如何量化交互强度？对时间序列分析中非参数推断的交互雅可比矩阵与评估方法的批判性反思
How to quantify interaction strengths? A critical rethinking of the interaction Jacobian and evaluation methods for non-parametric inference in time series analysis
AI总结：
针对交互雅可比矩阵在生物学解释与数值实现上偏离数学定义的问题，提出对角元素减1调整法及累积交互强度（CIS）基准，为生态时间序列非参数推断提供更严谨的评估框架。
摘要：
量化动力系统中状态变量间的交互强度对于理解生态网络至关重要。在经验动力学建模方法中，多元S-map在不假设特定动力学模型的情况下，从时间序列数据中推断交互雅可比矩阵。该方法通过状态空间重建实现了对种间交互的非参数统计推断。然而，交互雅可比矩阵在生物学解释和数值实现上偏离其数学定义，带来了挑战。我们使用微分商在数学上重新引入交互雅可比矩阵，揭示了两个问题：(1) 交互雅可比矩阵与其生物学含义不匹配，使得种间和种内交互的比较复杂化；(2) 交互雅可比矩阵在由给定参数模型（特别是使用常微分方程）数值导出的参数雅可比矩阵中未被完全实现。因此，基于模型的S-map方法评估变得不合适。为解决这些问题，(1) 我们提议将交互雅可比矩阵的对角元素减去1进行调整，以解决种间与种内交互强度的可比性问题。种群动力学模拟表明，此调整防止了种内交互强度的过高估计。(2) 我们引入了一种替代的参数雅可比矩阵，进而提出累积交互强度（CIS），为评估S-map方法提供了更严格的基准。此外，我们证明了在现实场景中CIS与现有参数雅可比矩阵之间的数值差距很大，建议将CIS作为首选基准。这些解决方案为在生态时间序列分析中开发非参数方法提供了更清晰的框架。
英文摘要：
Quantifying interaction strengths between state variables in dynamical systems is essential for understanding ecological networks. Within the empirical dynamic modeling approach, multivariate S-map infers the interaction Jacobian from time series data without assuming specific dynamical models. This approach enables the non-parametric statistical inference of interspecific interactions through state space reconstruction. However, deviations in the biological interpretation and numerical implementation of the interaction Jacobian from its mathematical definition pose challenges. We mathematically reintroduce the interaction Jacobian using differential quotients, uncovering two problems: (1) the mismatch between the interaction Jacobian and its biological meaning complicates comparisons between interspecific and intraspecific interactions; (2) the interaction Jacobian is not fully implemented in the parametric Jacobian numerically derived from given parametric models, especially using ordinary differential equations. As a result, model-based evaluations of S-map methods become inappropriate. To address these problems, (1) we propose adjusting the diagonal elements of the interaction Jacobian by subtracting 1 to resolve the comparability problem between inter- and intraspecific interaction strengths. Simulations of population dynamics showed that this adjustment prevents overestimation of intraspecific interaction strengths. (2) We introduce an alternative parametric Jacobian and then cumulative interaction strength (CIS), providing a more rigorous benchmark for evaluating S-map methods. Furthermore, we demonstrated that the numerical gap between CIS and the existing parametric Jacobian is substantial in realistic scenarios, suggesting CIS as preferred benchmark. These solutions offer a clearer framework for developing non-parametric approaches in ecological time series analysis.
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