A novel distance correlation entropy and Auto-distance correlation function for measuring the complexity of time series data

•NDCE is proposed as an improvement of DCE, considering the time lag effects.•ADCF is introduced to measure the complexity of time series globally.•NDCE and ADCF could recognize the complexity of time series effectively.•Our methods can estimate time lags precisely.•Our methods are applied to stock market and get some interesting results. In the field of time series analysis, the assessment of complexity is a pivotal area of research, revealing the unique properties and structures inherent in time series data. However, current statistical-based approaches for complexity often rely on simple statistical measurements, which may not accurately capture the intricacies of time series data. In our article, we propose a novel distance correlation entropy based on distance correlation and weight function, validating that it is effective for measuring the complexity of time series. Besides, as the length of time series grows, the proposed novel distance correlation entropy is shown to converge gradually. Moreover, the auto-distance correlation function (ADCF) is introduced to globally consider the complexity of time series, overcoming the limitation of the proposed novel distance correlation entropy that only considers the complexity locally. The novel distance correlation entropy and ADCF are then applied to time lag analysis, and the result indicates that they are all effective in reflecting the time lag property of the simulated data. In real-world data analysis, we use these two measurements to Chinese and US stock index closing prices, where we innovatively analyze Chinese and American stocks as two distinct entities and get more significant results than that of individual analysis. In time lag analysis, we obtain that the US stocks are mainly influenced by domestic stocks and the US stock market influences the Chinese stock market periodically.