Inferring Non-Normal Amplification Geometry from Multivariate Time Series
Computational Geometry
Summary
The authors explain that some systems can temporarily amplify small disturbances even though they eventually calm down, and this behavior can't be seen by usual methods that look only at long-term trends. They developed a new data-driven way to detect this hidden amplification by analyzing patterns in time series data without knowing the exact underlying rules. Their method focuses on a simplified two-dimensional view of the system's response and uses key measures to capture how much temporary boosting occurs. They tested it on various data types and showed it can reveal meaningful changes linked to physical or behavioral events without needing prior labeling.
Authors
V. R. Saiprasad, V. Troude, D. Sornette
Abstract
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.