## 1 Introduction
Learning from existing state-of-the-art approaches, a deep Gaussian process (DGP)-based framework is proposed to achieve the trajectory prediction, and further to implement a probabilistic based short-term conflict detection (STCD) algorithm. A stochastic probabilistic process is capable of building the temporal dependencies of flight trajectory by fitting a probabilistic distribution at each prediction instant. That is to say, a probabilistic process is the joint distribution of sequential probabilistic distributions with respect to the prediction instant [30; 31]. Based on the specificities of the Gaussian distribution, the mean and standard deviation of the prediction serve as the predicted nominal flight trajectory (NFT) and its confidence interval (CI), respectively. The hierarchical architecture of the deep neural network (DNN) is introduced to improve the capability on modeling nonlinear features of the flight trajectory when performing the controller's instructions, such as climb, speed regulating, etc. According to a previous work [32], a Gaussian distribution is suitable for illustrating the trajectory positions by analyzing a huge number of real flight trajectories. Therefore, combining the probabilistic process and Gaussian distribution, the Gaussian process (GP) is formulated to build the transition patterns of flight trajectory. The proposed DGP model is an improved DNN architecture, in which GPs are applied to compute the nonlinear mappings between neurons [33; 34]. By training with historical trajectories, the parameters of the DGP model can be optimized for flight prediction, in which the flight patterns are learned in an implicit manner. With the proposed techniques, the flight trajectory prediction for four typical operations, including cruise, heading regulating, speed regulating (acceleration/deceleration) and altitude regulating (climb/descent), are integrated into a single framework. The framework of the DGP-based trajectory prediction is sketched in Figure 1.  
Historical trajectories are applied to optimize the parameters of the DGP models, in which different components of flight trajectory are designed to fit the model for different flight operations. Basically, the positional components are applied to build the transition patterns of the flight cruising and turn operation. Similarly, the altitude and velocity components serve as the motivation for predicting the flight trajectory when regulating flight altitude and speed, respectively, which generates a two-stage DGP solution. The information of the flight plan, including the waypoint sequence, cruise speed and altitude, are also extracted to build the constraints of the proposed DGP model for selecting different training samples and components and further learning different trajectory patterns. Based on the proposed DGP trajectory prediction approach, a probabilistic conflict detection approach is implemented pair-wisely, in which the Monte Carlo method is applied to simplify the computation by sampling flight positions from the predicted DGP models. The proposed trajectory prediction models and conflict detection algorithm are finally validated on real operating data. In short, the main contributions of this work can be summarized as follows:  
1. A DGP framework built with a Gaussian process and deep neural network is proposed to address the flight trajectory prediction issue in air traffic study. The stochastic probabilistic process is applied to build the temporal correlations of the flight trajectory, which greatly improve the modeling accuracy between predicting instants. 2. The deep architecture is proposed to improve the feature representations by using GPs as the nonlinear activation. The machine learning mechanism is applied to learn frequent transition patterns from historical trajectory and further support the prediction of time-varying specificities.