Constructing Compact Causal Mathematical Models for Complex Dynamics

From microbial communities, human physiology to social and bio- logical/neural networks, complex interdependent systems display multi-scale spatio-temporal pa erns that are frequently classi ed as non-linear, non-Gaussian, non-ergodic, and/or fractal. Distin- guishing between the sources of nonlinearity, identifying the na- ture of fractality (space versus time) and encapsulating the non- Gaussian characteristics into dynamic causal models remains a ma- jor challenge for studying complex systems. In this paper, we pro- pose a new mathematical strategy for constructing compact yet ac- curate models of complex systems dynamics that aim to scrutinize the causal e ects and in uences by analyzing the statistics of the magnitude increments and the inter-event times of stochastic pro- cesses. We derive a framework that enables to incorporate knowl- edge about the causal dynamics of the magnitude increments and the inter-event times of stochastic processes into a multi-fractional order nonlinear partial di erential equation for the probability to nd the system in a speci c state at one time. Rather than follow- ing the current trends in nonlinear system modeling which pos- tulate speci c mathematical expressions, this mathematical frame- work enables us to connect the microscopic dependencies between the magnitude increments and the inter-event times of one stochas- tic process to other processes and justify the degree of nonlinearity. In addition, the newly presented formalism allows to investigate appropriateness of using multi-fractional order dynamical models for various complex system which was overlooked in the literature. We run extensive experiments on several sets of physiological pro- cesses and demonstrate that the derived mathematical models o er superior accuracy over state of the art techniques.