Beyond expectations: Residual Dynamic Mode Decomposition and Variance for Stochastic Dynamical Systems

• URL: http://arxiv.org/abs/2308.10697v3
• Date: Fri, 10 Nov 2023 13:19:10 GMT
• Title: Beyond expectations: Residual Dynamic Mode Decomposition and Variance for Stochastic Dynamical Systems
• Authors: Matthew J. Colbrook, Qin Li, Ryan V. Raut, Alex Townsend
• Abstract summary: Dynamic Mode Decomposition (DMD) is the poster child of projection-based methods. We introduce the concept of variance-pseudospectra to gauge statistical coherency. Our study concludes with practical applications using both simulated and experimental data.
• Score: 8.259767785187805
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Koopman operators linearize nonlinear dynamical systems, making their spectral information of crucial interest. Numerous algorithms have been developed to approximate these spectral properties, and Dynamic Mode Decomposition (DMD) stands out as the poster child of projection-based methods. Although the Koopman operator itself is linear, the fact that it acts in an infinite-dimensional space of observables poses challenges. These include spurious modes, essential spectra, and the verification of Koopman mode decompositions. While recent work has addressed these challenges for deterministic systems, there remains a notable gap in verified DMD methods for stochastic systems, where the Koopman operator measures the expectation of observables. We show that it is necessary to go beyond expectations to address these issues. By incorporating variance into the Koopman framework, we address these challenges. Through an additional DMD-type matrix, we approximate the sum of a squared residual and a variance term, each of which can be approximated individually using batched snapshot data. This allows verified computation of the spectral properties of stochastic Koopman operators, controlling the projection error. We also introduce the concept of variance-pseudospectra to gauge statistical coherency. Finally, we present a suite of convergence results for the spectral information of stochastic Koopman operators. Our study concludes with practical applications using both simulated and experimental data. In neural recordings from awake mice, we demonstrate how variance-pseudospectra can reveal physiologically significant information unavailable to standard expectation-based dynamical models.

Related papers

• Multiplicative Dynamic Mode Decomposition [4.028503203417233]
  We introduce Multiplicative Dynamic Mode Decomposition (MultDMD), which enforces the multiplicative structure inherent in the Koopman operator within its finite-dimensional approximation. MultDMD presents a structured approach to finite-dimensional approximations and can accurately reflect the spectral properties of the Koopman operator. We elaborate on the theoretical framework of MultDMD, detailing its formulation, optimization strategy, and convergence properties.
  arXiv  Detail & Related papers  (2024-05-08T18:09:16Z)
• Rigged Dynamic Mode Decomposition: Data-Driven Generalized Eigenfunction Decompositions for Koopman Operators [0.0]
  We introduce the Rigged Dynamic Mode Decomposition (Rigged DMD) algorithm, which computes generalized eigenfunction decompositions of Koopman operators. Rigged DMD addresses challenges with a data-driven methodology that approximates the Koopman operator's resolvent and its generalized eigenfunctions. We provide examples, including systems with a Lebesgue spectrum, integrable Hamiltonian systems, the Lorenz system, and a high-Reynolds number lid-driven flow in a two-dimensional square cavity.
  arXiv  Detail & Related papers  (2024-05-01T18:00:18Z)
• A Poisson-Gamma Dynamic Factor Model with Time-Varying Transition Dynamics [51.147876395589925]
  A non-stationary PGDS is proposed to allow the underlying transition matrices to evolve over time. A fully-conjugate and efficient Gibbs sampler is developed to perform posterior simulation. Experiments show that, in comparison with related models, the proposed non-stationary PGDS achieves improved predictive performance.
  arXiv  Detail & Related papers  (2024-02-26T04:39:01Z)
• Score-based Diffusion Models in Function Space [140.792362459734]
  Diffusion models have recently emerged as a powerful framework for generative modeling. We introduce a mathematically rigorous framework called Denoising Diffusion Operators (DDOs) for training diffusion models in function space. We show that the corresponding discretized algorithm generates accurate samples at a fixed cost independent of the data resolution.
  arXiv  Detail & Related papers  (2023-02-14T23:50:53Z)
• Gaussian Process Koopman Mode Decomposition [5.888646114353371]
  We propose a nonlinear probabilistic generative model of Koopman mode decomposition based on an unsupervised Gaussian process. Applying the proposed model to both synthetic data and a real-world epidemiological dataset, we show that various analyses are available using the estimated parameters.
  arXiv  Detail & Related papers  (2022-09-09T03:57:07Z)
• Learning Dynamical Systems via Koopman Operator Regression in Reproducing Kernel Hilbert Spaces [52.35063796758121]
  We formalize a framework to learn the Koopman operator from finite data trajectories of the dynamical system. We link the risk with the estimation of the spectral decomposition of the Koopman operator. Our results suggest RRR might be beneficial over other widely used estimators.
  arXiv  Detail & Related papers  (2022-05-27T14:57:48Z)
• Residual Dynamic Mode Decomposition: Robust and verified Koopmanism [0.0]
  Dynamic Mode Decomposition (DMD) describes complex dynamic processes through a hierarchy of simpler coherent features. We present Residual Dynamic Mode Decomposition (ResDMD), which overcomes challenges through the data-driven computation of residuals associated with the full infinite-dimensional Koopman operator. ResDMD computes spectra and pseudospectra of general Koopman operators with error control, and computes smoothed approximations of spectral measures (including continuous spectra) with explicit high-order convergence theorems.
  arXiv  Detail & Related papers  (2022-05-19T18:02:44Z)
• Mixed Effects Neural ODE: A Variational Approximation for Analyzing the Dynamics of Panel Data [50.23363975709122]
  We propose a probabilistic model called ME-NODE to incorporate (fixed + random) mixed effects for analyzing panel data. We show that our model can be derived using smooth approximations of SDEs provided by the Wong-Zakai theorem. We then derive Evidence Based Lower Bounds for ME-NODE, and develop (efficient) training algorithms.
  arXiv  Detail & Related papers  (2022-02-18T22:41:51Z)
• Inverting brain grey matter models with likelihood-free inference: a tool for trustable cytoarchitecture measurements [62.997667081978825]
  characterisation of the brain grey matter cytoarchitecture with quantitative sensitivity to soma density and volume remains an unsolved challenge in dMRI. We propose a new forward model, specifically a new system of equations, requiring a few relatively sparse b-shells. We then apply modern tools from Bayesian analysis known as likelihood-free inference (LFI) to invert our proposed model.
  arXiv  Detail & Related papers  (2021-11-15T09:08:27Z)
• Deep Learning Enhanced Dynamic Mode Decomposition [0.0]
  We use convolutional autoencoder networks to simultaneously find optimal families of observables. We also generate both accurate embeddings of the flow into a space of observables and immersions of the observables back into flow coordinates. This network results in a global transformation of the flow and affords future state prediction via EDMD and the decoder network.
  arXiv  Detail & Related papers  (2021-08-10T03:54:23Z)
• Dynamic Mode Decomposition in Adaptive Mesh Refinement and Coarsening Simulations [58.720142291102135]
  Dynamic Mode Decomposition (DMD) is a powerful data-driven method used to extract coherent schemes. This paper proposes a strategy to enable DMD to extract from observations with different mesh topologies and dimensions.
  arXiv  Detail & Related papers  (2021-04-28T22:14:25Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.