非线性状态空间系统辨识的稀疏贝叶斯方法
A Sparse Bayesian Approach to the Identification of Nonlinear State-Space Systems
摘要：
本技术说明考虑了具有加性过程噪声但没有测量噪声的非线性离散时间系统的识别。特别是，我们提出了一种方法及其相关算法来识别系统的非线性函数形式和它们的相关参数从有限数量的时间序列数据点。为此，我们把这个识别问题转化为一个稀疏的线性回归问题，并采用贝叶斯的观点来解决它。我们提出了一个凸化过程依赖于一个有效的迭代重新加权$\ell_1$ -最小化算法，使用一般稀疏诱导先验的系统参数和边际似然最大化。使用这种方法，我们还展示了如何凸约束的参数可以很容易地添加到我们提出的迭代重新加权$\ell_1$最小化算法。在补充材料\cite{appendix}中，我们说明了所提出的识别方法在生物学和物理学中的两个经典系统上的有效性，即遗传抑制子网络和大规模互连的Kuramoto振荡器网络。
英文摘要：
This technical note considers the identification of nonlinear discrete-time systems with additive process noise but without measurement noise. In particular, we propose a method and its associated algorithm to identify the system nonlinear functional forms and their associated parameters from a limited number of time-series data points. For this, we cast this identification problem as a sparse linear regression problem and take a Bayesian viewpoint to solve it. As such, this approach typically leads to nonconvex optimisations. We propose a convexification procedure relying on an efficient iterative re-weighted $\ell_1$-minimisation algorithm that uses general sparsity inducing priors on the parameters of the system and marginal likelihood maximisation. Using this approach, we also show how convex constraints on the parameters can be easily added to our proposed iterative re-weighted $\ell_1$-minimisation algorithm. In the supplementary material \cite{appendix}, we illustrate the effectiveness of the proposed identification method on two classical systems in biology and physics, namely, a genetic repressilator network and a large scale network of interconnected Kuramoto oscillators.
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