Variational Approach for Learning Markov Processes from Time Series Data

Abstract

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 crossvalidation for hyper-parameter optimization and model selection in VAMP. VAMP is valid for both reversible and nonreversible processes and for stationary and non-stationary processes or realizations.

The Dynamic Conformational Landscapes of the Protein Methyltransferase SETD8

“…All further steps until scoring were conducted by fitting the model to the training set only, then transforming the test set according to this model. Scoring was based on the sum of the top 10 squaredeigenvalues of the transition matrix (rank-10 VAMP-2 96 ). Model scores are reported below as means with standard deviations over five shuffle-splits.…”

“…To determine the optimal number of microstates, we again used variational scoring [94][95][96][97] combined with cross-validation 38 to evaluate model quality. The full dataset (4.931 ms, 0.5 ns/frame, 9,862,657 frames) was separately featurized with the top-scoring feature sets: 6,567 distances (featurization a above) and 920 dihedral angles (featurization c above).…”

“…MSMs were constructed with PyEMMA 2.5.1 103 from the discrete trajectories of the training set using a lag time of 50 ns and subsequently scored on the test set, using the rank-10 VAMP-2 96 score. The highest scoring model (Figure S21) had dihedral features (featurization c above) and 100 microstates (VAMP-2 = 9.25 (SD = 0.32)).…”

“…The code used for the generation and analysis of the molecular dynamics data is available via a Github repository at https://github.com/choderalab/SETD8-materials.Optimal hyperparameter selection for featurization and tICA. To select the optimal featurization of the data for subsequent Markov state model (MSM) analysis, we used variational scoring[94][95][96][97] combined with cross-validation 38 to evaluate model quality, consistent with modern MSM construction practice38 .…”

Molecular dynamics simulations of protein aggregation: protocols for simulation setup and analysis with Markov state models and transition networks

“…Next, find the best suited feature for capturing the slow dynamical processes. Here, the VAMP-2 score [38] is an established measure for ranking them. The aim is to extract the feature which preserves the most kinetic variance corresponding to the highest VAMP-2 score:…”