混沌序列的无损压缩与复杂度
Lossless Compression and Complexity of Chaotic Sequences
摘要：
研究了混沌动力系统短符号序列的无损压缩算法的复杂性。特别是，我们研究了非顺序递归对替换（NSRPS），一个无损压缩算法首先由W。Ebeling等人[Math. Biosc. 52，1980]和Jiménez-Montaño等人[arXiv：cond-mat/0204134，2002]），其随后被证明是最佳的。NSPRS也被用来估计书面英语的熵（P. Grassberger [arXiv：physics/0207023，2002]）。我们提出了一种新的复杂性度量-定义为将输入序列转换为常数序列所需的NSRPS迭代次数。我们在不同分叉参数值的逻辑斯蒂映射的符号序列上测试了该度量。所提出的复杂性的措施是很容易计算的，并观察到高度相关的原始非线性时间序列的李雅普诺夫指数，即使是非常短的符号序列（短至50个样本）。最后，我们从Skew-Tent映射构造符号序列，这些符号序列是不可压缩的流行压缩算法，如WinZip，WinRAR和7-Zip，但可压缩的NSRPS。
英文摘要：
We investigate the complexity of short symbolic sequences of chaotic dynamical systems by using lossless compression algorithms. In particular, we study Non-Sequential Recursive Pair Substitution (NSRPS), a lossless compression algorithm first proposed by W. Ebeling et al. [Math. Biosc. 52, 1980] and Jiménez-Montaño et al. [arXiv:cond-mat/0204134, 2002]) which was subsequently shown to be optimal. NSPRS has also been used to estimate Entropy of written English (P. Grassberger [arXiv:physics/0207023, 2002]). We propose a new measure of complexity - defined as the number of iterations of NSRPS required to transform the input sequence into a constant sequence. We test this measure on symbolic sequences of the Logistic map for various values of the bifurcation parameter. The proposed measure of complexity is easy to compute and is observed to be highly correlated with the Lyapunov exponent of the original non-linear time series, even for very short symbolic sequences (as short as 50 samples). Finally, we construct symbolic sequences from the Skew-Tent map which are incompressible by popular compression algorithms like WinZip, WinRAR and 7-Zip, but compressible by NSRPS.
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