Closed-form projection operator wavelet kernels in support vector learning for nonlinear dynamical systems identification

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As a special idempotent operator, the projection operator plays a crucial role in the Spectral Decomposition Theorem for linear operators in Hilbert space. In this paper, an innovative orthogonal projection operator wavelet kernel is developed for support vector learning. In the framework of multi-resolution analysis, the proposed wavelet kernel can easily fulfill the multi-scale, multidimensional learning to estimate complex dependencies. The peculiar advantage of the wavelet kernel developed in this paper lies in its expressivity in closed-form, which greatly facilitates its application in kernel learning. To our best knowledge, it is the first closed-form orthogonal projection wavelet kernel in the literature. In the scenario of linear programming support vector learning, the proposed closed-form projection operator wavelet kernel is used to identify a parallel model of a benchmark nonlinear dynamical system. A simulation study confirms its superiority in model accuracy and sparsity.