- 學位論文
運用經驗動態模型搭配巢狀訓練集分析偵測非線性系統的改變
Detecting shifts in nonlinear dynamics using Empirical Dynamic Modeling with Nested-Library Analysis
摘要
在自然界的系統中常可觀察到突然發生且影響持續之系統狀態上的改變，而這樣的現象被解釋為系統在不同吸子（attractor）之間的躍遷。出於對系統行為和結構轉變之關切，科學家提出諸多時間序列分析方法以偵測系統躍遷之發生時刻。然而，目前仍缺乏一個與系統控制方程有實質關聯的通用偵測方法。本研究中，我們運用以Takens嵌入定理（embedding theorem）為理論基礎的經驗動態模型，為此難題提出名為「訓練集分析」的解決方法。具體而言，此方法偵測的方式是循序地移除部分的訓練集並找出能夠最佳化吸子重建的資料切點。為驗證這個作法的效果，我們將該方法用在一個模式生態模型——一個受到隱藏因子影響系統機制的食物鏈模型。分析結果顯示，這個新方法可以偵測到系統動力的躍遷；相較之下，利用統計特徵的改變來偵測的傳統方法表現不盡理想。此外，本論文提出的新方法也被用在在太平洋震盪指數作為真實資料上之應用。分析的結果顯示了太平洋震盪的機制，在1970年代中期前後經歷了重大轉變，而此結果也與其他對太平洋震盪系統的研究認知相符。執得一提的，我們的方法對系統控制方程的假設有相當的一般性，且只需要一條時間序列資料。
並列摘要
Abrupt and persistent changes in system dynamics are often observed in natural systems; such phenomena are explained as transitions between alternative equilibria and/or attractors. Concerned with changes in the system behavior and structure, scientists have suggested methods to detect regime shifts in time series data. Nonetheless, a generic method virtually connecting with the system dynamics (i.e. the governing equation) is lacking. For this purpose, we advise a novel method named Nested-Library Analysis (NLA), using empirical dynamic modeling rooted in Takens' embedding theory of attractor reconstruction. Specifically, NLA determines the cutting point by sequentially deconstructing the historical library set that optimizes attractor reconstruction for prediction. We illustrate this method by employing it on a pedagogical model, where a hidden regime driver exerts an influence on the governing equation of a chaotic three-species food chain. Results show that NLA can detect the abrupt change in system dynamics; in comparison, the existing approaches based on statistical characteristics fail to detect the transition. Besides, NLA is applied to the Pacific Decadal Oscillation (PDO) index as an empirical example. Our result shows that the dynamics of PDO underwent an abrupt change around the mid-1970s, showing consistency with other studies on PDO regime shifts. Importantly, our method requires no assumption about the explicit form of the governing equation but only a single time series.