Physics Department, University of California at Merced, 5200 North Lake Road, Merced, California 95343, USA.
Chaos. 2011 Sep;21(3):037112. doi: 10.1063/1.3637502.
We investigate a stationary process's crypticity--a measure of the difference between its hidden state information and its observed information--using the causal states of computational mechanics. Here, we motivate crypticity and cryptic order as physically meaningful quantities that monitor how hidden a hidden process is. This is done by recasting previous results on the convergence of block entropy and block-state entropy in a geometric setting, one that is more intuitive and that leads to a number of new results. For example, we connect crypticity to how an observer synchronizes to a process. We show that the block-causal-state entropy is a convex function of block length. We give a complete analysis of spin chains. We present a classification scheme that surveys stationary processes in terms of their possible cryptic and Markov orders. We illustrate related entropy convergence behaviors using a new form of foliated information diagram. Finally, along the way, we provide a variety of interpretations of crypticity and cryptic order to establish their naturalness and pervasiveness. This is also a first step in developing applications in spatially extended and network dynamical systems.
我们使用计算力学的因果状态来研究一个平稳过程的隐秘性——一种衡量其隐藏状态信息与其观测信息之间差异的度量。在这里，我们将隐秘性和隐秘有序性作为物理上有意义的量来监测一个隐藏过程的隐藏程度，这是通过在一个更直观的几何设置中重新表述以前关于块熵和块状态熵收敛的结果而实现的，并且还得出了一些新的结果。例如，我们将隐秘性与观察者如何与过程同步联系起来。我们表明，块因果状态熵是块长度的凸函数。我们对自旋链进行了完整的分析。我们提出了一种分类方案，根据可能的隐秘性和马尔可夫阶对平稳过程进行分类。我们使用新形式的叶状信息图来说明相关的熵收敛行为。最后，一路走来，我们提供了隐秘性和隐秘有序性的各种解释，以确立它们的自然性和普遍性。这也是在空间扩展和网络动力系统中开发应用的第一步。
