On a grand scale, motion trajectories are usually defined by an origin, a sequence of waypoints, and a destination. A typical example is in air traffic management (ATM), where a flight from an origin passes several waypoints and arrives at a destination. The origin, the waypoints, and the destination contain useful information for trajectory modeling. On the other hand, due to trajectory design criteria and ATM restrictions and requirements, there are long-range dependencies in a flight trajectory. Such dependencies can be modeled by taking the origin, waypoints, and destination into account in trajectory modeling. In this article, we propose a class of conditionally Markov (CM) sequences to model such trajectories with long-range dependencies. First, we define a general CM sequence as a foundation. Then, we discuss its special cases for different scenarios. We derive dynamic models of these CM sequences in the Gaussian case. We show how parameters of the models can be learned from data or designed. We also justify the use of the proposed CM models for trajectory modeling. In addition, we obtain optimal filters and predictors for different models. Simulation demonstrations are given.

中文翻译：

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具有航路点和目的地的轨迹的条件马尔可夫建模和最优估计

在宏观上，运动轨迹通常由起点、一系列航点和目的地定义。一个典型的例子是空中交通管理 (ATM)，从始发地起飞的航班经过多个航路点并到达目的地。起点、航点和目的地包含用于轨迹建模的有用信息。另一方面，由于轨迹设计标准和 ATM 的限制和要求，飞行轨迹存在长期依赖关系。这种依赖可以通过在轨迹建模中考虑起点、航点和目的地来建模。在本文中，我们提出了一类条件马尔可夫 (CM) 序列来对具有长期依赖关系的此类轨迹进行建模。首先，我们定义一个通用的 CM 序列作为基础。然后，我们讨论它在不同场景下的特殊情况。我们在高斯情况下推导出这些 CM 序列的动态模型。我们展示了如何从数据中学习或设计模型的参数。我们还证明使用建议的 CM 模型进行轨迹建模是合理的。此外，我们获得了不同模型的最佳过滤器和预测器。给出了模拟演示。