Songklanakarin Journal of Science and Technology SJST R1 Sukparungsee

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1 Songklanakarin Journal of Science and Technology SJST-0-0.R Sukparungsee Bivariate copulas on the exponentially weighted moving average control chart Journal: Songklanakarin Journal of Science and Technology Manuscript ID SJST-0-0.R Manuscript Type: Original Article Date Submitted by the Author: 0-Jan-0 Complete List of Authors: Kuvattana, Sasigarn ; King Mongkut's University of Technology, North Bangkok, Applied Statistics Busababodhin, Piyapatr; Mahasarakham University, Mathematics Areepong, Yupaporn; King Mongkut's University of Technology, North Bangkok, Applied Statistics Sukparungsee, Saowanit; King Mongkut's University of Technology, North Bangkok, Applied Statistics Keyword: ARL, copula, EWMA control chart, Monte Carlo simulation

2 Songklanakarin Journal of Science and Technology SJST-0-0.R Sukparungsee Page of SJST MANUSCRIPT TEMPLATE FOR A TEXT FILE Original Article Bivariate copulas on the exponentially weighted moving average control chart Abstract Sasigarn Kuvattana, Piyapatr Busababodhin, Yupaporn Areepong, and Saowanit Sukparungsee * Department of Applied Statistics, Faculty of Applied Science, King Mongkut s University of Technology North Bangkok, Bang Sue, Bangkok, 000 Thailand. Department of Mathematics, Faculty of Science, Mahasarakham University, Kantarawichai, Mahasarakham, 0 Thailand. * Corresponding author, address: saowanit.s@sci.kmutnb.ac.th This paper proposes four types of copulas on the Exponentially Weighted Moving Average (EWMA) control chart when observations are from an exponential distribution by using a Monte Carlo simulation approach. The performance of the control chart is based on the Average Run Length (ARL ) which is compared for each copula. Copula functions for specifying dependence between random variables are used and measured by Kendall s tau. The results show that the Normal copula can be used for almost all shifts. Keywords: ARL, copula, EWMA control chart, Monte Carlo simulation.

3 Page of Songklanakarin Journal of Science and Technology SJST-0-0.R Sukparungsee Introduction Control charts are statistical and visual tools that are used in the monitoring of the quality of production from industry manufacturing processes. They are designed and evaluated under the assumption of observations by the process. Multivariate Statistical Process Control (MSPC) is a valuable method when several process variables are being monitored (Prabhu and Runger, ). It is used as the relationships between random variables that are sensitive to assignable causes and they are poorly detected by univariate control charts on individual variables. Multivariate control charts are generalizations of their univariate counterparts (Mahmoud and Maravelakis, 0), e.g. the Hotelling T control chart first introduced by Hotelling (); the Multivariate Exponentially Weighted Moving Average (MEWMA) control chart proposed by Lowry et al. () and the Multivariate Cumulative Sum (MCUSUM) control chart suggested by Crosier (). The MEWMA and MCUSUM control charts are commonly used to detect small or moderate shifts in the mean vectors (see Midi and Shabbak, 0; Runger et al., ). Most multivariate detection procedures are based on a multinormality assumption and independence but many processes are often non-normality and correlated. Many multivariate control charts have a lack of related joint distribution but copulas can specify this property. The copula approach is a method for modeling nonlinearity, asymmetricality and tail dependence in several fields; it can be used in the study of dependence or association between random variables. The copula approach is based on a representation from Sklar s theorem (Sklar, ) and copula theory is the formalization of the separated correlation of a multivariate distribution from the marginal distributions that

4 Songklanakarin Journal of Science and Technology SJST-0-0.R Sukparungsee Page of make up the multivariate distribution. If two or more variables are correlated, a joint distribution can be constructed from the marginal distributions of variables. A bivariate copulas approach is the simplest case for the description of dependent random variables and it can apply to control charts (Kuvattana et al., 0). Recent papers have used copulas on control charts including: copula- based bivariate ZIP control chart (Fatahi et al., 0; Fatahi et al., 0); copula Markov CUSUM chart (Dokouhaki and Noorossana, 0); Shewhart control charts for autocorrelated and normal data (Hryniewicz, 0), and copulas of non-normal multivariate cases for the Hotelling control chart (Verdier, 0). According to above papers, differences in distribution can be found in control charts. In the manufacturing process, the time is used to represent some attributes or variable measures which are observed as consecutive events of concern. When the probability of the event in the next small time interval does not vary through time, the distribution of the time for an event is known as an exponential distribution. It is a continuous distribution and it is widely known in the monitoring of time for successive occurrences of events. This paper therefore presents work on the Multivariate Exponentially Weighted Moving Average control chart when observation are generated by an exponential distribution with the mean shifts and uses bivariate copulas for specifying dependence between random variables.. The Multivariate Exponentially Weighted Moving Average Control Chart The Multivariate Exponentially Weighted Moving Average (MEWMA) chart is a standard tool in statistical quality control introduced by Lowry et al. (). Given T

5 Page of Songklanakarin Journal of Science and Technology SJST-0-0.R Sukparungsee observations W, W,... from a d-variate Gaussian distribution N ( µ, Σ ), define recursively, for i=,,..., Z =Λ W + ( Λ) Z () i i i where Z 0 is the vector of variable values from the historical data, and Λ is a diagonal matrix with entries λ,...,. λ d The quantity to be plotted is where / T i i i i = Z Z () λ λ i = ( λ) i () when λ =... = λ = λ (0,), as assumed in the present paper. The control chart signals a shift in the mean vector when a desired in-control d T > h, where h is the control limit chosen to achieve i The performance of the MEWMA in detecting changes in the mean is generally measured by the Average Run Length (ARL ) as a function of the difference µ µ 0 between the target mean µ 0 and its real value µ. The ARL further depends on the degree of dependence between the variables, measured by the covariance matrix Σ, and the scalar charting weight λ associated to the past observations. Note that. if λ= in (), the MEWMA control chart statistic reduces to / T i Z i i i = Montgomery, c00). Z, the statistic used in the Hotelling T control chart (Runger et al., ;. In this paper consider the bivariate Exponentially Weighted Moving Average control chart and will extend to the multivariate case for future work.. Copula Function

6 Songklanakarin Journal of Science and Technology SJST-0-0.R Sukparungsee Page 0 of Considering a bivariate case, If H( x, y ) is the joint distribution of the random vector ( X, Y ), with continuous marginal distributions F( x) = H ( x, ), G(y) = H (, y) of X, Y, respectively, then there exists a unique copula Cuv (, ) such that ( ) H ( x, y) = C F( x), G( y); θ where C (.,.) is determined by where F defined as C u v H F u G v (, ) = ( ( ), ( )) (.), G (.) are quantiles functions of F(.), G (.), respectively, which are F [ ] : 0, { } F ( u) = inf x : F( x) u For the purposes of the statistical method, it is desirable to parameterize the copula function. Let θ denote the association parameter of the bivariate distribution, and there exists a copula C (Trivedi and Zimmer, 00). This paper focuses on the Normal copula and three types of Archimedean copulas which are the Clayton, Frank and Gumbel copulas, respectively because these copulas are well-known. (Genest and MacKay, ).. Normal copula The Normal copula is an elliptical copula. From the bivariate Normal distribution with zero means, unit variances and correlation matrix Σ, the Normal copula determined by: C( u, v; Σ) = Φ( Φ ( u), Φ ( v); Σ ); θ, ()

7 Page of Songklanakarin Journal of Science and Technology SJST-0-0.R Sukparungsee where Φ(., Σ ) is the bivariate normal cumulative distribution function, univariate normal cumulative distribution function, Φ is the Φ is the univariate normal inverse cumulative distribution function or quantiles function and θ = ( µ, µ, σ, σ, ρ) (Joe, 0).. Archimedean copulas Suppose that a class Φ of functions φ :[ 0,] [ 0, ] X Y X Y with a continuous, strict decrease, such that φ() = 0, φ ( t) < 0 and φ ( t) > 0 for all 0< t< (Genest and MacKay, ; Genest and Rivest, ; Nelsen, 00). There are three types of Archimedean copulas and these types are generated as follows:.. Clayton copula θ θ / θ θ C( u, v) = maxu ( + v, 0), φ( t) = ( t ) / θ ; θ [, ) \ 0... Frank copula θu θv θt ( e )( e ) e C( u, v) = ln( + ), φ( t) = ln( ) ; θ (, ) \ 0. () θ θ θ e e.. Gumbel copula θ θ / θ θ Cuv (, ) = exp( [( ln u) + ( ln v) ] ), φ( t) = [ lnt ( )] ; θ [, ). (). Dependence Measures for Data Theoretically, a parametric measure of the linear dependence between random variables is measured by correlation coefficient whereas nonparametric measures of dependence are measured by Spearman s rho and Kendall s tau. According to the earlier literature, the copulas can be used in the study of dependence or association between ()

8 Songklanakarin Journal of Science and Technology SJST-0-0.R Sukparungsee Page of random variables and the values of Kendall s tau are easy to calculate so this measure is used for observation dependencies. Let X and Y be continuous random variables whose copula is C then τ = Kendall s tau for X and Y is given by Cuv (, ) dcuv (, ) - where τ c is Kendall s tau of copula C and the unit square c I Ι is the product Ι Ι where Ι = [ 0,] and the expected value of the function C( u, v) of uniform (0,) random variables U and V whose joint distribution function is C, i.e., τ = ECU [ (, V )] (Nelsen, 00). Archimedean copula C is generated by φ, then c τ Arch φ( t) = dt + where φ ( t) τ Arch is Kendall s tau of Archimedean copula C (Genest and MacKay, ). From all above literature about dependence measures for data, we can be applied copula function as shown in Table. Table. Kendall s tau of copula function. Average Run Length and Monte Carlo Simulation The performance of a control chart is measured by the Average Run Length ( ARL ). ARL is the average number of points that must be plotted before a point indicates an out-of-control condition. ARL is classified into ARL 0 and ARL, where 0 ARL 0 is the Average Run Length when the process is in-control and ARL is the Average Run Length when the process is out-of-control (Busaba et al., 0). In this paper we used a Monte Carlo simulation by using R statistical software (see Yan, 00; Hofert and M achler, 0; M achler and Zurich, 0; Hofert et al.,

9 Page of Songklanakarin Journal of Science and Technology SJST-0-0.R Sukparungsee ) with,000 simulations and a sample size of,000. Observations were from an exponential distribution with parameter α =. A shift size is reported in terms of the quantity δ = µ µ 0 and large values of δ correspond to bigger shifts in the mean. The value δ = 0 and the process mean µ = are in-control. The process means are defined as.,,.,, and for out-of-control processes. The simulation experiments were carried out to assess the performance of the MEWMA control chart with λ = 0.0 and 0.. Copula estimations are restricted to the cases of dependence (positive and negative dependence). For all copula models, the setting θ corresponds with Kendall s tau. The level of dependence is measured by Kendall s tau values ( τ ) which are defined to 0. and -0. for moderate dependence.. Research Results The simulation results are presented in Table to and the results are only empiric. The different values of exponential parameters denote µ for the variables X and µ for the variables Y. For in-control process, the MEWMA control chart was chosen by setting the desired ARL 0 = 0 for each copula. Tables and show moderate positive dependence ( τ = 0.), and Tables and show moderate negative dependence ( τ = 0.). Tables to show that the ARL values of λ= 0.0 give less than λ= 0. for all cases. The result in Table shows the mean shifts of τ = 0. when µ is fixed at, the ARL values of the Normal copula are less than the other copulas for small shifts ( µ =,. µ ) and the ARL values of the Gumbel copula are less than the other copulas for moderate and large shifts ( µ =,. µ ). When µ is fixed at, the ARL values of the Normal copula are less than the other copulas

10 Songklanakarin Journal of Science and Technology SJST-0-0.R Sukparungsee Page of for small shifts ( µ =., µ = ) and the ARL values of the Gumbel copula are less than the other copulas for moderate and large shifts ( µ, µ = ). In Table, in the case of the same shifts in both exponential parameters of τ = 0., the ARL values of the Normal copula are less than the other copulas for all shifts. Table shows that for the mean shifts of τ = 0., the ARL values of the Normal copula are less than the other copulas for all shifts. In Table, in the case of the same shifts in both exponential parameters of τ = 0., the ARL values of the Normal copula are less than the other copulas for small and moderate shifts (. µ,. µ ) and the ARL values of the Frank copula are less than the other copulas for large shifts ( µ, µ ). Table. ARL of the MEWMA control chart with Kendall s tau value equal to 0. in the case of one exponential parameter Table. ARL of the MEWMA control chart with Kendall s tau value equal to 0. in the case of two exponential parameters Table. ARL of the MEWMA control chart with Kendall s tau value equal to -0. in the case of one exponential parameter Table. ARL of the MEWMA control chart with Kendall s tau value equal to -0. in the case of two exponential parameters

11 Page of Songklanakarin Journal of Science and Technology SJST-0-0.R Sukparungsee Conclusions Dependence measures of two or more variables can be investigated in terms of various copulas. We consider types of copulas because these copulas are well-known for practician and compare with Elliptical and Archimedean copulas. No research can be found regarding the use of the copula function in the literature addressing MEWMA control charts. This paper shows multivariate exponentially weighted moving average (MEWMA) control charts for four types of copulas and the level of dependence is measured by Kendall s tau values. The results revealed that it is necessary to detect the dependence of the observation to indicate the copula which fits the observation. For moderate dependence, the Normal copula can be used for almost all shifts. Acknowledgments The authors would like to express their gratitude to the Ministry of Science and Technology, Thailand and the Graduate College, King Mongkut s University of Technology North Bangkok for their financial support. References Busaba, J., Sukparungsee, S. and Areepong, Y. 0. Numerical approximations of average run length for AR() on exponential CUSUM. Proceedings of the International MultiConference of Engineers and Computer Scientists, Hong Kong, March -, 0, -. Crosier, R.B.. Multivariate generalizations of cumulative sum qualitycontrol schemes. Technometrics. 0(), -0.

12 Songklanakarin Journal of Science and Technology SJST-0-0.R Sukparungsee Page of Dokouhaki, P. and Noorossana, R. 0. A copula markov CUSUM chart for monitoring the bivariate auto-correlated binary observation. Quality and Reliability Engineering International. (), -. Fatahi, A.A., Dokouhaki, P. and Moghaddam, B.F. 0. A bivariate control chart based on copula function. Proceeding of IEEE International Conference on Quality and Reliability (ICQR), Bangkok, Thailand, September -, 0, -. Fatahi, A.A., Noorossana, R., Dokouhaki, P. and Moghaddam, B.F. 0. Copula-based bivariate ZIP control chart for monitoring rare events. Communications in Statistics - Theory and Methods. (), -. Genest, C. and MacKay, J.. The joy of copulas: bivariate distributions with uniform marginals. The American Statistician. (), 0-. Genest, C. and Rivest, L.-P.. Statistical inference procedures for bivariate Archimedean copulas. Journal of the American Statistical Association. (), 0 0. Hofert, M. and M achler, M. 0. Nested Archimedean copulas meet R: the nacopula package. Journal of Statistical Software. (), -0. Hofert, M., M achler, M. and McNeil, A.J. 0. Likelihood inference for Archimedean copulas in high dimensions under known margins. Journal of Multivariate Analysis. 0, -.

13 Page of Songklanakarin Journal of Science and Technology SJST-0-0.R Sukparungsee Hotelling, H.. Multivariate Quality Control. In C. Eisenhart, M.W. Hastay and W.A. Wallis, eds., Techniques of Statistical Analysis, McGraw-Hill, New York, U.S.A., pp. -. Hryniewicz, O. 0. On the robustness of the Shewhart control chart to different types of dependencies in data. Frontier in Statistical Quality Control. 0, 0-. Joe, H., 0. Dependence modeling with copulas, Chapman and Hall/ CRC Press Monographs on statistics and applied probability, pp.. Kuvattana, S., Sukparungsee, S., Busababodhin, P. and Areepong, Y. 0. Efficiency of bivariate copula on the CUSUM chart. Proceedings of the International MultiConference of Engineers and Computer Scientists, Hong Kong, March -0, 0, -. Lowry, C.A., Woodall, W.H., Champ, C.W. and Rigdon, S.E.. A multivariate exponentially weighted moving average control chart. Technometrics. (), -. M achler, M. and Zurich, E. 0. Numerically stable Frank copula functions via multiprecision: R package Rmpfr. Avialable from: [March, 0]. Mahmoud, M.A. and Maravelakis, P.E. 0. The performance of multivariate CUSUM control charts with estimated parameters. Journal of Statistical Computation and Simulation. (), -.

14 Songklanakarin Journal of Science and Technology SJST-0-0.R Sukparungsee Page of Midi, H. and Shabbak, A. 0. Robust multivariate control charts to detect small shifts in mean. Mathematical Problems in Engineering. 0, -. Montgomery, D.C. c00. Statistical quality control: a modern introduction, th ed., Wiley, New York, U.S.A., pp. -. Nelsen, R.B. 00. An introduction to copulas, nd ed., Springer, New York, U.S.A., pp.0-0. Prabhu, S.S. and Runger, G.C.. Designing a multivariate EWMA control chart. Journal of Quality Technology. (), -. Runger, G.C., Keats, J.B., Montgomery, D.C. and Scranton, R.D.. Improving the performance of the multivariate exponentially weighted moving average control chart. Quality and Reliability Engineering. (), -. Sklar, A.. Random variables, joint distributions, and copulas. Kybernetica. (), -0. Trivedi, P.K. and Zimmer, D.M. 00. Copula modeling: an introduction for practitioners, Foundations and Trends in Econometrics. pp. -. Verdier, G. 0. Application of copulas to multivariate control charts. Journal of Statistical Planning and Inference. (), -. Yan, J. 00. Enjoy the joy of copulas: with a package copula. Journal of Statistical Software. (), -.

15 Page of Songklanakarin Journal of Science and Technology SJST-0-0.R Sukparungsee Table. Kendall s tau of copula function. Copula Kendall s tau Parameter space of θ Normal arcsin( θ ) / ( π / ) [,] Clayton θ / ( θ + ) [, ) \{0} Frank θ t + dt - / θ t θ (, ) \{0} e 0 Gumbel ( θ ) / θ [, ) Table. ARL of the MEWMA control chart with Kendall s tau value equal to 0. in the case of one exponential parameter. Parameters ARL 0 and ARL (τ = 0.) µ µ λ = 0.0 λ = 0. Normal Clayton Frank Gumbel Normal Clayton Frank Gumbel

16 Songklanakarin Journal of Science and Technology SJST-0-0.R Sukparungsee Page 0 of Table. ARL of the MEWMA control chart with Kendall s tau value equal to 0. in the case of two exponential parameters Parameters ARL 0 and ARL (τ = 0.) µ µ λ = 0.0 λ = 0. Normal Clayton Frank Gumbel Normal Clayton Frank Gumbel Table. ARL of the MEWMA control chart with Kendall s tau value equal to -0. in the case of one exponential parameter Parameters ARL 0 and ARL (τ = -0.) µ µ λ = 0.0 λ = 0. Normal Clayton Frank Normal Clayton Frank

17 Page of Songklanakarin Journal of Science and Technology SJST-0-0.R Sukparungsee Table. ARL of the MEWMA control chart with Kendall s tau value equal to -0. in the case of two exponential parameters Parameters ARL 0 and ARL (τ = -0.) µ µ λ = 0.0 λ = 0. Normal Clayton Frank Normal Clayton Frank