Permutation complexity and dependence measures of time series

It is an interesting area to analyze the complexity or dependence of time series. Many information-theoretic methods have been proposed for this purpose. In this letter, we adapt the permutation entropy to infer the complexity of short-time series by freely changing the time delay, and test it with Gaussian random series and random walks. We also propose a Renyi permutation entropy to characterize the rare events from frequent events. It successfully analyzes the temporal structure of the autoregressive (AR) model and also the daily closing prices in Shanghai stock market. Moreover, we introduce a permutation mutual information method to detect the dependence between two time series. We test it by the Henon map, autoregressive fractionally integrated moving average (ARFIMA) model and observe its significance by the randomization test. It is also applied to measure the dependence between air temperature and air humidity.