KALMAN FILTERING IN REPRODUCING KERNEL HILBERT SPACES By PINGPING ZHU A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy KALMAN FILTERING IN REPRODUCING KERNEL HILBERT SPACES By Pingping Zhu May 2013 Chair: José C. Prı́ncipe Major: Electrical and Computer Engineering There are numerous dynamical system applications that require estimation or prediction from noisy data, including vehicle tracking, channel tracking, time series denoising, prediction, estimation, and so on. Many linear algorithms have been developed to deal with these problems under different assumptions and approximations, such as the Kalman filter, the recursive least squares algorithm, and the least mean squares algorithm. However, these linear algorithms cannot solve nonlinear problems that often occur in real life. To address these nonlinear problems, some nonlinear algorithms have been recently proposed, like kernelized version of the linear algorithms. Our research follows this line and seeks to develop novel algorithms using kernel methods to deal with nonlinear problems. Specifically, our goal is to derive the Kalman filter in the reproducing kernel Hilbert space (RKHS), which is a space of functions, to implement signal denoising, prediction and estimation. In this dissertation, we first analyze and discuss in depth the extended kernel recursive least squares algorithm, and point out the limitation of this algorithm and its close relationship with the Kalman filter in RKHS. Next, we develop a novel extended kernel recursive least squares based on the nonlinear Kalman filters and kernel recursive least squares algorithms, to improve tracking and prediction performance. However, the nonlinear Kalman filter is implemented in the input space, not the RKHS. Then, we introduce the concepts of embeddings in RKHS and conditional embedding