Closed-Form Prediction of Nonlinear Dynamic Systems by Means of Gaussian Mixture Approximat.pdf

Closed-Form Prediction of Nonlinear Dynamic Systems by Means of Gaussian Mixture Approximation of the Transition Density Marco Huber, Dietrich Brunn, and Uwe D. Hanebeck Abstract—Recursive prediction of the state of a nonlinear stochastic dynamic system cannot be efficiently performed in general, since the complexity of the probability density function characterizing the system state increases with every prediction step. Thus, representing the density in an exact closed-form manner is too complex or even impossible. So, an appropriate approximation of the density is required. Instead of directly approximating the predicted density, we propose the approximation of the transition density by means of Gaussian mixtures. We treat the approximation task as an optimization problem that is solved offline via progressive processing to bypass initialization problems and to achieve high quality approximations. Once having calculated the transition density approximation offline, prediction can be performed efficiently resulting in a closed-form density representation with constant complexity. I. INTRODUCTION Estimation of uncertain quantities is a typical challenge in many engineering applications like information processing in sensor-actuator-networks, localization of vehicles or robotics and machine learning. One aspect that arises is the inference of a given uncertain quantity through time. Particularly the recursive processing of this so-called prediction requires an efficient implementation for practical applications. Typically, random variables are used to describe the quan- tities and their uncertainties. For such a representation the prediction problem is solved by the Bayesian estimator. In general, the probability density of the predicted quantity cannot be calculated in closed form and the complexity of the density representation increases with each time step. The consequence of this is an impractical computational effort. Only for some special cases full analytical