Binary spatio-temporal data are common in many application areas. Such data can be considered from many perspectives, including via deterministic or stochastic cellular automata (CA), where local rules govern the transition probabilities that describe the evolution of the 0 and 1 states across space and time. One implementation of a stochastic CA for such data is via a spatio-temporal generalized linear model (or mixed model), with the local rule covariates being included in the transformed mean response. However, in many applications we do have a complete understanding of the local rules and must instead explore the rules space, which can be accomplished through symbolic regression. Even with a learned rule space, the data-driven rules may be insufficient to describe the process behavior and it is helpful to augment the transformed linear predictor with a latent spatio-temporal dynamic process. Here, we demonstrate for the first time that an echo state network (ESN) latent process can be used to enhance symbolic regression-learned local rule covariates. We implement this in a hierarchical Bayesian framework with regularized horseshoe priors on the ESN output weight matrices, which extends the ESN literature as well. Finally, we gain added expressiveness from the ESNs by considering an ensemble of ESN reservoirs, which we accommodate through weighted model averaging, which is also new to the ESN literature. We demonstrate our methodology on a simulated process in which we assume we do not know all of the local CA rules, as well as on multiple environmental data sets.

中文翻译：

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时空二元随机元胞自动机的回声状态网络增强符号回归

二进制时空数据在许多应用领域都很常见。可以从多个角度考虑此类数据，包括通过确定性或随机元胞自动机 (CA)，其中局部规则控制描述 ​​0 和 1 状态在空间和时间上的演化的转移概率。此类数据的随机 CA 的一种实现是通过时空广义线性模型（或混合模型），其中局部规则协变量包含在变换后的平均响应中。然而，在许多应用中，我们确实对局部规则有完整的了解，并且必须探索规则空间，这可以通过符号回归来完成。即使有了学习的规则空间，数据驱动的规则也可能不足以描述过程行为，并且用潜在的时空动态过程来增强变换后的线性预测器是有帮助的。在这里，我们首次证明回声状态网络（ESN）潜在过程可用于增强符号回归学习的局部规则协变量。我们在分层贝叶斯框架中实现了这一点，在 ESN 输出权重矩阵上使用正则化马蹄先验，这也扩展了 ESN 文献。最后，我们通过考虑 ESN 存储库的集合来从 ESN 中获得额外的表达能力，我们通过加权模型平均来容纳它，这对于 ESN 文献来说也是新的。我们在模拟过程以及多个环境数据集上展示了我们的方法，其中我们假设我们不知道所有当地的 CA 规则。