Previously, we showed that computational mechanic's causal states -- predictively-equivalent trajectory classes for a stochastic dynamical system -- can be cast into a reproducing kernel Hilbert space. The result is a widely-applicable method that infers causal structure directly from very different kinds of observations and systems. Here, we expand this method to explicitly introduce the causal diffusion components it produces. These encode the kernel causal-state estimates as a set of coordinates in a reduced dimension space. We show how each component extracts predictive features from data and demonstrate their application on four examples: first, a simple pendulum -- an exactly solvable system; second, a molecular-dynamic trajectory of $n$-butane -- a high-dimensional system with a well-studied energy landscape; third, the monthly sunspot sequence -- the longest-running available time series of direct observations; and fourth, multi-year observations of an active crop field -- a set of heterogeneous observations of the same ecosystem taken for over a decade. In this way, we demonstrate that the empirical kernel causal-states algorithm robustly discovers predictive structures for systems with widely varying dimensionality and stochasticity.

本文研究了计算机制的因果状态与随机动力系统的预测等价轨迹类之间的关系，提出了一种从不同观察和系统中直接推断因果结构的方法。通过引入因果扩散成分，该方法展示了如何从数据中提取预测特征，应用于多种案例，证明了其在面对广泛的维度和随机性时的有效性和鲁棒性。

推断核 $ε$-机：在复杂系统中发现结构