The Robustness of the Multivariate EWMA Control Chart

Transcription

1 The Robustness of the Multivariate EWMA Control Chart Zachary G. Stoumbos, Rutgers University, and Joe H. Sullivan, Mississippi State University Joe H. Sullivan, MSU, MS Key Words: Elliptically symmetric, skewed, statistical process control. Control charts are graphical tools widely used to monitor manufacturing processes to quickly detect any change in a process that may result in a change in product quality. The statistic plotted on a control chart is based on samples of n 1 observations (rational subgroups) that may be taken at regular sampling intervals. However, there are numerous practical applications using individual observations (n = 1), as in many chemical and process industries or when the rate of production is slow. See, for example, Montgomery (1997, pp ) and Ryan (2000, p. 133) for many applications using individual observations. The standard approach to control charts has been univariate, with the most frequently investigated charts being those designed to monitor the mean µ of a single normally distributed process variable X. However, with rapidly evolving data-acquisition technology, it is now common practice to simultaneously monitor several, usually correlated, quality variables. Using separate univariate charts does not account for correlation between variables, so a multivariate control chart is more suitable. Two multivariate control charts that have received attention in the statistical process control (SPC) literature are the Shewhart-type χ 2 chart, originating in the work of Hotelling (1947), and the multivariate exponentially weighted moving average (MEWMA) chart, proposed much more recently by Lowry et al. (1992). Both of these charts were originally developed for the problem of monitoring the mean vector, say µ, of a continuous multivariate process variable (vector), say x. The χ 2 chart can be relatively easily applied and is helpful in identifying large shifts in µ, but is ineffective at detecting small and moderate-sized shifts. Unlike the χ 2 chart, the MEWMA chart accumulates information from past observations, making it more sensitive in detecting small, sustained shifts. The survey articles of Alt and Smith (1988), Wierda (1994), Lowry and Montgomery (1995), and Mason et al. (1997) give good discussions on multivariate control charts and extensive lists of references. When monitoring the mean vector, control chart design is usually predicated on the approximate multivariate normality (multinormality) of process data. Nonnormality generally is not a major concern with large subgroups, because the central limit theorem applies and the sample mean vector x is approximately multinormal, for any practical distribution of the individual observations. However, with small samples quite common in SPC practices from a non-normal population, x may be far from multinormal. The problem is most severe with the smallest sized subgroups, individual observations. The univariate non-normality problem was recently studied by Stoumbos and Reynolds (2000), Reynolds and Stoumbos (2001), and Borror, Montgomery, and Runger (1999) with the conclusion that non-normality can seriously degrade the performance of the Shewhart X chart, but the EWMA and cumulative sum (CUSUM) charts can be designed to be robust. The more general multivariate problem has not been investigated. We investigate the performance of the MEWMA chart and of Hotelling s χ 2 chart for multivariate nonnormal observations. We show that the in-control (IC) performance of the individuals χ 2 chart suffers from an excessive number of false alarms with many types of non-normality. This greatly limits the χ 2 chart s practical usefulness because of excessive process disruptions and adjustments, loss of confidence in the control procedure, and, ultimately, lost productivity. We recommend the MEWMA chart as a robust and effective alternative to the individuals χ 2 chart. We show that an individuals MEWMA control chart can be designed to have very nearly the designed IC performance regardless of the distribution of the individual observations. The same design also provides excellent out-of-control (OOC) performance across a wide range of process shifts regardless of the distribution of the process data. Note that while the MEWMA chart offers excellent performance in detecting sustained shifts in the mean vector, if designed for robustness it may not necessarily detect all isolated outlying observations. Data with outliers can be regarded as having a non-normal distribution in the sense that the observations can be modeled as a mixture of observations from two (or more) distributions or, alternatively, from a distribution with heavy tails. For example both the multivariate gamma and multivariate t distributions, for some parameters, have an increased probability for observations that could be regarded as outliers. Thus, in designing the chart to be robust against all non-normal distributions, one natural consequence is some reduction in sensitivity to process outliers. The MEWMA chart and Hotelling s χ 2 chart are presented in the next section, and then multivariate generalizations of the univariate t and gamma distributions are discussed. The MEWMA Chart and Hotelling s χ 2 Chart Let x 1, x 2, be a sequence of p-variate process ob- 3362

2 servations, with mean vectors µ 1 = µ 0 + δ 1, µ 2 = µ 0 + δ 2,, respectively, an IC mean vector µ 0, and constant covariance matrix Σ X. We will study the case in which there is a single, sustained shift from µ 0, so that δ i = 0 for i T, and δ i = δ (δ 0) for all subsequent observations; i.e., T is the unknown, random time after which the process mean vector shifts from µ 0 to µ 0 + δ. The IC parameters µ 0 and Σ X are assumed known, estimated with negligible error from a sufficiently large sample, but T and δ are unknown. The magnitude or size of the T 1 shift δ we consider is λ = δ Σ X δ. In the context of multivariate normality, λ is called the noncentrality parameter. Define the vector-accumulations of the observations as z i = r( xi µ 0 ) + (1 r) z i 1, for i = 1, 2, K, where 0 < r 1 is the smoothing parameter, and the starting value is usually chosen to be z 0 = 0. Then, the control 2 T 1 statistic of the MEWMA chart is Ti = z i Σ Z z i, for i = 1, 2,, where Σ Z is the asymptotic covariance matrix 2i r(1 (1 r) ) r of z i, Σ Z = lim i Σ X = Σ X. 2 r 2 r The MEWMA control chart signals at the first sampling time i for which T i 2 > h, where h > 0 denotes the chart s upper control limit (UCL), usually determined so that the MEWMA chart achieves a desired IC average run length (ARL). There is no lower control limit. The ARL is the expected number of sample subgroups (which may be individual observations) taken until the chart generates a signal. For r = 1, the MEWMA chart reduces to Hotelling s χ 2 chart. A large value of r renders the MEWMA chart effective at detecting large shifts. Small and moderate sustained shifts are detected much more effectively with a chart having a small r, but a small r may delay slightly the detection of very large shifts. These issues and the choice of a value for r are discussed later. Before evaluating the chart performance, we discuss possible generalizations of the univariate t and gamma distributions to multivariate process observations. The Multivariate t and Gamma Distributions Following the univariate analyses of Stoumbos and Reynolds (2000), Reynolds and Stoumbos (2001), and Borror, Montgomery, and Runger (1999), we considered distributions that were either elliptically symmetrical, such as the multinormal and the multivariate t distribution, or skewed such as the multivariate gamma distribution. First, consider the multivariate generalization of the univariate t distribution, which is the ratio of a standard-normal random variable, divided by the square root of an independent chi-squared random variable, in turn, divided by its degrees of freedom. One generalization would be simply a vector of univariate t distributions. However, more commonly, a single chisquared random variable is used to divide a vector of standard-normal random variables, the definition we used. This latter distribution is called the multivariate t with common denominator by Johnson and Kotz (1972, p. 134), who say it is often called the general multivariate t distribution. If x has a standardized p- variate normal distribution (with zero mean vector) and covariance matrix R, which is also the correlation matrix since the elements are individually standardized, and S 2 has an independent chi-squared distribution with ν degrees of freedom, then y =µ + x S 2 / ν has a p- variate t distribution with ν degrees of freedom, and with E (y) = µ and Var(y) = ν R/(ν 2). This p-variate t distribution will be denoted here by t p (ν ). With an elliptically symmetrical distribution, the statistical performance of the MEWMA chart depends on the shift and the covariance matrix only through the noncentrality parameter (see Sullivan and Stoumbos (2002)). There is less agreement in the literature on a specific multivariate generalization of the gamma distribution, with a number of possible alternative definitions. Often, the multivariate gamma distribution is given as one-half the diagonal elements of a matrix having a Wishart distribution, the distribution we used. A symmetric matrix has a Wishart distribution with n degrees of freedom and scale matrix Σ if it can be written as X T X, where X is a data matrix of n observations from the multivariate normal distribution with zero mean vector and covariance matrix Σ. Note that Σ is the covariance matrix of the implicit multinormal distribution and is not the covariance matrix of the actual multivariate gamma observations. This distribution is described by Krishnaiah (1985) as a (central or noncentral) multivariate gamma distribution with n/2 as the shape parameter and Σ as the covariance matrix of the accompanying multivariate normal. The p-variate gamma distribution with shape parameter α and scale parameter β will be denoted here by Gam p (α, β ). We take β = 1 with no loss of generality. The statistical performance with multivariate gamma observations is not fully specified by λ (see Sullivan and Stoumbos (2002)). Some shifts increase the ARL. In general, we wish to evaluate a chart s performance in detecting a shift in the mean vector when the higher moments are fixed. Ideally, a control chart would be distribution-free in the sense that its performance would be independent of the distribution s shape as determined by the higher moments. The second moment has no effect on the chart performance, provided it is known accurately from a sufficiently large number of IC data points. Our study concerns how the distribu- 3363

3 tion s higher moments, or more generally, its characteristic function, affect the chart s ability to detect a shift in the mean vector. We elected to study the multivariate t distribution, which differs from the multinormal in the fourth and higher moments, and the multivariate gamma distribution, because it differs in the third and higher moments. Some distributions, such as the gamma with shape parameter of 1, depart from normality to a great extent. Our approach has been to study the performance of nearly normal distributions as well as those that differ greatly with respect to the third or fourth moments (small shape parameter or small degrees of freedom). Thus, we considered Gam p (α,1) observations having shape parameter values of 1 (very large skewness and kurtosis), 4, 16, 64, 256, and 1024 (1024 being nearly normal). For the t p (ν ) observations we considered degrees of freedom values 3 (kurtosis very different from normal), 4, 6, 10, 20, 40, 100, and 1000 (1000 being nearly normal). With degrees of freedom below 3, the second moment is undefined. These parameter values give a comprehensive view of the performance by including pathologically non-multinormal observations, observations nearly multinormal and a generous assortment of intermediate distributions. Technically, the mean vector cannot be specified independently of the covariance matrix for the multivariate gamma distribution. The diagonal elements of a Wishart matrix have as their mean and variance, respectively, nσ 2 ii and 2nσ ii. Therefore, in generating IC observations we subtracted the mean vector from the vector of multivariate observations and generated the OOC observations by adding a shift vector. Specifically, if x i is a multivariate observation from a specified process distribution, we calculated the vector accumulation as zi = r( xi µ 0 + δi ) + (1 r) zi 1. We also wanted to study the effect of a wide range of values of r. We used 10 values that are evenly spaced / 3 logarithmically, as given by the formula r = 10 k, for k = 0, 1,, 9. In the following section, we discuss issues associated with the computation of the ARL for the MEWMA chart and for Hotelling s χ 2 chart and with the determination of their control limits. The ARL is traditionally used as the performance measure for control chart comparisons. The control limits are generally adjusted so that all charts have approximately the same IC ARL, and the control chart with the lowest OOC ARL is considered best at detecting that particular shift. Following the studies of Lowry et al. (1992), Rigdon (1995), and Prabhu and Runger (1997), among others, in order to study the robustness of the MEWMA chart, the UCL (h) was determined to give an IC ARL 200 under multivariate normality. In-Control Performance Comparisons We start with a detailed discussion of chart performance for bivariate observations, and then discuss higher dimensions using less detail, as the discussion generalizes. Table 1, given at the end of this article, provides a detailed tabulation of ARL values for bivariate MEWMA and Hotelling s χ 2 charts. The columns correspond to λ = 0, 0.05, 0.1, 0.5, 1, 1.5, 2, and 3, with λ = 0 corresponding to IC ARL values. Table 1 is organized into panels (sub-tables), each corresponding to a different distribution. On the left are gamma distributions with shape parameters α = 1 and 16. At the bottom left is the multinormal distribution. On the right are t distributions with ν = 3, 6, and 100. Within each panel are rows for the same set of values for r. For example, the top left panel corresponds to the highly skewed Gam 2 (1,1) distribution. For this distribution, as expected, the IC ARL is greatly reduced when r is large. The IC ARL of the χ 2 chart is only Similarly, with the t 2 (3) distribution the IC ARL is only 41.3, which indicates many more false alarms than with normal observations. On the other hand, as the shape parameter increases or the degrees of freedom increase, the distributions approach normality, and the IC ARL of the MEWMA chart approaches 200, regardless of the value of r. One of the major findings of our study is that, although the IC ARL of Hotelling s χ 2 chart is highly sensitive to violations of multivariate normality, for an appropriate choice of r, the IC ARL of the MEWMA chart is robust even to highly non-normal distributions. For example, consider the row for r = , with IC ARL values of and for the highly skewed Gam 2 (1,1) distribution and the extremely heavy-tailed t 2 (3) distribution, respectively. Values larger than show some reduction in the IC ARL for these extreme distributions, with the difference decreasing as the distribution approaches normality. For example, with r = 0.1 and Gam 2 (16,1) or t 2 (20) the IC ARL is or 192.6, respectively, both very close to the multinormal IC ARL. However, where the process data may follow a distribution that is extremely non-normal, the value of r = 0.1 may not be sufficiently small to completely guard against an excessive number of false alarms. Thus, for bivariate data, we recommend using a value for r between 0.02 and 0.05 when there is concern about non-normality. Out-of-Control Performance Comparisons We now proceed to the OOC chart performance. The multivariate t distribution is elliptically symmetrical, and the MEWMA chart s performance depends on the shift only through λ. This issue is discussed in Lowry, et al. (1992) for the multinormal case. Without loss of generality, the shift was in the direction e 1 = (1, 0,, 0) T, with the identity covariance matrix. 3364

4 With non-elliptically symmetrical distributions, the shift direction matters. As shown in Figure 1, the directional sensitivity is evident with skewed univariate data, a point not noted in some previous robustness studies. For example, with and r = 1 and an IC ARL of 200 with normal data, with Gam 1 (1,1) data the IC ARL decreases to The OOC ARL increases to an average of 69.6 for a shift of ±1, and further increases to for a shift of ±1.5, before decreasing to 5.8 for a shift of ±2. Note that the OOC ARL can exceed even the IC ARL with normal observations ARL Shift Figure 1. Univariate ARL versus Shift, r = 1, Gam 1 (1,1) We averaged the ARL over all possible shift directions, using a uniform prior distribution for the direction. The ARL for a specific direction can differ considerably from the average, sometimes exceeding the IC ARL. However, the purpose of this article is not so much to chronicle poor performance with non-normal data, as it is to identify the way to consistently achieve excellent performance, so reporting the average suffices for the latter purpose. While the performance is invariant to a change in the covariance matrix of the actual multivariate gamma observations, it is not invariant to the covariance matrix of the implicit Wishart distribution. Since we are not aware of any reasonable model for the distribution of this Wishart covariance matrix, especially in higher dimensions, we did not average the results over such a distribution. Instead, we used the identity matrix for the simulations, realizing the details, but not the general conclusions, depend on the choice. In addition to IC ARL values, Table 1 gives OOC ARL values for bivariate observations. The panel for multinormal observations confirms that the MEWMA chart is much more effective than Hotelling s χ 2 chart at detecting small and moderate-sized shifts. It is well documented in the SPC literature that the MEWMA chart can be designed to give excellent performance over a range of shifts. Therefore, the main objective of our analysis is to compare the non-normal ARL values with the respective ones for the normal distribution. We now consider the choice of r to best detect shifts of various sizes in the context of various distributions. For a given distribution and shift size, there is a unique optimal r, for which the corresponding MEWMA chart has the smallest OOC ARL among all MEWMA charts with the same IC ARL. Using a non-optimal r increases the ARL, the increase depending on the size of the shift and the distribution. For the values of λ and r used in the bivariate simulations, the increases are tabulated in Table 2 for multinormal observations. For example, for a 0.05 shift, the best r (among those shown) is , with an ARL of Increasing r to increases the ARL to 182.1, a very small increase of 4.8 for that combination of r and λ. For each value of r, there is a unique λ for which the increase is worst. For example, with r = the increase is greatest (among those shown) at 11.0 for a shift size of 0.1. In practice, the exact shift size is seldom known in advance. Instead, a range of shift sizes should be effectively detected, and often the range is not bounded above. From this perspective, the r = 0.1 chart detects all shifts of 0.5 or greater with the minor increase in ARL of no more than 1.5. At 0.046, the upper end of our recommended range for r, the maximum increase is only about 1.8. Thus, although this value is smaller than typically recommended, the inertia problem only amounts to at most 1.8 observations, which seems very small. At 0.022, the lower end of our recommended range, the maximum increase is merely about 3.2. Since the chart is robust to non-normality, the same results generally apply regardless of the distribution of the observations. This is the basis for our recommendation, namely that performance is very near optimal regardless of (1) the distribution of the data, (2) the size of the subgroup, or (3) the magnitude of the shift. Table 2. Out-of-Control ARL Increase Shift Magnitude λ r A major conclusion is that a suitable choice of r gives robust OOC as well as robust IC performance. Consider the values for r = For shifts of any size, the ARL is very nearly the same regardless of the distribution, depending only on the size of the shift. Even with extremely non-normal observations, this value of the smoothing parameter gives nearly the same performance regardless of the third or fourth moment of the distribution, distribution-free performance in a 3365

5 sense. Thus, we recommend using a value for r between 0.02 and 0.05 for a bivariate MEWMA chart, where there is concern about non-multinormal process data. The univariate case and some higher dimensional cases are discussed in the following section. Other Dimensions We also conducted simulations for various other dimensions besides the bivariate case, although space does not allow tabulation of these results. In the univariate case, for r = 1, the decrease in IC ARL with t 1 (3) data is only 70%, compared to 79% for t 2 (3) data. On the other hand, with λ = 1, the OOC ARL increases by 19% compared to a 29% decrease for t 2 (3) data. With ten-dimensional data from the highly skewed Gam 10 (1,1) distribution, the IC ARL decreases by 19% for r = 0.46, the upper end of the range we recommend for bivariate data. For r = 0.22, the decrease reduces to only 3.5%, and becomes negligible a smaller r. With t 10 (3) data, the changes in the IC ARL are, respectively, a 22% decrease and an increase of merely 0.9%. Thus, our recommended range of r between 0.02 and 0.05 offers robust performance to the MEWMA control chart up to p = 10, even with extremely non-multinormal process data. This is completely unlike Hotelling s χ 2 chart, which generates an excessive number of false alarms with non-multinormal data of all dimensions. Concluding Remarks This article has demonstrated that Hotelling s χ 2 chart is not effective at detecting small and moderatesized shifts in the mean vector of a multivariate quality variable x. Moreover, designing the individuals χ 2 chart under the assumption of multivariate normality, when the process observations actually follow a multivariate t or gamma distribution, results in a much smaller in-control ARL, which leads to a much larger number of false alarms than anticipated. The out-ofcontrol performance also deteriorates, with the ARL increasing compared with that of the same sized shift and multinormal observations. For practical applications, the MEWMA control chart offers much better performance than the χ 2 chart. Furthermore, unlike the χ 2 chart, the MEWMA chart that is based on the asymptotic covariance matrix can be designed to be robust to non-normality for both incontrol and out-of-control data across a wide range of shifts. Although in most cases, a smoothing parameter of r between 0.02 and 0.05 gives robust performance, for applications in excess of five dimensions, a value for r of 0.02 or smaller will ensure performance comparable to that with multinormal data, even for highly skewed and extremely heavy-tailed non-normal distributions. Furthermore, since the use of individual observations represents the case most sensitive to nonnormality, by extension, the MEWMA chart can be made to be robust for subgroups of any size. The need for robust, distribution-free multivariate SPC has been noted in a number of recent articles (see, e.g., Woodall and Montgomery (1999), Stoumbos et al. (2000), and references therein). In particular, Coleman (1997) noted that he would never believe the multivariate normal assumption for industrial data, and that even if he wanted to, he cannot believe that there are tests for multivariate normality with sufficient power for practical sample sizes that he would even bother to use them. He concluded that, distribution-free multivariate SPC is what we need. The performance of an appropriately designed MEWMA control chart comes very close to being distributionfree. In many practical applications, the MEWMA chart would be more appealing than multivariate nonparametric control charts, which are less powerful, more computationally intensive, and generally do not apply to skewed distributions (see, e.g., Stoumbos and Jones (2000) and Stoumbos et al. (2000)). Acknowledgement Zachary G. Stoumbos work was funded in part by the Law School Admission Council (LSAC) and by 2000 and 2001 Rutgers Faculty of Management Research Fellowships. The opinions and conclusions contained in this publication are those of the authors and do not necessarily reflect the position or policy of LSAC. References Alt, F.B. and Smith, N.D. (1988), Multivariate Process Control, in Handbook of Statistics, Vol. 7, Eds. P. R. Krishnaiah and C. R. Rao, Amsterdam: Elsevier Science. Borror, C. M., Montgomery, D. C., and Runger, G. C. (1999), Robustness of the EWMA Control Chart to Nonnormality, Journal of Quality Technology, 31, Coleman, D. E. (1997), Individual Contributions, in A Discussion on Statistically-Based Process Monitoring and Control, Eds. D. C. Montgomery and W. H. Woodall, Journal of Quality Technology, 29, Hotelling H. (1947), Multivariate Quality Control, in Techniques of Statistical Analysis, Eds. C. Eisenhart, M. W. Hastay, and W. A. 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6 Weighted Moving Average Control Chart, Technometrics, 34, Mason, R. L., Champ, C. W., Tracy, N. D., Wierda, S. J., and Young J. C. (1997), Assessment of Multivariate Process Control Techniques, Journal of Quality Technology, 29, Montgomery, D. C. (1997), Introduction to Statistical Quality Control (3rd ed.), New York: Wiley. Reynolds, M. R., Jr., and Stoumbos, Z. G. (2002), Control Charts and the Efficient Allocation of Sampling Resources, preprint. Prabhu, S. S., and Runger, G. C. (1997), Designing a Multivariate EWMA Control Chart, Journal of Quality Technology, 29, Rigdon, S. E. (1995), An Integral Equation for the In- Control Average Run Length of a Multivariate Exponentially Weighted Moving Average Control Chart, Journal of Statistical Computation and Simulation, 52, Ryan, T. P. (2000), Statistical Methods for Quality Improvement (2nd ed.), New York: Wiley. Stoumbos, Z. G., and Jones, L. A. (2000), On the Properties and Design of Individuals Control Charts Based on Simplicial Depth, Nonlinear Studies, 7, Table 1. ARL Values for Bivariate MEWMA and Hotelling s χ 2 (r = 1) Charts Stoumbos, Z. G., Jones, L. A., Woodall, W. H., and Reynolds, M. R., Jr. (2000), On Shewhart-Type Nonparametric Multivariate Control Charts Based on Data Depth, in Frontiers in Statistical Quality Control 6, eds. H.-J. Lenz and P.-Th. Wilrich, Heidelberg, Germany: Springer-Verlag, pp Stoumbos, Z. G., and Reynolds, M. R., Jr. (2000), Robustness to Non-normality and Autocorrelation of Individuals Control Charts, Journal of Statistical Computation and Simulation, 66, Stoumbos, Z. G., Reynolds, M. R., Jr., Ryan, T. P., and Woodall W. H. (2000), The State of Statistical Process Control as We Proceed into the 21st Century, Journal of the American Statistical Association, 95, Stoumbos, Z. G., and Sullivan, J. H. (2002), Robustness to Non-normality of the Multivariate EWMA Control Chart, Journal of Quality Technology, 34, Wierda, S. J. (1994), Multivariate Statistical Process Control Recent Results and Directions for Future Research, Statistica Neerlandica, 48, Woodall W. H., and Montgomery, D. C. (1999), Research Issues and Ideas in Statistical Process Control, Journal of Quality Technology, 31, Shift Magnitude λ Shift Magnitude λ r r Gam 2(1,1) t 2(3) Gam 2(16,1) t 2(6) Bivariate Normal t 2(100)