預測區間的合併：一般化的數學規劃

Combining prediction intervals: a generic mathematical program

温雅筑(Wen Ya-Chu)

指導教授 ： 莊皓鈞 ； 周彥君

國立政治大學/商學院/資訊管理學系/碩士(2021年)

https://doi.org/10.6814/NCCU202101054

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。

關鍵字

預測區間 ； 數學規劃 ； 區間預測合併

並列摘要

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.

並列關鍵字

Prediction interval ； Interval forecast ； Combining prediction interval ； Mathematical program

參考文獻

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