Distance correlation entropy and ordinal distance complexity measure: efficient tools for complex systems

Many distance measures are used to quantify the relationship between components in complex systems, such as the commonly used Euclidean distance and cos similarity. These distances have a strong connection with the Pearson coefficient. However, the Pearson coefficient sometimes ignores important nonlinear relations. Compared with the Pearson coefficient, the distance correlation coefficient can be a reliable measure of linear and nonlinear relationships. Inspired by this, we propose distance correlation entropy for uncertainty quantification and classifying different states. Unlike other entropy, the distribution of distance correlation is utilized to evaluate complexity. DCE retains the advantages of the previous methods, such as high consistency. The ordinal distance complexity measure is proposed as a supplement to DCE for quantifying information about pattern transitions in time series. Both DCE and ODCM are insensitive to the length of time series. Moreover, distance rank entropy derived from DCE and ODCM can be used to detect abnormal data. Experiments show that DCE and ODCM can distinguish periodic and chaotic behavior as well as different states in nonlinear dynamic systems, such as financial time series, while ODCM and distance rank entropy can be well combined with the uniform manifold approximation and projection method for data classification and visualization. The application in bearing data illustrates that they can be applied to fault diagnosis and get satisfactory results.