The DOI matches its Crossref record; 63 of the 63 references it cites were found in a registry or an index.
Variational Approach for Learning Markov Processes from Time Series Data
Open accessAbstract
Inference, prediction, and control of complex dynamical systems from time series is important in many areas, including financial markets, power grid management, climate and weather modeling, or molecular dynamics. The analysis of such highly nonlinear dynamical systems is facilitated by the fact that we can often find a (generally nonlinear) transformation of the system coordinates to features in which the dynamics can be excellently approximated by a linear Markovian model. Moreover, the large number of system variables often change collectively on large time- and length-scales, facilitating a low-dimensional analysis in feature space. In this paper, we introduce a variational approach for Markov processes (VAMP) that allows us to find optimal feature mappings and optimal Markovian models of the dynamics from given time series data. The key insight is that the best linear model can be obtained from the top singular components of the Koopman operator. This leads to the definition of a family of score functions called VAMP-r which can be calculated from data, and can be employed to optimize a Markovian model. In addition, based on the relationship between the variational scores and approximation errors of Koopman operators, we propose a new VAMP-E score, which can be applied to cross-validation for hyper-parameter optimization and model selection in VAMP. VAMP is valid for both reversible and nonreversible processes and for stationary and nonstationary processes or realizations.
Overview
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Integrity signals
Registry
Cites
Statistics
Verification
Claims the paper makes
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
The self-check needs the paper’s full text.
Citations
Cites
Regression shrinkage and selection via the lasso
Tibshirani, R. (1996) · Journal of the Royal Statistical Society Series B: Statistical Methodology · doi:10.1111/j.2517-6161.1996.tb02080.x · Cited by 51,482
The Elements of Statistical Learning
Hastie, T., Friedman, J., & Tibshirani, R. (2001) · Springer Series in Statistics · doi:10.1007/978-0-387-21606-5 · Cited by 24,162
Numerical Recipes, The Art of Scientific Computing
Press, W. H., Flannery, B. P., Teukolsky, S. A., Vetterling, W. T., & Gould, H. (1987) · American Journal of Physics · doi:10.1119/1.14981 · Cited by 10,024
Numerical Recipes: The Art of Scientific Computing
Ziegel, E. (1987) · Technometrics · doi:10.1080/00401706.1987.10488304 · Cited by 6,702
Practical Bayesian Optimization of Machine Learning Algorithms
Snoek, J., Larochelle, H., & Adams, R. P. (2012) · arXiv · doi:10.48550/arxiv.1206.2944 · Cited by 6,691
Dynamic mode decomposition of numerical and experimental data
SCHMID, P. J. (2010) · Journal of Fluid Mechanics · doi:10.1017/s0022112010001217 · Cited by 5,915
Inequalities: Theory of Majorization and its Applications.
Arnold, B. C., Marshall, A. W., & Olkin, I. (1981) · Journal of the American Statistical Association · doi:10.2307/2287859 · Cited by 4,870
Discovering governing equations from data by sparse identification of nonlinear dynamical systems
Brunton, S. L., Proctor, J. L., & Kutz, J. N. (2016) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.1517384113 · Cited by 4,796
A survey of cross-validation procedures for model selectionPreprint
Arlot, S., & Célisse, A. (2010) · Cited by 3,625
A survey of cross-validation procedures for model selection
Arlot, S., & Celisse, A. (2010) · Statistics Surveys · doi:10.1214/09-ss054 · Cited by 3,522
Research works printed but not linked: 9
- Bollt, E.M., Santitissadeekorn, N.: Applied and Computational Measurable Dynamics. SIAM (2013) · doi:10.1137/1.9781611972641
- Froyland, G., González-Tokman, C., Watson, T.M.: Optimal mixing enhancement by local perturbation. SIAM Rev. 58(3), 494–513 (2016) · doi:10.1137/15m1023221
- Hsing, T., Eubank, R.: Theoretical Foundations of Functional Data Analysis, with an Introduction to Linear Operators. Wiley, Amsterdam (2015) · doi:10.1002/9781118762547
- Klus, S., Schütte, C.: Towards tensor-based methods for the numerical approximation of the perron-frobenius and koopman operator (2015). arXiv:1512.06527 · arXiv
- Klus, S., Koltai, P., Schütte, C.: On the numerical approximation of the perron-frobenius and koopman operator (2015). arXiv:1512.05997 · arXiv
- Konrad, A., Zhao, B.Y., Joseph, A.D., Ludwig, R.: A Markov-based channel model algorithm for wireless networks. In: Proceedings of the 4th ACM International Workshop on Modeling, Analysis and Simulation of Wireless and Mobile Systems, pp. 28–36. ACM (2001) · doi:10.1145/381591.381602
- Korda, M., Mezić, I.: On convergence of extended dynamic mode decomposition to the Koopman operator. J. Nonlinear Sci. 28(2), 687–710 (2018) · doi:10.1007/s00332-017-9423-0
- Kurebayashi, W., Shirasaka, S., Nakao, H.: Optimal parameter selection for kernel dynamic mode decomposition. In: Proceedings of the International Symposium NOLTA, volume 370, p. 373 (2016)
- Ma, Y., Han, J.J., Trivedi, K.S.: Composite performance and availability analysis of wireless communication networks. IEEE Trans. Veh. Technol. 50(5), 1216–1223 (2001) · doi:10.1109/25.950322
Cited by
Data-driven discovery of coordinates and governing equations
Champion, K., Lusch, B., Kutz, J. N., & Brunton, S. L. (2019) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.1906995116 · Cited by 909
Time-lagged autoencoders: Deep learning of slow collective variables for molecular kinetics
Wehmeyer, C., & Noé, F. (2018) · The Journal of Chemical Physics · doi:10.1063/1.5011399 · Cited by 401
Self-citationEffect of natural mutations of SARS-CoV-2 on spike structure, conformation, and antigenicity
Gobeil, S. M.-C., Janowska, K., McDowell, S., Mansouri, K., Parks, R., Stalls, V., Kopp, M. F., Manne, K., Li, D., Wiehe, K., et al. (2021) · Science · doi:10.1126/science.abi6226 · Cited by 394
Saheed, Y. K., Omole, A. I., & Sabit, M. O. (2025) · Sensors International · doi:10.1016/j.sintl.2024.100297 · Cited by 88
A kinetic ensemble of the Alzheimer’s Aβ peptide
Löhr, T., Kohlhoff, K., Heller, G. T., Camilloni, C., & Vendruscolo, M. (2021) · Nature Computational Science · doi:10.1038/s43588-020-00003-w · Cited by 57
Metadata
- Authors
- Wu, H., & Noé, F.
- Venue
- Journal of Nonlinear Science
- Published
- 2019-08-05
- Volume
- 30
- Issue
- 1
- Pages
- 23-66
- ISSN
- 0938-8974 (ISSN-L)
- 1432-1467
- OpenAlex
- W2964917500
- Cited by
- 304
- References
- 63
- FWCI
- 14.30
- Type
- journal article
- Language
- en
- Open access
- green
- Licence
- www.springer.com/tdm (tdm)
- www.springer.com/tdm (vor)
- Funders
- Deutsche Forschungsgemeinschaft (SFB 1114/A4)
- European Research Commission (ERC StG 307494 “pcCell”)
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review
How to cite
Cite CoChat for the checks on this page, and the paper for its own findings.
- This page
- CoChat. (2026). Citation check: Variational Approach for Learning Markov Processes from Time Series Data. https://app.cochat.ai/papers/10.1007/s00332-019-09567-y
- The paper
- Wu, H., & Noé, F. (2020). Variational Approach for Learning Markov Processes from Time Series Data. Journal of Nonlinear Science. https://doi.org/10.1007/s00332-019-09567-y
Data sources
Bibliographic data: Semantic Scholar, OpenAlex, Crossref.
Contains information from the Semantic Scholar Academic Graph Datasets, made available under the ODC Attribution License (ODC-BY 1.0).
Semantic Scholar’s data is described in: Kinney, R., et al. (2023). The Semantic Scholar Open Data Platform. arXiv:2301.10140.
