Integrated Gaussian Processes for Tracking

In applications such as tracking and localisation, a dynamical model is typically specified for the modelling of an object's motion. An appealing alternative to the traditional parametric Markovian dynamical models is the Gaussian Process (GP). GPs can offer additional flexibility and represent non-Markovian, long-term, dependencies in the target's kinematics. However, a standard GP with constant or zero mean is prone to oscillating around its mean and not sufficiently exploring the state space. In this paper, we consider extensions of the common GP framework such that a GP acts as the driving disturbance term that is integrated over time to produce a new Integrated GP (iGP) dynamical model. It potentially provides a more realistic modelling of agile objects' behaviour. We prove here that the introduced iGP model is, itself, a GP with a non-stationary kernel, which we derive fully in the case of the squared exponential GP kernel. Thus, the iGP is straightforward to implement, with the usual growth over time of the computational burden. We further show how to implement the model with fixed time complexity in a standard sequential Bayesian updating framework using Kalman filter-based computations, employing a sliding window Markovian approximation. Example results from real radar measurements and synthetic data are presented to demonstrate the ability of the proposed iGP modelling to facilitate more accurate tracking compared to conventional GP.