We merge computational mechanics' definition of causal states (predictively-equivalent histories) with reproducing-kernel Hilbert space (RKHS) representation inference. The result is a widely-applicable method that infers causal structure directly from observations of a system's behaviors whether they are over discrete or continuous events or time. A structural representation -- a finite- or infinite-state kernel $\epsilon$-machine -- is extracted by a reduced-dimension transform that gives an efficient representation of causal states and their topology. In this way, the system dynamics are represented by a stochastic (ordinary or partial) differential equation that acts on causal states. We introduce an algorithm to estimate the associated evolution operator. Paralleling the Fokker-Plank equation, it efficiently evolves causal-state distributions and makes predictions in the original data space via an RKHS functional mapping. We demonstrate these techniques, together with their predictive abilities, on discrete-time, discrete-value infinite Markov-order processes generated by finite-state hidden Markov models with (i) finite or (ii) uncountably-infinite causal states and (iii) a continuous-time, continuous-value process generated by a thermally-driven chaotic flow. The method robustly estimates causal structure in the presence of varying external and measurement noise levels.

中文翻译：

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通过复制内核希尔伯特空间$ε$-机器发现因果结构

我们将计算力学对因果状态（预测等效历史）的定义与再现内核希尔伯特空间（RKHS）表示推论相结合。结果是一种广泛应用的方法，该方法可以直接根据对系统行为的观察来推断因果结构，而无论它们是在离散事件还是连续事件或时间之上。结构化表示-有限状态或无限状态内核$ \ epsilon $-机器-通过降维变换提取，该变换给出因果状态及其拓扑的有效表示。这样，系统动力学由作用于因果状态的随机（普通或部分）微分方程表示。我们引入一种算法来估计相关的进化算子。平行于福克-普朗克方程，它可以有效地演化因果状态分布，并通过RKHS功能映射在原始数据空间中进行预测。我们在具有（i）有限或（ii）无穷无限因果状态和（iii）的有限状态隐马尔可夫模型生成的离散时间，离散值无穷马氏阶过程上证明了这些技术及其预测能力。由热驱动的混沌流产生的连续时间，连续值过程。在存在变化的外部和测量噪声水平的情况下，该方法可稳健地估计因果结构。由具有（i）有限或（ii）无限数因果状态和（iii）由热驱动混沌产生的连续时间，连续值过程的有限状态隐藏Markov模型产生的离散值无限Markov级过程流。在存在变化的外部和测量噪声水平的情况下，该方法可稳健地估计因果结构。由具有（i）有限或（ii）无限数因果状态和（iii）由热驱动混沌产生的连续时间，连续值过程的有限状态隐藏Markov模型产生的离散值无限Markov级过程流。在存在变化的外部和测量噪声水平的情况下，该方法可稳健地估计因果结构。