We introduce a new class of probabilistic cellular automata that are capable of exhibiting rich dynamics such as synchronization and ergodicity and can be easily inferred from data. The system is a finite-state locally interacting Markov chain on a circular graph. Each site's subsequent state is random, with a distribution determined by its neighborhood's empirical distribution multiplied by a local transition matrix. We establish sufficient and necessary conditions on the local transition matrix for synchronization and ergodicity. Also, we introduce novel least squares estimators for inferring the local transition matrix from various types of data, which may consist of either multiple trajectories, a long trajectory, or ensemble sequences without trajectory information. Under suitable identifiability conditions, we show the asymptotic normality of these estimators and provide non-asymptotic bounds for their accuracy.

翻译：我们提出了一类新型概率元胞自动机，该类系统能够展现同步性与遍历性等丰富动力学特性，并可基于数据轻松实现推断。该系统是在环形图上定义的有限状态局部交互马尔可夫链。每个格点的后续状态具有随机性，其分布由邻域经验分布与局部转移矩阵的乘积确定。我们建立了局部转移矩阵实现同步性与遍历性的充分必要条件。此外，我们引入新型最小二乘估计量，用于从多类数据（包括多条轨迹、单条长轨迹或无轨迹信息的系综序列）中推断局部转移矩阵。在适当的可辨识性条件下，我们证明了这些估计量的渐近正态性，并给出了其精度的非渐近界。