Support Vector Machines for Nonlinear Kernel ARMA System Identification

Nonlinear system identification based on support vector machines (SVM) has been usually addressed by means of the standard SVM regression (SVR), which can be seen as an implicit nonlinear autoregressive and moving average (ARMA) model in some reproducing kernel Hilbert space (RKHS). The proposal of this letter is twofold. First, the explicit consideration of an ARMA model in an RKHS (SVM-ARMA<formula><tex>$_{2K}$</tex></formula>) is proposed. We show that stating the ARMA equations in an RKHS leads to solving the regularized normal equations in that RKHS, in terms of the autocorrelation and cross correlation of the (nonlinearly) transformed input and output discrete time processes. Second, a general class of SVM-based system identification nonlinear models is presented, based on the use of composite Mercer's kernels. This general class can improve model flexibility by emphasizing the input–output cross information (SVM-ARMA<formula><tex>$_{4K}$</tex></formula>), which leads to straightforward and natural combinations of implicit and explicit ARMA models (SVR-ARMA<formula><tex>$_{2K}$</tex></formula> and SVR-ARMA<formula> <tex>$_{4K}$</tex></formula>). Capabilities of these different SVM-based system identification schemes are illustrated with two benchmark problems.