论文标题
有效的贝叶斯估计由相关噪声驱动的非马克维亚兰格文模型
Efficient Bayesian estimation of a non-Markovian Langevin model driven by correlated noise
论文作者
论文摘要
非马克维亚动力学的数据驱动建模是在许多领域（例如气候研究，分子动力学，生物物理学或风能建模）中应用的最新研究主题。在经常使用的标准Langevin方程中，可以通过附加的隐藏组件来实现内存效应，该组件可作为相关的噪声起作用，从而导致非马克维亚模型。它可以看作是部分观察到的扩散的模型类别的一部分，通常通过贝叶斯估计将观察到的数据适应，从而通过Gibbs采样器解决了未知噪声值的难度。但是，当关于$ 10^6 $或$ 10^7 $的大型数据集关于数据点时，对相同数量的潜在变量的分布进行采样是不可行的。对于这项工作中讨论的模型，我们通过直接推导了模型的欧拉 - 玛鲁山近似后分布通过潜在变量的分析边缘化来解决此问题。然而，在非线性噪声过程的情况下，模型估计的逆问题被证明是不适合的，并且在数值上仍然很昂贵。我们通过将噪声限制为Ornstein-Uhlenbeck过程来处理这些并发症，从而大大降低了估计的歧义。此外，在这种情况下，如果以分段常数方式近似观察到的分量的漂移和扩散函数，则可以非常有效地执行估计。我们通过湍流的示例说明了有效的贝叶斯对被考虑的非马克维亚兰格文模型的贝叶斯估算的过程。
Data-driven modeling of non-Markovian dynamics is a recent topic of research with applications in many fields such as climate research, molecular dynamics, biophysics, or wind power modeling. In the frequently used standard Langevin equation, memory effects can be implemented through an additional hidden component which functions as correlated noise, thus resulting in a non-Markovian model. It can be seen as part of the model class of partially observed diffusions which are usually adapted to observed data via Bayesian estimation, whereby the difficulty of the unknown noise values is solved through a Gibbs sampler. However, when regarding large data sets with a length of $10^6$ or $10^7$ data points, sampling the distribution of the same amount of latent variables is unfeasible. For the model discussed in this work, we solve this issue through a direct derivation of the posterior distribution of the Euler-Maruyama approximation of the model via analytical marginalization of the latent variables. Yet, in the case of a nonlinear noise process, the inverse problem of model estimation proves to be ill-posed and still numerically expensive. We handle these complications by restricting the noise to an Ornstein-Uhlenbeck process, which considerably reduces the ambiguity of the estimation. Further, in this case, the estimation can be performed very efficiently if the drift and diffusion functions of the observed component are approximated in a piecewise constant manner. We illustrate the resulting procedure of efficient Bayesian estimation of the considered non-Markovian Langevin model by an example from turbulence.
