基于熵的复杂系统马尔可夫划分的生成
Entropy-based Generating Markov Partitions for Complex Systems
摘要：
为一般动力系统的轨迹找到正确的编码是一项复杂的任务：符号序列需要保留系统轨迹的不变属性。在理论上，这个问题的解决方案是在得到生成马尔可夫划分（GMP）时找到的，该生成马尔可夫划分仅在不稳定流形和稳定流形以无限精度已知并且始终存在时才定义。然而，这些流形通常形成高度卷积的欧几里德集，是\emph{a priori}未知的，并且，正如在任何真实世界的实验中发生的那样，测量是在有限的时间跨度上以有限的分辨率进行的。如果系统是由相互作用的动力学单元组成的网络，即高维复杂系统，则任务变得更加复杂。在这里，我们解决这个任务，并解决它定义了一种方法来近似构建任何复杂系统的有限分辨率和有限时间轨迹的GMP。我们严格测试我们的方法在网络上的耦合映射，编码成符号序列的轨迹。我们表明，这些序列是最佳的，因为它们最大限度地减少了信息损失，也增加了任何虚假的信息。因此，我们的方法允许我们近似计算的不变概率测度的复杂系统从观测数据。因此，我们可以有效地定义适用于各种复杂现象的复杂性度量，例如从不同大脑区域测量的EEG信号表征大脑活动，或从不同地球区域测量的温度异常表征气候变化。
英文摘要：
Finding the correct encoding for a generic dynamical system's trajectory is a complicated task: the symbolic sequence needs to preserve the invariant properties from the system's trajectory. In theory, the solution to this problem is found when a Generating Markov Partition (GMP) is obtained, which is only defined once the unstable and stable manifolds are known with infinite precision and for all times. However, these manifolds usually form highly convoluted Euclidean sets, are \emph{a priori} unknown, and, as it happens in any real-world experiment, measurements are made with finite resolution and over a finite time-span. The task gets even more complicated if the system is a network composed of interacting dynamical units, namely, a high-dimensional complex system. Here, we tackle this task and solve it by defining a method to approximately construct GMPs for any complex system's finite-resolution and finite-time trajectory. We critically test our method on networks of coupled maps, encoding their trajectories into symbolic sequences. We show that these sequences are optimal because they minimise the information loss and also any spurious information added. Consequently, our method allows us to approximately calculate the invariant probability measures of complex systems from observed data. Thus, we can efficiently define complexity measures that are applicable to a wide range of complex phenomena, such as the characterisation of brain activity from EEG signals measured at different brain regions or the characterisation of climate variability from temperature anomalies measured at different Earth regions.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
