Identification of non-linear time series via kernels

Given a time series arising as the observations of some dynamical system, it is possible to reconstruct the qualitative behaviour of the dynamical system. Given such a reconstruction, this provides a general approach for making accurate short-term predictions. The time series reconstruction problem can be viewed as a function approximation problem which can be addressed by reproducing kernel Hilbert spaces (RKHS). Such an approach is described and related to Bayesian estimation with Gaussian priors. It is also shown that an explicit formulation of the model as an input-dependent autoregressive model is possible. This linearization is, however, unlike other approaches, exact and the autoregressive coefficients can be calculated explicitly at each time step. A particular example of RKHS approximation is applied to two benchmark problems, one synthetic and one real. The results demonstrate various aspects related to the optimization of the models and also show that the RKHS method is competitive against the best alternatives.