Improving Estimation of the Koopman Operator with Kolmogorov-Smirnov Indicator Functions

Area of Science:

• Computational Chemistry
• Statistical Mechanics
• Data Science

Background:

• Kinetic analysis often employs approximate Koopman operators to simplify high-dimensional time series data into dynamical modes.
• The effectiveness of Koopman operator methods hinges on selecting appropriate observables that accurately represent slow relaxation processes.
• Identifying optimal observables a priori is challenging, and suboptimal choices can lead to inaccurate estimations of system dynamics and timescales.

Purpose of the Study:

• To develop a computationally efficient method for inferring surrogate observables that form a good basis for slow dynamical modes.
• To leverage Hidden Markov Models (HMM) for identifying key states and transition timescales in stochastic systems.
• To improve the accuracy of kinetic analysis using Koopman operators, particularly when optimal observables are not readily available.

Main Methods:

• Proposed a clustering procedure based on the Hidden Markov Model (HMM) representation of slow dynamics.
• Inferred surrogate observables designed to form a suitable basis for expanding slow relaxation modes.
• Applied the method to an analytically solvable model and three protein systems of varying complexity.

Main Results:

• Demonstrated significant improvement in estimating the leading eigenvalues of Koopman operators using the inferred observables.
• Successfully identified key states and transition timescales in stochastic systems.
• Showcased the method's effectiveness even when optimal observables were not known a priori.

Conclusions:

• The proposed HMM-based clustering approach provides a simple and efficient way to infer optimal observables for Koopman operator analysis.
• This method enhances the accuracy of kinetic analysis by improving the estimation of dynamical modes and characteristic time scales.
• The findings are applicable to a range of stochastic systems, including complex biomolecular systems.