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題名 預測區間的合併：一般化的數學規劃
Combining prediction intervals: a generic mathematical program作者 温雅筑
Wen, Ya-Chu貢獻者 莊皓鈞<br>周彥君
Chuang, Hao-Chun<br>Chou, Yen-Chun
温雅筑
Wen, Ya-Chu關鍵詞 預測區間
數學規劃
區間預測合併
Prediction interval
Interval forecast
Combining prediction interval
Mathematical program日期 2021 上傳時間 4-Aug-2021 14:46:20 (UTC+8) 摘要 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.參考文獻 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. 描述 碩士
國立政治大學
資訊管理學系
108356003資料來源 http://thesis.lib.nccu.edu.tw/record/#G0108356003 資料類型 thesis dc.contributor.advisor 莊皓鈞<br>周彥君 zh_TW dc.contributor.advisor Chuang, Hao-Chun<br>Chou, Yen-Chun en_US dc.contributor.author (Authors) 温雅筑 zh_TW dc.contributor.author (Authors) Wen, Ya-Chu en_US dc.creator (作者) 温雅筑 zh_TW dc.creator (作者) Wen, Ya-Chu en_US dc.date (日期) 2021 en_US dc.date.accessioned 4-Aug-2021 14:46:20 (UTC+8) - dc.date.available 4-Aug-2021 14:46:20 (UTC+8) - dc.date.issued (上傳時間) 4-Aug-2021 14:46:20 (UTC+8) - dc.identifier (Other Identifiers) G0108356003 en_US dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/136338 - dc.description (描述) 碩士 zh_TW dc.description (描述) 國立政治大學 zh_TW dc.description (描述) 資訊管理學系 zh_TW dc.description (描述) 108356003 zh_TW dc.description.abstract (摘要) 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。 zh_TW dc.description.abstract (摘要) 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. en_US dc.description.tableofcontents 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 zh_TW dc.format.extent 2822078 bytes - dc.format.mimetype application/pdf - dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0108356003 en_US dc.subject (關鍵詞) 預測區間 zh_TW dc.subject (關鍵詞) 數學規劃 zh_TW dc.subject (關鍵詞) 區間預測合併 zh_TW dc.subject (關鍵詞) Prediction interval en_US dc.subject (關鍵詞) Interval forecast en_US dc.subject (關鍵詞) Combining prediction interval en_US dc.subject (關鍵詞) Mathematical program en_US dc.title (題名) 預測區間的合併：一般化的數學規劃 zh_TW dc.title (題名) Combining prediction intervals: a generic mathematical program en_US dc.type (資料類型) thesis en_US dc.relation.reference (參考文獻) 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. zh_TW dc.identifier.doi (DOI) 10.6814/NCCU202101054 en_US
