Variational Approach for Learning Markov Processes from Time Series Data

By Hao Wu, Frank No'e. 2017

323 citations recorded by the source corpus.

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.

Subfields: Artificial Intelligence, Statistical and Nonlinear Physics, Statistics and Probability

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Citations in the corpus

Deep Generative Markov State Models (2018, methodology): “When the VAMPnet has a probabilistic output (e.g. SoftMax layer), the Markovian model conserves probability, but is not guaranteed to be a valid transition probability matrix with nonnegative elements.”
On Convergence of Extended Dynamic Mode Decomposition to the Koopman Operator (2017, methodology): “This line of research was already investigated in the context of stochastic systems in [28], providing an interesting and actionable method for selecting FN .”
Error bounds for kernel-based approximations of the Koopman operator (2023, methodology): “Koopman methods have found widespread application in system identification [5], control [29, 49, 30, 22, 56], sensor placement [38], molecular dynamics [57, 51, 42, 43, 23, 64], and many other fields.”
Machine learning for molecular simulation (2019, methodology): “If no specific constraints are imposed on P, the minimum regression error of yt+τ = Pyt, the variational approach of Markov processes (VAMP) [85, 73], or maximum likelihood will all lead to the same estimator [86]: P = C−100 C0τ , (14) using the latent space covariance matrices C00 = 1 T ∑T−τ t=1 yt”
VAMPnets for deep learning of molecular kinetics (2017, methodology): “[47], that can be equivalently formulated as the following subspace version: VAMP variational principle : For any two sets of linearly independent functions χ 0 ( x ) and χ 1 ( x ) , let us call This variational theorem shows that the VAMP-2 score measures the consistency between sub-spaces of…”