Statistical Learning of Some Complex Systems

A complex system is one with many parts, whose behaviors are strongly dependent on each other. There are two interesting questions about complex systems. One is to understand how to recover the true structure of a complex system from noisy data. The other is to understand how the system interacts with its environment. In this thesis, we address these two questions by studying two distinct complex systems: dynamic systems and market microstructure. To address the first question, we focus on some nonlinear dynamic systems. We develop a novel Bayesian statistical method, Gaussian Emulator, to estimate the parameters of dynamic systems from noisy data, when the data are either fully or partially observed. Our method shows that estimation accuracy is substantially improved and computation is faster, compared to the numerical solvers. To address the second question, we focus on the market microstructure of hidden liquidity. We propose some statistical models to explain the hidden liquidity under different market conditions. Our statistical results suggest that hidden liquidity can be reliably predicted given the visible state of the market.