具有长程关联的符号随机动力系统的可加N步Markov链原型模型
Additive N-Step Markov Chains as Prototype Model of Symbolic Stochastic Dynamical Systems with Long-Range Correlations
摘要：
一个理论的符号动态系统与长程相关的基础上考虑的二进制N步马尔可夫链早期开发的物理。90，110601（2003）被推广到有偏的情况下（非相等数量的零和单位链）。在该模型中，链中第i个符号等于零（或单位）的条件概率是前N个符号中单位（零）数量的线性函数。解析地得到了任意长度L的字的相关函数、分布函数以及符号数的方差，并通过数值模拟进行了验证。揭示了所研究的随机过程的自相似性，给出了链参数的相似群变换。讨论了控制L字分布函数的扩散Fokker-Planck方程。如果持续相关性不是非常强，则分布函数是高斯分布，方差与L呈非线性相关。给出了可加马氏链的记忆函数和关联函数之间的关系式。该等式允许使用系统的关联函数来重构记忆函数。通过简单的模型实例验证了该方法的有效性和鲁棒性。发现了具体的粗粒度文学文本的记忆功能，揭示了它们在长距离下的普遍幂律行为。
英文摘要：
A theory of symbolic dynamic systems with long-range correlations based on the consideration of the binary N-step Markov chains developed earlier in Phys. Rev. Lett. 90, 110601 (2003) is generalized to the biased case (non equal numbers of zeros and unities in the chain). In the model, the conditional probability that the i-th symbol in the chain equals zero (or unity) is a linear function of the number of unities (zeros) among the preceding N symbols. The correlation and distribution functions as well as the variance of number of symbols in the words of arbitrary length L are obtained analytically and verified by numerical simulations. A self-similarity of the studied stochastic process is revealed and the similarity group transformation of the chain parameters is presented. The diffusion Fokker-Planck equation governing the distribution function of the L-words is explored. If the persistent correlations are not extremely strong, the distribution function is shown to be the Gaussian with the variance being nonlinearly dependent on L. An equation connecting the memory and correlation function of the additive Markov chain is presented. This equation allows reconstructing a memory function using a correlation function of the system. Effectiveness and robustness of the proposed method is demonstrated by simple model examples. Memory functions of concrete coarse-grained literary texts are found and their universal power-law behavior at long distances is revealed.
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