A comparative study of pre-image techniques: The kernel autoregressive case

The autoregressive (AR) model is one of the most used techniques for time series analysis, applied to study stationary as well as non-stationary processes. However, being a linear technique, it is not adapted for nonlinear systems. Recently, we introduced the kernel AR model, a straightforward extension of the AR model to the nonlinear case. It is based on the concept of kernel machines, where data are nonlinearly mapped from the input space to a feature space. The AR model can thus be applied on the mapped data. Nevertheless, in order to predict future samples, one needs to go back to the input space, by solving the pre-image problem. The prediction performance highly depends on the considered pre-image technique. In this paper, a comparative study of several state-of-the-art pre-image techniques is conducted for the kernel AR model, investigating the prediction error with the optimal model parameters, as well as the computational complexity. The conformal map approach presents results as good as the well known fixed-point iterative method, with less computational time. This is shown on unidimensional and multidimensional chaotic time series.