setting, the parameter vaues are unknown and must be estimated. Because of the compex form of the OSFEs and the possibe presence of an OC condition, it appears to be very difficut if not impossibe) to obtain the anaytica properties and make use of the OSFEs. Thus in this artice we 9 88 do not deveop a test and estimation method for a mutistage =,...,m change point mode based on OSFEs. Aternativey, we may focus directy on origina observations from each stage. As we discuss ater, the method based on origina observations can convenienty make fu use of the underying information in the mutistage mode. Thus we propose constructing a proper test for the mutistage change point mode by investigating the origina product observations, y,j,...,y p,j, directy. Let y j = y,j,...,y p,j ) for j =,,...,m. Because the product observations are independent, we can regard y,..., y m as a sequence of independent p-dimensiona mutivariate norma vectors with unknown mean μ 0 and unknown variance matrix. Therefore, the test for a mean shift in the data set, y j = y,j,...,y p,j ), j =,,...,m}, may be considered a change point detection probem, which concerns the mean vector change under the assumption of an unknown but constant covariance matrix, when mutivariate individua observations are avaiabe. Such a change point probem has been studied by Srivastava and Worsey 986), who proposed a method designated as the SW method hereinafter) to detect a change point in the mean vector. James, James, and Siegmund 99) proposed a simiar method to the SW method but with a different approximation of the critica vaue for the test statistics. Suivan and Wooda 000) proposed a preiminary anaysis method for detecting a shift in the mean vector, the covariance matrix, or both, for mutivariate individua observations and used the SW method as the benchmark. Reated asymptotic resuts of the SW method aso have been given by Chen and Gupta 000). Because we are concerned mainy with the shift in the mean vector rather than the change in the variation, we use the SW method as a benchmark for comparisons with the proposed scheme in next section. Here we briefy describe the SW method. Denote the means of the first and ast m observations as ȳ = j= y j, ȳ m = m mj=+ y j, and the pooed sampe variance matrix as W = y j ȳ )y j ȳ ) m m y j ȳ m )y j ȳ m ) }. Obviousy, under the OC mode 3), W is an unbiased estimator of ony when τ< +. From an asymptotic viewpoint, it is consistent if /τ = + o); see the arguments in the proof of Proposition. Then the standardized difference between the observations before and after the change point is m ) t = m ȳ ȳ m ), and the corresponding Hoteing T statistic is T = t W t, =,...,m. 4) Then H 0 is rejected if max m T is arge enough. The ML estimator for the ocation of change τ is given by τ SW = arg max T. 5) The foregoing SW method is a genera approach for testing and estimating the change point in mutivariate observation cases. But using this method for a mutistage process may cause us to overook some important process information. We iustrate this point and propose our approach in the next section. 3. THE DIRECTIONAL MULTIVARIATE CHANGE POINT METHOD Approaches to testing and estimating a change point, such as the SW method, are usuay based on the two-sampe ikeihood ratio test by means of the binary segmentation procedure Chen and Gupta 000). Therefore, before studying the change point probem of the mutistage mode, we first consider corresponding one-sampe mutivariate hypothesis testing probems in Section 3.. Then the anaogous arguments can be readiy extended to the two-sampe mutivariate testing probems with unknown parameters and ead to our proposed change point method for mutistage processes.

4 TECH asa v.007/0/3 Prn:5/07/008; 5:3 F:tech07057.tex; Roberta) p. 4 4 CHANGLIANG ZOU, FUGEE TSUNG, AND YUKUN LIU 3. Mutivariate Hypothesis Testing Probems 60 6 Let y j = y,j,...,y p,j ), j =,...,m, be a random sampe of size m from a p-variate norma popuation with mean μ and covariance matrix. We consider the probem in testing the foowing hypothesis test: 5 64 i r H 0 : μ = μ 0 H : μ μ 0. For simpicity, we suppose that the covariance matrix,, is positive definite and known. Aso, without oss of generaity, we assume that μ 0 = 0. For this probem, the most popuar test is the generaized ikeihood ratio test GLRT), for which the test statistic Anderson 984) is mȳ m ȳ m χ p, 6) where χp denotes the chi-squared distribution with p degrees of freedom. The GLRT has some optima properties, incuding admissibiity, minimax properties and being the optima invariant test against H. When is estimated, a Hoteing T distribution shoud be used instead. Sometimes we do have some knowedge about the aternative hypothesis, such as its specific direction, μ = δd, where the d ζ,k = C k A i if k ζ 8 3 i=ζ + 8 direction, d, is a known vector but the scae, δ, is an unknown constant. In such a case the hypothesis test woud be H 0 : μ = 0 H : μ = δd. For this probem, the ikeihood ratio test is considered to be the uniformy most powerfu unbiased test Lehmann 99), which has a rejection region 9 88 md ȳ m ) /d d >χ α), where α is the prespecified type I error and χ α) is the upper α percentie with a χ distribution. Furthermore, in a more practica situation, we may consider the foowing hypothesis test: H 0 : μ = 0 H : μ = δd or μ = δd... or μ = δd r, where the aternative hypothesis has severa possibe directions, d, d,...,d r, with known vectors. To consider the case where r is finite, the GLRT wi reject the nu hypothesis if the test statistic, max i=,...,r md i ȳ m ) /d i d i }, 7) is arger than a prespecified critica vaue. To the best of our knowedge, there is no pubished theoreti- U = max 47 ca optima property of such a test method under the aternative k p d k W t ) /d k W d k, =,...,m, 06 hypothesis. However, the efficiency of this test can be expected to be superior to test 6) because of its more sharpy focused rejection region. Moreover, when the nu hypothesis is rejected, we naturay can use arg maxmd i ȳ m ) /d i d i } 8) d i =d,...,d r as an estimator of the true mean vector direction. 55 <m 4 Next we present a proposition that estabishes some good as- ymptotic properties of test 7) and estimator 8). Proposition. For testing the nu hypothesis, H 0,against a. For any finite set of directions, test 7) is asymptoticay no ess powerfu than test 6). b. Without oss of generaity, we assume that μ = δd k, where k r. Then, the estimator of k, is consistent. ˆk = arg maxmd i ȳ m ) /d i d i }, We wi find this proposition quite hepfu in constructing our test and estimation of the change point in a mutistage mode in the foowing sections. 3. The Directiona Mutivariate Change Point Method Here we propose a mutivariate change point scheme that makes fu use of the directiona information from a mutistage process. Suppose that a shift occurs at some stage, ζ, as represented by mode 3). The expectation of y j wi change from μ 0 to μ = μ 0 + δd ζ, where d ζ = d ζ,, d ζ,,...,d ζ,p ), and k 0 otherwise. Thus in the mutistage process considered here, the process shift occurs ony in one of p known directions d ζ,ζ=,...,p.this a priori shift direction information is a particuar characteristic in mutistage processes. If we coud construct a mutivariate change point scheme that makes use of this a priori knowedge propery and sufficienty, then the testing and estimation of a change point in the mutistage process woud be enhanced compared with the SW method based ony on the GLRT, which does not take such constraining information into consideration. The SW method combines a binary segmentation procedure with the two-sampe mean test Chen and Gupta 000). On the other hand, the SW test statistic aso can be obtained from the ikeihood ratio procedure approach by assuming that is known and then using an estimator to repace. Anaogousy, for the mutistage mode change point probem, by comparing testing statistics 6) and 7), it is natura to obtain the foowing test from 4): H 0 is rejected if max U > c, 9) m 59 the aternative, H τ t τ ) /d k W 8, the foowing hod: k p where 0) and c is a constant for attaining the prespecified type I error probabiity α. This testing method integrates the binary segmentation procedure and the two-sampe GLR mean test with directiona information. Moreover, estimators of ζ and τ are given by and ζ = arg τ = arg maxd k W maxu } ) τ d k }. )

5 TECH asa v.007/0/3 Prn:5/07/008; 5:3 F:tech07057.tex; Roberta) p. 5 A CHANGE POINT APPROACH FOR PHASE I ANALYSIS 5 Remark. Note that the foregoing directiona mutivariate change point DMCP) test can be rewritten in the foowing form: 60 6 } max max 5 k p m d k W t ) /d k W d k > c. μ 0 can be represented as 64 On the other hand, the statistic V k max m d k W t ) / 8 d ) k W d k is simiar to the test statistic based on the ikeihood k p 67 9 ratio approach for a change point probem set before and after = C A a 0,...,C k A i a 0,...,C p A i a the change point of the mean vector, μ = μ 0 and μ = μ 0 + i= i= 69 δd k see the proof of Prop. 4 in the App.). The DMCP aso can be seen as the combination of p specia mutivariate change point tests, each one being the simpe hypothesis versus simpe hypothesis. This can hep us obtain an approximate significance eve of the DMCP test, which we iustrate ater Remark. Note that can be represented as vary,j ) covy,j, y,j ) covy,j, y p,j ) covy,j, y,j ) vary,j ) covy,j, y p,j ) =...., covy,j, y p,j ) covy,j, y p,j ) vary p,j ) where [ k ) vary k,j ) = Ck A i ε k k ) ] A i σw + σv 6 i= = i=+ 85 and 9 88 covy,j, y k,j ) = C k k A i C A i ε i= i= k 33 + C k A i C A i σw m, < k. for k =,...,p. The GLRT statistic wi be 9 34 m= i=m+ i=m+ 93 k p m Thus to estimate, we may estimate ε, σv, and σ w k instead of using a pooed sampe variance matrix. In fact, this prob- 37 em wi ead to the parameter estimation in a variance com- 96 d k d k W ς) ponent mode. The typica statistica estimation agorithms are ML estimation, restricted ML estimation REML), and minimum norm quadratic unbiased estimation MINQUE) see Rao and Keffe 988). Zhou et a. 004) suggested using MINQUE as an approximation of the ML estimate for mutistage modes because the computation oad of MINQUE is much ower than that of ML estimation or REML. But we do not beieve that MINQUE is suitabe for phase I anaysis, for two reasons. First, because MINQUE is obtained by soving p + noninear equations, nonconvergence of the iterative agorithm can occur. To reduce the frequency of nonconvergence, the use of good starting vaues is recommended Rao 997). But choosing such starting vaues is rather difficut in the presence of OC condition observations. In fact, the convergence of MINQUE may be questionabe under the OC condition. Second, our change point method is based on the binary segmentation procedure, so that m segments are needed. For each segment and each d k, the MINQUE of must be obtained; that is, p + noninear equations are soved. When the sampe size, m, and number of stages, p, are arge, this is not a trivia computationa task. Thus we suggest using the pooed sampe variance matrix, W,toestimate. This estimator is easy to cacuate and has a cosed form, so that we can obtain some anaytica properties for the proposed DMCP. Remark 3. It is noteworthy that for a mutistage mode ), μ 0 = Ey j ) a 0 ς; that is, in the p-variate vector, μ 0, there is ony one unknown parameter, a 0. Note that we do not incorporate this point into the proposed test 9), because we assume that we have no prior knowedge about which part of the data set may be in contro, that is, mode 3). But sometimes we indeed have some additiona information about which part of the data is in contro. In such a case, without oss of generaity, we assume that the observations before some unknown change point are IC, and it is equivaent to consider the test against the aternative H 0 : Ey ) = Ey ) = =Ey m ) = a 0 ς H A : there is an integer τ such that a 0 ς = Ey ) = =Ey τ ) Ey τ+ ) = =Ey m ) = a 0 ς + δd k ς d k W max max m ) ς W ȳ m ς W ȳ m ς W / [ ς W d k m ] m ς W d k ) )} ς W ς. For testing the preceding change point hypothesis, appying this test statistic may improve the efficiency compared with the test 9). To end this section, we present an asymptotic resut on the consistency of the change point estimator ) and the estimated stage where the shift occurs [eq. )]. Proposition. Suppose that 0 < im m τ/m = θ<, where τ is the true change point in the observations. When the process is statisticay OC as mode 3) and the size of shift δ is a fixed constant, we then have the foowing: a. τ τ =O p ), where τ is as defined in ) and O p ) means bounded in probabiity. b. ζ ζ in probabiity as m, where ζ is as defined in ).

6 TECH asa v.007/0/3 Prn:5/07/008; 5:3 F:tech07057.tex; Roberta) p. 6 6 CHANGLIANG ZOU, FUGEE TSUNG, AND YUKUN LIU This proposition ensures that the proposed estimators ) and ) are asymptoticay effective. Athough whether or not τ outperforms τ SW remains undemonstrated anayticay at this point in the artice, the simuations in Section 4 show that τ indeed improves the accuracy of the change point estimates for the mutistage processes considered in this artice Approximate Critica Vaue 9 68 Srivastava and Worsey 986) and James et a. 99) presented two different methods for approximating the significance eve of the SW test. In this section we study some properties of DMCP and then propose a suitabe method to approximate its significance eve or, equivaenty, the critica vaue. First, we give an approximation of PrV k > c), k =,...,p. The statistic T in 4) is known to foow Hoteing s T distribution, which does not depend on the parameters μ 0 and.to deveop the approximations of PrV k > c), we need to show that V k has a simiar affine-invariant property. Denoting the statistics as d k W t ) /d k W d k by G,k, we then have the foowing et a. 987, 99) considered approximation through a d- proposition. 80 Proposition 3. When a process is statisticay IC, for any given vector d k 0, the distribution of G,k does not depend on d k, μ 0,or. 3 8 Based on the foregoing proposition, PrV k > c) can be convenienty obtained by Monte Caro simuations, because we can choose any set of vaues, d k, μ 0,or. Without oss of 9 generaity, et d k = 0, 0,...,) e p, μ h 0 = 0, and be the 88 identity matrix. This proposition aso makes approximating PrV k > c), k =,...,p easy, because we need ony study each specific choice for d k. To approximate PrV k > c), we need to expore the distributiona properties of G,k. By Proposition 3, we have the foowing coroary. Coroary. When the process is statisticay IC, the statistic G,k has the same distribution as m m p F + p ) m p F, where F and F are two independent random variabes with F distributions with, m p ) and p, m p) degrees of freedom. This coroary indicates that G,k has asymptoticay a chisquared distribution with degree of freedom as m.in fact, when is known, the random variabe with repacing W in G,k wi foow exacty the χ distribution. Furthermore, by a modification of theorem.3. of Csorgo and Horvath 997), we have the foowing proposition, which estabishes the asymptotic nu distribution of V k. Proposition 4. When the process is statisticay IC, for every k =,...,p,wehave im Pr Aog m)[v k ] / t + Dog m) } = exp e t ) m ) for a t, where Ax) = ogx) / and Dx) = ogx + og og x og π. We can use this proposition to get asymptotic critica vaues of V k ; however, in change point probems, the rate of convergence of the distribution of the test statistic based on binary segmentation is beieved to be usuay sow see sec..3 of Csorgo and Horvath 997 for some discussions). Consequenty, when m is not arge enough, the approximation of 3) yieds somewhat conservative resuts for sma vaues of p. Moreover, empiricay speaking, when the number of stages p is fairy arge but m is sma, using 3) may ead to a much arger type I error than the nomina vaue, due to inaccurate estimation of the high-dimensiona nuisance parameter ). The resut of some simuations that we conducted not reported here) support these assertions. In fact, the asymptotic distribution of V k given by 3) equas that of the cassica ikeihood ratio, Z m,fortesting ony a mean change see Csorgo and Horvath 997; Chen and Gupta 000; and references therein). When there is no other unknown nuisance parameter, the critica vaue can be cacuated using the recursion technique of Hawkins 977). James dimensiona Brownian bridge. A simiar but more expicit suggestion was given by Csorgo and Horvath 997, sec..3); their recommended approximation is Pr Zm / > x) x exp x /) ns) π x ns) + 4 }, x 4) where s = h) ) and h = = nm)) 3/ /m. Thisisfairy accurate if x is not too sma. Appying 4) to V k wi yied a arge bias when m is not arge enough, because the statistic G,k is not exacty χ distributed due to the variation in estimating ; that is, it is usuay necessary to take a arger critica vaue for V k than what comes from 4). Later we describe a simuation study that which demonstrates this point. But the earier interpretation of 3) suggests a heuristic approximation of PrV k > c) and, consequenty, an appropriate approximation of c, ĉ. Given a desired type I error, α, say, we may first find a vaue c, defined as the vaue of x where x soves 4) with the repaced by =. Then we can determine ĉ by the equation Fĉ) PrG,k < ĉ) = F χ c ), 5) where F χ ) is the cumuative distribution function cdf) of the χ distribution and F ) is the cdf of G,k, which can be computed by numerica integration according to the distribution given in Coroary. A Fortran program for cacuating the vaue ĉ is avaiabe from the authors on request.) In contrast, PrV k > c) can be approximated by first finding c from 5). and then substituting c / for x in 4). To iustrate the effectiveness of the proposed approximation method, in Tabe we tabuate some simuated probabiities, PrV k > ĉ), for various vaues of m and p.hereα is the desired type I error and α is the estimated vaue from using ĉ estimated by 30,000 repetitions of a Monte Caro simuation. It can be seen that the accuracy of the approximation, ĉ, usuay increases as m increases and decreases as p increases. The vaues of α are cose to the desired vaues, α, even with a sma number of sampes m = 0) and a arge number of stages p = 6). Generay, ĉ yieds a nice approximation in most cases. 59 8

7 TECH asa v.007/0/3 Prn:5/07/008; 5:3 F:tech07057.tex; Roberta) p. 7 A CHANGE POINT APPROACH FOR PHASE I ANALYSIS 7 Tabe. Performance of the approximation based on 4) and 5) 60 6 p = p = 4 p = 6 4 m α c ĉ α ĉ α ĉ α test statistics are independent Simes 986). Some efforts have distributions, such as those of Samue-Cahn 996), Sarkar and Chang 997), and Sarkar 998). A of these efforts rey on the assumption that the nu distributions of test statistics sat isfy some specific conditions. See Sarkar 998 for a thorough of order two, which is satisfied by a arge famiy of mutivariate Remark 4. As one of the referees pointed out, there may exist two or more soutions of 4) for some arge vaues of m. Empiricay speaking, we can simpy take the argest soution as c in such a situation. But if some very mid conditions are imposed on m and α, then we may obtain ony one soution of 4). For exampe, suppose that m 0 and α., which are very common in industria practice and can be satisfied in most appications. By some direct cacuations, we can show that the soution of 4) exists, is unique, and is within the range of [, ) Finay, we turn to deveoping the approximation of the significance vaue of the DMCP test statistic max k p V k.the probabiity, Prmax k p V k > c), seems rather difficut to derive exacty, because V k, k =,...,p, are usuay positivey correated, and the correations depend on the unknown parameter. A simpe and natura approach to tacking this probem is to use the cassica Bonferroni procedure, that is, Prmax k p V k > c) p k= PrV k > c). We woud expect this to maintain a fase aarm rate cose to its nomina anaog when the number of stages, p, is not arge; nevertheess, this approximation is conservative for arge p and acks power if severa V k s are highy correated. Note that the form of the proposed DMCP test statistic is simiar in spirit to that of a goba test statistic in a mutipe-testing probem. Thus here we recommend appying the we-known Simes modified Bonferroni procedure see Simes 986 for detais) instead of the Bonferroni approximation. Given the strong empirica evidence of the modified procedure s superiority, researchers have begun using it in various mutipe-testing appications. Formay, we summarize this procedure for the present context as foows: 9 88 Step. Compute the vaues of statistics V k for k =,...,p. Step. Use 4) and 5) to obtain the corresponding approximate p vaues for each V k,say P k, k =,...,p. Step 3. Given a desired fase aarm the rate α, reject nu hypothesis if P i) iα/p for at east one i, where P ) P p) are the ordered vaues of P,..., P p. Our extensive numerica simuations demonstrate that Simes s procedure works very we and usuay provides ess conservative approximations compared the Bonferroni method. Some simuation resuts are given in Section 4. But the theoretica conservativeness of this procedure for our considered test statistics remains unknown to us. In fact, the Simes method for testing the intersection of more than two hypotheses is known to contro the probabiity of type I error when the underying been made to prove that the conservativeness of Simes s procedure may hod for dependent statistics with various mutivariate review and discussion.) For exampe, Sarkar 998) considered a quite reaxed condition, say the mutivariate totay positive distributions. Nevertheess, the joint distribution of V k,eventhe margina distribution for each V k not the asymptotic one), is compicated and remains a chaenge. Thus appying Sarkar s 998) resuts to the DMCP test statistics seems very difficut. An ongoing effort of our group is to theoreticay prove the conservativeness of Simes s procedure for the present mutistage process probem. 4. PERFORMANCE COMPARISONS In this section we first investigate the test and estimation performance of the proposed scheme through Monte Caro simuation and compare the resuts with the SW method under various combinations of the mutistage state-space parameters. We then conduct a sensitivity anaysis to evauate the effectiveness and robustness of the proposed scheme under misspecification of vaues of A k and C k. 4. Powers of the Two Tests For simpicity, we consider ony the case of type I error when α =.05 and fix the number of stages to p = 5 and the number of observations to m = 50. Without oss of generaity, we use the parameters a 0 = 0 and ε = σw k = σv = throughout this section. We compare the proposed DMCP method with the aternative scheme, the SW method. As we know, the performance of the method for testing change points in a mutistage process depends on the parameters A k and C k for k =,,...,p; the different ocations, ζ, where the shift initiay occurs; the magnitude of the shift, δ; and the change point, τ. In this section, various eves of δ =.5,.0,.5,.0, 3.0, 4.0 are considered. A k, C k ) =.0,.0), A k, C k ) =.,.8), and A k, C k ) =.8,.) for ζ =, 3, 5 and τ = 0 have been found to be consistent with the numerica comparison settings of Xiang and Tsung 006). Tabe gives the powers of the tests obtained from 0,000 repications, with the critica vaues for the two methods isted in the ast row. Note that these vaues are obtained by simuations, so the two methods can be fairy compared. The tabe aso gives the simuated type I errors for the DMCP method, determined from the Simes and Bonferroni procedures. Apparenty, both the Simes and Bonferroni procedures perform we in controing the fase aarm rate, but the Simes approach indeed provides ess conservative performance. With respect to the powers, Tabe shows that the proposed DMCP is uniformy superior to the aternative SW method in detecting any magnitude of shifts. Aso, it can be ceary seen that

8 TECH asa v.007/0/3 Prn:5/07/008; 5:3 F:tech07057.tex; Roberta) p. 8 8 CHANGLIANG ZOU, FUGEE TSUNG, AND YUKUN LIU Tabe. Performance comparisons between DMCP and SW with α =.05, τ = 0, and m = 50 for a five-stage process 60 A k, C k ) =.0,.0) A k, C k ) =.,.8) A k, C k ) =.8,.) 6 4 δ DMCP SW DMCP SW DMCP SW 63 5 ζ = ζ = ζ = Simes Bonferroni Critica vaue the DMCP method s superiority over its counterpart sti hods when the vaues of the parameters, A k, C k, and ζ,vary.here we do not tabuate the performance comparisons of these two methods for shifts at other change points, because simiar resuts were obtained. It is worth noting that the difference in performance between the DMCP and SW methods wi be more prominent in higherdimensiona probems. To verify this point, we consider a mutistage process of p = 0 and fix ζ = 0, τ = 40, and m = 00 for simpicity. The other parameters are the same as those given in Tabe. Tabe 3 gives the simuation resuts. Compared with the power resuts given in Tabe, the superiority of the DMCP method is more remarkabe in the case where p = 0, especiay for moderate magnitudes of change, say δ =.5 or.0. Based on the resuts given in Tabes and 3 and other simuations of various mutistage mode parameters avaiabe from the authors on request), we concude that the DMCP method, by taking advantage of incorporating shift direction information, is amost aways better than the SW method in detecting mean process change. In the foowing section, we assess the effectiveness of the corresponding estimation methods as we Performance of the Estimators ) and ) Here we investigate the performance of the proposed approach in estimating the change point after the nu hypothesis is rejected. We compare the performance of two change point estimators, τ SW in 5) and τ in ). The number of stages, p = 5, and the number of observations, m = 50, are considered again. A tota of 50,000 independent series are generated in the simuations. Note that any series for which no signa was trigged is discarded. The process change point is simuated at τ = 0 and τ = 0. Again the parameter combinations Tabe 3. Performance comparisons between DMCP and SW with α =.05, τ = 40, and m = 00, and ζ = 0 for a 0-stage process 48 A k, C k ) =.0,.0) A k, C k ) =.,.8) A k, C k ) =.8,.) 07 δ DMCP SW DMCP SW DMCP SW Simes Bonferroni Critica vaue

9 TECH asa v.007/0/3 Prn:5/07/008; 5:3 F:tech07057.tex; Roberta) p. 9 A CHANGE POINT APPROACH FOR PHASE I ANALYSIS 9 Tabe 4. The averages AVE), standard deviations SD), and precisions of change point estimates for shift at the third stage 60 in a five-stage process 6 3 AVE SD P P P 3 6 δ τ τ 5 SW τ τ SW τ τ SW τ τ SW τ τ SW P ζ 64 6 τ = τ = A k, C k ) =.0,.0) and ζ = 3 are considered. To quantify the diagnostic precision of the estimators, in addition to cacuating the averages and the standard deviations of the change point estimates, Tabe 4 gives the probabiities, Pr τ τ ), Pr τ τ ), and Pr τ τ 3) denoted as P, P, and P 3 ). For a sma shift of.0, the two estimators both appear to be biased in estimating the process change point, especiay in the case where τ = 0. The proposed τ performs uniformy better in terms of the average and standard deviation for any magnitude of shift. Moreover, τ has better performance in terms of diagnostic precision. Other extensive simuations conducted for various parameter combinations aso demonstrated that τ appears to be more accurate in estimating τ. In addition, the probabiities Pr ζ = ζ) P ζ ), given in the ast coumn of Tabe 4, aso indicate that the estimates of ζ ) are fairy accurate Sensitivity Anaysis In genera, more powerfu mode-based strategies come at the price of being more adversey affected by deviations from the mode assumptions. The performance of a mode-based method depends on the nature and degree of the misspecification of the assumption. As mentioned in Section, A k and C k usuay are derived or estimated from engineering knowedge. In practica appications, we may have inaccurate estimations of these parameters, especiay if there are many stages. Estimation errors in A k and C k wi yied incorrect direction vectors and adversey affect the performance of the DMCP method. Here we conduct some simuations to check how the DMCP method performs when A k and C k are misspecified. For simpicity, here we consider α =.05, p = 0, ζ = 0, τ = 40, and m = 00, the same vaues as in Tabe 3. Suppose that the true vaues of A k and C k are fixed as.0. Let  k and Ĉ k denote the incorrect vaues for the stage k.we consider two situations, I) k = 5, 0, and 5 and II) k = i for i =,,...,0, which correspond to ow and high degrees of parameter misspecification. For each situation, the foowing  k and Ĉ k are considered: a)  k =.8, b) Ĉ k =.8, c)  k =., d) Ĉ k =., e)  k =. and Ĉ k =.8, f)  k =.8 and Ĉ k =., g)  k =.8 and Ĉ k =.8, and h)  k =. and Ĉ k =.. Tabe 5 gives the powers of the tests, obtained from 0,000 repications. Note that the vaues in the third and fourth coumns of the tabe are from Tabe 3, where the vaues of A k and C k are prespecified correcty. We can see that when the degree of misspecification is ow, such as in I), the DMCP approach sti generay performs better than the SW method except for very sma δ, but its superiority is reduced compared with the power performance in the fourth coumn. For the situation II), where many parameters are estimated incorrecty, as we woud expect, misspecification of A k and C k has a significanty adverse effect on the performance of the DMCP approach, especiay in the ast case, h). Tabe 5. Performance comparison with misspecified parameters A 46 k and C k DMCP with misspecified vaues of A 06 k and C DMCP k δ SW true a) b) c) d) e) f) g) h) 50 I) II)

10 TECH asa v.007/0/3 Prn:5/07/008; 5:3 F:tech07057.tex; Roberta) p. 0 0 CHANGLIANG ZOU, FUGEE TSUNG, AND YUKUN LIU Thus athough the proposed approach can sti work quite we even when vaues of A k and C k are misspecified, we beieve that it is critica to estimate A k and C k accuratey when using a mode-based method CONCLUSIONS Phase I anaysis of mutistage processes remains a chaeng ing probem that has not been thoroughy investigated in the are consistent and unbiased for testing H 0. Note that under H, 69 iterature. In this artice we have focused on the probem of testing and estimating the change point in a phase I reference sampe. We have proposed an approach that combines the cassica binary segmentation procedure and the mutivariate twosampe test. This nove approach fuy incorporates the directiona information based on the mutistage state-space mode for testing the stabiity of mutistage processes. We give an appropriate approximation of the threshod for the test statistic. We aso expore an estimation approach to identify an OC stage and ocate the change point, with the estimators shown to be consistent. Our simuation resuts aso show that the proposed scheme consistenty outperforms conventiona mutivariate approaches to mutistage processes, uness the process mode is grossy misspecified. Note that athough the focus of this artice is on a univariate process, many mutistage processes are actuay mutivariate in practice. Extension of our proposed DMCP scheme to a mutivariate mutistage process warrants further research. Moreover, it shoud be pointed out that our proposed approach can be readiy extended to detect mutipe change points by using the binary segmentation method recursivey Vostrikova 98; Yao ). independent. Then, using the fact that x) φx) 3 x [where 9 ACKNOWLEDGMENTS The authors thank the editor, associate editor, and two anonymous referees for their many hepfu comments that have resuted in significant improvements in the artice. This work was competed when Zou was a research assistant at the Hong Kong University of Science and Technoogy, whose hospitaity is appreciated and acknowedged. This research was supported by RGC Competitive Earmarked Research grants 60606, Nationa Science Foundation of Tianjin grant 07JCYBJC04300, and??? NNSF) of China Grant Liu s work aso was supported by NNSF of China grant and SRFDP of China grant APPENDIX: PROOFS Proof of Proposition a. Under the nu hypothesis, we have Prmȳ m ȳ m > C }=α and } Pr max i=,...,r md i ȳ m ) /d i d i } > C = α, 55 4 where C is the upper α percentie of the χp distribution and C is some chosen constant. Here we denote mȳ m ȳ m as Vp) and md i ȳ m ) /d i d i as V i, which foow a χ distribution under H 0. As is we known, the GLRT is equivaent to max i=,..., md i ȳ m ) /d i d i } > C, where d i, i =,..., } is a countabe set dense in the parameter space Roy 953). It immediatey foows that C > C. Without oss of generaity, we suppose that μ = δd and δ>0 under H. It is obvious that these two mutivariate tests Prmax i=,...,r V i > C} PrV > C}. Thus it suffices to prove that PrV < C} PrVp)<C } 0 as m, where V χ mδ ) and Vp) χp mδ ).Hereχp mδ ) is the noncentra chi-squared distribution with the noncentra parameter mδ ) /. For simpicity, denote ξ = mδ as m. Note that Vp) can be partitioned into Up ) and V), which are distributed as χp and χ ξ ). Consequenty, by C > C, there exists ε>0, so that C C + ε) > 0. We then have PrVp)<C } PrV < C} = PrUp ) + V)<C } PrV < C} PrV)<C + ε} PrUp )<C C + ε)} PrV < C} The inequaity derives from the fact that Up ) and V) are ) and φ ) are the cdf and density function of standard norma distribution] for arge x,wehave PrV)<C + ε} PrV < C} = Pr[z + ξ] < C + ε} Pr[z + ξ] < C} ξ C + ε) ξ C) exp C + ε C)ξ ε } as m, where z is a standard norma random variabe. By noting that the probabiity PrUp )<C C + ε)} is a vaue greater than 0 for the fixed ε, it foows that exp C + ε C)ξ ε } PrUp )<C C + ε)} as m. b. Without oss of generaity, we assume that δ>0. For notationa convenience, et u i = d i / d i d i, i =,...,r. Then the estimator 8) can be rewritten as arg maxmu i ȳ m ) }. u i =u,...,u r

11 TECH asa v.007/0/3 Prn:5/07/008; 5:3 F:tech07057.tex; Roberta) p. A CHANGE POINT APPROACH FOR PHASE I ANALYSIS Thus proving the consistency of this estimator is equivaent to showing that } Pr [u k ȳ m ) <u i ȳ m ) ] 0 as m i k 64 Because } Pr [u k ȳ m ) <u i ȳ m ) ] 9 i k 68 i k Pru k ȳ m ) <u i ȳ m ) }, 70 7 ] it suffices to prove that Pru 5 j ȳ m ) u k ȳ m ) > 0} 0 asm. d k d k 74 i k Denote z i = u j ȳ m ) u k ȳ m ). For any ε>0, 8 θ )θ 77 = δ 9 Prz i Ez i ) + Ez i )>ε} d 78 k d d k d k + Op m / )), k 0 i k 79 = [ Pr z i Ez i ) ε ] [ + Ez i ) ε ] } > 0 i k Pr z i Ez i )> ε } + Pr Ez i )> ε }. A.) i k i k [γ d ζ d ζ ) d ζ d ζ ] Under μ = δd 8 k, we have that ȳ m N p δ u k, m ), where δ = 87 δd 9 k d k ) /. It is not difficut to verify that z i Ez i ) as = d ζ d ζ ) / 88 m and varz i ) = Om ). Thus, by the Chebychev in- 3 equaity, the first term of A.) tends 0 as m. [ I d ζ d ζ ) / d ζ d ζ /] / 90 On the other hand, u i u i =, i =,...,r, yied Ez i ) =[Eu i ȳ m )] [Eu k ȳ m )] 35 = δ [u i u k ) u k u k ) ]. θθ 0 and the second inequaity comes from the 94 From u k u i ) u k u i ) 0 and u k +u i ) u k +u i ) 0, we have that Ez i )<0. It immediatey foows that the second term of A.) equas 0. Therefore, A.) tends to 0 as m, which competes the proof. Proof of Proposition a. Bhattacharya 987) studied the asymptotic distribution of the ML estimate of the change point in a genera mutiparameter case. Gombay and Horvath 996) aso considered the asymptotic behavior of the ML estimate of a change point in a exponentia famiy case. For the sake of simpicity, they assumed no nuisance parameter Csorgo and Horvath 997, sec..5). Because our suggested testing statistics are somewhat specia, the methods in the preceding artices cannot be directy extended to prove the consistency of the estimator τ.herewe give a short proof. We first consider τ< < m. Assume that when m, /m θ, where θ 0, ) and θ >θ. By the aw of arge numbers, we can obtain 50 G,k = d k W t ) /d k W d k = d k W t ) / d k W d k, m / t 56 = δθ θ d ζ + O p m / ), 5 m / t τ = δ θ θ)d ζ + O p m / ), W τ = + O p m / ), 59 8 and W = + O p m / ), where = + θ θθ )d ζ d ζ. Consequenty, it is easy to verify that m G τ,ζ = O p )>0 and m G,k = O p ). Thus to prove part a, it is sufficient to show that when m,for any k p, τ< < m, m G τ,ζ G,k )>0, a.e. It then foows from the definition that m G τ,ζ G,k ) [ θ θ)d = δ ζ d ζ ) d θ )θ d ζ d k) ζ d ζ + O p m / )) where = γ d ζ d ζ d ζ d ζ and γ = θ )/θ ) >. Now we just need to show that the matrix is positive definite, which can be seen from = d ζ d ζ ) >d ζ d ζ ) [ d ζ d ζ ) d ζ d ζ ] 0, where the first inequaity comes from = θ )d ζ d ζ matrix d ζ d ζ ) / d ζ d ζ / being symmetric and idempotent. Here for two symmetric matrixes, C and D, C D means that C D is semipositive definite. The proof for < <τ is anaogous to the foregoing arguments and is omitted here. b. By τ τ =O p ) and a simiar argument as in the proof of Proposition.b, the proof of consistency is straightforward and is omitted here. Proof of Proposition 3 First, et Z i = / y i, i =,...,m. Then it foows that where d k = / d k and W and t are given by substituting Z i into W and t instead of y i. Obviousy, Z i, i =,...,m, is a random sampe of size m from a p-variate standard norma popuation. Thus t and m ) W are independenty distributed as a p-variate standard mutivariate norma distribution and a Wishart distribution with m degrees of freedom. Thus it suffices to show that the distribution of G,k does not depend on d k, which is some unknown vector.