Dai Xiaowu, Li Lexin
Department of Economics and Simons Institute for the Theory of Computing, the University of California, Berkeley, Berkeley, CA.
Department of Biostatistics and Epidemiology, the University of California, Berkeley, Berkeley, CA.
J Am Stat Assoc. 2022;117(540):1711-1725. doi: 10.1080/01621459.2021.1882466. Epub 2021 Apr 27.
Ordinary differential equation (ODE) is widely used in modeling biological and physical processes in science. In this article, we propose a new reproducing kernel-based approach for estimation and inference of ODE given noisy observations. We do not assume the functional forms in ODE to be known, or restrict them to be linear or additive, and we allow pairwise interactions. We perform sparse estimation to select individual functionals, and construct confidence intervals for the estimated signal trajectories. We establish the estimation optimality and selection consistency of kernel ODE under both the low-dimensional and high-dimensional settings, where the number of unknown functionals can be smaller or larger than the sample size. Our proposal builds upon the smoothing spline analysis of variance (SS-ANOVA) framework, but tackles several important problems that are not yet fully addressed, and thus extends the scope of existing SS-ANOVA as well. We demonstrate the efficacy of our method through numerous ODE examples.
常微分方程（ODE）在科学中对生物和物理过程建模方面有着广泛应用。在本文中，我们提出了一种基于再生核的新方法，用于在有噪声观测的情况下对常微分方程进行估计和推断。我们不假定常微分方程中的函数形式已知，也不将其限制为线性或可加形式，并且允许成对相互作用。我们进行稀疏估计以选择单个泛函，并为估计的信号轨迹构建置信区间。我们在低维和高维设置下都建立了核常微分方程的估计最优性和选择一致性，其中未知泛函的数量可以小于或大于样本量。我们的提议基于方差平滑样条分析（SS - ANOVA）框架构建，但解决了一些尚未得到充分解决的重要问题，从而也扩展了现有SS - ANOVA的范围。我们通过众多常微分方程示例证明了我们方法的有效性。
