Kernel autoregressive models using Yule–Walker equations

• Nonlinear autoregressive models using the formalism based on kernel machines.
• The pre-image problem for time series prediction.
• The Yule–Walker equations, from linear to nonlinear models.

This paper proposes nonlinear autoregressive (AR) models for time series, within the framework of kernel machines. Two models are investigated. In the first proposed model, the AR model is defined on the mapped samples in the feature space. In order to predict a future sample, this formulation requires to solve a pre-image problem to get back to the input space. We derive an iterative technique to provide a fine-tuned solution to this problem. The second model bypasses the pre-image problem, by defining the AR model with an hybrid model, as a tradeoff considering the computational time and the precision, by comparing it to the iterative, fine-tuned, model. By considering the stationarity assumption, we derive the corresponding Yule–Walker equations for each model, and show the ease of solving these problems. The relevance of the proposed models is studied on several time series, and compared with other well-known models in terms of accuracy and computational complexity.