Control Chart For Monitoring Nonparametric Profiles With Arbitrary Design

Transcription

1 Contro Chart For Monitoring Nonparametric Profies With Arbitrary Design Peihua Qiu 1 and Changiang Zou 2 1 Schoo of Statistics, University of Minnesota, USA 2 LPMC and Department of Statistics, Nankai University, China Abstract Nonparametric profie monitoring (NPM) is for monitoring over time a functiona reationship between a response variabe and one or more expanatory variabes when the reationship is too compicated to be specified parametricay. It is widey used in industry for the purpose of quaity contro of a process. Existing NPM approaches require a fundamenta assumption that design points within a profie are deterministic (i.e., non-random) and they are unchanged from one profie to another. In practice, however, different profies often have different design points, and in some cases they might even be random. NPM is particuary chaenging in such cases because it is difficut to combine data in different profies propery for data smoothing and for process monitoring. In this paper, we propose a nove exponentiay weighted moving average (EWMA) contro chart for handing this probem, based on oca inear kerne smoothing. In the proposed chart, the exponentia weights used in the EWMA scheme at different time points are integrated into a nonparametric smoothing procedure for smoothing individua profies. Because of certain good properties of the charting statistic, this contro chart is fast to compute, easy to impement, and efficient to detect profie shifts. Some numerica resuts show that it works we in appications. Keywords: Bandwidth Seection; EWMA; Loca Linear Kerne Smoothing; Nonparametric Regression; Profie Monitoring; Sef-Starting; Statistica Process Contro. 1

2 1 Introduction In many appications, quaity of a process is characterized by the functiona reationship between a response variabe and one or more expanatory variabes. Profie monitoring is mainy for checking stabiity of this reationship (or profie) over time. In some caibration appications, the profie can be described adequatey by a inear regression mode, whie in other appications more fexibe modes are necessary. This paper focuses on nonparametric profie monitoring (NPM) when the profie is too compicated to be specified parametricay. In the iterature, some existing references focus on inear profie monitoring. See, for instance, Kang and Abin (2000), Kim et a. (2003), Mahmoud and Wooda (2004), Zou et a. (2006, 2007b), and Mahmoud et a. (2007), among severa others. Extensions to mutipe and/or poynomia profie modes are discussed by Zou et a. (2007a), and Kazemzadeh et a. (2008). Recenty, noninear profie modes have been considered by some statisticians, incuding Lada et a. (2002), Ding et a. (2006), Coosimo and Pacea (2007) and Wiiams et a. (2007a,b). NPM is discussed by Zou et a. (2008, 2009). For an overview on profie monitoring, see Wooda et a. (2004). A the contro charts mentioned above require the fundamenta assumptions that design points within a profie are deterministic (i.e., non-random) and that they are the same from one profie to another. These assumptions are (approximatey) vaid in certain caibration appications of the manufacturing industry. In some other appications, however, they may be invaid. For instance, when data acquisition takes the random design scheme, design points within a profie woud be i.i.d. random variabes from a given distribution. Another commony seen exampe occurs when observations within different profies have missing vaues at different time points (e.g., the vertica-density profie (VDP) data considered in Waker and Wright 2002). Furthermore, we wi demonstrate in this paper by both theoretica and empirica resuts that, even for appications where an equa design scheme (i.e., design points are the same from profie to profie and they are deterministic) is possibe, one may get a better profie monitoring by using a random design scheme, as ong as the two design schemes invove simiar measurement effort. In this artice, we propose a nove contro chart for handing the NPM probem when the profie design points are arbitrary. The proposed chart is based on oca inear kerne smoothing of individua profie data and on the exponentiay weighted moving average (EWMA) process contro scheme as we. It incorporates the exponentia weights used in the EWMA scheme at different time 2

3 points into the oca inear kerne smoother. We show that this chart is effective in detecting profie shifts when profie design points are arbitrary. It is aso fast to compute and easy to impement. This chart is described in detai in Section 2. Its numerica performance is investigated in Section 3. In Section 4, we demonstrate the method using a rea-data exampe from the semiconductor manufacturing industry. Severa remarks concude the artice in Section 5. Some technica detais are provided in the Appendix. 2 Methodoogy Our proposed methodoogy is described in seven parts. In Section 2.1, we briefy introduce the statistica process contro (SPC) probem and the we known EWMA contro chart. Then, in Section 2.2, an EWMA contro chart accommodating nonparametric regression of individua profies is introduced for monitoring nonparametric profies with arbitrary design. Adaptive seection of its weighting parameter and bandwidth parameter, used in EWMA and nonparametric regression, are respectivey discussed in Sections 2.3 and 2.4. Certain computationa issues are addressed in Section 2.5. A sef-starting version is given in Section 2.6. Finay, some practica guideines regarding design and impementation of the proposed contro chart are provided in Section Statistica process contro and the EWMA contro chart SPC is for monitoring sequentia processes (e.g., production ines in manufacturing industry) to make sure that they work staby. When the process works staby, it is in the in-contro (IC) state, and it becomes out-of-contro (OC) otherwise. In the iterature, SPC is often divided into two phases. In Phase I, a set of process data is gathered and anayzed. Any unusua patterns in the data ead to adjustments and fine tuning of the process. Once a such assignabe causes are accounted for, we are eft with a cean set of data, gathered under stabe operating conditions and iustrative of the actua process performance. This set is then used for estimating the IC distribution of the process measurements. In Phase II, the estimated IC measurement distribution from Phase I data is used, and the major goa of this phase is to detect any shift in the measurement distribution from the IC distribution after an unknown time point. Performance of a Phase II SPC procedure is often measured by the average run ength (ARL), which is the average number of sampes obtained at sequentia time points that are needed for the procedure to signa a shift 3

4 in the measurement distribution. The IC ARL vaue of the procedure is usuay controed at a certain eve, and the procedure performs better if its OC ARL is smaer when detecting a given shift, which is in parae to the type-i and type-ii error probabiities in hypothesis testing. In the iterature, most SPC contro charts are for Phase II process monitoring which is aso the focus of the current paper. Let {X k,k = 1,2,...} be the sequentia, Phase II, univariate, process measurements. Then, the we-known EWMA contro chart is based on the foowing sequence of statistics S k = (1 λ)s k 1 + λx k, for k = 1,2,..., where S 0 = 0 and λ [0,1] is a weighting parameter. It signas a shift at the k-th time point if S k > L, where L is a contro imit chosen to achieve a given IC ARL vaue. Obviousy, S k = λx k + λ(1 λ)x k λ(1 λ) k 1 X 1. So, S k is a weighted average of a observations, more recent observations woud receive more weight in the average, and the weight changes exponentiay over time. 2.2 Monitoring nonparametric profies when design points are arbitrary In this paper, we are concerned about Phase II profie monitoring. At the k-th time point, the profie is assumed to foow the nonparametric mode y kj = g (x kj ) + ε kj, j = 1,...,n k, k = 1,2,..., (1) where {x kj,y kj } n k j=1 are the k-th profie data, x kj is the j-th design point in the k-th profie, g is a smooth nonparametric profie function, and ε kj s are i.i.d. random errors with mean 0 and variance σ 2. Without oss of generaity, we assume that x kj [0,1], for a k and j. In cases when the design points X k = {x k1,x k2,...,x knk } are unchanged from one profie to another, the nonparametric EWMA chart by Zou et a. (2008), which is caed the NEWMA chart hereafter, first averages observed responses y kj s across different profies at each design point and then detects potentia profie shifts using the generaized ikeihood ratio (GLR) test statistic. This idea can not be appied to the current probem because the response is observed at different design points in different profies in the current setup of the probem. A naive modification to Zou et a. s method is to first obtain a nonparametric estimate of g from each profie data, and then predict response vaues using the estimated g at some points {z 1,z 2,...,z n } in the design interva which are unchanged 4

5 from one profie to another. Then, the NEWMA chart can be appied to the predicted response vaues. However, this naive approach, which is caed the NAEWMA chart hereafter, may not be efficient, due to the facts that ony n k observations are used for estimating g in the k-th profie, n k coud be very sma, and thus the predicted response vaues coud have arge bias and variance. As an aternative, in this paper, we consider using the foowing weighted oca ikeihood at any point z [0, 1], which combines the exponentia weighting scheme used in EWMA at different time points (i.e., through the term (1 λ) t k in the expression beow) with the oca inear kerne smoothing procedure (cf., Fan and Gijbes 1996): WL(a,b;z,λ,t) = t n k [y kj a b(x kj z)] 2 K h (x kj z) (1 λ) t k, k=1 j=1 where t is the current time point for profie monitoring, K h ( ) = K( /h)/h, K is a symmetric density kerne function, λ [0,1] is a weighting parameter, and h is a bandwidth. Then, the oca inear kerne estimator of g(z), defined as the soution to a in the minimization probem min a,b WL(a,b;z,λ,t), has the expression ĝ t,h,λ (z) = t n k k=1 j=1 U (t,h,λ) kj (z)y kj / t n k k=1 j=1 U (t,h,λ) kj (z), (2) where [ ] U (t,h,λ) kj (z) = (1 λ) t k K h (x kj z) m (t,h,λ) 2 (z) (x kj z)m (t,h,λ) 1 (z), t m (t,h,λ) n k (z) = (1 λ) t k (x kj z) K h (x kj z), = 0,1,2. (3) k=1 j=1 From the expression of WL(a,b;z,λ,t) above, we can see that this estimator makes use of a avaiabe observations up to the current (i.e., the t-th) time point, and different profies are weighted as in a conventiona EWMA chart (i.e., more recent profies get more weight and the weight changes exponentiay over time). If the process under monitoring is IC up to the t-th time point, then ĝ t,h,λ in (2) shoud be cose to the IC profie function denoted as g 0. Therefore, a charting statistic for profie monitoring can be defined based on the difference between ĝ t,h,λ and g 0. For simpicity, et us first assume that g 0 and the error variance σ 2 are both known. In such cases, a more convenient way to define the charting statistic is to use ξ t,h,λ (z), which is the estimator defined by (2), after y kj are repaced by ξ kj = [y kj g 0 (x kj )]/σ, for a k and j. Then, ξ t,h,λ (z) shoud be uniformy cose to 0 when the 5

6 process is IC up to the t-th time point. A natura charting statistic for profie monitoring is then defined by T t,h,λ = c 0,t,λ [ ξ t,h,λ (z)] 2 Γ 1 (z) dz, where c t0,t 1,λ = a 2 t 0,t 1,λ /b t 0,t 1,λ, a t0,t 1,λ = t 1 k=t 0 +1 (1 λ) t 1 k n k, b t0,t 1,λ = t 1 k=t 0 +1 (1 λ) 2(t 1 k) n k, and Γ 1 is some pre-specified positive density function. In the expression of T t,h,λ, the scae parameter c 0,t,λ is used for unifying its asymptotic variance (see Theorem 1 beow and its proof in the Appendix). In practice, we can use the foowing approximation: T t,h,λ c 0,t,λ n 0 n 0 i=1 ] 2 [ ξt,h,λ (z i ), (4) where z i, for i = 1,...,n 0, are some pre-specified i.i.d. design points from Γ 1. Then, the contro chart triggers a signa if T t,h,λ > L, where L > 0 is a contro imit chosen to achieve a specific IC ARL. Hereafter, this chart is referred to as the nonparametric profie contro (NPC) chart. It shoud be pointed out that it is computationay faster to use points z i rather than the origina design points x kj in approximating the statistic T t,h,λ. As shown in Section 2.5 beow, T t,h,λ can be cacuated in a recursive manner when z i are used in the approximation, and it does not enjoy such a feature when x kj are used. Further, from theoretica properties of T t,h,λ given in Theorem 2 beow and certain empirica resuts presented in Section 3, seection of z i and n 0 has itte effect on the performance of the NPC chart as ong as n 0 is not too sma. See reated discussion in Section 2.7 about practica guideines on seection of certain procedure parameters. As a remark, one may define T t,h,λ aternativey by c 0,t,λ n 0 n 0 i=1 [ĝ t,h,λ (z i ) g 0 (z i )] 2. (5) Namey, we can first compute profie estimators ĝ t,h,λ from the origina data and then construct the contro chart accordingy. It can be shown that (5) and (4) are asymptoticay equivaent under some reguarity conditions given in Appendix A. However, in finite-sampe cases, properties of (5) depend on g 0. As a comparison, formua (4) transforms the testing probem of H 0 : g = g 0 versus H 1 : g g 0 to the one of H 0 : g = 0 versus H 1 : g 0. Therefore, it is invariant to g 0. Its IC distribution and a quantities reated to this distribution do not depend on g 0 either. A direct 6

7 benefit of this property is that the contro imit L can be simpy searched from a process with zero IC profie and unity error standard deviation. Next, we give some asymptotic properties of the charting statistic T t,h,λ which can justify the performance of the NPC chart to a certain degree and shed some ight on practica design of the chart as we. The foowing theorem estabishes the asymptotic nu distribution of T t,h,λ, in which design points x kj s are assumed to be i.i.d. with a density Γ 2 in each IC profie. Theorem 1 Under conditions (C1) (C5) and (C7) given in Appendix A, when the process is IC, we have (T t,h,λ µ h )/ σ h L N(0,1), where µ h = [K(u)] 2 du h Γ1 (x) Γ 2 (x) dx, σ2 h = 2 [K K(u)] 2 du Γ 2 1 (x) h Γ 2 2 (x)dx. The next theorem investigates the asymptotic behavior of T t,h,λ under the OC mode g 0 (x kj ) + ε kj, if 1 k τ y kj = g 1 (x kj ) + ε kj, if k > τ, (6) where τ is an unknown shift time point, and g 1 (x) = g 0 (x) + δ(x) is the unknown OC profie function. Denote ζ δ = 1 [ ] 2 σ 2 δ(u) + h2 η 1 2 δ (u) Γ 1 (u)du, η 1 = ζ 1 = δ 2 (u)γ 1 (u)du, ζ 2 = [δ (u)] 2 Γ 1 (u)du. K(t)t 2 dt, Theorem 2 Under conditions (C1)-(C4), (C5 ), (C6) and (C7) given in Appendix A and the extra condition that ζ 2 < M for some constant M > 0, we have (i) If c 0,t,λ hζ 1 0, then (T t,h,λ µ h c 0,t,λ ζ δ )/ σ h converges in distribution to N(0,1). (ii) If ζ 2 0, then T t,h,λ has a nontrivia power (i.e., the power wi not converge to zero) when δ c 4/9 0,t,λ and h = O(c 2/9 0,t,λ ). From Theorem 2, we notice that the asymptotic power of the test statistic T t,h,λ depends on δ and its second order derivative. The charting statistic of the NEWMA chart has simiar eading 7

8 terms in its asymptotic expression. However, compared to NEWMA, T t,h,λ has the uxury to use a smaer bandwidth in oca inear kerne smoothing because it uses observations from different profies in its smoothing process, as described above. Therefore, the NPC chart based on T t,h,λ woud be more effective when the profie shift has arge curvature (i.e., δ is arge) since sma h woud diminish the effect of h2 η 1 2 δ (u) in the above expression of ζ δ. This expains why we can get a better profie monitoring by using a random design scheme, instead of an equa design scheme, when the curvature of δ is arge. In Section 2.4, we wi discuss how to seect the bandwidth h adaptivey in the NPC chart to accommodate different magnitudes of δ. 2.3 Adaptive seection of the weighting parameter It is we known that optima seection of the weighting parameter λ used in EWMA charts depends on the target shift: sma λ woud be effective for detecting sma shifts and arge λ is effective for detecting arge shifts. So, an EWMA chart with a given λ cannot have a neary minimum ARL for both sma and arge shifts (cf., e.g., Lucas and Saccucci 1990). This assertion is aso vaid for our proposed contro chart by noting the foowing resut. Proposition 1 Under conditions (C1)-(C4) given in Appendix A and the extra conditions that h 0, n k, n 0 h 3 2, n k h 3, and n k h 5 0, if a2 τ,t,λ b 0,t,λ hζ 1 0 and ζ 2 < M for some constant M > 0, we have T t,h,λ D µh + h 1 2 w + a 2 τ,t,λ b 0,t,λ ζ δ, where D denotes asymptotic equaity in distribution, and w is a norma random variabe with mean zero and variance σ 2 h. The proof of this proposition is anaogous to that of Theorem 2 given in Appendix B. So, it is omitted. For simpicity, et us discuss the case when n k = n. In such a case, from expressions of a τ,t,λ and b 0,t,λ, we have a 2 τ,t,λ b 0,t,λ = 2 λ λ[1 (1 λ) 2t ] [1 (1 λ)t τ ] 2. By Proposition 1 and the above expression, intuitivey, if ζ δ is sma, then it woud require a arge vaue of t τ to signa and this aso depends heaviy on the factor (2 λ)/λ. When λ is chosen smaer, (2 λ)/λ becomes arger. Consequenty, the sma shift woud be detected quicker. On the 8

9 other hand, if ζ δ is arge, then we can expect the run ength t τ to be reativey sma as ong as λ is chosen reativey arge. If λ is chosen sma in such a case, then 1 (1 λ) t τ woud approach 1 too sow to detect shifts effectivey. Motivated by the AEWMA chart suggested by Capizzi and Masarotto (2003), in this section we suggest an adaptive procedure for choosing the weighting parameter λ. The underying idea is to adapt weights used for past profies to the goodness-of-fit of the current profie, so that the reated chart can detect shifts of different sizes in a more efficient way. To be specific, et ψ be a score function used for determining the adaptive weights. Capizzi and Masarotto (2003) propose severa candidates for ψ. For simpicity, we suggest using the foowing one: 1 (1 λ 0 ) 0 /u, if u 0 ψ 0,λ 0 (u) = λ 0, if u < 0, where 0 < λ 0 1 and 0 > 0 are two parameters, λ 0 defines the minimum weight, and 0 is used for baancing detection abiity of the contro chart for arge and sma shifts. Apparenty, a arge (sma) 0 woud generate a sma (arge) adaptive weight, making the contro chart more sensitive to sma (arge) shifts. Further discussion on seection of λ 0 and 0 wi be given in Section 2.7. Then, the NPC chart with the adaptive weight, denoted as NPC-W, signas when T t,h,ψ0,λ 0 (T t,h ) > L, (7) where the contro imit L > 0 is chosen to achieve a specific IC ARL, and T t,h is defined in the same way as T t,h,λ except that ony the current profie data {(x tj,y tj ),j = 1,...,n t } are used here. It is easy to check that T t,h is actuay T t,h,1; it is therefore easy to compute. So, the NPC-W chart essentiay combines the EWMA and Shewhart procedures in a natura way. It is worth mentioning that impementation of this adaptive contro chart doesn t require much extra computationa effort, compared to that of the NPC chart, because recursive formuas given in Section 2.5 for computing T t,h,λ ony require nonparametric regression of individua profies. From numerica exampes in Section 3, we can see that, after choosing λ 0 and 0 propery, the NPC-W chart provides we baanced protection against various shifts. 2.4 Adaptive seection of the bandwidth parameter Like many other smoothing-based tests, performance of the NPC chart depends on seection of the bandwidth parameter h. Optima seection of h remains an open probem in this area, and it is 9

10 widey recognized that optima h for nonparametric curve estimation is generay not optima for testing (cf., e.g., Hart 1997). A uniformy most powerfu test usuay does not exist due to the fact that nonparametric regression functions have infinite dimensions. But the term h2 η 1 2 δ ( ) in Theorem 2 tes us that appropriate seection of h woud improve the testing power. Intuitivey, a smaer h woud be more effective in detecting shifts with arge curvature (i.e., arge δ ), and a arger h woud perform better when shifts are fat or smooth (i.e., sma δ ). This intuition motivates us to use an adaptive seection procedure, briefy described beow. For the ack-of-fit testing probem, Horowitz and Spokoiny (2001) suggested choosing a singe h based on the maximum of a studentized conditiona moment test statistic over a sequence of smoothing parameters, and proved that the resuting test woud have certain optimaity properties. Because this method is easy to use and has good performance in various cases, we use it here for choosing h. Let H be a set of admissibe smoothing parameter vaues defined to be the foowing geometric grid: H = {h j = h max γ j : h j h min, j = 0,...,J n }, (8) where 0 < h min < h max are the ower and upper bounds, and γ > 1 is a parameter. Ceary, the number of vaues in H is J n og γ (h max /h min ). Then, the charting statistic of the NPC chart with adaptive bandwidth, denoted as NPC-B, becomes T t,h,λ µ h T t,h,λ = max, (9) h H σ h where µ h and σ 2 h are respectivey the asymptotic expectation and variance of T t,h,λ, defined in Theorem 1. The next proposition estabishes the consistency of T t,h,λ against smooth aternatives. Proposition 2 Under conditions (C1)-(C4), (C6), and the extra conditions that h min and h max both satisfy condition (C5), ζ 1 > M 1 (c 1 0,t,λ n n c 0,t,λ) 8 9, and ζ 2 < M 2, where M 1 and M 2 are two positive constants, then T t,h,λ is consistent under mode (6) in the sense that its power converges to 1 when t increases. From the proof of this proposition given in Appendix B, we can see that T t,h,λ woud automaticay maximize the asymptotic power function c 0,t,λ h2ζ 1 δ. Thus, it adapts to different magnitudes of δ ; consequenty, Tt,H,λ woud be more robust to various potentia shifts, compared to T t,h,λ with a given bandwidth. 10

11 2.5 Some computationa issues Athough computing power gets improved fast and it is computationay trivia to do nonparametric function estimation for individua profies, for on-ine process monitoring, which generay handes a arge amount of profies, fast impementation is important and some computationa issues are worth our carefu examination. For the proposed charts, computation of the test statistic T t,h,λ might be time-consuming, and it requires a substantia amount of storage for past profie observations as we. In this part, we provide updating formuas for computing the charting statistic, which wi greaty simpify the computation and essen the storage requirement. Let m (t,h) (z) = q (t,h) (z) = n k (x tj z) K h (x tj z), = 0,1,2, j=1 n k (x tj z) K h (x tj z)y tj, = 0,1. j=1 Then, m (t,h,λ) (z) in (3) can be recursivey updated by m (t,h,λ) (z) = (1 λ)m (t 1,h,λ) (z) + m (t,h) (z), = 0,1,2, where m (0,h,λ) by the recursive formua (z i ) = 0, for = 0,1,2. Let q (t,h,λ) (z), for = 0,1, be two working functions defined q (t,h,λ) (z) = (1 λ)q (t 1,h,λ) (z) + q (t,h) (z), = 0,1, where q (0,h,λ) (z) = 0, for = 0,1. Then, we have [ { ĝ t,h,λ (z) = M (t,h,λ)] 1 [ ] (1 λ) 2 M (t 1,h,λ) ĝ t 1,h,λ + q (t,h) 0 m (t,h,λ) 2 q (t,h) 1 m (t,h,λ) 1 [ ] } + (1 λ) q (t 1,h,λ) 0 m (t,h) 2 q (t 1,h,λ) 1 m (t,h) 1, (10) where M (t,h,λ) (z) = m (t,h,λ) 2 (z)m (t,h,λ) 0 (z) [m (t,h,λ) 0 (z)] 2. On the right hand side of the above equation, dependence on z in each function is not made expicit in notation for simpicity, which shoud not cause any confusion. Using the above updating formuas, impementation of the NPC chart can be briefy described as foows. At time point t, we first compute quantities m (t,h) (z), for = 0,1,2, and q (t,h) (z), for = 0,1, at n 0 pre-determined z ocations (see reated discussion in Sections 2.2 and 2.7 about seection of n 0 and {z i,i = 1,2,...,n 0 }). Then, m (0,h,λ) (z i ), for = 0,1,2, and q (0,h,λ) (z i ), for 11

12 = 0,1, are updated by the above formuas. Finay, ĝ t,h,λ (z) is computed from (10), and the test statistic T t,h,λ is computed from ĝ t,h,λ (z), after repacing y kj by ξ kj. This agorithm ony requires O(n 0 n k h) operations for monitoring each profie, which is in the same order as the computation invoved in conventiona oca inear kerne smoothing. If n k and n 0 are both arge, we coud further decrease the computation to the order of O(n k h), using the updating agorithm proposed by Seifert et a. (1994). See Fan and Marron (1994) for a simiar agorithm. Ceary, using the proposed updating formuas, required computer storage does not grow with time t. In addition, compared to the NPC chart with fixed weight and bandwidth parameters, impementation of the NPC-W chart does not require much extra computationa effort, and impementation of the NPC-B chart requires J n times both computationa effort and computer storage. 2.6 A sef-starting version The NPC chart makes expicit use of the IC regression function g 0 and the error variance σ 2 (cf., mode (1)). In practice, both g 0 and σ 2 might be unknown. In such cases, they need to be estimated from an IC data set. If such IC data are of sma to moderate size, then there woud be considerabe uncertainty in the estimates, which in turn woud distort the IC run ength distribution of the contro chart. Even if the contro imit of the chart is adjusted propery to attain a desired IC run ength behavior, its OC run ength woud sti be severey compromised (cf., e.g., Jones 2002). To avoid such probems, a arge and thus costy coection of IC profie sampes woud be necessary (see Jensen et a for reated discussion). Zou et a. (2008) provide a genera guideine on how many IC profie sampes are necessary to obtain good run ength behavior for the NEWMA chart, according to which at east forty IC profie sampes with more than fifty observations in each profie sampe are required to obtain satisfactory resuts in various cases. In this section, we present a sef-starting version of the NPC chart, which can substantiay reduce the required IC profie sampes. The basic idea of the sef-starting version is to repace g 0 and σ 2, both of which are used in defining ξ kj, with some appropriate estimators constructed from past profie data. If the chart does not give a signa of profie shift at time point t, then g 0 (x) can be estimated by the conventiona oca inear kerne estimator constructed from t historica profie sampes, which is denoted as ĝ (t) 0 (x). 12

13 The variance σ 2 can be estimated recursivey by { t 1 n t σ t 2 = n k σ t }/ t [y tj ĝ (t 1) 0 (x tj )] 2 n k. k=1 k=1 j=1 Then, the sef-starting version, denoted as NPC-S, is the contro chart based on the charting statistic T t,h,λ, which is constructed in the same way as T t,h,λ (cf., (4)) except that ξ kj need to be repaced by ξ kj = [y kj ĝ (k 1) 0 (x kj )]/ σ k 1, for a k and j. It is worth mentioning that, in practice, it is not necessary to update ĝ (t) 0 (x kj) and σ 2 t after t is arge enough. As a matter of fact, it is straightforward to show that, when the process is IC, where N t = t k=1 ĝ (t) 0 (x) = g 0(x) + O p ((N t h) 1 2) + O(h 4 ), σ 2 t = σ2 (1 + O p (N 1 2 t ) + O p ((N t h) 1 ) n k. Thus, when t is sufficienty arge, say t t 0, the approximations of ĝ (t) 0 (x) and σ 2 t to g 0 and σ 2 woud be good enough, and we coud simpy use ĝ (t 0) 0 (x) and σ 2 t 0 for a profies with t t 0 in process monitoring. There are two benefits with this modification. First, it reduces much computation and storage requirement with very itte oss of efficiency. Second, it may reduce the masking-effect (cf., Hawkins 1987) to certain extent. That is, when the potentia shift occurs after time t 0, the estimates ĝ (t 0) 0 (x) and σ 2 t 0 woud not be contaminated by the OC observations, which is not the case if these estimates are updated at every time point. From the description in this and previous two subsections, it can be seen that the NPC-S chart can accommodate adaptive seection of the weight and bandwidth parameters. The resuting chart, denoted as NPC-SWB, shoud be abe to offer a baanced protection against shifts of different magnitudes and adapt to the smoothness of the IC and OC profie functions as we. Formuation of the NPC-SWB chart can be readiy obtained by incorporating (7) and (9) into T t,h,λ ; hence, it is not eaborated here. Its performance wi be investigated in Section 3. ), 2.7 Certain practica guideines On choosing n k and x kj : In certain appications, design points x kj are determined by the industria process itsef, and we can not do much in choosing them. In some others (cf., Zou et a. 2008), they need to be specified before process monitoring. As demonstrated by Theorem 2, 13

14 random design has some benefits, compared to the design in which different profies share the same design points, because observations from different profies woud provide information about more detais of the regression function g in the former case. So, if we can choose the design points, then random design woud be a good choice. This amounts to determining a proper design distribution Γ 2, from which design points x kj are generated for individua profies. The number of design points n k can aso be random, athough in many appications n k = n woud be the most convenient scheme to use. The vaue of n can be chosen smaer than the one used in the NEWMA chart by Zou et a. (2008), because, in computing T t,h,λ, roughy c 0,t,λ = 2 λ λ n observations are actuay used. On choosing n 0 and z i s: Based on our numerica experience, seection of n 0 and z i s does not affect the performance of the NPC chart much, as ong as n 0 is not too sma and z i s cover a the key parts (e.g., peaks/vaeys or osciating regions) of g 0 we. In our numerica exampes presented in Section 3, we find that resuts do not change much when n On choosing 0 and λ 0 used in the NPC-W chart: As described in Section 2.3, λ 0 is the minimum weight used by the chart NPC-W, and 0 is the parameter that contros the chance for the chart to use that minimum weight. The chart woud use the minimum weight λ 0 if T t,h,1 < 0. We suggest using the upper α 0 percentie of T t,h,1 as the vaue of 0, which coud be obtained by simuation before profie monitoring. An appropriate method for determining α 0 is to set α 0 = c/arl 0, where c > 1 is a constant and ARL 0 is the desired IC ARL vaue. Since the reciproca of ARL 0 can be regarded as a rough estimate of the fase aarm probabiity, it woud be reasonabe to choose α 0 to be c/arl 0 so that the chart NPC-W can achieve the given IC ARL vaue. With respect to λ 0, it shoud be chosen smaer than the commony used vaue 0.2 in the EWMA iterature (cf., Lucas and Saccucci 1990). Based on our numerica experience and the resuts in Capizzi and Masarotto (2003), we recommend using λ 0 [0.05,0.1] and 5 c 15. On choosing H used in the NPC-B chart: Theoreticay speaking, the parameters γ, J n, h max and h min shoud satisfy certain conditions to obtain the corresponding asymptotic resuts. See Appendix A for reated discussion. Based on our simuations, we notice that the proposed contro chart is actuay quite robust to them, which is consistent with the findings in Horowitz and Spokoiny (2001). By both theoretica arguments and numerica studies, we recommend using the choices that 1 < γ < 2, J n coud be 4, 5 or 6, h max = Mc 1/7 0,t,λ, and h j = h max γ j, for j = 1,...,J n, where 0.5 M 2 is a constant. Note that the recommended vaue h max = Mc 1/7 0,t,λ due to condition (C5) given in Appendix A. is partiay 14

15 On the NPC-S chart: As suggested by Hawkins et a. (2003), we recommend coecting three to ten IC profie sampes before using the NPC-S chart. These preiminary profie sampes are mainy for stabiizing the variation of T t,h,λ. 3 A Simuation Study We present some simuation resuts in this section regarding the numerica performance of the proposed NPC chart. Throughout this section, the kerne function is chosen to be the Epanechnikov kerne function K(x) = 0.75(1 x 2 )I( 1 x 1). The IC ARL is fixed at 200. The error distribution is assumed to be Norma. For simpicity, we assume that n k = n = 20 for a k, x kj Uniform(0,1), for j = 1,...,n, z i = (i 0.5)/n 0, for i = 1,...,n 0, and n 0 = 40. A the ARL resuts in this section are obtained from 50,000 repications uness indicated otherwise. In addition, we focus on the steady-state OC ARL behavior of each chart (cf., Hawkins and Owe 1998), and assume that τ = 30 (cf., the OC mode (6)). When computing the ARL vaues, any simuation runs in which a signa occurs before the (τ + 1)-th profie are discarded. As a side note, our numerica resuts (not reported here to save some space) show that steady-states of the reated charts considered in this section are reached when τ is as sma as 10 in a cases considered. To compare the NPC chart with aternative methods turns out to be difficut, due to ack of an obvious comparabe method. One possibe aternative method is the NEWMA chart proposed by Zou et a. (2008) in which design points are assumed to be equay spaced in each profie and they are unchanged from profie to profie. To make the procedures comparabe, for the NEWMA chart, we assume that design points are (i 0.5)/n, for i = 1,...,n, in each profie sampe. Another possibe method to compare is the naive modification of the NEWMA chart described in Section 2.2, which is caed the NAEWMA chart beow. By the NAEWMA chart, profie functions are first estimated from individua profie data, then response vaues are predicted from these estimated profie functions at some common points {z 1,z 2,...,z n } in the design interva for different profies, and finay the NEWMA chart is appied to the predicted response vaues. For this chart, we take z i = (i 0.5)/n, for i = 1,...,n, as in the NEWMA chart. To appreciate the benefits of different versions of the NPC chart, we investigate numerica performance of the NPC, NPC-W, NPC-B, NPC-S, and NPC-WBS charts separatey. Note that, for charts NEWMA and NAEWMA, the bandwidth h and the weighting parameter λ shoud both be 15

16 pre-specified. Therefore, we first compare the charts NPC, NEWMA and NAEWMA in such cases. Foowing the recommendations by Zou et a. (2008), h is chosen to be either h 1 = 1.5n 1/5 Var(x) or h 2 = 1.5[n(2 λ)/λ] 1/5 Var(x). The smaer bandwidth h 2 is considered here because the actua number of observations used in the NPC chart at each time point is c 0,t,λ which is roughy n(2 λ)/λ. In a three charts, λ is chosen to be 0.1 or 0.2. The IC mode used is g 0 (x) = 1 exp( x), and the foowing two representative OC modes are considered here: (I) : g 1 (x) = 1 exp( x) + θx; (II) : g 1 (x) = 1 exp( x) + θ sin(2π(x 0.5)). In case (I), g 1 (x) g 0 (x) = θx is a straight ine; and g 1 (x) g 0 (x) = θ sin(2π(x 0.5)) osciates a ot in case (II). Tabe 1 presents the OC ARL vaues of the three charts in various cases. Their contro imits L are aso incuded in the tabe. From the tabe, we can have the foowing resuts. First, in case (I), arger h (i.e., h 1 ) yieds better performance for both NPC and NEWMA charts, because δ(x) = g 1 (x) g 0 (x) is straight in this case. For a given bandwidth, the NPC chart outperforms the NEWMA chart uniformy. This is because the effective number of observations used in the NPC chart at each time point is arger than that used in the NEWMA chart, and consequenty the charting statistic of the NPC chart woud converge to c 0,t,λ ζ 1 faster (cf., Theorem 1). Second, in case (II) where δ(x) osciates a ot, the NPC chart with the smaer h has better performance, which is intuitivey reasonabe. However, this is not the case for the NEWMA chart, because smaer bandwidth in the NEWMA chart woud resut in arge bias in estimating the regression function and thus reduce its abiity in detecting profie shift. This exampe confirms the fact that the NPC chart aows us to use a smaer bandwidth to better detect osciating profie shifts (cf., reated discussion at the end of Section 2.2). Third, in case (II), the NPC chart outperforms the NEWMA chart uniformy. Fourth, the NPC chart outperforms the NAEWMA chart by a quite arge margin in most cases, and the NAEWMA chart aso performs uniformy worse than the NEWMA chart. These resuts confirm our argument made in Section 2.2 that the naive chart NAEWMA woud be inefficient because profie estimates constructed from individua profie data woud have reativey arge bias and variance. By the way, our simuations (not reported here) show that, when n 0 is chosen arger than 40, performance of the NPC chart woud not change much. Next we consider the NPC-W chart in which the weight parameter λ is adaptivey chosen so that the chart woud be robust to shift size. Its other parameters are chosen according to the practica guideines given in Section 2.7. More specificay, we choose λ 0 = 0.1 and α 0 =

17 Tabe 1: OC ARL comparison of the NPC, NEWMA, and NAEWMA charts when IC ARL=200, λ = 0.1 or 0.2, and n = 20. θ NPC NEWMA NAEWMA h 1 h 2 h 1 h 2 h 1 h 2 λ = (.357) 87.4 (.486) 89.5 (.393) 102 (.452) 101 (.455) 103 (.464) (.106) 33.1 (.152) 32.7 (.118) 38.5 (.142) 39.7 (.154) 41.0 (.160) (.046) 17.4 (.066) 17.2 (.050) 19.7 (.059) 20.5 (.066) 20.9 (.067) (.026) 11.4 (.037) 11.2 (.027) 12.6 (.031) 13.3 (.036) 13.6 (.037) (I) (.013) 6.68 (.018) 6.64 (.013) 7.33 (.015) 7.64 (.016) 7.76 (.017) (.008) 4.80 (.011) 4.74 (.008) 5.19 (.009) 5.38 (.010) 5.44 (.010) (.005) 3.12 (.007) 3.12 (.005) 3.37 (.005) 3.46 (.006) 3.50 (.006) (.004) 2.37 (.004) 2.38 (.004) 2.56 (.004) 2.62 (.004) 2.65 (.004) (.313) 65.3 (.343) 72.4 (.316) 81.6 (.348) 94.2 (.419) 95.6 (.424 ) (.082) 22.2 (.090) 24.5 (.080) 27.6 (.092) 31.7 (.112) 32.2 (.114 ) (.035) 12.0 (.039) 13.0 (.034) 14.3 (.037) 16.0 (.044) 16.2 (.045 ) (II) (.020) 8.03 (.022) 8.76 (.019) 9.53 (.021) 10.5 (.024) 10.5 (.024 ) (.010) 4.96 (.012) 5.32 (.009) 5.75 (.010) 6.15 (.011) 6.20 (.011 ) (.007) 3.65 (.008) 3.89 (.006) 4.16 (.007) 4.42 (.008) 4.43 (.008 ) (.004) 2.42 (.005) 2.62 (.004) 2.78 (.004) 2.94 (.004) 2.95 (.004 ) (.003) 1.90 (.003) 2.04 (.003) 2.15 (.003) 2.25 (.003) 2.26 (.003 ) L λ = (.452) 108 (.501) 112 (.519) 125 (.595) 132 (.625) 135 (.648) (.148) 41.0 (.174) 43.2 (.188) 51.9 (.228) 60.0 (.269) 62.2 (.281) (.063) 19.6 (.076) 20.1 (.076) 23.7 (.089) 28.4 (.116) 29.6 (.120) (.031) 11.7 (.036) 11.7 (.036) 13.4 (.045) 16.1 (.056) 16.6 (.058) (I) (.013) 6.05 (.013) 6.03 (.013) 6.64 (.018) 7.69 (.020) 7.94 (.022) (.009) 4.09 (.009) 4.09 (.009) 4.44 (.009) 4.98 (.011) 5.15 (.011) (.004) 2.55 (.004) 2.56 (.004) 2.74 (.004) 3.00 (.005) 3.04 (.005) (.003) 1.93 (.003) 1.94 (.003) 2.06 (.003) 2.21 (.003) 2.26 (.003) (.461) 85.7 (.407) 93.9 (.429) 105 (.479) 133 (.630) 135 (.639) (.130) 27.6 (.112) 31.3 (.125) 35.4 (.148) 48.6 (.235) 55.3 (.242) (.049) 13.0 (.045) 14.2 (.049) 16.0 (.054) 22.5 (.081) 23.2 (.086) (II) (.022) 7.91 (.022) 8.57 (.022) 9.38 (.027) 12.3 (.036) 12.5 (.038) (.009) 4.37 (.009) 4.68 (.009) 5.00 (.009) 5.98 (.012) 6.05 (.013) (.004) 3.07 (.004) 3.26 (.004) 3.46 (.004) 4.00 (.007) 4.04 (.007) (.004) 2.01 (.004) 2.14 (.004) 2.25 (.004) 2.50 (.004) 2.53 (.004) (.003) 1.56 (.003) 1.66 (.003) 1.74 (.003) 1.91 (.003) 1.92 (.003) L NOTE: Standard errors are in parentheses. 17

18 Figure 1: OC ARL comparison of the NPC-W chart and three NPC charts with λ = 0.1,0.2 and 0.4. Since we are mainy concerned about the robustness of the NPC-W chart to shift size, ony the OC mode (I) is considered here. The bandwidth h is chosen to be h 1, because it is more appropriate to use in this case than h 2, by Tabe 1. For comparison purpose, the OC ARL vaues of three NPC charts when λ = 0.1,0.2 and 0.4 are aso computed. To measure robustness of a chart T to shift size, the reative mean index (RMI) originay proposed by Han and Tsung (2006) is used, which is defined by RMI(T) = 1 m m i=1 ARL θi (T) MARL θi MARL θi, where ARL θi (T) is the OC ARL of T for detecting a shift of size θ i, and MARL θi is the smaest vaue among such OC ARL vaues of a charts considered. In this exampe, θ i ranges from 0.1 to 2 with a step 0.1. Obviousy, sma RMI(T) impies that T has a robust performance in detecting shifts of various sizes. Figure 1 shows the OC ARL vaues (in og scae) and the RMI vaues of the four charts considered. It can be seen that performance of the three NPC charts depends heaviy on their pre-specified λ vaues, as expected, and the NPC-W chart offers a baanced protection against various shift sizes. In terms of RMI, the NPC-W chart performs the best. After taking into account its convenient impementation, we think that the NPC-W chart is a vauabe improvement of the NPC chart. 18

19 Tabe 2: OC ARL comparison of the NPC and NPC-B charts when IC ARL=200, λ = 0.2 and n = 20. θ NPC-B NPC h = 0.6 h = 0.3 h = (.031) 7.84 (.022) 8.27 (.031) 10.8 (.036) (.031) 8.95 (.027) 9.45 (.031) 12.2 (.040) (.036) 11.4 (.036) 10.7 (.036) 13.9 (.040) (.054) 16.2 (.058) 14.1 (.049) 15.9 (.058) (.112) 86.3 (.398) 35.0 (.125) 26.9 (.103) (.098) 49.0 (.224) 50.6 (.224) 24.4 (.094) (.286) 166 (.814) 174 (.832) 48.0 (.206) (.344) 120 (.577) 125 (.581) 81.4 (.376) L NOTE: Standard errors are in parentheses. Next, we consider the NPC-B chart. By the guideines in Section 2.7, we choose γ = 1.4, h max = 1.0[(2 λ/λ)n] 1/7, and h j = h max γ j, for j = 1,...,4. We consider the foowing OC mode g 1 (x) = 1 exp( x) cos(θπ(x 0.5)). By changing θ, this mode can cover various cases with different smoothness of δ( ). For comparison purposes, we aso consider three NPC charts with bandwidth 0.6, 0.3, and 0.15, respectivey. Their other parameters are chosen as in the exampe of Tabe 1. OC ARL vaues of reated charts are shown in Tabe 2, with their contro imits L isted in the bottom ine. From the tabe, it can be seen that the NPC chart with a fixed bandwidth outperforms the NPC-B chart in certain ranges of θ, but they can aso be much worse in other ranges of θ. As a comparison, the NPC-B chart is aways cose to the best chart in a cases, because it can adapt to the unknown smoothness of δ(x) and pick up an appropriate bandwidth accordingy from H. We now investigate the numerica performance of the NPC-S chart. First, we study its IC run ength distribution. As recognized in the iterature, it is often insufficient to summarize run ength behavior by ARL, especiay for sef-starting contro charts (cf., Jones 2002). As an aternative, here we use the hazard function H 1 (r)/h 2 (r) recommended by Hawkins and Maboudou-Tchao (2007), where H 1 (r) is the probabiity that the run ength equas r and H 2 (r) is the probabiity that the run ength equas r or a arger number. Note that, if the run ength foows a geometric distribution, then the corresponding hazard is a constant whose inverse is the ARL. In the exampe of Tabe 1 when h = h 1 and L = 10.47, which corresponds to an IC ARL of 200 when the IC mode is 19

20 assumed known, the IC hazard function of the NPC-S chart based on 250,000 repications is shown in Figure 2. When computing the IC hazard function, we foow the suggestion given in Section 2.7 that process monitoring starts after five IC profies are coected beforehand. From the pot, we can see that the IC hazard starts around , then drops quicky to vaues around = 1/200, and gets stabiized at that eve for good. This pot shows that, except for short run-engths, the geometric distribution is an exceent fit to the IC run ength of the NPC-S chart, which is consistent with the findings in Hawkins and Maboudou-Tchao (2007) about a sef-starting chart for monitoring mutivariate Norma processes. Furthermore, sampe mean and sampe standard deviation of the run engths are 196 and 194, respectivey, which are amost identica and which further confirms that the NPC-S chart works we under the IC condition. We conducted some other simuations with various combinations of n, h and Γ 2 to check whether the above concusions are true in other settings. These simuation resuts, not reported here but avaiabe from the authors, show that the NPC-S chart has quite satisfactory performance in other cases as we, except certain extreme cases such as the ones when n is too sma (e.g., n 5). Figure 2: Hazard curve of the NPC-S chart. Next, we examine the OC performance of the NPC-S chart. As demonstrated in the iterature, OC performance of sef-starting charts is generay affected by the shift time point (cf., e.g., Hawkins et a. 2003). In this exampe, we consider two shift times τ = 40 and τ = 80. The simuation 20

21 Tabe 3: OC ARL performance of the NPC-S and NPC-SWB charts when IC ARL=200, λ = 0.2 and n = 20. NPC-S NPC-SWB θ τ = 40 τ = 80 τ = 40 τ = 80 h 1 h 2 h 1 h (.903) 176 (.939) 143 (.818) 161 (.899) 151 (.859) 132 (.859) (.738) 116 (.778) 68.2 (.505) 88.9 (.626) 82.5 (.626) 54.7 (.474) (.456) 60.4 (.523) 24.9 (.192) 34.5 (.268) 32.2 (.313) 19.6 (.130) (.188) 26.3 (.291) 11.7 (.054) 15.0 (.085) 14.0 (.098) 10.8 (.040) (I) (.018) 7.25 (.027) 5.66 (.018) 6.63 (.018) 6.20 (.018) 5.87 (.018) (.009) 4.37 (.009) 3.77 (.009) 4.22 (.009) 4.14 (.009) 3.95 (.009) (.004) 2.64 (.004) 2.38 (.004) 2.62 (.004) 2.38 (.004) 2.35 (.004) (.004) 1.95 (.004) 1.81 (.004) 1.96 (.004) 1.64 (.004) 1.64 (.004) (.872) 161 (.926) 146 (.823) 142 (.814) 150 (.836) 132 (.827) (.581) 82.2 (.631) 55.2 (.402) 54.5 (.398) 68.0 (.483) 48.1 (.367) (.206) 26.5 (.237) 18.3 (.103) 17.6 (.103) 22.9 (.165) 17.2 (.080) (II) (.063) 10.6 (.067) 9.44 (.036) 8.86 (.031) 10.7 (.040) 9.75 (.031) (.013) 4.68 (.013) 4.77 (.009) 4.56 (.009) 5.34 (.013) 5.10 (.013) (.004) 3.19 (.004) 3.24 (.004) 3.15 (.004) 3.62 (.009) 3.58 (.009) (.004) 2.05 (.004) 2.09 (.004) 2.03 (.004) 2.11 (.004) 2.10 (.004) (.004) 1.56 (.004) 1.61 (.004) 1.56 (.004) 1.43 (.004) 1.42 (.004) L NOTE: Standard errors are in parentheses. resuts in various cases considered in Tabe 1 are presented in Tabe 3. From the tabe, it can be seen that the NPC-S chart performs amost equay we for both vaues of τ when the shift size is arge. For detecting sma to moderate shifts, it generay performs better with a arger τ, because the updated parameter estimates woud be more accurate in such a case under the IC condition, which is confirmed by the tabe. As a comparison, in Tabe 3, we aso present the OC ARLs of the NPC-SWB chart, which is a combination of the sef-starting chart and adaptive seection of the weight and bandwidth parameters. Its parameters are chosen to be the same as those used in the exampes of Figure 1 and Tabe 2. From the tabe, we can see that the NPC-SWB chart outperforms both NPC-S charts using h 1 and h 2 in a cases except certain cases with moderate shifts. Thus, in practice, the NPC-SWB chart is recommended, if the extra computation invoved is not a major concern (cf., the ast paragraph of Section 2.5 for reated discussion about computation). 21

22 4 A Rea-Data Appication In this section, we demonstrate the proposed NPC chart by appying it to a dataset obtained from the semiconductor manufacturing industry for monitoring a deep reactive ion etching (DRIE) process which is critica to the output wafer quaity and requires carefu contro and monitoring. In the DRIE process, the desired profie is the one with smooth and straight sidewas and fat bottoms, and ideay the sidewas of a trench are perpendicuar to the bottom of the trench with a certain degree of smoothness around the corners (cf., the midde shape shown in Figure 3). Various other profie shapes, such as positive and negative ones (cf., the two eft-hand-side and two right-hand-side shapes shown in Figure 3) due to underetching and overetching, are considered to be unacceptabe. More detaied discussion about the DRIE exampe can be found in Rauf et a. (2002) and Zhou et a. (2004). Negative Positive Figure 3: Iustrations of various etching profies from a DRIE process. The DRIE data considered here have 21 profies. The origina data incude their images, ike the ones shown in Figure 3. To monitor the DRIE process, we need to obtain sampes from individua profies, which can be acquired by the scanning eectron microscope (SEM). Since the profies are usuay symmetric, we can focus on one haf of each profie (e.g., the eft haf) for profie monitoring purposes. To make that part of the profie convenient to describe by a mathematica function, it is rotated by 45 degrees aong a reference point in a pre-specified coordinate system, before dimensiona readings of the profie are coected by SEM. Among the 21 profies, based on engineering knowedge, the first 18 profies are IC and the remaining 3 profies are OC. Since the number of IC profies is not arge, the IC profie function g 0 and the error standard deviation σ may not be accuratey estimated from the IC data. Therefore, we consider using the sef-starting chart NPC-S in this exampe. As pointed out at the end of Section 2.2 and confirmed numericay in the exampe of Tabe 1, for appications such as the current one, a better profie monitoring can be 22

23 achieved by using a random design scheme, instead of a fixed design scheme for a profies. Next, we iustrate how to design a reasonabe random design scheme in this exampe and how to appy the NPC-S chart to the resuting data. To design a random design scheme, we first need to specify the density Γ 1 of the design points. In this exampe, it seems that the corner part of the profie contains critica information regarding whether the profie is OC; it shoud receive enough attention. A reasonabe design distribution that takes this into account is x Norma(0,2.5), where the origina point of x is ocated at the center of the corner. This random design ensures that most design points fa within [-4,4] that covers haf of the bottom trench, and about 65% design points are ocated in [-1.5,1.5] that covers the corner part. For each profie, we fix n = 20, and dimensiona readings are coected by SEM at n design points generated from Γ 1. Using eectronic sensor and information technoogies, this entire data acquisition process can be finished automaticay by a computer. In the NPC-S chart, we fix the IC ARL at 200, n 0 at 40, and z i s to be equay spaced in [-3.5,3.5]. A other parameters of the NPC-S chart are chosen to be the same as those used in the exampe of Tabe 3. The contro imit is computed to be L = by simuation. Foowing the practica guideines given in Section 2.7, we take the first 10 IC profies as preiminary data, and profie monitoring starts at the 11th profie. The charting statistic T t,h,λ, for t = 11,...,21, is shown in Figure 4, aong with its contro imit shown by the soid horizonta ine. In that figure, we aso present the NAEWMA chart and its contro imit by the dashed ines. Parameters of the NAEWMA chart are chosen to be λ = 0.2, IC ARL=200, n = 20, and z 1,...z n are equay spaced in [ 3.5,3.5]. From the pot, it can be seen that the NPC-S chart gives a signa of profie shift at the 20th time point which corresponds to the 2nd OC profie. The NAEWMA chart does not give any signa, even after the 3rd OC profie is coected. As a side note, it takes about 3.4/1000 seconds to compute a vaues of the charting statistic T t,h,λ that are potted in Figure 4, by a Pentium 2.4MHz CPU. Therefore, the proposed procedure shoud be quite convenient to use for on-ine automatic profie monitoring. 5 Summary and Concuding Remarks In this paper, we propose a contro chart for monitoring nonparametric profies with arbitrary design. Our proposed contro chart effectivey combines the EWMA contro chart and a nonpara- 23