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Multi-Resolution Dynamic Mode Decomposition
J. Nathan Kutz, Xing Fu, Steven L. Brunton
TL;DR
Multi-scale spatio-temporal systems require methods that separate features across differing time and space scales. The paper introduces mrDMD by combining DMD with multi-resolution analysis, demonstrating hierarchical decompositions that recover phenomena such as the 1997 El Niño mode and distinguish translating objects at different rates. Its present time-bin sifting uses a hard cutoff that may introduce artificial high-frequency oscillations.
Problem
Existing multi-resolution analyses typically operate in space or time separately, motivating a method that separates multi-scale spatio-temporal dynamics.
Method
mrDMD recursively combines DMD with multi-resolution analysis to decompose nonlinear dynamical data into low-rank terms with known temporal coefficients.
Results
The method separates multi-resolution dynamical components, including the 1997 El Niño mode and translating structures moving at speeds v and 10v.
Takeaways & Limitations
mrDMD provides a data-driven, equation-free tool for reconstructing multi-resolution spatio-temporal data and supporting dynamical-systems discovery.
Takeaways & Limitations
The present hard-cutoff time-bin sifting may introduce artificial high-frequency oscillations; wavelet-based sifting is left for future work.
Abstract
from arXiv · showhide
We demonstrate that the integration of the recently developed dynamic mode decomposition (DMD) with a multi-resolution analysis allows for a decomposition method capable of robustly separating complex systems into a hierarchy of multi-resolution time-scale components. A one-level separation allows for background (low-rank) and foreground (sparse) separation of dynamical data, or robust principal component analysis. The multi-resolution dynamic mode decomposition is capable of characterizing nonlinear dynamical systems in an equation-free manner by recursively decomposing the state of the system into low-rank terms whose temporal coefficients in time are known. DMD modes with temporal frequencies near the origin (zero-modes) are interpreted as background (low-rank) portions of the given dynamics, and the terms with temporal frequencies bounded away from the origin are their sparse counterparts. The multi-resolution dynamic mode decomposition (mrDMD) method is demonstrated on several examples involving multi-scale dynamical data, showing excellent decomposition results, including sifting the El Niño mode from ocean temperature data. It is further applied to decompose a video data set into separate objects moving at different rates against a slowly varying background. These examples show that the decomposition is an effective dynamical systems tool for data-driven discovery.
1 Introduction
The paper integrates multi-resolution analysis with dynamic mode decomposition to separate multi-scale spatio-temporal features and construct approximate dynamical models. It extends equation-free, data-driven analysis toward hierarchical time-scale decomposition.
- Motivation: Multi-resolution analysis addresses systems with spatial and temporal scales separated by potentially orders of magnitude.Wavelet and windowed Fourier methods recursively remove features through refined sampling.
- Motivation: Conventional multi-resolution analysis typically operates in either space or time, rather than both simultaneously.
- Contribution: The proposed mrDMD integrates multi-resolution analysis with DMD to separate multi-scale spatio-temporal features.It also supports construction of approximate dynamical models.
- DMD foundation: DMD provides equation-free, data-driven assessments of spatio-temporal coherent structures and can support short-time future estimates.The method is connected to Koopman spectral analysis and is applicable to nonlinear dynamical systems.
- Paper scope: The paper develops mrDMD after introducing DMD theory, then evaluates it on dynamical-system and video-analysis applications.
2 Dynamic Mode Decomposition
DMD extracts low-dimensional spatio-temporal modes by fitting a low-rank linear mapping to snapshot data. Its eigenvalues and modes provide temporal growth rates, frequencies, and future-state reconstructions without requiring an equation-based model.
- Data and objective: DMD forms a spatio-temporal decomposition from time-resolved snapshots or measurements of a dynamical system.The data matrix contains state measurements collected at successive sampling times.
- Data and objective: DMD can accommodate regularly spaced, sparse, or irregularly spaced spatial and temporal data collection.
- Linear operator approximation: The method approximates nonlinear dynamics through the eigendecomposition of a best-fit linear operator relating successive state snapshots.This operator is chosen to minimize the Frobenius-norm prediction error.
- Low-rank computation: A rank-reduced SVD projects the high-dimensional operator onto POD modes before eigenvalues and DMD modes are computed.The reduced SVD uses U, Σ, and V, with U containing the POD modes; singular-value decay can guide truncation.
- Reconstruction: DMD reconstructs future states from low-rank eigenvalues, eigenvectors, temporal coefficients, and initial modal amplitudes.The amplitudes are obtained by solving the initial-state relation with a Moore-Penrose pseudoinverse in the least-squares sense.
- Interpretation and use: The resulting approximation is an equation-free low-rank model that provides a predicted state for any future time where the approximation holds.Unlike POD-Galerkin prediction, no additional dynamical solve is required after constructing the DMD representation.
- Applications: DMD also supports robust foreground/background separation in video data and can perform low-rank/sparse separation faster than standard ℓ1 optimization methods.The cited comparison reports a speed advantage of 3-4 orders of magnitude.
3 Multi-Resolution Dynamic Mode Decomposition
mrDMD recursively separates slow-frequency content from residual dynamics, then reapplies DMD across progressively shorter time windows to represent modes by level and time bin.
- mrDMD first separates slow-mode dynamics from fast-mode content, retaining the latter as a residual matrix for further decomposition.The slow modes are removed from the full-snapshot DMD approximation before analyzing the residual.
- The residual is partitioned into two consecutive snapshot intervals, and DMD is reapplied separately to each interval.At the next level, each half contains M/2 snapshots and yields its own slow-DMD modes.
- The process recursively removes slow-frequency components and forms matrices with M/2, M/4, M/8, and subsequent snapshot counts.Decomposition continues until the prescribed multi-resolution structure has been achieved.
- Each extracted mode is indexed by decomposition level, time bin, and mode number, locating it within the mrDMD hierarchy.The expansion uses L levels, J = 2^(ℓ−1) time bins per level, and retained modes at each level.
- At each level and time bin, mrDMD provides a least-square linear-dynamical fit, with an indicator function selecting the corresponding time interval.The matrix A^(ℓ,j) captures dynamics in time bin j at level ℓ, while fℓ,j(t) performs the time-bin sifting.
- Hard time-bin cutoffs can introduce artificial high-frequency oscillations, motivating future use of wavelet functions for the sifting operation.The current formulation uses the indicator function rather than Haar, Daubechies, Mexican Hat, or other wavelet bases.
4 Application of Method
The mrDMD applications demonstrate multi-resolution separation across video, ocean-temperature, and moving-object data. It reconstructs intermittent dynamics, isolates the 1997 El Niño mode, and separates objects moving at distinct rates.
- Spatio-temporal filtering of video: The mrDMD accurately reconstructs multi-scale video dynamics, whereas standard DMD mixes modes and misses on-off behavior.The mrDMD reconstruction closely matches the true snapshots, while DMD shows inconsistencies, especially when signals turn on and off.
- Spatio-temporal filtering of video: The recursive decomposition removes slow frequency components and organizes remaining dynamics across successive levels and sampling windows.The video example retains dominant eigenvalues at each level and uses mode-removal regions to identify progressively faster components.
- Multi-scale time separation of complex system: El Niño, Southern Oscillation: In sea-surface-temperature data, level 1 extracts the 20-year average ocean temperature as a zero mode and a yearly cycle with period T = 52 weeks.The zero mode has period T = ∞, while the yearly cycle is the slowest mode extracted at level 1.
- Multi-scale time separation of complex system: El Niño, Southern Oscillation: The level-4 decomposition discovers a strong 1997 El Niño mode in the central and east-central equatorial Pacific, absent from the corresponding 1999 window.The 1997 mode appears as a warm-water band, while the shifted 1999 sampling window produces no El Niño mode.
- Translating and/or Rotating Structures: The moving-object example addresses a limitation of SVD-based methods and supports separating foreground objects at different speeds for video processing and surveillance.The authors describe mrDMD as effective for separating, for example, a slowly walking pedestrian from a rapidly moving vehicle.
- Translating and/or Rotating Structures: For moving objects with speeds v and 10v, mrDMD extracts the slow object at level 2 and the fast object at level 13.The levels at which objects appear can support reconstruction of their velocity and direction, although residual shadows remain between extracted modes.
5 Discussion and Outlook
The paper presents mrDMD as a principled, equation-free framework for decomposing multiscale spatio-temporal data and demonstrates its effectiveness across example datasets. It also outlines extensions involving compressed measurements, control, and machine learning.
- mrDMD integrates DMD with wavelet theory and multi-resolution analysis to reconstruct multiscale spatio-temporal datasets.
- The method is demonstrated on several datasets as a tool for extracting critical information and supporting data-driven discovery.
- Recursive removal of slow modes filters data for progressively higher-frequency content while supporting reconstruction of nonlinear dynamics.
- Compressed DMD is proposed as a direction for obtaining the same multi-resolution decomposition from considerably fewer measurements.The authors identify expensive or prohibitive data acquisition as a motivating setting.
- Potential extensions include DMD with control for input-output modeling and DMD-mode libraries combined with machine-learning and compressive-sensing strategies.