C0复杂度的数学基础

对于许多同时具有强烈非线性和非平稳性的连续生物医学信号来说，计算其复杂度往往要求：1）在数据长度比较短的情况下也可以得出比较鲁棒的估计值；2）无需对原始信号作像二值化这样的过分的粗粒化，我们以前所提出的C0复杂度就是这样的一种度量，但是这种度量缺乏严格的数学基础，因而影响到它的应用，提出了一种改进形式，并严格证明了它的重要性质。从而表明这个量在一定条件下可以作为时间序列随机程度的指标，因而在随机性复杂度的意义下也可作为复杂性的一个定量指标，由于这个量有计算速度快的优点，因此特别适合于一些需要大量计算复杂度的场合，例如计算长时间过程中滑动窗口中复杂度的动态变化。

Mathematical Foundation of C0 Complexity

For many continuous bio-medical signals with both strong nonlinearity and non-stationarity, two criterions were proposed for their complexity estimation:1) Only short data set is enough for robust estimation;2) No over-coarse graining preprocessing, such as transferring the original signal into a binary time series, is needed.C-0 complexity measure proposed by us previously is one of such measures. However, it lacked solid mathematical foundation and thus its use was limited.A modified version of this measure is proposed, and some important properties are proved rigorously. According to these properties, this measure can be considered as an index of randomness of time series in some senses, and thus also a quantitative index of complexity under the meaning of randomness finding complexity. Compared with other similar measures, this measure seems more suitable for estimating a large quantity of complexity measures for a given task, such as studying the dynamic variation of such measures in sliding windows of a long process, owing to its fast speed for estimation.