過去學者與預測人員進行預測時多著重於點預測的產出，然而點預測並沒有提供任何預測不確定性的資訊，若我們只產出沒有區間的點預測，在實務運用上將沒有任何價值，近日預測區間所獲的關注量提升，其最重要的價值為能夠呈現預測中的不確定性，決策者便可依照預測結果與其可能的準確程度與區間範圍做出決策。而預測區間產出與估計的方式有許多種，過去研究與預測競賽中亦發現合併後的區間能夠提升整體的準確率與校準度，因此本研究之研究問題為「要採用何種合併方法才能夠得到最佳的合併預測區間？」，研究將設計數學規劃模型找尋最佳化的合併方法，解決在實務上常面臨的選擇合併方法的問題。本研究以最小化 MSIS (Mean Scaled Interval Score) 指標為目的設計數學規劃模型，且將原為非線性的目標函式經過線性化處理，加快尋找最佳化權重效率，亦設計實驗流程找尋最佳化權重，實驗使用資料涵蓋線性與非線性時間序列資料以及實際的微處理器需求資訊，實驗流程首先將時間序列透過 Maxiumun Entorpy Boostrap 方式重複抽樣，再使用重複抽取的時間序列訓練預測模型與產出樣本外的預測誤差，藉由樣本外的預測誤差搭配不同估計方式生成多組的預測區間，再透過數學規劃模型找出最佳權重組合，實驗最後比較最佳化權重與常見的簡易合併區間方法之表現，發現使用最佳化權重合併後的區間表現良好且穩定，尤其當預測期數愈遠，愈能突顯最佳化權重與簡易合併方法之差距。

In the past, scholars and forecasters paid more attention on point forecasts, but point forecasts do not provide information on forecast uncertainty. If we only produce point forecasts without intervals, they are of no value in practical application. The most important value of prediction interval is to present the uncertainty in the forecast so that one can make decisions based on the forecast results and the likely accuracy and range. There are many ways to generate and estimate forecast intervals. Past studies and forecasting competitions have found that combining intervals can improve the overall accuracy and calibration. However, we still have question about "What combining method should be used to obtain the best combining forecast interval? In this study, a mathematical program is designed to find the optimal combining method to solve the practical problem of choosing the combining method. We design a mathematical program with the objective of minimizing the MSIS (Mean Scaled Interval Score). Also, we linearize the original non-linear objective function to speed up the efficiency of finding optimal weights. The experimental process starts with repeated sampling of time series by Maxiumun Entorpy Boostrap method. We use the bootstrapped time series to train the prediction models and generate the out-of-sample prediction errors. Then, generate multiple sets of prediction intervals by the combination of the out-of-sample prediction errors and different estimation methods in order to find the optimal weights by mathematical program. We finally compare the performance of the optimal weights with the simple combining approaches. It shows that the performance of the combining intervals with the optimal weights is good and stable. Especially when we take farther ahead forecast, the difference between the optimal weights and the simple combining approaches becomes more obvious.

第一章 緒論 1
第二章 文獻回顧與探討 4
第一節 簡易區間預測合併方法 4
第二節 區間預測評估指標 6
第三章 研究方法 8
第四章 資料與模擬實驗設計 12
第一節 資料與產出方法 12
第二節 模擬實驗設計 14
第五章 實驗結果與進階研究 22
第一節 實驗結果分析 22
第二節 進階模擬實驗 27
第六章 結論 35
第七章 參考文獻 37

Brockwell, P. J., Brockwell, P. J., Davis, R. A., & Davis, R. A. (2016). Introduction to time series and forecasting. Springer.
Chatfield, C. (1993). Calculating interval forecasts. Journal of Business & Economic Statistics, 11(2), 121-135.
Clemen, R. T. (1989). Combining forecasts: A review and annotated bibliography. International Journal of Forecasting, 5(4), 559-583.
Elliott, G. (2011). Averaging and the optimal combination of forecasts.
Gaba, A., Tsetlin, I., & Winkler, R. L. (2017). Combining interval forecasts. Decision Analysis, 14(1), 1-20.
Gardner Jr, E. S., & McKenzie, E. (1985). Forecasting trends in time series. Management Science, 31(10), 1237-1246.
Grushka-Cockayne, Y., & Jose, V. R. R. (2020). Combining prediction intervals in the m4 competition. International Journal of Forecasting, 36(1), 178-185.
Grushka-Cockayne, Y., Jose, V. R. R., & Lichtendahl Jr, K. C. (2017). Ensembles of overfit and overconfident forecasts. Management Science, 63(4), 1110-1130.
Guo, X., Lichtendahl, K. C., & Grushka-Cockayne, Y. (2019). An Exponential Smoothing Model with a Life Cycle Trend.
Haran, U., & Moore, D. A. (2014). A better way to forecast. California Management Review, 57(1), 5-15.
Holt, C. C. (2004). Forecasting seasonals and trends by exponentially weighted moving averages. International Journal of Forecasting, 20(1), 5-10.
Hora, S. C. (2004). Probability judgments for continuous quantities: Linear combinations and calibration. Management Science, 50(5), 597-604.
Hyndman, R., Koehler, A. B., Ord, J. K., & Snyder, R. D. (2008). Forecasting with Exponential Smoothing: the State space approach. Springer Science & Business Media.
Hyndman, R. J., & Khandakar, Y. (2008). Automatic time series forecasting: the forecast package for R. Journal of Statistical Software, 27(1), 1-22.
Kumar, S., & Srivastava, A. (2012). Bootstrap prediction intervals in non-parametric regression with applications to anomaly detection. Proc. 18th ACM SIGKDD Conf. Knowl. Discovery Data Mining,
Lee, Y. S., & Scholtes, S. (2014). Empirical prediction intervals revisited. International Journal of Forecasting, 30(2), 217-234.
Li, G., Wu, D. C., Zhou, M., & Liu, A. (2019). The combination of interval forecasts in tourism. Annals of Tourism Research, 75, 363-378.
Lichtendahl Jr, K. C., Grushka-Cockayne, Y., & Winkler, R. L. (2013). Is it better to average probabilities or quantiles? Management Science, 59(7), 1594-1611.
Makridakis, S., & Winkler, R. L. (1989). Sampling distributions of post‐sample forecasting errors. Journal of the Royal Statistical Society: Series C (Applied Statistics), 38(2), 331-342.
Manary, M. P., & Willems, S. P. (2021). Data Set: 187 Weeks of Customer Forecasts and Orders for Microprocessors from Intel Corporation. Manufacturing & Service Operations Management, forthcoming.
Matsypura, D., Thompson, R., & Vasnev, A. L. (2018). Optimal selection of expert forecasts with integer programming. Omega, 78, 165-175.
Park, S., & Budescu, D. V. (2015). Aggregating multiple probability intervals to improve calibration. Judgment and Decision Making, 10(2), 130.
Tong, H. (2012). Threshold models in non-linear time series analysis (Vol. 21). Springer Science & Business Media.
Vinod, H. D., & López-de-Lacalle, J. (2009). Maximum entropy bootstrap for time series: the meboot R package. Journal of statistical software, 29(1), 1-19.
Zivot, E., & Wang, J. (2007). Modeling financial time series with S-Plus® (Vol. 191). Springer Science & Business Media.