A novel way to detect correlations on multi-time scales, with temporal evolution and for multi-variables

In this paper, two new methods, Temporal evolution of Detrended Cross-Correlation Analysis (TDCCA) and Temporal evolution of Detrended Partial-Cross-Correlation Analysis (TDPCCA), are proposed by generalizing DCCA and DPCCA. Applying TDCCA/TDPCCA, it is possible to study correlations on multi-time scales and over different periods. To illustrate their properties, we used two climatological examples: i) Global Sea Level (GSL) versus North Atlantic Oscillation (NAO); and ii) Summer Rainfall over Yangtze River (SRYR) versus previous winter Pacific Decadal Oscillation (PDO). We find significant correlations between GSL and NAO on time scales of 60 to 140 years, but the correlations are non-significant between 1865–1875. As for SRYR and PDO, significant correlations are found on time scales of 30 to 35 years, but the correlations are more pronounced during the recent 30 years. By combining TDCCA/TDPCCA and DCCA/DPCCA, we proposed a new correlation-detection system, which compared to traditional methods, can objectively show how two time series are related (on which time scale, during which time period). These are important not only for diagnosis of complex system, but also for better designs of prediction models. Therefore, the new methods offer new opportunities for applications in natural sciences, such as ecology, economy, sociology and other research fields.