Beyond expectations: residual dynamic mode decomposition and variance for stochastic dynamical systems Article Swipe
Matthew J. Colbrook
,
Qin Li
,
Ryan V. Raut
,
Alex Townsend
·
YOU?
·
· 2023
· Open Access
·
· DOI: https://doi.org/10.1007/s11071-023-09135-w
YOU?
·
· 2023
· Open Access
·
· DOI: https://doi.org/10.1007/s11071-023-09135-w
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 Topics
Concepts
Dynamic mode decomposition
Residual
Observable
Dynamical systems theory
Applied mathematics
Variance (accounting)
Mathematics
Operator (biology)
Spurious relationship
Nonlinear system
Projection (relational algebra)
Computer science
Algorithm
Statistics
Quantum mechanics
Chemistry
Accounting
Business
Transcription factor
Machine learning
Repressor
Physics
Gene
Biochemistry
Metadata
- Type
- article
- Language
- en
- Landing Page
- https://doi.org/10.1007/s11071-023-09135-w
- https://link.springer.com/content/pdf/10.1007/s11071-023-09135-w.pdf
- OA Status
- hybrid
- Cited By
- 17
- References
- 88
- Related Works
- 10
- OpenAlex ID
- https://openalex.org/W4390144100
All OpenAlex metadata
Raw OpenAlex JSON
- OpenAlex ID
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https://openalex.org/W4390144100Canonical identifier for this work in OpenAlex
- DOI
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https://doi.org/10.1007/s11071-023-09135-wDigital Object Identifier
- Title
-
Beyond expectations: residual dynamic mode decomposition and variance for stochastic dynamical systemsWork title
- Type
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articleOpenAlex work type
- Language
-
enPrimary language
- Publication year
-
2023Year of publication
- Publication date
-
2023-12-23Full publication date if available
- Authors
-
Matthew J. Colbrook, Qin Li, Ryan V. Raut, Alex TownsendList of authors in order
- Landing page
-
https://doi.org/10.1007/s11071-023-09135-wPublisher landing page
- PDF URL
-
https://link.springer.com/content/pdf/10.1007/s11071-023-09135-w.pdfDirect link to full text PDF
- Open access
-
YesWhether a free full text is available
- OA status
-
hybridOpen access status per OpenAlex
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https://link.springer.com/content/pdf/10.1007/s11071-023-09135-w.pdfDirect OA link when available
- Concepts
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Dynamic mode decomposition, Residual, Observable, Dynamical systems theory, Applied mathematics, Variance (accounting), Mathematics, Operator (biology), Spurious relationship, Nonlinear system, Projection (relational algebra), Computer science, Algorithm, Statistics, Quantum mechanics, Chemistry, Accounting, Business, Transcription factor, Machine learning, Repressor, Physics, Gene, BiochemistryTop concepts (fields/topics) attached by OpenAlex
- Cited by
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17Total citation count in OpenAlex
- Citations by year (recent)
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2025: 11, 2024: 5, 2023: 1Per-year citation counts (last 5 years)
- References (count)
-
88Number of works referenced by this work
- Related works (count)
-
10Other works algorithmically related by OpenAlex
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