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Home > Vol 16, No 2 (Apr.–Jun. 2023) > Bualuang

Performance of the CUSUM Control Chart Using Approximation to ARL for Long- Memory Fractionally Integrated Autoregressive Process with Exogenous Variable

Direk Bualuang, Wilasinee Peerajit

Abstract


The cumulative sum (CUSUM) control chart is a well-known process monitoring tool that is sensitive to small-to-moderate changes in process parameters. In this paper, we propose an approximated average run length (ARL) method based on the numerical integral equation (NIE) method for monitoring the mean of a long-memory autoregressive fractionally integrated process with an exogenous variable (ARFIX) running on a CUSUM control chart. The approximated ARL based on the NIE method is realized by solving a system of linear equations and integration based on the partitioning and summation of the area under the curve of a function derived by using the Gauss-Legendre quadrature. In a comparative study with the ARL based on analytical formulas, the proposed approximated ARL method could detect shifts of various sizes in the process mean of an ARFIX process running on a CUSUM control chart. In addition, the proposed method was compared with their analytical formulas in terms of the relative percentage change (r%) to verify the accuracy of the ARL results. The results revealed that the ARL results obtained from the NIE method are an approach to analytical formulas with r% of less than 0.25. Hence, the NIE method is very accurate and in excellent agreement with the analytical formulas approach. Apparently, the NIE method is an alternative as efficiently as the analytical method for this situation. It also performed well in comparison with the approximated ARL for the same process running on an exponentially weighted moving average control chart. In addition, real datasets are also used to demonstrate the efficacy of the proposed method.

Keywords



[1] E. S. Page, “Continuous inspection schemes,” Biometrika, vol. 41, no. 1–2, pp. 100–115, 1954.

[2] S. W. Robert, “Control chart test based on geometric moving averages,” Technometrics, vol. 1, pp. 239– 250, 1959.

[3] D. C. Montgomery, Introduction to Statistical Quality Control, 6th ed. New York: John Wiley & Sons, 2009.

[4] M. Riaz, S. A. Abbasi, S. Ahmad, and B. Zaman, “On efficient phase II process monitoring charts,” The International Journal of Advanced Manufacturing Technology, vol. 70, pp. 2263– 2274, 2014.

[5] R. A. Johnson and M. Bagshaw, “The effect of serial correlation on the performance of CUSUM tests,” Technometrics, vol. 16, no. 1, pp. 103–112, 1974.

[6] C. W. Lu and M. R. J. Reynolds, “CUSUM charts for monitoring an autocorrelated process,” Journal of Quality Technology, vol. 33, no. 3, pp. 316–334, 2001.

[7] S. H. Kim, C. Alexopoulos, K. L. Tsui, and J. R. Wilson, “A distribution-free tabular CUSUM chart for autocorrelated data,” IIE Transactions, vol. 39, no. 3, pp. 317–330, 2007.

[8] W. Jiang, “Average run length computation of ARMA charts for stationary process,” Communications in Statistics, vol. 30, no. 3, pp. 699–717, 2001.

[9] C. W. Lu and M. R. Reynolds, “EWMA control charts for monitoring the mean of autocorrelated process,” Journal of Quality Technology, vol. 31, pp. 166–188, 1999.
[10] S. Yue and P. Pilon, “Interaction between deterministic trend and autoregressive process,” Water Resources Research, vol. 39, no. 4, 2003, doi: 10.1029/2001WR001210.
[11] K. H. Hamed, “Enhancing the effectiveness of prewhitening in trend analysis of hydrologic data,” Journal of Hydrology, vol. 368, no. 1, pp. 143–155, 2009.

[12] A. D. Karaoglan and G. M. Bayhan, “ARL performance of residual control charts for trend AR(1) process: A case study on peroxide values of stored vegetable oil,” Scientific Research and Essays, vol. 7, no. 13, pp. 1405–1414, 2012.

[13] R. Zhang and S. Ling, “Asymptotic inference for AR models with heavy-tailed G-GARCH noises,” Econometric Theory, vol. 31, no. 4, pp. 880–890, 2015.

[14] M. Novkovic, “On exponential autoregressive time series models,” Journal of Mathematics, vol. 29, no. 1, pp. 97–101, 1999.

[15] M. Ibazizen and H. Fellag, “Bayesian estimation of an AR(1) process with exponential white noise,” Statistics, vol. 37, no. 5, pp. 365–372, 2003.

[16] I. M. S. Pereira and M. A. Amaral-Turkman, “Bayesian prediction in threshold autoregressive models with exponential white noise,” Sociedad de Estadística e Investigación Operativa, vol. 13, no. 1, pp. 45–64, 2004.

[17] L. Larbi and H. Fellag, “Robust bayesian analysis of an autoregressive model with exponential innovations,” Afrika Statistika, vol. 11, no. 1, pp. 955–964, 2016.

[18] S. Suparman, “A new estimation procedure using a reversible jump MCMC algorithm for AR models of exponential white noise,” International Journal of GEOMATE, vol.15, no. 49, pp. 85–91, 2018.

[19] C. W. J. Granger and R. Joyeux, “An introduction to long memory time series models and fractional differencing.” Journal of Time Series Analysis, vol. 1, no. 1, pp. 15–29, 1980.

[20] J. R. M. Hosking, “Fractional differencing,” Biometrika, vol. 68, no. 1, pp. 165–176, 1981.

[21] H. Ebens, “Realized stock index volatility,” Ph.D. dissertation, Department of Economics, Johns Hopkins University, USA,1999.

[22] R. Ramjee, “Qualitycontrol charts and persistent processes,” Ph.D. dissertation, Department of Mathematical Sciences, Stevens Institute of Technology, USA, 2000.

[23] R. Ramjee, N. Crato, and B. K. A. Ray, “Note on moving average forecasts of long memory processes with an application to quality control,” International Journal of Forecasting, vol. 18, pp. 291–297, 2002.

[24] J. N. Pan and S. T. Chen, “Monitoring longmemory air quality data using ARFIMA model,” Environmetrics, vol. 19, pp. 209–219, 2008.

[25] L. Rabyk and W. Schmid, “EWMA control charts for detecting changes in the mean of a longmemory process,” Metrika, vol. 79, pp. 267–301, 2016.

[26] A. Muhammad, “A mixed EWMA-CUSUM control chart for weibull-distributed quality characteristics,” Quality and Reliability Engineering International, vol. 32, no. 8, pp. 2987–2994, 2016.

[27] A. Muhammad, A. Muhammad, and J. Chi-Hyuck, “A HEWMA-CUSUM control chart for the weibull distribution,” Communications in Statistics - Theory and Methods, vol. 47, no. 24, pp. 5973– 5985, 2018.

[28] D. Brook and D.A. Evans, “An approach to the probability distribution of the CUSUM run length,” Biometrika, vol. 59, no. 3, pp. 539–549, 1972.

[29] D. M. Hawkins, “Evaluation of average run lengths of cumulative sum charts for an arbitrary data distribution,” Communications in Statistics Simulation and Computation, vol. 21, pp. 1001– 1020, 1992.

[30] C. W. Champ and S. E. Rigdon, “A comparison of the markov chain and the integral equation approaches for evaluating the run length distribution of quality control charts,” Communications in Statistics Simulation and Computation, vol. 20, pp. 191–204. 1991.
[31] J. E. Wieringa, “Statistical process control for serially correlated data,” Ph.D. dissertation, Faculty of Economics and Business, University of Groningen, Netherlands, 1999.

[32] C. A. Acosta-Mejía, J. J. Pignatiello, and B. V. Rao, “A comparison of control charting procedures for monitoring process dispersion,” IIE Transactions, vol. 31, pp. 569–579, 1999.

[33] S. Knoth, “Computation of the ARL for CUSUMS2 schemes,” Computational Statistics & Data Analysis, vol. 51, pp. 499–512, 2006.

[34] W. Peerajit, Y. Areepong, and S. Sukparungsee, “Numerical integral equation method for ARL of CUSUM chart for long-memory process with non-seasonal and seasonal ARFIMA models,” Thailand Statistician, vol. 6, no. 1, pp. 26–37, 2018.

[35] W. Peerajit and Y. Areepong, “The performance of CUSUM control chart for monitoring process mean for autoregressive moving average with exogenous variable model,” Applied Science and Engineering Progress, vol. 15, no. 1, 2022, Art. no. 3490, doi: 10.14416/j.asep.2020.11.007.

[36] W. Peerajit, “Cumulative sum control chart applied to monitor shifts in the mean of a longmemory ARFIMAX(p,d*,q,r) process with exponential white noise,” Thailand Statistician, vol. 20, no. 1, pp. 144–161, 2022.

[37] P. Phanthuna and Y. Areepong, “Analytical solutions of ARL for SAR(p)L model on a modified EWMA chart,” Mathematics and Statistics, vol. 9, no. 5, pp. 685–696, 2021.

[38] R. Sunthornwat, Y. Areepong, and S. Sukparungsee, “Average run length with a practical investigation of estimating parameters of the EWMA control chart on the long memory AFRIMA process,” Thailand Statistician, vol. 16, no. 2, pp. 190–202, 2018.

[39] A. Muhammad, S. Ambreen, A. Mohammed, M.-M. Jean-Claude, and C. S. Sandile, “A new CUSUM control chart under uncertainty with applications in petroleum and meteorology,” Plos One, vol. 16, no. 2, 2021, Art. no. e0246185.

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DOI: 10.14416/j.asep.2022.05.003

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