Kernel adaptive filtering with confidence intervals

Since its introduction, kernel adaptive filtering (KAF) has attracted considerable attention in recent years. Its main advantages include universal nonlinear approximation using kernel methods, linearity with convex learning in the Reproducing Kernel Hilbert Space (RKHS), and online adaptation with moderate complexity. Among its applications, the kernel least mean square (KLMS) algorithm deserves particular attention due to its simplicity and effectiveness for learning complex systems. A major drawback of current implementations of KAF is the lack of a simple determination of the certainty of each estimate. In this paper, we present a novel kernel adaptive filtering architecture with confidence intervals. By introducing an auxiliary filter, the variance of each estimate can be computed using stochastic gradient descent in O(N). Results show that the proposed algorithm produces comparable estimates of the mean and the variance functions, using only a fraction of the computation associated with the Gaussian process (GP) prediction, and is more versatile in the cases of time-varying noise variance or heteroskedasticity.