  In the last decade, a considerable research effort has been devoted to
developing adaptive algorithms based on kernel functions. One of the main
features of these algorithms is that they form a family of universal
approximation techniques, solving problems with nonlinearities elegantly. In
this paper, we present data-selective adaptive kernel normalized least-mean
square (KNLMS) algorithms that can increase their learning rate and reduce
their computational complexity. In fact, these methods deal with kernel
expansions, creating a growing structure also known as the dictionary, whose
size depends on the number of observations and their innovation. The algorithms
described herein use an adaptive step-size to accelerate the learning and can
offer an excellent tradeoff between convergence speed and steady state, which
allows them to solve nonlinear filtering and estimation problems with a large
number of parameters without requiring a large computational cost. The
data-selective update scheme also limits the number of operations performed and
the size of the dictionary created by the kernel expansion, saving
computational resources and dealing with one of the major problems of kernel
adaptive algorithms. A statistical analysis is carried out along with a
computational complexity analysis of the proposed algorithms. Simulations show
that the proposed KNLMS algorithms outperform existing algorithms in examples
of nonlinear system identification and prediction of a time series originating
from a nonlinear difference equation.
