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

Obtaining an accurate description of complex systems is challenging, particularly when their elements exhibit intricate interactions. We propose a data-driven multivariate Langevin equation (LE) to approximate real-world complex systems' observables. By reconstructing the drift and diffusion terms of the LE through a nonparametric technique following the definition of the Kramers-Moyal coefficients, our approach unravels the main features of a complex system without requiring a priori knowledge about the underlying governing mechanisms. We illustrate our framework's adaptability, reliability, and capability to extract pertinent information through three case studies. First, we benchmark the framework with a simple prototypical example from mechanics, a particle confined by a bistable potential energy well. We then turn to two more involved examples from financial markets, the electricity day-ahead prices and currency-exchange rates, where the nonparametric multivariate LE has not previously been applied. In all cases, our framework accurately identifies the equilibrium values, metastabilty regions, and distinct diffusion behaviors, in a functional agnostic manner, as opposed to price-equation models that require specific domain knowledge.