隨著航運活動日益頻繁，船舶燃油消耗的即時監控對營運效率與環境永續日益重要。然而，船舶航行資料多具有時間相依性與殘差厚尾特性，傳統以常態分佈為假設之管制圖方法，可能導致管制界限低估與誤警風險增加，且多數研究仍採取建模與監控分離的兩階段分析架構。
本研究提出一套整合式時間序列管制圖方法，結合ARIMAX模型與位置計量t分佈，將管制界限視為未知參數，並透過引入潛在指標變數與EM演算法，在最大概似估計架構下同步估計模型參數與管制界限，以同時處理自相關結構與殘差厚尾現象。
為評估方法效能，本研究進行蒙地卡羅模擬，比較多種誤差分佈與ARIMA結構下不同估計策略之參數估計表現與監控效果，並進一步以實際船舶資料進行實證分析。結果顯示，傳統兩階段估計方法比聯合估計模型，更能提供可靠的監控表現。本研究之比較結果，可作為未來在時間序列管制圖設計與船舶燃油消耗監控實務上的參考。

With increasingly frequent shipping activities, real-time monitoring of ship fuel consumption is crucial for operational efficiency and environmental sustainability. However, ship navigation data often exhibits time dependence and heavy-tailed residuals. Traditional control chart methods, which assume normal distributions, may lead to underestimated control limits and increased false alarm risks. Furthermore, most studies still employ a two-stage analysis framework that separates modeling and monitoring.
This study proposes an integrated time-series control chart method that combines the ARIMAX model with the location-scale t distribution. It treats control limits as unknown parameters and, by introducing latent indicator variables and the EM algorithm, simultaneously estimates model parameters and control limits within the maximum likelihood estimation framework, thus addressing both autocorrelation structure and heavy-tailed residuals.
To evaluate the method's effectiveness, this study conducts Monte Carlo simulations, comparing the parameter estimation performance and monitoring effectiveness of different estimation strategies under various error distributions and ARIMA structures. Further empirical analysis using actual ship data is then performed. The results show that the traditional two-stage estimation method provides more reliable monitoring performance than the joint estimation model. The comparative results of this study can serve as a reference for future practice in time series control chart design and ship fuel consumption monitoring.

1. 交通部運輸研究所（2013），國際海運物流完整流向資料收集與分析之研究，https://www.iot.gov.tw/zh_tw/archive/pub/reports/-98868387。
2. Bocchetti, D., Lepore, A., Palumbo, B., & Vitiello, L. (2013, November). A statistical control of the ship fuel consumption. In Proceedings of the International Conference on the Design, Construction and Operation of Passenger Ships, London, UK (pp. 20-21).
3. Bocchetti, D., Lepore, A., Palumbo, B., & Vitiello, L. (2015). A statistical approach to ship fuel consumption monitoring. Journal of Ship Research, 59(03), 162-171.
4. Capezza, C., Coleman, S., Lepore, A., Palumbo, B., & Vitiello, L. (2019). Ship fuel consumption monitoring and fault detection via partial least squares and control charts of navigation data. Transportation Research Part D: Transport and Environment, 67, 375-387.
5. Díaz-Cao, J. M., Liu, X., Kim, J., Clavijo, M. J., & Martínez-López, B. (2023). Evaluation of the application of sequence data to the identification of outbreaks of disease using anomaly detection methods. Veterinary Research, 54(1), 75.
6. Farag, Y. B., & Ölçer, A. I. (2020). The development of a ship performance model in varying operating conditions based on ANN and regression techniques. Ocean Engineering, 198, 106972.
7. Gkerekos, C., Lazakis, I., & Theotokatos, G. (2019). Machine learning models for predicting ship main engine Fuel Oil Consumption: A comparative study. Ocean Engineering, 188, 106282.
8. Hsu, C. R., & Wang, H. (2024). EWMA control chart integrated with time series models for COVID-19 surveillance. Mathematics, 13(1), 115.
9. International Maritime Organization. (2020). Fourth IMO greenhouse gas study 2020 (Final report). International Maritime Organization.
10. Imro'ah, N., & Huda, N. A. M. (2022). Control chart as verification tools in time series model. BAREKENG: Jurnal Ilmu Matematika dan Terapan, 16(3), 995-1002.
11. Imro'ah, N., Huda, N. A. M., & Pratama, A. Y. (2023). The Implementation of Control Charts as a Verification Tool in a Time Series Model for COVID-19 Vaccine Participants in Pontianak. ComTech: Computer, Mathematics and Engineering Applications, 14(1), 55-67.
12. Imro'ah, N., Huda, N. A. M., Utami, D. S., Umairah, T., & Arini, N. F. (2024). Control Chart for Correcting the ARIMA Time Series Model of GDP Growth Cases. JTAM (Jurnal Teori dan Aplikasi Matematika), 8(1), 312-323.
13. Kim, Y. R., Jung, M., & Park, J. B. (2021). Development of a fuel consumption prediction model based on machine learning using ship in-service data. Journal of Marine Science and Engineering, 9(2), 137.
14. Li, X., Zuo, Y., & Jiang, J. (2022). Application of regression analysis using broad learning system for time-series forecast of ship fuel consumption. Sustainability, 15(1), 380.
15. Montgomery, D. C. (2020). Introduction to statistical quality control (8th ed.). John wiley & sons.
16. Page, E. S. (1954). Continuous inspection schemes. Biometrika, 41(1/2), 100-115.
17. Roberts, S. W. (1959). Control Chart Tests Based on Geometric Moving Averages. Technometrics, 1(3), 239-250.
18. Shaw, H. J., & Lin, C. K. (2021). Marine big data analysis of ships for the energy efficiency changes of the hull and maintenance evaluation based on the ISO 19030 standard. Ocean Engineering, 232, 108953.
19. Shewhart, W. A. (1930). Economic quality control of manufactured product 1. Bell System Technical Journal, 9(2), 364-389.