Dynamical Origins of Distribution Functions

Many real-world problems are time-evolving in nature, such as the progression of diseases, the cascading process when a post is broadcasting in a social network, or the changing of climates. The observational data characterizing these complex problems are usually only available at discrete time stamps, this makes the existing research on analyzing these problems mostly based on a cross-sectional analysis. In this paper, we try to model these time-evolving phenomena by a dynamic system and the data sets observed at different time stamps are probability distribution functions generated by such a dynamic system. We propose a theorem which builds a mathematical relationship between a dynamical system modeled by differential equations and the distribution function (or survival function) of the cross-sectional states of this system. We then develop a survival analysis framework to learn the differential equations of a dynamical system from its cross-sectional states. With such a framework, we are able to capture the continuous-time dynamics of an evolutionary system.We validate our framework on both synthetic and real-world data sets. The experimental results show that our framework is able to discover and capture the generative dynamics of various data distributions accurately. Our study can potentially facilitate scientific discoveries of the unknown dynamics of complex systems in the real world.