生物系統中存在許多物種，物種間的交互作用相當複雜，使得理解其潛在機制以及其對整個系統行為變得具有挑戰性。傳統文獻中，線性方法是理解生物網絡的基本途徑之一，然而這很可能無法捕捉到這些複雜系統中的重要非線性交互作用。為了解決這個限制，基於狀態空間重構 (SSR) 的經驗動態建模(EDM)(Chang et al., 2017) 技術被開發，以用於量化 動態系統中的非線性交互作用。最近，Chang et al. (2021) 提出了一種多視圖距離正則化的 MDR S-map 方法，用於重構具有高維度、時間變化和非線性動態系統特徵的交互作用網絡。 然而，該方法的計算時間較長。為了解決這個問題，我們提出了一種核範數正則化的 S-map 方法，透過隨機森林中的鄰近測量 (proximity measure) 考慮物種異質性，並使用核範數正則化的預測損失來維持模型穩定性。本研究旨在有效推論交互作用效應並準確預測生態系統中 的時間序列數據。所提出的方法應用於四個模擬數據集，以證明其有效性。結果顯示:核範 數正則化的 S-map 方法相對 Chang 等人所提出的 MDR S-map 方法節省超過 70%的計算時 間，同時保持一定程度的估計性能。

Biological systems are highly complex due to their numerous interacting components. This complexity makes it challenging to understand their underlying mechanisms and their contri- butions to overall system behavior. Linear methods are one basic approach to understanding biological networks, but they may fail to capture important nonlinear interactions in these com- plex systems. To address this limitation, Empirical Dynamic Modeling (EDM) (Chang et al., 2017) based on State Space Reconstruction (SSR) (Takens, 2006) has been developed as a tech- nique to quantify nonlinear interactions in dynamical systems. Recently, Chang et al. (2021) proposed a multiview distance regularized MDR S-map approach for reconstructing interac- tion networks characterized by high dimensionality, temporal variations, nonlinear dynamical systems. However, this method is computationally time-consuming. To solve this issue, we propose a nuclear-norm regularized (NR) S-map approach that considers species heterogeneity using proximity in random forest and maintains model stability using prediction loss by nuclear- norm regularization. The study aims to effectively solve for interaction effects and accurately predict time series data in ecological systems. The proposed methodology is applied to four simulation datasets to demonstrate its effectiveness, and the results show that the proposed method achieves similar estimation performance as Chang’s approach while saving more than 70% of the computational time.

摘要 ................ i
Abstract ................ ii
List of Figures ................ v
List of Tables ................ vi
1 Introduction ............... 1
2 Data and Notations ............... 4
2.1 Data1 ................ 5
2.2 Data2 ................ 6
2.3 Data3 ................ 7
3 Overview of MDR S-map ................ 9
3.1 Measure Multiview Distances ................ 11
3.2 Estimate the Jacobian Interaction Matrix ................ 12
4 A Joint Estimation for Jacobian Interaction Across Species ... 14
4.1 Alternative Weights by Proximity Defined in Random Forest ..14
4.2 Nuclear Norm Loss ................15
4.3 Prediction ................. 18
5 Proposed NR S-map ................ 20
5.1 NR S-map with W ̂_t^((i))................ 20
5.2 NR S-map with W ̃_t^((i)) ................ 20
5.3 NR S-map with W ̌_t ................ 21
6 Simulation ................ 22
6.1 Without Regularization ................ 22
6.2 With Nuclear-Norm Regularization ................ 27
7 Discussion ................ 30
References ................ 31

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