表观复杂性的谱简单性，II：精确复杂性和复杂性谱
Spectral Simplicity of Apparent Complexity, Part II: Exact Complexities and Complexity Spectra
摘要：
在第一部分中发展的亚纯泛函演算克服了线性算子的非对角化问题，这种问题经常出现在复杂系统的时间演化中，并且对于预测其行为的元代数学是通用的。使用由此产生的谱分解，我们推导出封闭形式的表达式关联函数，有限长度香农熵率近似，渐近熵率，过剩熵，瞬态信息，瞬态和渐近状态不确定性，和同步信息的有限状态隐马尔可夫模型产生的随机过程。这介绍了分析易处理调查信息处理的离散事件随机过程，符号动力学和混沌动力系统。比较揭示了数学的复杂性措施之间的相似性，最初认为捕捉不同的信息和计算属性。我们还介绍了一种新的光谱分析，通过日冕光谱图和过去-未来互信息的频率相关光谱。我们分析了一些例子来说明的方法，强调过程中的多变量依赖性超越两两相关。附录给出了一个例子的频谱分解计算的全部细节。
英文摘要：
The meromorphic functional calculus developed in Part I overcomes the nondiagonalizability of linear operators that arises often in the temporal evolution of complex systems and is generic to the metadynamics of predicting their behavior. Using the resulting spectral decomposition, we derive closed-form expressions for correlation functions, finite-length Shannon entropy-rate approximates, asymptotic entropy rate, excess entropy, transient information, transient and asymptotic state uncertainty, and synchronization information of stochastic processes generated by finite-state hidden Markov models. This introduces analytical tractability to investigating information processing in discrete-event stochastic processes, symbolic dynamics, and chaotic dynamical systems. Comparisons reveal mathematical similarities between complexity measures originally thought to capture distinct informational and computational properties. We also introduce a new kind of spectral analysis via coronal spectrograms and the frequency-dependent spectra of past-future mutual information. We analyze a number of examples to illustrate the methods, emphasizing processes with multivariate dependencies beyond pairwise correlation. An appendix presents spectral decomposition calculations for one example in full detail.
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