Adaptive Split/Merge-Based Gaussian Mixture Model Approach for Uncertainty Propagation

This paper presents an adaptive splitting and merging scheme for dynamic selection of Gaussian kernels in a Gaussian mixture model. The Gaussian kernel in the Gaussian mixture model is split into multiple components if the Kolmogorov equation error exceeds a prescribed threshold. Two different splitting mechanisms are presented in this work. The first splitting mechanism corresponds to splitting one Gaussian kernel in all directions, whereas the second splitting mechanism corresponds to splitting in only the direction of maximum nonlinearity. The state transition matrix in conjunction with unscented transformation is used to compute the departure from linearity, and hence approximate the direction of maximum nonlinearity. The merging mechanism exploits the angle between eigenvectors corresponding to the maximum eigenvalue of covariance matrices corresponding to two different Gaussian kernels to find candidate components for merging. Finally, a sparse approximation problem is defined to provide a mechanism to trade off between the number of Gaussian kernels and the Kolmogorov equation error in a mixture model. The uncertainty propagation problem for a satellite motion in a low Earth orbit is considered to show the efficacy of the proposed ideas.