Variational approach for learning Markov processes from time series data (1707.04659v3)

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 cross-validation 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 paper presents a variational approach (VAMP) that uses the Koopman operator's SVD to extract optimal linear representations of complex dynamics.
• It introduces the VAMP-E score to enhance model selection and hyper-parameter tuning across reversible, irreversible, and non-stationary processes.
• Applications on systems like the double-gyre and Lorenz attractor demonstrate VAMP's effectiveness in capturing dominant stochastic and nonlinear behaviors.

Variational Approach for Learning Markov Processes from Time Series Data

The paper "Variational approach for learning Markov processes from time series data" by Hao Wu and Frank Noé introduces a novel framework, VAMP, for deriving optimal Markovian representations from time series data. The analysis of complex dynamical systems is a pivotal task in various fields such as finance, molecular dynamics, and climate science. A common approach is to represent these systems by Markov models, especially when a linear transformation of the system can approximately capture its dynamics as Markovian.

Core Contributions and Methodology

The primary contribution of this paper is the development of the variational approach for Markov processes (VAMP). The authors achieve this by leveraging the Koopman operator's singular value decomposition (SVD), allowing for the identification of optimal linear models that can map the complex dynamics of a system. This approach extends the applicability beyond reversible and stationary processes, offering a unified method for reversible, irreversible, stationary, and non-stationary processes.

Key points of the methodology include:

• Koopman Operator and Singular Components: The optimal linear model for capturing the dynamics is obtained from the singular components of the Koopman operator. The SVD provides the foundation for defining a family of variational scores, termed VAMP-$r$, which guide the optimization of feature mappings.
• Score Functions and Model Selection: A novel VAMP-E score is introduced, offering advantages in model selection through cross-validation, addressing issues such as hyper-parameter tuning.
• Nonlinear and Feature Time-lagged Canonical Correlation Analysis (TCCA): This framework allows for the optimization of feature transformations, even with complex nonlinear mappings such as those provided by deep neural networks, which were unexplored in prior reversible approaches.

Implications and Results

The authors present rigorous numerical simulations and theoretical underpinnings to validate their approach. Using examples like the double-gyre system and the Lorenz attractor, they demonstrate that VAMP successfully identifies the dominant dynamics in systems with different characteristics (e.g., stochastic influence, nonlinear behavior).

Contributions to AI and Future Directions

VAMP is a significant stride in the data-driven analysis of Markov processes. It broadens the scope for dynamically characterizing systems without the limitation to reversibility or stationarity, a restriction present in many traditional methodologies. Furthermore, its applicability to irreversible processes and capacity for model selection and hyper-parameter tuning offers substantial utility in real-world scenarios.

The research hints at potential future explorations, particularly in leveraging advanced machine learning architectures like deep learning to discover higher-dimensional feature spaces more efficiently. This direction could prove instrumental in overcoming challenges posed by large-scale systems, enabling the development of models that maintain fidelity across various dynamical regimes.

In conclusion, the introduction of VAMP provides an advanced platform for accurately modeling and understanding complex temporal dynamics from empirical data, marking a critical inflection point in the application of variance-based Markov modeling. The theoretical robustness combined with practical applicability ensures its relevance across diverse scientific and engineering disciplines dealing with time series analysis.