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Random Jolts with Memory: How to Extract a System's Laws from Data ✛ ✎

Original: "Colored-LIM: A Data-Driven Method for Studying Dynamical Systems with Temporally Correlated Stochasticity"
· Justin Lien, Yan-Ning Kuo, Hiroyasu Ando, Shoichiro Kido
arXiv:2402.15184 · 2024-02-23 · arXiv non-exclusive · independent analysis · 1 min
math.NA
A new method called Colored-LIM reveals how random impacts with memory drive a complex system, using only observational records.

A translator gets distracted by every rustle in the hall, and a one-hour meeting stretches into two. Likewise, a complex system, such as the ocean, receives random jolts that don't vanish instantly but trail behind like an echo. Scientists used to think that these jolts don't remember each other, like raindrops hitting a roof. But in reality, they have memory, and old methods got it wrong.

White noise is random jolts without memory, like raindrops. Colored noise is jolts with memory, like gusts of wind that arrive in bursts.

The new Colored-LIM method learns to hear this echo. It looks at observational records, such as ocean temperature, and calculates how random jolts and system behavior are related. It turns out that even a tiny memory in the noise changes the picture dramatically. The method was tested on El Niño data, the warm current that affects weather. It showed that atmospheric influences remember themselves for about two months, and forecasts became more accurate.

The method was also applied to a university's energy consumption. It identified a daily cycle with a period of 22.7 hours, matching human activity, and found buildings with unusual consumption. This helps manage energy better and understand complex systems, from epidemics to earthquakes. Now random noise doesn't hinder; it helps reveal how the world works.

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In the white-noise limit (τ → 0), Colored-LIM does not reduce to classical LIM: the correlation function of colored noise has zero derivative at zero, whereas for white noise the derivative jumps. This means that even arbitrarily small but finite noise memory fundamentally changes the system's behavior near zero lag.

0 = A C_{xx} + C_{xx} A^T + Q B^T + B Q
Relates the dynamical matrix A, covariance Cxx, diffusion matrix Q, and resolvent B = (I - τA)^{-1}, generalizing the classical relation to the case of colored noise.
K''(0) = \frac{1}{2} \left[ A \left( K'(0) + \frac{1}{\tau} C_{xx} \right) + \left( K'(0)^T + \frac{1}{\tau} C_{xx} \right) A^T \right]
Expresses the curvature of the correlation function at zero in terms of the dynamical matrix, covariance, and first derivative, allowing estimation of A when τ is known.
K'''(0) = \frac{1}{2} \left[ \frac{1}{\tau^2} K'(0) + A \left( K''(0) - \frac{1}{\tau^2} C_{xx} \right) - \frac{1}{\tau^2} K'(0)^T - \left( K''(0) - \frac{1}{\tau^2} C_{xx} \right) A^T \right]
Used together with K'(0) and K''(0) to uniquely determine the dynamical matrix A when A is not negative definite.
Concepts
linear inverse model colored noise Stochastic modeling biological rhythm modeling labeled data stochastic dynamical system Dynamical Systems nonlinear dynamics nonlinear laplacian spectral analysis Contrastive Learning linear regression probabilistic model statistical method noninvasive dynamic imaging simulated data noise cancellation self-supervised learning physics informed neural network regularization methods Large Language Models