MONITORING THE PROCESS MEAN OF A MODIFIED EWMA CHART FOR ARMA(1,1) PROCESS AND ITS APPLICATION
Keywords:
Modified EWMA, average run length, ARMA process, explicit formulasAbstract
The performance of a control chart monitoring a statistical process is regularly measured in terms of the average run length (ARL) which guarantees that a comparative study can be conducted between several monitoring and detecting procedures. Explicit formulas are derived and evaluated for an exact ARL based on a first-order autoregressive moving average (ARMA(1,1)) process with exponential white noise through a modified exponentially weighted moving average (EWMA) control chart. The analytical results are more accurate in comparison with the numerical integral equation method as it compares the performance of the control chart between the EWMA and modified EWMA. The superiority of the modified EWMA control chart is illustrated by applying it to a dataset of changes in Earth’s rotation.
References
Agarwal, G.C., Goodarzi, S.M., O’Neill, W.D., and Gottlieb, G.L. (1982). Autoregressive moving average (ARMA) models of neuromuscular system. IFAC Proceedings Volumes, 15(4):1,205-1,210.
Areepong, Y. (2012). Explicit formulas of average run length for a moving average control chart monitoring the number of defective products. Int. J. Pure Appl. Math., 80:331-343.
Bogusz, J., Brzezinski, A., Kosek, W., and Nastula J. (2015). Earth rotation and geodynamics. Geodesy and Cartography, 64(2):201-242.
Busaba, J., Sukparungsee, S., Areepong, Y., and Mititelu, G. (2012). Numerical approximations of average run length for AR(1) on exponential CUSUM. Proceedings of the International MutiConference of Engineers and Computer Scientists 2012; March 14-16, 2012; Kowloon, Hong Kong, p. 1,268-1,273.
Candella, R.N., Rabinovich, A.B., and Thomson, R.E. (2008). The 2004 Sumatra tsunami as recorded on the Atlantic coast of South America. Adv. Geosci., 14:117-128.
Chadha, R.K., Latha, G., Yeh, H., Peterson, C., and Katada, T. (2005). The tsunami of the great Sumatra earthquake of M 9.0 on 26 December 2004: impact on the east coast of India. Curr. Sci. India, 88:1,297-1,301.
Changyi, X. and Wenke, S. (2012). Co-seismic Earth's rotation change caused by the 2012 Sumatra earthquake. Geodesy and Geodynamics, 3(4):28-31.
Chattopadhyay, S., Jhajharia, D., and Chattopadhyay, G. (2011). Trend estimation and univariate forecast of the sunspot numbers: Development and comparison of ARMA, ARIMA and Autoregressive Neural Network models. CR. Geosci., 343:433-442.
Chua, L.H.C. and Wong, T.S.W. (2011). Runoff forecasting for an asphalt plane by artificial neural networks and comparisons with kinematic wave and autoregressive moving average models. J. Hydrol., 397:191-201.
Dilling, S. and MacVicar, B.J. (2017). Cleaning high-frequency velocity profile data with autoregressive moving average (ARMA) models. Flow Meas. Instrum., 54:68-81.
Fu, J.C., Spiring, F.A., and Xie, H. (2002). On the average run length of quality control schemes using a Markov chain approach. Stat. Probabil. Lett., 56:369-380.
Gill, S.J. and Biging, G.S. (2002). Autoregressive moving average models of crown profiles for two California hardwood species. Ecol. Model., 152:213-226.
Herdiani, E.T., Fandrilla, G., and Sunusi, N. (2018). Modified exponential weighted moving average (EWMA) control chart on autocorrelation data. J. Phys. Conf. Ser., 979(1):1-7.
Jiang, W., Tsui, K., and Woodall, W.H. (2000). A new SPC monitoring method: the ARMA chart. Technometrics, 42:399-410.
Kim, J. and Kim, M. (2015). ARMA prediction of SBAS ephemeris and clock corrections for low earth orbiting satellites. Int. J. Aerospace Eng., 2015:1-8.
Kozin, F. (1988). Autoregressive moving average models of earthquake records. Probabilist. Eng. Mech., 3(2):58-63.
Lei, Y., Cai, H., and Zhao, D. (2016). A simple differencing technology to improve prediction accuracy of Earth rotation parameters. Lect. Notes Electr. Eng., 390:201-211.
Liu, Z., Cai, Z., and Tan, X. (2011). Forecasting research of aero-engine rotate speed signal based on ARMA model. Procedia Engineer., 15:115-121.
Mastrangelo, C.M. and Montgomery, C.M. (1995). SPC with Correlated Observations for the Chemical and Process Industries. Qual. Reliab. Eng. Int., 11: 79-89.
Mititelu, G., Areepong, Y., Sukparungsee S., and Novikov, A.A. (2010). Explicit analytical solutions for the average run length of CUSUM and EWMA charts. East-West Journal of Mathematics, 2010:253-265.
Mobarakeh, A.A., Rofooei, F.R., and Ahmadi, G. (2002). Simulation of earthquake records using time-varying Arma(2,1) model. Probabilist. Eng. Mech., 17(1):15-34
Monecke, K., Finger, W., Klarer, D., Kongko, W., McAdoo, B.G., Moore, A.L., and Sudrajat, S.U. (2008). A 1,000-year sediment record of tsunami recurrence in northern Sumatra. Nature, 455:1,232-1,234.
Page, E.S. (1954). Continuous inspection schemes. Biometrika, 41:100-115.
Pappas, S.S., Ekonomou, L., Karamousantas, D.C., Chatzarakis, G.E., Katsikas, S.K., and Liatsis, P. (2008). Electricity demand loads modeling using autoregressive moving average (ARMA) models. Energy, 33:1,353-1,360.
Patel, A.K. and Divecha, J. (2011). Modified exponentially weighted moving average (EWMA) control chart for an analytical process data. Journal of Chemical Engineering and Materials Science, 2(1):12-20.
Peerajit, W., Areepong, Y., and Sukparungsee, S. (2017). Analytical explicit formulas of average run length for long memory process with ARFIMA model on CUSUM control chart. Songklanakarin J. Sci. Technol., 6:779-786.
Petcharat, K., Areepong, Y., and Sukparungsee, S. (2013). Exact solution of average run length of EWMA chart for MA(q) processes. Far East Journal of Mathematical Sciences, 78:291-300.
Petcharat, K., Sukparungsee S., and Areepong, Y. (2015). Exact solution of average run length for the cumulative sum chart for a moving average process of order q. ScienceAsia, 41:141-147.
Phantu, S., Sukparungsee, S., and Areepong, Y. (2016). Explicit expressions of average run length of moving average control chart for Poisson integer valued autoregressive model. Proceedings of the International MultiConference of Engineering and Computer Scientists 2016; March 16-18, 2016; Kowloon, Hong Kong, p. 892-895.
Phanyaem, S., Areepong, Y., and Sukparungsee, S. (2014). Numerical integral of average run length of CUSUM control chart for ARMA process. Int. J. Appl. Phys. Math., 4:232-235.
Reynolds Jr, M.R. and Lu, C. (1997). Control charts for monitoring processes with autocorrelated data. Nonlinear Anal. Theor., 30:4,059-4,067.
Roberts, S.W. (1959). Control chart tests based on geometric moving average. Technometrics, 1:239-250.
Sivavaraprasad, G. and Ratnam, V. (2017). Performance evaluation of ionospheric time delay forecasting models using GPS observations at a low-latitude station. Adv. Space Res., 60:475-490.
Sukparungsee, S. and Areepong, Y. (2009). A study of the performance of EWMA chart with transformed Weibull observations. Thailand Statistician, 7(2):179-191.
Sunthornwat, R., Areepong, Y., and Sukparungsee, S. (2018). Average run length with a practical investigation of estimating parameters of the EWMA control chart on the long memory ARFIMA process. Thailand Statistician, 16(2):190-202.
Suriyakat, W., Areepong, Y., Sukparungsee S., and Mititelu, G. (2012). On EWMA procedure for AR(1) observations with exponential white noise. Int. J. Pure Appl. Math., 77:73-83.
Ubeyli, E.D. and Guler, I. (2004). Spectral analysis of internal carotid arterial Doppler signals using FFT, AR, MA, and ARMA methods. Comput. Biol. Med., 34:293-306.
Valipour, M., Banihabib, M.E., and Behbahani, S.M.R. (2013). Comparison of ARMA, ARIMA, and the autoregressive artificial neural network models in forecasting the monthly inflow of Dez dam reservoir. J. Hydrol., 476:433-441.
Wang, C., Wu, C., and Tzang, S. (2012). Implementing option pricing models when asset returns follow an autoregressive moving average process. Int. Rev. Econ. Finance, 24:8-25.
Yokoyama, Y. (2002). Excitation Sompi method and its application to an analysis of the Earth’s polar motion. Geophys. J. Int., 150:467-482.
Zhang, J., Li, Z., Chen, B., and Wang, Z. (2014). A new exponentially weighted moving average control chart for monitoring the coefficient of variation. Comput. Ind. Eng., 78:205-212.
Zhang, N.F. (2012). A Statistical Control Chart for Stationary Process Data. Technometrics, 40:24-38.
Zhou, J., Sun, W., Sun, H., Xu, J., and Cui, X. (2014). Co-seismic change of length of day based on the point dislocation theory for a SNREI Earth. J. Geodyn., 79:18-22.
