The radial basis function network-based state-dependent autoregressive (RBF-AR) model has been widely used in modeling and prediction of nonlinear time series. The parameter identification of RBF-AR model can be reformulated as a separable nonlinear least squares problem. The variable projection (VP) algorithm has been proven to be valuable in solving such problems. However, for ill-posed problems, the classical VP algorithm usually yields unstable models. In this paper, we consider a novel regularized separable algorithm that takes advantage of the VP method and the expectation–maximization (EM) method. The proposed algorithm utilizes the VP algorithm to optimize the nonlinear parameters and automatically picks out the regularization parameters during the search process. Numerical results on real-world data and synthetic time series confirm the effectiveness of the proposed algorithm.

基于径向基函数网络的状态相关自回归（RBF-AR）模型已广泛应用于非线性时间序列的建模和预测。RBF-AR 模型的参数识别可以重新表述为可分离的非线性最小二乘问题。变量投影 (VP) 算法已被证明在解决此类问题中很有价值。然而，对于不适定问题，经典的 VP 算法通常会产生不稳定的模型。在本文中，我们考虑了一种利用 VP 方法和期望最大化 (EM) 方法的新型正则化可分算法。该算法利用VP算法对非线性参数进行优化，并在搜索过程中自动挑选出正则化参数。