We provide some basic and sensible definitions of different types of digital twins and recommendations on when and how to use them. Following up on our recent publication of the Learning Causal Digital Twin, this article reports on a stochastic formulation and solution of the problem. Structural Vector Autoregressive Model (SVAR) for Causal estimation is recast as a state-space model. Kalman filter (and smoother) is then employed to estimate causal factors in a system of connected machine bearings. The previous neural network algorithm and Kalman Smoother produced very similar results; however, Kalman Filter/Smoother may show better performance for noisy data from industrial IoT sources.

中文翻译：

---

因果数字孪生的随机表示：卡尔曼滤波算法

我们为不同类型的数字孪生提供了一些基本且合理的定义，并提供了有关何时以及如何使用它们的建议。在我们最近出版的《学习因果数字孪生》之后，本文报道了该问题的随机表述和解决方案。因果估计的结构矢量自回归模型（SVAR）被重铸为状态空间模型。然后，使用卡尔曼滤波器（以及平滑器）来估计所连接的机器轴承系统中的因果因子。先前的神经网络算法和Kalman Smoother产生了非常相似的结果。但是，对于来自工业物联网来源的嘈杂数据，卡尔曼滤波器/平滑器可能会表现出更好的性能。