Autocorrelated SPC for Non-Normal Situations. Clear Water Bay, Kowloon, Hong Kong

Transcription

1 QUALITY AND RELIABILITY ENGINEERING INTERNATIONAL Qual. Reliab. Engng. Int. 2005; 21: Published online 27 January 2005 in Wiley InterScience ( DOI: /qre.612 Research Autocorrelated SPC for Non-Normal Situations Philippe Castagliola 1 and Fugee Tsung 2,, 1 Department of Automatic Control and Production Systems, École des Mines de Nantes & IRCCyN, Nantes, France 2 Department of Industrial Engineering and Engineering Management, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong The importance of statistical process control (SPC) techniques in quality improvement is well recognized in industry. However, most conventional SPC techniques have been developed under the assumption of independent, identically and normally distributed observations. With advances in sensing and data capturing technologies, large volumes of data are being routinely collected from individual units in manufacturing industries. These data are often autocorrelated and skewed. Conventional SPC techniques can lead to false alarms or other types of poor performance monitoring of such data. There is a great need for process control techniques for variation reduction in these environments. Much recent research has focused on the development of appropriate SPC techniques for autocorrelated data, but few studies have considered the impact of non-normality on these techniques. This paper investigates the effect of skewness on conventional autocorrelated SPC techniques, and provides an effective approach based on a scaled weighted variance approach to improve SPC performance in such an environment. Copyright c 2005 John Wiley & Sons, Ltd. KEY WORDS: residual-based control chart; non-normality; scaled weighted variance 1. INTRODUCTION As quality has become a decisive factor in global market competition, statistical process control (SPC) techniques are becoming critical in both manufacturing and service industries that aim at 6σ excellence. With modern measurement and inspection technologies, it is common to collect large volumes of data from individual units routinely on very short time intervals. Such nearly continuous measurement inevitably results in data that tend to be autocorrelated and non-normally distributed. However, most existing SPC techniques were not designed for such environments. It is known that conventional SPC techniques are affected by autocorrelated and skewed data. Specifically, false alarm rates are so high that true alarms are often ignored. Correspondence to: Fugee Tsung, Department of Industrial Engineering and Engineering Management, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong. season@ust.hk Contract/grant sponsor: EGIDE-PROCODE; contract/grant number: 04792UE Contract/grant sponsor: RGC; contract/grant number: F-HK20/01T Copyright c 2004 John Wiley & Sons, Ltd. Received 4 December 2002 Revised 9 August 2003

2 132 P. CASTAGLIOLA AND F. TSUNG Since the primary purpose of SPC is to detect quickly unusual sources of variability so that their root causes can be properly addressed, data autocorrelation and skewness have severe adverse impacts on the economic benefits of implementing SPC. Much recent research has considered SPC for autocorrelated processes (see Tsung and Apley 1 and references therein). The most popular existing SPC methods for autocorrelated processes are residual-based control charts. They first represent the autocorrelation using standard time series models. Then, the on-line data are whitened or uncorrelated by subtracting the model s predictions from the actual data, with the difference called the residual. If the time series model adequately represents the process behavior, the residuals will be uncorrelated. Thus, conventional SPC methods, such as Shewhart charts and exponentially weighted moving average (EWMA) charts, which were developed for uncorrelated data, can be applied directly to the residuals for detecting process changes. However, these autocorrelated SPC methods have rarely considered another critical assumption of a modern industrial process, the normally distributed assumption, which may also have a serious impact on the quality of the model s outcome. The design of most autocorrelated SPC methods supposes that the distribution of the quality characteristic has to be normal or approximately normal. However, in many situations, it may happen that this condition does not hold. Especially in residual-based control charts, actively whitening the non-normal process may make the violation of the normal distribution assumption even more serious. In this case, the usual methods for computing the control limits are inefficient and their misuse can lead to erroneous decisions in terms of control limits and in-control and out-of-control average run lengths (ARLs). In this research, we first evaluate the impacts of nonnormality on existing autocorrelated SPC methods. We then provide guidelines on when it is appropriate to use existing autocorrelated SPC methods if the distribution skewness is within a tolerable range. It is important to notice that one of the main goals of this paper is not to find a new approach that minimizes the out-of-control ARL of the residual-based control charts for non-normal residuals, but to find an appropriate approach that makes the residual-based control charts less sensitive and consequently more robust to possible non-normal residuals. 2. LITERATURE REVIEW SPC is traditionally applied to processes in which successive observations are assumed to be independent and identically distributed (i.i.d.). Unfortunately, the i.i.d. assumption is often violated in practice, and the presence of autocorrelation has a serious impact on the performance of the control charts (see, for example, Montgomery and Mastrangelo 2 and Apley and Tsung 3 ). To deal with autocorrelated data, various charting techniques have been proposed. The residual-based chart or the special cause chart (SCC) is one of the most widely discussed methods 4,5. The key idea of the residual-based charts is to fit a time series model to subtract the autocorrelation and then to monitor the residuals. Assuming that the time series model is accurate, the residuals are statistically independent, and thus conventional control charts, such as the Shewhart, cumulative sum (CUSUM) or EWMA chart, can be applied to the residuals. The residual behavior of an autocorrelated process has gained extensive attention. Moskowitz et al. 5 studied the residual-based charting performance for ARMA(1,1) processes, and Vander Wiel 6 studied it for IMA(0,1,1) processes. Hu and Roan 7 investigated the change patterns of the residuals for ARMA(1,1) and ARMA(2,1) processes. Various authors have used theoretical derivation, Markov chain approximation and Monte Carlo simulation to evaluate the run length properties of the conventional residualbased control charts (see, for example, Superville and Adams 8 and Moskowitz et al. 5 ). These results, however, are all based on the assumption that the disturbance process is normally distributed. In industrial practice, very few quality characteristics of the processes are actually normally distributed, especially in modern data-rich manufacturing processes where large volumes of data are being routinely collected from individual units. Several approaches that may handle the non-normal situations in autocorrelated SPC are as follows. One is to use conventional control charts if the effect of non-normality is not severe. Many authors studied the effect of non-normality (the effects of skewness and kurtosis) on Shewhart control charts. They include Burrows 9,Burr 10, Schilling and Nelson 11, Balakrishnan and Kocherlakota 12, Hapuarachchi et al. 13, Yourstone and Zimmer 14. Borror et al. 15, Stumbos and Sullivan 16. Testik et al. 17 investigated

3 AUTOCORRELATED SPC FOR NON-NORMAL SITUATIONS 133 the robustness of both univariate and multivariate EWMA charts to the normality assumption. Even if the conclusions of these studies are sometimes contradictory and depend on the point of view adopted by the authors themselves, it seems clear that the impact of skewness and kurtosis on the performance of control charts, especially Shewhart charts, can be substantial. Another approach is to assume that the distribution of the underlying population is known and then derive specific control limits that verify the type I error, α. Such an approach was chosen by Ferrell 18 and Nelson 19. Ferrell assumed a log-normal distribution for the underlying population and proposed control limits for the geometric midrange and the geometric range, whereas Nelson assumed a Weibull distribution and derived control limits for the median, range, scale, and location. We will investigate how to modify the conventional residual-based methods if the non-normality (i.e. the skewness and kurtosis) exceeds a tolerable range. One can also use distribution-free control charts that provide a type I risk close enough to the theoretical one. This approach was first considered by Cowden 20, who proposed to split the skewed distribution into two parts at its mode, and to consider the two new distributions as two half normal distributions having the same mean, but with different standard deviations. Another very similar approach, called the weighted variance control chart (WV control chart), was proposed by Choobineh and Ballard 21, who suggested splitting the skewed distribution into two parts at its mean, instead of its mode, and computing the standard deviations of the two new distributions using the semivariance approximation of Choobineh and Branting 22. Bai and Choi 23 have computed tables that make use of this method for X and R control charts easier. Finally, we can cite the recent works of Plante et al. 24 and Jones and Woodall 25, who suggest using the Bootstrap in the computation of control limits. Additionally, Willemain and Runger 26 propose the notion of statistically equivalent blocks to design non-parametric control charts. Alloway and Raghavachari 27 and Pappanastos and Adams 28 use a non-parametric approach based on the Hodges Lehmann statistic. Castagliola 29 proposes an extension of the WV method called the scaled weighted variance (SWV) method. These distribution-free and non-parametric methods may be modified for use in residual-based control charts when the effect of non-normality is substantial. If possible, one may transform the data in order to make them quasi-normal. This approach was chosen by Pyzdek 30, Farnum 31 and Castagliola 32, who used the Johnson system of distributions as a general tool for transforming the data to normal. 3. RESIDUAL-BASED CONTROL CHARTS In this study, we consider a process, X t, that follows an ARMA(1, 1) model, because most autocorrelated manufacturing processes can be modeled by the popular first-order ARMA model 33. We also assume that only one observation is available at each time period, since, in modern manufacturing processes such as automobile manufacturing, 100% individual inspection has become commonplace. The ARMA(1, 1) model is as follows X t = (1 φ)µ 0 + φx t 1 + a t + θa t 1 where X t is the observation at time t = 1, 2,..., a t is the random noise at time t = 1, 2,... assumed to be i.i.d. normal (0,σ a ), φ is the autoregressive parameter, θ is the moving average parameter and µ 0 is the nominal mean of the process. In this paper, the ARMA parameters φ, θ and µ 0 are assumed to be known. Also, the relationship between the standard deviation, σ x,ofthearma(1, 1) process, X t, and the standard deviation, σ a, of the random noise, a t, is known to be σ 2 x = 1 + θ 2 2φθ 1 φ 2 σ 2 a Here, the SCC is merely an individual chart for the random noise, a t,i.e. a t = X t (1 φ)µ 0 φx t 1 θa t 1

4 134 P. CASTAGLIOLA AND F. TSUNG where a 1 = X 1 µ 0. Because a t isassumedtobeani.i.d.normal(0,σ a ) random variable when the process is in-control, the control limits of the SCC chart are simply CL =±Kσ a where K is a constant that is usually chosen to be 3 (see Montgomery 34 ). Let R be the discrete random variable representing the run length of the SCC chart after a constant mean shift, δ = µ 1 µ 0 /σ x, occurs in the process, where µ 1 is the new out-of-control mean of the process. Note that this shift definition does not apply to a non-stationary process where the σ x is not finite. In the case of an ARMA(1, 1) model, Moskowitz et al. 5 demonstrated that the probability density function (pdf) f R (r δ) of R can be computed using r 1 f R (r δ) = P(R = r) = p r (1 p k ) k=0 where { [ p k = 1 F Kσ a δ 1 + (θ φ)(1 θ ]} k 1 ) 1 θ { [ + F Kσ a δ 1 + (θ φ)(1 θ ]} k 1 ) 1 θ (1) where F() is the cumulative density function (cdf) of the random variable a t.asa t is i.i.d. normal (0,σ a ), we have F(u)= (u/σ a ),where () is the cdf of the normal (0, 1) distribution. The ARL of the SCC chart is simply E(R), i.e. ARL = + r=1 rf R (r δ) (2) In practice, the previous infinite sum can be replaced by a finite sum when the product rf R (r δ) becomes small enough. It is known that very few industrial processes are actually normally distributed and that non-normality may impact the performance of conventional SCC charts. This will be demonstrated by an illustrated example in the next section. 4. IMPACT OF NON-NORMALITY ON RESIDUAL-BASED CONTROL CHART ThedatainFigure1(a) consist of 100 observations, X t, that correspond to an in-control process. These data have been simulated by generating an ARMA(1, 1) process with µ 0 = 20, φ = 0.4, θ = 0.8, where the a t have a Lognormal distribution with E(a t ) = 0, σ a = 0.1 and a skewness coefficient, γ 3 = 1. We have estimated the first autocorrelation coefficient, ˆρ 1 = Because this value is quite large, we may suspect some important autocorrelation in these data. Consequently, we decided to fit the data to an ARMA(1, 1) model. The estimated values of the ARMA(1, 1) parametersare ˆφ = 0.525, ˆθ = 0.721, µ ˆ 0 = and ˆσ a = Using these parameters, we compute and plot the corresponding residuals, â t, in Figure 1(b). The estimated skewness coefficient for the residuals is ˆγ 3 = The p-values of the Anderson Darling test and the skewness test are respectively and These values strongly suggest that the residuals are not normallydistributed at all. In Figure 1(b), we plot the classical 3σ control limits for the SCC chart, with CL =± =± We can see that even if the process is supposed to be in-control, two points are above the upper limit and some other points are very close to it. It is worthwhile to note that the residuals may also be sensitive to errors in the parameter estimates, especially when the historical data set is relatively small 35.

5 AUTOCORRELATED SPC FOR NON-NORMAL SITUATIONS (a) X t (b) t SWV < a t t SWV 3 Figure 1. An illustrated example: (a) a process with autocorrelated data; (b) an SCC chart with 3σ and SWV control limits 5. THE EFFECT OF NON-NORMALITY ON RESIDUAL-BASED METHODS Without loss of generality, we will assume µ 0 = 0andσ x = 1. In order to test the effect of non-normality on the SCC chart, we first compute the ARL of this control chart for different selected values, (φ, θ), for different values of the shift, δ, and for different kinds of skewed distributions and skewness coefficients, γ 3,forthe random noise, a t. Note that these skewed distributions must also verify E(a t ) = 0andV(a t ) = σ 2 a.wethen compare the results with the ARL in the normal case. Concerning the distributions used in this test, we choose to use the three most commonly used skewed distributions. One is the Gamma distribution. Its pdf for u>cis defined as f(u,b,c,d)= (u c)d 1 exp((c u)/b) b d Ɣ(d) Another is the Lognormal distribution. Its pdf for u>cis defined as f(u,b,c,d)= bϕ{d + b ln(u c)} u c

6 136 P. CASTAGLIOLA AND F. TSUNG The other is the Weibull distribution. Its pdf for u>cis defined as d{(u c)/b}d 1 f(u,b,c,d)= exp{ {(u c)/b} d } b Each of those distributions depends on three parameters, b (scale), c (position) and d (shape), which can be computed from specified µ, σ a and the skewness coefficient, γ 3 (see Appendix A). For the same skewness coefficient, these distributions have very different kurtosis coefficients, γ 4 = µ 4 /µ We consider 25 possible combinations of (φ, θ), with φ { 0.95, 0.475, 0, 0.475, 0.95} and θ { 0.9, 0.45, 0, 0.45, 0.9}, for the shifts δ {0, 0.5,...,2.5}, under the Lognormal, Gamma and Weibull distributions with the skewness coefficients γ 3 { 2, 1, 0, 1, 2}. We compute (b,c,d)from µ = 0, σ a and γ 3 (see Appendix A), and compute ARL using relations (1)and(2) with F(u)= F(u,b,c,d),whereF(u,b,c,d) is the cdf of the Lognormal, Gamma or Weibull distributions. The results are given in Table I. In the table we have also added the ARL in the case of a normal distribution. For example, for φ = 0.95, θ = 0.45, δ = 0.5, the Lognormal case with a skewness coefficient of γ 3 = 1, we have ARL = 109, while in the normal case, for the same shift, δ = 0.5, we have ARL = 350. Some important conclusion can be drawn from these results. It can been seen that, whatever the value of (φ, θ), when there is no shift in the mean (δ = 0), the in-control ARL 0 is the same for similar distributions having the same absolute value for the skewness coefficient. For example, when δ = 0, for the Gamma distribution with a skewness coefficient γ 3 = 1orγ 3 = 1, we always have ARL 0 = 97. This result is logical since, in (1), when δ = 0, the value of p k no longer depends on (φ, θ). For a given size of δ, thearl performance with γ 3 = 1 is in general better or at least no worse than that with γ 3 = 1. Also, their difference becomes more significant for a smaller shift. In many cases, the ARL values actually increase as the shift size increases when γ 3 < 0. Thus, we may conclude from Table II that, for the shift in the positive direction, negative skewness will deteriorate the power to detect a meanshift, while positive skewness will enhance the meanshift detection. 6. TOLERABLE RANGES TO USE RESIDUAL-BASED METHODS Based on the above computational results, we suggest continuing to use conventional residual-based methods when the actual ARL 0 due to the distributional skewness is still within a certain percentage, say 10% of the designed ARL 0. Here, tolerable ranges for different skewed distributions are recommended, and the ARL performance of residual-based methods under such situations are studied. One of the conclusions drawn in the previous section is that the value of ARL 0, the in-control ARL, only depends on the distribution and the skewness coefficient, but not on the ARMA(1, 1) parameters (φ, θ) (assuming that these parameters are known or estimated with little error). This suggests that we investigate, for each distribution, the skewness coefficient range in which the ARL 0 does not exceed a certain percentage, say 10% and 20%, of the normal case with incontrol ARL 0 = 370. In Figure 2, we plot the in-control ARL 0, in the cases of Lognormal, Gamma and Weibull distributions, for γ 3 (0, 0.5). In the Lognormal case, we can observe that for γ 3 = 0, the ARL 0 = 370, as in the normal case. This is logical, since the asymptotic distribution of the Lognormal distribution when γ 3 0 is the normal distribution. For any other values of γ 3,wehaveARL 0 < 370. When γ 3 ( 0.147, 0.147) the ARL 0 (333, 370) and when γ 3 ( 0.223, 0.223) the ARL 0 (296, 370). In the Gamma case, we can also observe that for γ 3 = 0, the ARL 0 = 370, as in the normal case. This is logical as well, since the asymptotic distribution of the Gamma distribution when γ 3 0 is the normal distribution. For any other values of γ 3,wehaveARL 0 < 370. When γ 3 ( 0.158, 0.158) the ARL 0 (333, 370) and when γ 3 ( 0.238, 0.238) the ARL 0 (296, 370). On the other hand, we can observe in the Weibull case that for γ 3 = 0, the ARL 0 = 1321, which is very different from the normal case. This can be explained by the fact that the asymptotic distribution of the Weibull distribution when γ 3 0 is not the normal distribution. When γ 3 ( 0.371, 0.305) (0.305, 0.371) the ARL 0 (333, 407),andwhenγ 3 ( 0.412, 0.279) (0.279, 0.412) the ARL 0 (296, 444).

7 AUTOCORRELATED SPC FOR NON-NORMAL SITUATIONS 137 Table I. ARL for (φ, θ), with φ { 0.95, 0.475, 0, 0.475, 0.95} and θ { 0.9, 0.45, 0, 0.45, 0.9}, for δ {0, 0.5,..., 2.5}, for the Lognormal, Gamma and Weibull distributions, and for the skewness coefficients γ 3 { 2, 1, 0, 1, 2} Normal Lognormal Gamma Weibull δ φ = 0.950,θ= 0.900,σ x /σ a = 1.013,ρ 1 = φ = 0.475,θ= 0.900,σ x /σ a = 1.111,ρ 1 = φ = 0.000,θ= 0.900,σ x /σ a = 1.345,ρ 1 = φ = 0.475,θ= 0.900,σ x /σ a = 1.855,ρ 1 = φ = 0.950,θ= 0.900,σ x /σ a = 6.009,ρ 1 = φ = 0.950,θ= 0.450,σ x /σ a = 1.888,ρ 1 = φ = 0.475,θ= 0.450,σ x /σ a = 1.000,ρ 1 =

8 138 P. CASTAGLIOLA AND F. TSUNG Table I. Continued Normal Lognormal Gamma Weibull δ φ = 0.000,θ= 0.450,σ x /σ a = 1.097,ρ 1 = φ = 0.475,θ= 0.450,σ x /σ a = 1.451,ρ 1 = φ = 0.950,θ= 0.450,σ x /σ a = 4.594,ρ 1 = φ = 0.950,θ= 0.000,σ x /σ a = 3.203,ρ 1 = φ = 0.475,θ= 0.000,σ x /σ a = 1.136,ρ 1 = φ = 0.000,θ= 0.000,σ x /σ a = 1.000,ρ 1 = φ = 0.475,θ= 0.000,σ x /σ a = 1.136,ρ 1 =

9 AUTOCORRELATED SPC FOR NON-NORMAL SITUATIONS 139 Table I. Continued Normal Lognormal Gamma Weibull δ φ = 0.950,θ= 0.000,σ x /σ a = 3.203,ρ 1 = φ = 0.950,θ= 0.450,σ x /σ a = 4.594,ρ 1 = φ = 0.475,θ= 0.450,σ x /σ a = 1.451,ρ 1 = φ = 0.000,θ= 0.450,σ x /σ a = 1.097,ρ 1 = φ = 0.475,θ= 0.450,σ x /σ a = 1.000,ρ 1 = φ = 0.950,θ= 0.450,σ x /σ a = 1.888,ρ 1 = φ = 0.950,θ= 0.900,σ x /σ a = 6.009,ρ 1 =

10 140 P. CASTAGLIOLA AND F. TSUNG Table I. Continued Normal Lognormal Gamma Weibull δ φ = 0.475,θ= 0.900,σ x /σ a = 1.855,ρ 1 = φ = 0.000,θ= 0.900,σ x /σ a = 1.345,ρ 1 = φ = 0.475,θ= 0.900,σ x /σ a = 1.111,ρ 1 = φ = 0.950,θ= 0.900,σ x /σ a = 1.013,ρ 1 = Table II. Skewness coefficient ranges SR 10% and SR 20% in which the ARL 0 does not exceed, respectively, 10% and 20% of the normal case in-control ARL 0 = 370, when traditional 3σ limits and SWV limits are used SR 10% SR 20% 3σ limits Lognormal ( 0.147, 0.147) ( 0.223, 0.223) Gamma ( 0.158, 0.158) ( 0.238, 0.238) Weibull ( 0.371, 0.305) (0.305, 0.371) ( 0.412, 0.279) (0.279, 0.412) SWV limits Lognormal ( 0.225, 0.225) ( 0.343, 0.343) Gamma ( 0.265, 0.265) ( 0.395, 0.395) Weibull ( 0.569, 0.455) (0.455, 0.569) ( 0.645, 0.411) (0.411, 0.645) The results concerning the skewness coefficient ranges SR 10% and SR 20% are summarized in the upper part of Table II. For the Lognormal, Gamma and Weibull distributions, we have computed the minimum and maximum out-of-control ARL when the skewness coefficient is within the skewness coefficient ranges of SR 10% and SR 20%, for shifts δ {0.5,...,2.5}. The results are given in Table III for φ { 0.95, 0.475, 0, 0.475, 0.95} and θ { 0.9, 0.45, 0, 0.45, 0.9}. For example, we can observe that in the Lognormal case, when φ = 0.475, θ = 0.9, δ = 1, the minimum ARL = 50 and the maximum ARL = 76 if γ 3 SR 10%,andthe minimum ARL = 46 and the maximum ARL = 88 if γ 3 SR 20%, while the normal case ARL = 60.

11 AUTOCORRELATED SPC FOR NON-NORMAL SITUATIONS LOGNORMAL ARL GAMMA ARL WEIBULL ARL Figure 2. In-control ARL 0, in the case of a Lognormal, Gamma and Weibull distribution, for γ 3 (0, 0.7), usingthe traditional approach (i.e. 3σ limits) It is also interesting to notice in Table III that, in the case of small to moderate shifts, the minimum out-ofcontrol ARL is roughly the same magnitude for the Gamma, Lognormal and Weibull distributions, whereas, for the maximum out-of-control ARL, we can see a huge difference between the Gamma and Lognormal case on one side, and the Weibull case on the other side. In the case of the Weibull distribution, the out-of-control ARLs are so large (e.g. φ = 0.475, θ = 0.9, δ = 1, the maximum ARL = 1276 if γ 3 SR 20% ) that any monitoring schemes will probably be of little value.

12 142 P. CASTAGLIOLA AND F. TSUNG Table III. Minimum and maximum out-of-control ARL when the skewness coefficient is within the skewness coefficient ranges SR 10% and SR 20% of Table II, for shifts δ {0.5,...,2.5}, φ { 0.95, 0.475, 0, 0.475, 0.95} and θ { 0.9, 0.45, 0, 0.45, 0.9}, traditional 3σ limits Lognormal Gamma Weibull δ Normal 10% 20% 10% 20% 10% 20% φ = 0.950,θ= 0.900,σ x /σ a = 1.013,ρ 1 = φ = 0.475,θ= 0.900,σ x /σ a = 1.111,ρ 1 = φ = 0.000,θ= 0.900,σ x /σ a = 1.345,ρ 1 = φ = 0.475,θ= 0.900,σ x /σ a = 1.855,ρ 1 = φ = 0.950,θ= 0.900,σ x /σ a = 6.009,ρ 1 = φ = 0.950,θ= 0.450,σ x /σ a = 1.888,ρ 1 = φ = 0.475,θ= 0.450,σ x /σ a = 1.000,ρ 1 =

13 AUTOCORRELATED SPC FOR NON-NORMAL SITUATIONS 143 Table III. Continued Lognormal Gamma Weibull δ Normal 10% 20% 10% 20% 10% 20% φ = 0.000,θ= 0.450,σ x /σ a = 1.097,ρ 1 = φ = 0.475,θ= 0.450,σ x /σ a = 1.451,ρ 1 = φ = 0.950,θ= 0.450,σ x /σ a = 4.594,ρ 1 = φ = 0.950,θ= 0.000,σ x /σ a = 3.203,ρ 1 = φ = 0.475,θ= 0.000,σ x /σ a = 1.136,ρ 1 = φ = 0.000,θ= 0.000,σ x /σ a = 1.000,ρ 1 = φ = 0.475,θ= 0.000,σ x /σ a = 1.136,ρ 1 =

14 144 P. CASTAGLIOLA AND F. TSUNG Table III. Continued Lognormal Gamma Weibull δ Normal 10% 20% 10% 20% 10% 20% φ = 0.950,θ= 0.000,σ x /σ a = 3.203,ρ 1 = φ = 0.950,θ= 0.450,σ x /σ a = 4.594,ρ 1 = φ = 0.475,θ= 0.450,σ x /σ a = 1.451,ρ 1 = φ = 0.000,θ= 0.450,σ x /σ a = 1.097,ρ 1 = φ = 0.475,θ= 0.450,σ x /σ a = 1.000,ρ 1 = φ = 0.950,θ= 0.450,σ x /σ a = 1.888,ρ 1 = φ = 0.950,θ= 0.900,σ x /σ a = 6.009,ρ 1 =

15 AUTOCORRELATED SPC FOR NON-NORMAL SITUATIONS 145 Table III. Continued Lognormal Gamma Weibull δ Normal 10% 20% 10% 20% 10% 20% φ = 0.475,θ= 0.900,σ x /σ a = 1.855,ρ 1 = φ = 0.000,θ= 0.900,σ x /σ a = 1.345,ρ 1 = φ = 0.475,θ= 0.900,σ x /σ a = 1.111,ρ 1 = φ = 0.950,θ= 0.900,σ x /σ a = 1.013,ρ 1 = MODIFIED RESIDUAL-BASED METHODS FOR NON-NORMALITY The objective of this paper is not to find a new approach that minimizes the out-of-control ARL of the SCC for non-normal residuals, but to suggest an appropriate approach that makes the SCC less sensitive and consequently more robust to possible non-normal residuals. To take into account the possible skewness of the residuals, a t, we suggest a modification of the existing residual-based methods by using a SWV method. The SWV method proposed by Castagliola 29 is easy and straightforward to apply and has proven to have better performance in comparison with classical 3σ control charts and the WV method 23. In this approach, we assume that f(x)is the unknown pdf of the process random variable, X, µ = E(X) is the mean, σ 2 = V(X)is the variance, and π = P(X µ) is the probability that X is less than or equal to the mean, µ. The asymmetrical control limits derived by Castagliola 29 using the SWV method are 1 π ( LCL = µ nπ 1 1 α 4π ( UCL = µ + 1 π n(1 π) 1 ) σ ) α σ 4(1 π)

16 146 P. CASTAGLIOLA AND F. TSUNG with α = The corresponding control chart is called the SWV X control chart. It is easy to see that when π = 1 2, the previous control limits reduce to the classical LCL = µ 3σ n UCL = µ + 3σ n This method can be straightforwardly applied to the classical residual-based method by assuming µ = 0, σ 2 = σ 2 a = V(a t), π = P(a t 0) and n = 1, i.e. LCL = UCL = 1 π π π (1 π) 1 ( 1 1 α ) σ a (3) 4π ( 1 ) α σ a (4) 4(1 π) The ARL for the residual-based method using the SWV control limits can be easily deduced from Equations (1)and(2) by replacing Equation (1) with { [ p k = 1 F UCL δ 1 + (θ φ)(1 θ ]} k 1 ) 1 θ { [ + F LCL δ 1 + (θ φ)(1 θ ]} k 1 ) 1 θ where LCL and UCL are given by Equations (3)and(4). As an example, we compute the control limits of the SWV SCC chart for the data plotted in Figure 1(b). We estimate ˆπ = 0.61 and thus we have LCL = and UCL = With these modified asymmetrical control limits, all the points remain inside the control limits, as expected. Using the same approach as with the 3σ control limits case, we compute the ARL of the residual-based method using the SWV control limits for the Lognormal, Gamma and Weibull distributions. The results are given in Table IV. Asinthe3σ control limits case, whatever the value of (φ, θ), when there is no shift in the mean, (δ = 0), the in-control ARL 0 is the same for similar distributions having the same absolute value for the skewness coefficient. For example, when δ = 0, for the Lognormal distribution with a skewness coefficient γ 3 = 1orγ 3 = 1, we always have ARL 0 = 156. For the Lognormal, Gamma and Weibull distributions, we have computed the minimum and the maximum out-of-control ARL when the skewness coefficient is within the skewness coefficient ranges SR 10% and SR 20% of Table II, the3σ control limits case, for shifts δ {0.5,...,2.5}. The results are given in Table V for φ { 0.95, 0.475, 0, 0.475, 0.95} and θ { 0.9, 0.45, 0, 0.45, 0.9}. For example, we can observe that in the Lognormal case, when φ = 0.475, θ = 0.9, δ = 1, the minimum ARL = 56 and the maximum ARL = 66 if γ 3 SR 10%, and the minimum ARL = 54 and the maximum ARL = 69 if γ 3 SR 20%, while the normal case ARL = 60. If we compare these results with the ones obtained in the case of the classical 3σ limits (minimum ARL = 50 and maximum ARL = 76 if γ 3 SR 10%, and minimum ARL = 46 and maximum ARL = 88 if γ 3 SR 20% ), we may conclude that, inside the same skewness range, the use of the SWV limits give out-of-control minimum and maximum ARLs that are closer to the normal ARLs than the minimum and maximum values obtained with the classical 3σ limits. We can also observe that, in the case of the Weibull distribution for small to moderate shifts, the maximum out-of-control ARL can still be very large.

17 AUTOCORRELATED SPC FOR NON-NORMAL SITUATIONS 147 Table IV. ARL for (φ, θ), with φ { 0.95, 0.475, 0, 0.475, 0.95} and θ { 0.9, 0.45, 0, 0.45, 0.9}, for δ {0, 0.5,..., 2.5}, for the Lognormal, Gamma and Weibull distributions, and for the skewness coefficients γ 3 { 2, 1, 0, 1, 2}, using the SWV control limits Normal Lognormal Gamma Weibull δ φ = 0.950,θ= 0.900,σ x /σ a = 1.013,ρ 1 = φ = 0.475,θ= 0.900,σ x /σ a = 1.111,ρ 1 = φ = 0.000,θ= 0.900,σ x /σ a = 1.345,ρ 1 = φ = 0.475,θ= 0.900,σ x /σ a = 1.855,ρ 1 = φ = 0.950,θ= 0.900,σ x /σ a = 6.009,ρ 1 = φ = 0.950,θ= 0.450,σ x /σ a = 1.888,ρ 1 = φ = 0.475,θ= 0.450,σ x /σ a = 1.000,ρ 1 =

18 148 P. CASTAGLIOLA AND F. TSUNG Table IV. Continued Normal Lognormal Gamma Weibull δ φ = 0.000,θ= 0.450,σ x /σ a = 1.097,ρ 1 = φ = 0.475,θ= 0.450,σ x /σ a = 1.451,ρ 1 = φ = 0.950,θ= 0.450,σ x /σ a = 4.594,ρ 1 = φ = 0.950,θ= 0.000,σ x /σ a = 3.203,ρ 1 = φ = 0.475,θ= 0.000,σ x /σ a = 1.136,ρ 1 = φ = 0.000,θ= 0.000,σ x /σ a = 1.000,ρ 1 = φ = 0.475,θ= 0.000,σ x /σ a = 1.136,ρ 1 =

19 AUTOCORRELATED SPC FOR NON-NORMAL SITUATIONS 149 Table IV. Continued Normal Lognormal Gamma Weibull δ φ = 0.950,θ= 0.000,σ x /σ a = 3.203,ρ 1 = φ = 0.950,θ= 0.450,σ x /σ a = 4.594,ρ 1 = φ = 0.475,θ= 0.450,σ x /σ a = 1.451,ρ 1 = φ = 0.000,θ= 0.450,σ x /σ a = 1.097,ρ 1 = φ = 0.475,θ= 0.450,σ x /σ a = 1.000,ρ 1 = φ = 0.950,θ= 0.450,σ x /σ a = 1.888,ρ 1 = φ = 0.950,θ= 0.900,σ x /σ a = 6.009,ρ 1 =

20 150 P. CASTAGLIOLA AND F. TSUNG Table IV. Continued Normal Lognormal Gamma Weibull δ φ = 0.475,θ= 0.900,σ x /σ a = 1.855,ρ 1 = φ = 0.000,θ= 0.900,σ x /σ a = 1.345,ρ 1 = φ = 0.475,θ= 0.900,σ x /σ a = 1.111,ρ 1 = φ = 0.950,θ= 0.900,σ x /σ a = 1.013,ρ 1 = In the case where the SWV control limits are used, we also compute the skewness coefficient range in which the ARL 0 does not exceed 10% and 20% of the normal case in-control ARL 0 = 370. In Figure 3, weplotthe in-control ARL 0, in the case of a Lognormal, Gamma and Weibull distribution, for γ 3 (0, 0.7). We see that the three curves decrease less rapidly than in the 3σ limits case, showing that the use of the SVW approach gives limits that are less sensitive to the skewness. The results concerning the skewness coefficient ranges, SR 10% and SR 20%, are summarized in the lower part of Table II. We can observe that the range for the skewness coefficient obtained using SWV control limits is larger than the range obtained with the traditional 3σ limits approach. For the Lognormal, Gamma and Weibull distributions, we have finally computed the minimum and the maximum out-of-control ARL when the skewness coefficient is within the skewness coefficient ranges SR 10% and SR 20% of Table II, the SWV control limits case, for shifts δ {0.5,...,2.5}. The results are in Table VI for φ { 0.95, 0.475, 0, 0.475, 0.95} and θ { 0.9, 0.45, 0, 0.45, 0.9}. For example, we can observe that in the Lognormal case, when φ = 0.475, θ = 0.9, δ = 1, the minimum ARL = 54 and the maximum ARL = 69 if γ 3 SR 10% and the minimum ARL = 52 and the maximum ARL = 76 if γ 3 SR 20%, while the normal case ARL = 60.

21 AUTOCORRELATED SPC FOR NON-NORMAL SITUATIONS 151 Table V. Minimum and maximum out-of-control ARL when the skewness coefficient is within the skewness coefficient ranges SR 10% and SR 20% of Table II, for shifts δ {0.5,...,2.5}, φ { 0.95, 0.475, 0, 0.475, 0.95} and θ { 0.9, 0.45, 0, 0.45, 0.9}, SWV limits Lognormal Gamma Weibull δ Normal 10% 20% 10% 20% 10% 20% φ = 0.950,θ= 0.900,σ x /σ a = 1.013,ρ 1 = φ = 0.475,θ= 0.900,σ x /σ a = 1.111,ρ 1 = φ = 0.000,θ= 0.900,σ x /σ a = 1.345,ρ 1 = φ = 0.475,θ= 0.900,σ x /σ a = 1.855,ρ 1 = φ = 0.950,θ= 0.900,σ x /σ a = 6.009,ρ 1 = φ = 0.950,θ= 0.450,σ x /σ a = 1.888,ρ 1 =

22 152 P. CASTAGLIOLA AND F. TSUNG Table V. Continued Lognormal Gamma Weibull δ Normal 10% 20% 10% 20% 10% 20% φ = 0.475,θ= 0.450,σ x /σ a = 1.000,ρ 1 = φ = 0.000,θ= 0.450,σ x /σ a = 1.097,ρ 1 = φ = 0.475,θ= 0.450,σ x /σ a = 1.451,ρ 1 = φ = 0.950,θ= 0.450,σ x /σ a = 4.594,ρ 1 = φ = 0.950,θ= 0.000,σ x /σ a = 3.203,ρ 1 = φ = 0.475,θ= 0.000,σ x /σ a = 1.136,ρ 1 = φ = 0.000,θ= 0.000,σ x /σ a = 1.000,ρ 1 =

23 AUTOCORRELATED SPC FOR NON-NORMAL SITUATIONS 153 Table V. Continued Lognormal Gamma Weibull δ Normal 10% 20% 10% 20% 10% 20% φ = 0.475,θ= 0.000,σ x /σ a = 1.136,ρ 1 = φ = 0.950,θ= 0.000,σ x /σ a = 3.203,ρ 1 = φ = 0.950,θ= 0.450,σ x /σ a = 4.594,ρ 1 = φ = 0.475,θ= 0.450,σ x /σ a = 1.451,ρ 1 = φ = 0.000,θ= 0.450,σ x /σ a = 1.097,ρ 1 = φ = 0.475,θ= 0.450,σ x /σ a = 1.000,ρ 1 = φ = 0.950,θ= 0.450,σ x /σ a = 1.888,ρ 1 =