Data-driven reconstruction of a multivariate Langevin equation to model complex systems

Area of Science:

• Complex Systems Analysis
• Statistical Mechanics
• Quantitative Finance

Background:

• Modeling complex systems with intricate interactions is challenging.
• Existing methods often require a priori knowledge of underlying mechanisms.
• Accurate description of system observables is crucial for understanding behavior.

Purpose of the Study:

• To propose a data-driven multivariate Langevin equation (LE) for approximating complex system observables.
• To unravel key features of complex systems without requiring prior knowledge.
• To demonstrate the framework's adaptability and reliability across diverse applications.

Main Methods:

• Reconstruction of drift and diffusion terms in the LE using a nonparametric technique.
• Application of Kramers-Moyal coefficients for LE term identification.
• Benchmarking with a mechanical system (particle in a bistable potential) and financial data (electricity prices, currency exchange rates).

Main Results:

• The framework accurately identifies equilibrium values, metastability regions, and distinct diffusion behaviors.
• Successful application to financial markets (electricity day-ahead prices, currency-exchange rates) where LE has not been previously used.
• Demonstrated functional-agnostic approach, contrasting with domain-specific price-equation models.

Conclusions:

• The proposed nonparametric multivariate LE framework offers a reliable, data-driven approach to modeling complex systems.
• The method effectively extracts pertinent information and system features without necessitating a priori domain knowledge.
• This approach provides a powerful tool for analyzing diverse complex systems, including financial markets.