A Multivariate EWMA Control Chart for Skewed Populations using Weighted Variance Method

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1 OPEN ACCESS Int. Res. J. of Science & Engineering, 04; Vol. (6): 9-0 ISSN: RESEARCH ARTICLE A Multivariate EMA Control Chart for Skewed Populations using eighted Variance Method Atta AMA *, Shoraim MHA and Yahaya SSS Department of Economics and Policy Sciences, College of Commerce and Economics, Hodeida University, Hodeida, Yemen. Department of Mathematic and Statistics, School of Quantitative Sciences, Universiti Utara Malaysia, Sintok, Malaysia *Corresponding author abduatta@yahoo.com Manuscript Details Received : Revised : Accepted :..04 Published: ISSN: Editor: Dr. Arvind Chavhan Cite this article as: Atta AMA, Shoraim MHA and Yahaya SSSA. Multivariate EMA Control Chart for Skewed Populations using eighted Variance Method Int. Res. J. of Sci. & Engg., 04; (6):9-0. ABSTRACT This article proposes Multivariate Exponential eighted Moving Average control chart for skewed population using heuristic eighted Variance (V) method, obtained by decomposing the variance into the upper and lower segments according to the direction and degree of skewness. This method adusts the variance-covariance matrix of quality characteristics. The proposed chart, called V- hereafter, reduces to standard multivariate Exponential eighted Moving Average control chart (standard ) when the underlying distribution is symmetric. Incontrol and out-of-control ARLs of the proposed V- control chart are compared with those of the weighted standard deviation Exponential eighted Moving Average (SD-) and (standard ) control charts for multivariate normal, lognormal and gamma distributions. In general, the simulation results show that the performance of the proposed V- chart is better than SD- and charts when the underlying distributions are skewed. Keywords: ; skewed populations; average run length (ARLs); eighted Variance.. INTRODUCTION Copyright: Author(s), This is an open access article under the terms of the Creative Commons Attribution Non-Commercial No Derivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is noncommercial and no modifications or adaptations are made. In the quality of the process, there are many cases determined by the oint level of several quality characteristics. For example the quality of chemical process may depend on the oint level of viscosity, concentration, molecular weight, and ph (Chang,007). Since these quality characteristics are clearly correlated and monitoring them separately will be misleading. Thus it is desirable to have control charts that can monitor multivariate measurement simultaneously. The most widely used multivariate control charts are the Hotelling s T, multivariate CUSUM (MCUSUM) and multivariate EMA () charts. Many researchers have developed the chart, however a potential disadvantage of the control charts is the normality assumption of the underlying process distribution. 04 All right reserved 9

2 Atta et al., 04 In practice, the normality assumption is usually difficult to ustify and is often violated. For instance, the measurements from chemical process, filling processes, and semiconductor processes are often skewed. See (Khoo et al., 009). As skewness increases, the incontrol ARL of conventional chart decreases. Here, we will discuss some extensions of the chart; (Yumin, 996) investigated the chart with the generalized smoothing parameter matrix, where the principal components of the original variables are used to construct the chart. (Sullivan and oodall, 996) recommended the use of a chart for a preliminary analysis of multivariate observations. (Prabhu and Runger,997) provided recommendations for the selection of parameters for a chart. They used the Markov chain method to provide design recommendations for a chart that parallels many of the results provided by (Lucas and Saccucci, 990) for the univariate EMA chart. (Runger et al., 999) showed how the shift detection capability of the chart can be significantly improved by transforming the original process variables to a lower dimensional subspace through the use of a U transformation. The U transformation is similar to the option of principal components, but it allows the user to define the specific subspace to be monitored. (Stoumbos and Sullivan, 00) investigated the effects of non-normality on the statistical performance of the chart. They showed that the chart can be designed to be robust to non-normality and very effective in detecting process shifts of any size or direction, even for highly skewed and extremely heavy-tailed multivariate distributions. (Kim and Reynolds, 005) investigated the chart for monitoring the mean vector when the sample sizes are unequal. (Pan, 005) proposed and examined the sensitivity of a chart when the distribution of the chart s statistic is derived from the Box quadratic form. (Yeh et al., 005) proposed a chart that monitors changes in the variance-covariance matrix of a multivariate normal process for individual observations. (Reynolds and Kim,005) showed that the chart based on sequential sampling is more efficient in detecting changes in the mean vector than standard control charts based on nonsequential sampling. (Lee and Khoo, 006) proposed the optimal statistical designs of the chart based on ARL and MRL. (Hawkins et al.,007) proposed a general chart in which the smoothing matrix is full, rather than having only diagonal elements. In additional to these extensions, (Chang, 007) proposed MCUSUM and charts based on weighted standard deviation method (SD) to solve the problem of multivariate skewness. hile in this article, another chart is developed by using weighted variance method (V) suggested by (Bai and Choi,995) and this newly developed chart (V-) is a multivariate extension of the chart proposed by (Khoo and Atta,008). The section and 3 review the multivariate eighted Deviation (SD) and the proposed multivariate eighted Variance (V) methods respectively, followed by SD- and proposed V- charts in section 4 and 5. In section 6, the performance of the proposed chart is compared with SD- and standard charts in terms of in-control and out-of-control ARLs for multivariate lognormal and gamma distributions. Finally, conclusion is drawn in section 7.. eighted Deviation (SD) Method Chang and Bai (004) proposed the SD method to modify the variance-covariance matrix of each quality characteristic. Assume that p-variate random vector ( p ) ( µ, L, µ ) T p,, T = L is distributed with mean µ = and variance-covariance σ ρ σσ L ρ σσ M σp p p σ L ρ pσσ p =. () O Here, T denotes the transpose of a vector or matrix, σ is the standard deviation of, and ρ i is the correlation coefficient of iand. The modified variance-covariance matrix of P multivariate normal distribution for approximation, is. ( ) σ ρ σ σ L ρ pσ σp ( σ ) = = () O M L ρ pσ σp ( σp ) where = diag {,,..., p }, 9

3 A Multivariate EMA Control Chart for Skewed Populations using eighted Variance Method, if i = θ >µ ( θ ), otherwise σ = σ, θ = Pr ( µ ). (3) Note that, if i is greater than µ, the PDF related to i is modified by adusting the ith row and th column of variance-covariance matrix using θ σ in place of θ σ. σ. Otherwise, the ith row and th column of variancecovariance matrix is adusted by ( ) However, the correlation matrix ρ = { ρ i } ρ = does not change even though the variance-covariance matrix is adusted. The SD method approximates the original PDF with segments from p multivariate normal distributions. See (Chang and Bai 004) for more details. 3. eighted Variance (V) Method Bai and Choi (995) proposed the eighted Variance (V) method to setting up control limits of R and R charts for skewed populations. (Khoo et al.,009) proposed S and S charts for skewed populations using eighted Variance (V) method. An EMA chart using eighted Variance (V) method of skewed population was suggested by (Khoo and Atta, 008). The V method decomposes the variance into two parts while the SD method decomposes the standard deviation into two parts. All the three charts used V method for univariate case (involve one quality characteristic) only. However, in this paper we will extend the univariate V method to multivariate version by adusting the variance-covariance matrix with Vs of each quality characteristic. Since the correlation matrix represents the dependent structure of quality characteristics, we should not tamper with it, and a multivariate control chart must reflect this dependency. In multivariate quality characteristics, the V method uses the same approach as the SD method, except that equation 3 will be modified by using square root, because the V method decomposes the variance into two parts. Consider that the p-variate random vector = ( p ),, T L is distributed as in (), with the same mean and variance-covariance matrix.the modified variance-covariance matrix of P multivariate normal distribution for approximation, is ( ) Q Q Q Q Q σ ρ σ σ L ρ pσ σ p Q ( σ ) = Q Q = (4) O M Q ( σp ) Q Q Q L ρ pσ σp { Q Q Q } Q= diag,,..., p, θ, if i >µ Q = ( θ ), otherwise (5) Note that, in variance-covariance matrix in 4, σ Q = Q σ and θ Pr ( i µ ) = is the probability that i is less than or equal to its mean. If i is greater than µ, the PDF related to i is modified by adusting the ith row and th column of variance-covariance matrix using θ σ in place of σ. Otherwise, the ith row and th column of variancecovariance matrix is adusted by ( ) the correlation matrix ρ = { ρ i } θ σ. Also, ρ = does not change even though the variance-covariance matrix is adusted. 4. The SD- Chart (Chang, 007) developed the SD- chart to improve the performance of the standard chart for skewed populations. The SD- chart s statistic (Chang, 007) is defined as M = λ Z + ( I λ) M i i i where 0 =, for i =,,,(6) M 0 and = { λ, λ,..., } 0 < λ, for =,,, p. λ diag λ p, Since Z i follows a multivariate normal distribution with mean vector 0 and variance-covariance matrix ρ approximately, the asymptotic variance-covariance matrix of M is Σ = λ ( λ) i M ρ, if we assume that there is no a priori reason to weight past observations differently for the p variables. Note that in equation 6, thezi for the th. element as defined by (Chang, 007) is. Int. Res. J. of Science & Engineering, 04; Volume, No. 6, November December,

4 Atta et al., 04 Z i µ θ = = σ n In Equation (7), and Z ( θ ) Z, if i i Z, otherwise i >µ. (7) i i = (8) σ µ n, if i = θ > µ ( θ ), otherwise (9) here i represents the mean of quality characteristic, in sample i, n is the sample size, I denotes the identity matrix and ρ the correlation matrix of. An out-ofcontrol signal is detected when the chart s statistic, M M (0) ( ) E i = i Σ M i exceeds the control limit, h E. The value of h E is determined based on a desired in-control ARL when the underlying process is assumed to follow a multivariate normal distribution. Therefore, the control limit of the chart can be used approximately for the chart. The SD chart reduces to the chart for symmetric distributions (Chang,007). 5. The Proposed eighted Variance (V) Chart (Khoo and Atta,008) proposed the univariate V- EMA chart to improve the performance of the univariate SD-EMA and standard EMA charts for skewed populations. In this article we will use V method to improve the performance of the SD- and standard charts for skewed populations. The chart statistic is defined as differently for the p variables, the asymptotic variancecovariance matrix of M is Σ = λ ( λ) TheZ in equation for the th element is Z Q i Q i i µ = = Q σ n such that Z and Q i θ ( θ ) M ρ. Z, if i i Z, otherwise i >µ () i i = (3) σ µ n θ, if i >µ Q = ( θ ), otherwise (4) In Equation (), i is the mean of quality characteristic in sample i and n is the sample size, =, µ and σ are the target while θ Pr ( i µ ) values of the mean and standard deviation of quality characteristic, while, I denotes the identity matrix and ρ the correlation matrix of. An out-of-control signal is detected when the chart s statistic, ( M ) Q Q Q E i = i MM i Σ (5) exceeds the control limit, h E, where h E is determined based on a desired in-control ARL when the underlying process is assumed to follow a multivariate normal distribution. Therefore, the control limit of the chart can be used approximately for the chart. As in SD chart, the V chart also reduces to the chart for symmetric distributions (Chang, 007). ( λ) M = λ Z + I M, fori =,,, () Q Q Q i i i M and = { λ, λ,..., } where Q 0 =0 λ diag λ p, 0 < λ, for =,,, p. As long as, Z follows a multivariate normal distribution with mean vector 0 and variance-covariance matrixρ,if we assume that there is no a priori reason to weight past observations 94 Q i 6. Performance of the Proposed V- control chart In this section, we compare the performance of the proposed V- chart with that of chart and SD- chart developed by (Chang, 007). These charts are compared based on the in-control and out-of-control Average Run Length

5 A Multivariate EMA Control Chart for Skewed Populations using eighted Variance Method (ARLs). The out-of-control (ARL) is computed under zero state mode, where a mean shift is assumed to occur immediately after inspecting an entire sample or at the beginning of a new sample. In this paper,the bivariate normal, bivariate lognormal and Cheriyan and Rambhadran s bivariate gamma distribution (Cheriyan, 94; Rambhadran, 95) are considered in the computation of the in-control ARLs. As for the out-ofcontrol ARLs computation, only bivariate lognormal distribution is considered because of space constraint. Note that, the lognormal distribution can only represent various skewnesses and correlations, but the gamma distribution can only represent some positive correlations, see (Kotz et al., 000) for more details about these distributions. Here, for the convenience location parameters of (0, 0) are chosen for lognormal distribution, since the skewness does not depend on it, while the scale parameters of (, ) are chosen for gamma distribution. It is also assumed that the nominal in-control ARL is equal to 370.4, the same as in the univariate 3- sigma Shewhart control chart. The limits of the, SD- and V- control charts are designed to be optimal for a µ. Note that the process mean shift of size d ( ) = optimal =diag{ 0.5, 0,5} λ and the corresponding limit for bivariate case of these charts is h = 0.4 E. These optimal parameters are determined using the method discussed by (Lee and Khoo,006). The correlation coefficient ρ = 0.3, 0.5 and 0.8 are considered for the in-control ARLs computations, while for the out-of-control computations, ρ = 0.5 is considered. It should also be noted, the selected parameters of the quality characteristics (, ) from the bivariate lognormal distribution are chosen to give the desired skewness( α, α ). For the bivariate gamma distribution, the shape parameters of (, ) (, ) are determined so that the desired skewness α α are achieved. The in-control ARLs are computed for (, ) α α ϵ {(,), (,), (,3), (,), (,3), (3,3)} with sample size, n ϵ {,3,5,7}, while the out-ofcontrol ARLs are computed for (, ) α α ϵ {(,), (,), (3,3)} with sample size ofn = 5. Six direction of the process mean shifts from = (, ) (, ) = µ + δ σ µ + δ σ µ to µ µ 0 µ are considered and the size of the mean shift is d µ µ µ 0 µ µ 0. ( µ ) = ( µ µ ) Σ ( µ µ ) The six directions of mean shifts are similar to those considered by (Chang, 007) (see Figure). Note that, the process is in-control if δ = δ = 0, and it will be out-of-control if at least δ 0 for = and. The incontrol and out-of-control ARLs are obtained from Monte Carlo simulations using SAS program version 9.3, where all the results are averages of 0000 replications. The in-control ARLs of the control charts are given in tables, and 3. cells marked as (*) in tables and 3 for Cheriyan and Rambhadran s bivariate gamma distribution cannot be computed because the corresponding shape parameters of one of the gamma distributed components used in the transformation to compute variate have negative values (a) (b) Figure : Directions of a mean shift when (a) only µ is shifted (b) both µ and µ are shifted Int. Res. J. of Science & Engineering, 04; Volume, No. 6, November December,

6 Atta et al., Table : In-Control ARL for bivariate distributions when ρ = 0.3. Distribution ( γ, γ ) Sample size, n Normal (0, 0) Log normal (, ) (, ) (, 3) (, ) (, 3) (3, 3) Gamma (, ) (, ) (, 3) (, ) (, 3) (3, 3) Atta et al.,

7 A Multivariate EMA Control Chart for Skewed Populations using eighted Variance Method Int. Res. J. of Science & Engineering, 04; Volume, No. 6, November December, Table : In-Control ARL for bivariate distributions when ρ = 0.5. Distribution ( γ, γ ) Sample size, n Int. Res. J. of Science & Engineering, 04; Volume, No. 6, November December, Normal (0, 0) Lognormal (, ) (, ) (, 3) (, ) (, 3) (3, 3) Gamma (, ) (, ) (, 3) * * * * * * * * * * * * (, ) (, 3) (3, 3) A Multivariate EMA Control Chart for Skewed Populations using eighted Variance Method

8 Atta et al., Table 3. In-Control ARL for bivariate distributions when ρ = 0.8 Distributio n ( γ, γ ) Sample size, n Normal (0, 0) Lognormal (, ) (, ) (, 3) (, ) (, 3) (3, 3) Gamma (, ) (, ) * * * * * * * * * * * * (, 3) * * * * * * * * * * * * (, ) (, 3) * * * * * * * * * * * * (3, 3) Atta et al.,

9 A Multivariate EMA Control Chart for Skewed Populations using eighted Variance Method Int. Res. J. of Science & Engineering, 04; Volume, No. 6, November December, Table 4: Out of control ARL for Cases and when only µ is shifted, ρ = 0.5 and n =5 d ( µ ) Case ( ) γ,γ ( ) d µ Case ( ) Int. Res. J. of Science & Engineering, 04; Volume, No. 6, November December, γ,γ 0.5 (, ) (, ) (, ) (, ) (3, 3) (3, 3) (, ) (, ) (, ) (, ) (3, 3) (3, 3) (, ) (, ) (, ) (, ) (3, 3) (3, 3) (, ) (, ) (, ) (, ) (3, 3) (3, 3) (, ) (, ) (, ) (, ) (3, 3) (3, 3) (, ) (, ) (, ) (, ) (3, 3) (3, 3) (, ) (, ) (, ) (, ) (3, 3) (3, 3) (, ) (, ) (, ) (, ) (3, 3) (3, 3) A Multivariate EMA Control Chart for Skewed Populations using eighted Variance Method

10 Atta et al., 04 Table 5. Out Of Control ARL for Case 3,4,5 and 6 when µ and µ are shifted,ρ = 0.5 and n =5 d ( µ ) ( ) Case γ,γ d ( µ ) Case ( γ,γ ) (, ) (, ) (, ) (, ) (3, 3) (3, 3) (, ) (, ) (, ) (, ) (3, 3) (3, 3) (, ) (, ) (, ) (, ) (3, 3) (3, 3) (, ) (, ) (, ) (, ) (3, 3) (3, 3) (, ) (, ) (, ) (, ) (3, 3) (3, 3) (, ) (, ) (, ) (, )..0.3 (3, 3) (3, 3) (, ) (, ) (, ) (, ) (3, 3) (3, 3) (, ) (, ).4..5 (, ) (, )..0.4 (3, 3) (3, 3) (, ) (, ) (, ) (, ) (3, 3) (3, 3) (, ) (, ) (, ) (, ) (3, 3) (3, 3) (, ) (, ) (, ) (, ) (3, 3) (3, 3) (, ) (, ) (, ) (, ) (3, 3) (3, 3) (, ) (, ) (, ) (, ) (3, 3) (3, 3) (, ) (, ) (, ) (, ) (3, 3) (3, 3) (, ) (, ) (, ) (, ) (3, 3) (3, 3) (, ) (, ) (, ) (, )

11 A Multivariate EMA Control Chart for Skewed Populations using eighted Variance Method Additionally, when ρ = 0.8, table 3 presents that the V- chart outperforms the SD- and standard charts in terms of in-control ARLs for all of the skewnesses levels and sample sizes, except for the case when the sample size of lognormal distribution equals, where SD- chart has large in-control ARLs. In general, the proposed V- chart has the largest in-control ARLs than the SD- and standard charts for all levels of skewnesses (, ) α α and sample sizes, n, and also all of the correlations,ρ = 0.3, 0.5 and 0.8. Tables 4 and 5 show the out-of-control ARLs of the control charts. As could be observed, the proposed V- chart outperforms the SD- chart in cases and 3, and it has lower out-of-control ARLs than the standard chart in cases,4,5and 6 when d ( µ ) ò ( 0.5, 0.75,,.5,.5,.75, and 3). The tables also show that the V- chart has about the same out-of-control ARLs as compared to SD- chart in cases,4,5 and 6. Overall, the proposed V- chart is found to have the most favourable performance among the charts considered in this study. The proposed V- chart gives large in-control ARLs compared to the other charts for skewed distributions. The proposed V- chart also provides the comparable performance in terms of out-of-control ARLs, for positive and negative directions of shifts (cases,,3,5 and 6). CONCULSION d ( µ ) ò ( 0.5, 0.75,,.5,.5,.75, and 3). From the findings, we can assure that this chart (V- ) can be a good alternative to the SD- and standard charts for skewed populations. REFERENCES. Bai DS and Choi IS. and R control charts for skewed populations, Journal of Quality Technology, 995;7: Chang YS and Bai DS. A multivariate T control chart for skewed populations using weighted standard deviations, Quality and Reliability Engineering International, 004;0: Chang YS. Multivariate CUSUM and EMA control charts for skewed populations using weighted standard deviations, Communications in Statistics Simulation and Computation, 007;36: Cheriyan KC. A bivariate correlated gamma type distribution function, Journal of the Indian Mathematical Society, 94;5: Hawkins DM, Choi S and Lee S. A general multivariate exponentially weighted moving average control chart, Journal of Quality Technology, 007;39: Khoo MBC, Atta AMA and Zhang u. A Multivariate Synthetic control chart for monitoring the process mean vector of skewed populations using weighted standard deviations, Communications in Statistics Simulation and Computation, 009; 38: This article proposed a simple heuristic V- chart for skewed populations to overcome the problem of low in-control ARLs due to the violation of normality assumptions. The proposed method adusts the variance-covariance matrix to reflect the skewness. This chart reduces to the standard chart when the underlying distribution is symmetric. This study showed how the standard and SD- control charts performance can significantly enhanced by using the V approach in the construction of V- chart. The simulation study showed that the V- chart has larger in-control ARLs as compared to SD- and standard charts for almost all levels of skewnesses (, ) α α and sample sizes, n =,3,5 and 7 under lognormal and gamma distributions. The simulation study also showed that the proposed V- chart has good out-ofcontrol ARLs regardless of the levels of skewnesses (, ) α α and magnitude of shifts, 7. Khoo MBC, Atta AMA. and Chen C-H. Proposed and S control charts forskewed distributions, Proceedings of the International Conference on Industrial Engineering and Engineering Management (IEEM 009), Dec. 009 pp , Hong Kong. 8. Khoo MBC. and Atta AMA. An EMA control chart for monitoring the mean of skewed populations using weighted variance, Proceedings of the International Conference on Industrial Engineering and Engineering Management (IEEM 008), Dec. 008 pp. 8-3, Singapore. 9. Kim K and Reynolds Jr M R. Multivariate monitoring using an control chart with unequal sample sizes, Journal of Quality Technology, 005;37: Kotz S, Balakrishnan N and Johnson NL.Continuous multivariate distributions, Vol. ( nd edition). John iley and Sons, Inc., New York Lee MH and Khoo MBC. Optimal statistical design of a multivariate EMA chart based on ARL and MRL, Communications in Statistics Simulation and Computation, 006;35 : Int. Res. J. of Science & Engineering, 04; Volume, No. 6, November December, 04. 0

12 Atta et al., 04. Lucas JM and Saccucci MS. Exponentially weighted moving average control schemes: properties and enhancements, Technometrics, 990;3:. 3. Pan. An alternative approach to multivariate EMA control chart, Journal of Applied Statistics, 005;3: Prabhu SS and Runger GC. Designing a multivariate EMA control chart, Journal of Quality Technology, 997;9: Rambhadran VR. A multivariate gamma type distributions, Sankhya, 95;: Reynolds MR. and Kim K. Multivariate monitoring of the process mean vector with sequential sampling, Journal of Quality Technology, 005;37: Runger GC, Keats B, Montgomery DC and Scanton RD.Improving the performance of the multivariate exponentially weighted moving average control chart, Quality and Reliability Engineering International, 999;5: Stoumbos ZG and Sullivan JH. Robustness to nonnormality of the multivariate EMA control chart, Journal of Quality Technology, 00;34: Sullivan JH and oodall H. A control chart for preliminary analysis of individual observations, Journal of Quality Technology, 996;8: Yeh AB, Huwang Land u C.A multivariate EMA control chart for monitoring process variability with individual observations, IIE Transactions, 005;37: Yumin L. An improvement for in multivariate process control, Computers and Industrial Engineering, 996;3: Published by IRJSE 0