论文标题
部分可观测时空混沌系统的无模型预测
Optimal-$k$ difference sequence in nonparametric regression
论文作者
论文摘要
储层计算是预测湍流的有力工具，其简单的架构具有处理大型系统的计算效率。然而，其实现通常需要完整的状态向量测量和系统非线性知识。我们使用非线性投影函数将系统测量扩展到高维空间，然后将其输入到储层中以获得预测。我们展示了这种储层计算网络在时空混沌系统上的应用，该系统模拟了湍流的若干特征。我们表明，使用径向基函数作为非线性投影器，即使只有部分观测并且不知道控制方程，也能稳健地捕捉复杂的系统非线性。最后，我们表明，当测量稀疏、不完整且带有噪声，甚至控制方程变得不准确时，我们的网络仍然可以产生相当准确的预测，从而为实际湍流系统的无模型预测铺平了道路。
Difference-based methods have been attracting increasing attention in nonparametric regression, in particular for estimating the residual variance.To implement the estimation, one needs to choose an appropriate difference sequence, mainly between {\em the optimal difference sequence} and {\em the ordinary difference sequence}. The difference sequence selection is a fundamental problem in nonparametric regression, and it remains a controversial issue for over three decades. In this paper, we propose to tackle this challenging issue from a very unique perspective, namely by introducing a new difference sequence called {\em the optimal-$k$ difference sequence}. The new difference sequence not only provides a better balance between the bias-variance trade-off, but also dramatically enlarges the existing family of difference sequences that includes the optimal and ordinary difference sequences as two important special cases. We further demonstrate, by both theoretical and numerical studies, that the optimal-$k$ difference sequence has been pushing the boundaries of our knowledge in difference-based methods in nonparametric regression, and it always performs the best in practical situations.
