AI 中文总结
研究多元时间序列中渐近稳定动力学的非正态放大几何结构，提出非正态方向响应推断方法，通过估计局部线性算子投影到二维子空间，用特征值分裂等总结动力学，能从有限数据恢复结构，应用于多种记录揭示相关变化。
AI 中文摘要
在流体动力学、生态学、神经科学、网络动力学、非厄米物理和社会经济系统中，渐近稳定动力学可展现出基于特征值分析不可见的大瞬态放大。其机制是几何而非光谱的：沿一个方向进入的扰动可能沿另一方向瞬时表达，使渐近衰减与强瞬态或噪声驱动放大共存。我们引入非正态方向响应推断，一种在控制算子未知时从多元时间序列检测此几何结构的数据驱动方法。通过滑动窗口估计局部线性算子并投影到主导二维输入 - 响应子空间。用特征值分裂$\Delta$、特征向量非正交性$K$和无标度比$R = K / K_c(\Delta)$总结简化动力学，其中$K_c(\Delta)$是瞬态放大的二维阈值。控制基准表明即使全高维算子估计不佳，也可从有限数据恢复简化几何结构，特别是$R$。跨样本大小、维度、训练范围、光谱结构和非平稳性的测试证实相关响应几何结构所需观测远少于全矩阵恢复。应用于移动窗口的子宫电图、癫痫脑电图、步态冻结和不稳定俯卧撑惯性记录，该方法通过$R$的变化、$\Delta$的变化或波动在推断响应方向上更强投影揭示已知生理或行为事件周围的系统变化。从而揭示局部响应几何结构中可解释的变化，而无需将问题框定为监督事件检测。
英文摘要
Across hydrodynamics, ecology, neuroscience, network dynamics, non-Hermitian physics, and socio-economic systems, asymptotically stable dynamics can exhibit large transient amplifications that are invisible to eigenvalue-based analyses. The mechanism is geometric rather than spectral: perturbations entering along one direction may be expressed transiently along another, allowing asymptotic decay to coexist with strong transient or noise-driven amplification. We introduce non-normal directional response inference, a data-driven method for detecting this geometry from multivariate time series when the governing operator is unknown. A local linear operator is estimated from sliding windows and projected onto the dominant two-dimensional input-response subspace. The reduced dynamics are summarized by the eigenvalue splitting $Δ$, eigenvector non-orthogonality $K$, and the scale-free ratio $R=K/K_c(Δ)$, where $K_c(Δ)$ is the two-dimensional threshold for transient amplification. Controlled benchmarks show that the reduced geometry, particularly $R$, can be recovered from finite data even when the full high-dimensional operator is poorly estimated. Tests across sample size, dimension, training horizon, spectral structure, and non-stationarity confirm that the relevant response geometry requires far fewer observations than full-matrix recovery. Applied in moving windows to electrohysterogram, seizure EEG, freezing-of-gait, and unstable push-up inertial recordings, the method reveals systematic changes around known physiological or behavioral episodes through shifts in $R$, changes in $Δ$, or stronger projection of fluctuations onto the inferred response direction. It thus exposes interpretable changes in local response geometry without framing the problem as supervised event detection.
Comments35 pages, 17 figures
