Granger causality based on vector time series and quaternion algebra with possible applications to molecular dynamics data analysis.

Causal analysis plays a significant role in physics, chemistry, and biology. Dynamics of complex (bio)molecular and nanosystems, from the microscopic to the macroscopic scale, are characterized by time-dependent vectors such as positions, forces, momenta, angular momenta, or torques. Identification and analysis of causal relationships between these time-dependent signals is an important problem in the multidimensional time-series analysis and is of great practical importance in describing the properties of such dynamical systems, and to understanding their functionality. For linear stochastic systems characterized by multidimensional scalar signals, Granger proposed a simple procedure to detect causal relationships, called Granger causality. In this study we extended this formalism to vector signals representing physical vector quantities. For this purpose, we used quaternion algebra, where vector signals are treated as time-dependent quaternions. The developed analytical model is based on the autoregressive formalism. This formalism (Q-MVAR) and its numerical implementation were validated using two simple dynamic models: a rigid body model represented by a benzenelike molecular fragment, interacting with a short-range harmonic potential with a wall, as well as a system of three model atomic balls moving inside a soft spherical surface and interacting with long range electrostatic forces. Although the motivation to these studies was the analysis of classical motions in complex (bio)molecular systems, described with a mechanical model and based on molecular dynamics (MD) simulations, in particular coarse-grained ones, it should be noted that the developed extended formalism can be applied to any system composed of many rigid elements that interact with arbitrary potentials and are characterized by complex internal motions. A description of the detailed procedure for calculating causality measures is provided in the Appendices of the Supplemental Material. This formalism and the prototype of its numerical implementation can be further developed and applied in many different fields of physical, natural, and engineering sciences.