The Performance of CUSUM Control Chart for Monitoring Process Mean for Autoregressive Moving Average with Exogenous Variable Model
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Published:
Oct 20, 2021
Keywords:
Explicit formulas Average Run Length (ARL) ARMAX(p,q,r)) process Exponential white noise
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Abstract
The objective of this study was to derive explicit formulas for the average run length (ARL) of an autoregressive moving average with an exogenous variable (ARMAX(p,q,r)) process with exponential white noise on a cumulative sum (CUSUM) control chart. To check the accuracy of the ARL derivations, the efficiency of the proposed explicit formulas was compared with a numerical integral equation (NIE) method in terms of the absolute percentage error. There was excellent agreement between the two methods, but when comparing their computational times, the explicit formulas only required 1 second whereas the NIE method required 599.499–835.891 s. In addition, real-world application of the derived explicit formulas was illustrated using Hong Kong dollar exchange rates data with an exogenous variable (the US dollar) to evaluate the ARL of an ARMAX (p,q,r) process on a CUSUM control chart.
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How to Cite
Peerajit, W., & Areepong, Y. (2021). The Performance of CUSUM Control Chart for Monitoring Process Mean for Autoregressive Moving Average with Exogenous Variable Model. Applied Science and Engineering Progress, 15(1). https://doi.org/10.14416/j.asep.2020.11.007
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Research Articles
References
[1] W. A. Shewhart, Economic Control of Quality of Manufacturing Product. New York: Van Nostrand, 1931.
[2] E. S. Page, “Continuous inspection schemes,” Biometrika, vol. 41, no. 1–2, pp. 100–115, Jun. 1954.
[3] S. W. Robert, “Control chart test based on geometric moving averages,” Technometrics, vol. 1, pp. 239– 250, 1959.
[4] W. D. Ewan and K. W. Kemp, “Sampling inspection of continuous processes with no autocorrelation between successive results,” Biometrika, vol. 47, no. 3–4, pp. 363–380, Dec. 1960.
[5] O. A. Adeoti, “Application of CUSUM control chart for monitoring HIV/AIDS patients in Nigeria,” International Journal of Statistics and Applications, vol. 3, no. 3, pp. 77–80, 2013.
[6] C. S. Van Dobben de Bruyn, Cumulative Sum Tests: Theory and Practice. London, England: Griffin, 1968.
[7] 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.
[8] 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, Feb. 2007.
[9] Y. M. Chang and T. L. Wu, “On average run lengths of control charts for autocorrelated processes,” Methodology and Computing in Applied Probability, vol. 13, no. 2, pp. 419–431, Jun. 2011.
[10] P. A. Jacob and P. A. W. Lewis, “A mixed autoregressive-moving average exponential sequence and point process (EARMA 1,1),” Advances in Applied Probability, vol. 9, no. 1, pp. 87–104, Mar. 1977.
[11] I. Mohamed and F. Hocine, “Bayesian estimation of an AR(1) process with exponential white noise,” A Journal of Theoretical and Applied Statistics, vol. 37, no. 5, pp. 365–372, 2003.
[12] I. M. S. Pereira and M. A. Turkrman, “Bayesian prediction in threshold autoregressive models with exponential white noise,” Sociedad de Estadisticae Investigacion Operativa Test, vol. 13, no. 1, pp. 45–64, Jun. 2004.
[13] 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, Sep. 2018.
[14] 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, Dec. 1972.
[15] W. H. Woodall, “The distribution of the run length of one-sided CUSUM procedures for continuous random variables,” Technometrics, vol. 25, no. 3, pp. 295–301. 1983.
[16] P. Theepprasan, “A comparison on efficiency of control charts for autocorrelated process, M.S. thesis, Graduate School, Chulalongkorn University, 2004.
[17] D. K. Aslan and M. B. Gunhan, “Performance comparison of residual control charts for trend stationary first order autoregressive processes,” Gazi University Journal of Science, vol. 24, no. 2, pp. 329–339, 2011.
[18] A. L. Goel and S. M. Wu, “Determination of A.R.L. and a contour nomogram for CUSUM charts to control normal mean,” Technometrics, vol. 13, no. 2, pp. 221–230, May. 1971.
[19] J. M. Lucas, “Counted data CUSUMs,” Journal of Quality Technology, vol. 27, no. 2, pp. 129–144, May 1985.
[20] S. Vardeman and D. Ray, “Average run lengths for CUSUM schemes when observations are exponentially distributed,” Technometrics, vol. 27, no. 2, pp. 145–150, May 1985.
[21] C. H. Jun and M. S. Choi, “Simulating the average run length for CUSUM schemes using variance reduction techniques,” Communications in Statistics B: Simulation, vol. 22, no. 3, pp. 877– 887, 1993.
[22] S. V. Crowder, “A simple method for studying runlength distributions of exponentially weighted moving average charts,” Technometrics, vol. 29, no. 4, pp. 401–407, Nov. 1987.
[23] G. Mititelu, Y. Areepong, and S. Sukparungsee, “Explicit analytical solutions for the average run length of CUSUM and EWMA charts,” East-West Journal of Mathematics, vol. 1, pp. 253–265, 2010.
[24] K. Petcharat, S. Sukparungsee, and Y. Areepong, “Exact solution of the average run length for the cumulative sum charts for a moving average process of order q,” Science Asia, vol. 41, pp. 141– 147, 2015.
[25] P. Paichit, “Exact expression for average run length of control chart of ARX(p) procedure,” KKU Science Journal, vol. 45, no. 4, pp. 948–958, 2017.
[26] 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.
[27] M. Sofonea, W. Han, and M. Shillor, Analysis and Approximation of Contact Problems with Adhesion or Damage. New York: Chapman & Hall/CRC Pure and Applied Mathematics, 2005.
[28] Investing. (2015, Jan.). Hong Kong Dollar Exchange Rate. FUSION MEDIA CO.,LTD, Bangkok, Thailand [Online]. Available: https:// th.investing.com/currencies.
[2] E. S. Page, “Continuous inspection schemes,” Biometrika, vol. 41, no. 1–2, pp. 100–115, Jun. 1954.
[3] S. W. Robert, “Control chart test based on geometric moving averages,” Technometrics, vol. 1, pp. 239– 250, 1959.
[4] W. D. Ewan and K. W. Kemp, “Sampling inspection of continuous processes with no autocorrelation between successive results,” Biometrika, vol. 47, no. 3–4, pp. 363–380, Dec. 1960.
[5] O. A. Adeoti, “Application of CUSUM control chart for monitoring HIV/AIDS patients in Nigeria,” International Journal of Statistics and Applications, vol. 3, no. 3, pp. 77–80, 2013.
[6] C. S. Van Dobben de Bruyn, Cumulative Sum Tests: Theory and Practice. London, England: Griffin, 1968.
[7] 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.
[8] 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, Feb. 2007.
[9] Y. M. Chang and T. L. Wu, “On average run lengths of control charts for autocorrelated processes,” Methodology and Computing in Applied Probability, vol. 13, no. 2, pp. 419–431, Jun. 2011.
[10] P. A. Jacob and P. A. W. Lewis, “A mixed autoregressive-moving average exponential sequence and point process (EARMA 1,1),” Advances in Applied Probability, vol. 9, no. 1, pp. 87–104, Mar. 1977.
[11] I. Mohamed and F. Hocine, “Bayesian estimation of an AR(1) process with exponential white noise,” A Journal of Theoretical and Applied Statistics, vol. 37, no. 5, pp. 365–372, 2003.
[12] I. M. S. Pereira and M. A. Turkrman, “Bayesian prediction in threshold autoregressive models with exponential white noise,” Sociedad de Estadisticae Investigacion Operativa Test, vol. 13, no. 1, pp. 45–64, Jun. 2004.
[13] 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, Sep. 2018.
[14] 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, Dec. 1972.
[15] W. H. Woodall, “The distribution of the run length of one-sided CUSUM procedures for continuous random variables,” Technometrics, vol. 25, no. 3, pp. 295–301. 1983.
[16] P. Theepprasan, “A comparison on efficiency of control charts for autocorrelated process, M.S. thesis, Graduate School, Chulalongkorn University, 2004.
[17] D. K. Aslan and M. B. Gunhan, “Performance comparison of residual control charts for trend stationary first order autoregressive processes,” Gazi University Journal of Science, vol. 24, no. 2, pp. 329–339, 2011.
[18] A. L. Goel and S. M. Wu, “Determination of A.R.L. and a contour nomogram for CUSUM charts to control normal mean,” Technometrics, vol. 13, no. 2, pp. 221–230, May. 1971.
[19] J. M. Lucas, “Counted data CUSUMs,” Journal of Quality Technology, vol. 27, no. 2, pp. 129–144, May 1985.
[20] S. Vardeman and D. Ray, “Average run lengths for CUSUM schemes when observations are exponentially distributed,” Technometrics, vol. 27, no. 2, pp. 145–150, May 1985.
[21] C. H. Jun and M. S. Choi, “Simulating the average run length for CUSUM schemes using variance reduction techniques,” Communications in Statistics B: Simulation, vol. 22, no. 3, pp. 877– 887, 1993.
[22] S. V. Crowder, “A simple method for studying runlength distributions of exponentially weighted moving average charts,” Technometrics, vol. 29, no. 4, pp. 401–407, Nov. 1987.
[23] G. Mititelu, Y. Areepong, and S. Sukparungsee, “Explicit analytical solutions for the average run length of CUSUM and EWMA charts,” East-West Journal of Mathematics, vol. 1, pp. 253–265, 2010.
[24] K. Petcharat, S. Sukparungsee, and Y. Areepong, “Exact solution of the average run length for the cumulative sum charts for a moving average process of order q,” Science Asia, vol. 41, pp. 141– 147, 2015.
[25] P. Paichit, “Exact expression for average run length of control chart of ARX(p) procedure,” KKU Science Journal, vol. 45, no. 4, pp. 948–958, 2017.
[26] 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.
[27] M. Sofonea, W. Han, and M. Shillor, Analysis and Approximation of Contact Problems with Adhesion or Damage. New York: Chapman & Hall/CRC Pure and Applied Mathematics, 2005.
[28] Investing. (2015, Jan.). Hong Kong Dollar Exchange Rate. FUSION MEDIA CO.,LTD, Bangkok, Thailand [Online]. Available: https:// th.investing.com/currencies.