Destination-Directed Trajectory Modeling, Filtering, and Prediction Using Conditionally Markov Sequences

Abstract

In some problems, there is information about the destination of a moving object. An example is a flight from an origin to a destination. Such problems have three main components: an origin, a destination, and motion in between. We call such trajectories ending up at the destination destination-directed trajectories (DDTs). Described by an evolution law and an initial probability density, a Markov sequence is not flexible enough to model DDT well. The future (including destination) of a Markov sequence is completely determined probabilistically by its initial density and evolution law. One class of conditionally Markov (CM) sequences, called the CM <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</sub> sequence (it includes the Markov sequence as a special case), has the following main components: a joint density of two endpoints and a Markov-like evolution law. This article proposes modeling DDT as CM <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</sub> sequences. We study the CM <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L </sub> sequence, its dynamic model, and its realizations, all for DDT modeling. We demonstrate that CM <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</sub> sequences naturally model DDT, enjoy several desirable properties for DDT modeling, and can be easily and systematically generalized if necessary. In addition, we study DDT filtering and trajectory prediction based on a CM <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</sub> model and compare them with those based on a Markov model. Several simulation examples are presented to illustrate DDT modeling and inference.