Mathematical foundation of a new complexity measure

For many continuous bio-medical signals with both strong nonlinearity and non-stationarity, two criterions were proposed for their complexity estimation: (1) Only a 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.C0 complexity measure proposed by us previously is one of such measures. However, it lacks the solid mathematical foundation and thus its use is 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.