How to quantify interaction strengths? A critical rethinking of the interaction Jacobian and evaluation methods for non-parametric inference in time series analysis

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 multivariate 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 unique mathematical definition pose challenges. We mathematically reintroduce the interaction Jacobian by starting our derivation with differential quotients, uncovering two key problems: (1) the mismatch between the interaction Jacobian and its biological meaning complicates comparisons between interspecific and intraspecific interaction strengths; (2) the interaction Jacobian is not fully implemented in the parametric Jacobian numerically derived from given parametric models, especially ordinary differential equation models. 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 interspecific and intraspecific interaction strengths. Simulations of population dynamics showed that this adjustment prevents overestimation of intraspecific interaction strengths, allowing for meaningful comparisons. (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 a preferred benchmark for future evaluations. These solutions offer a clearer framework for developing non-parametric approaches in ecological time series analysis.