Theoretical Study and Numerical Experiment on the Influence of Trend Changes on Correlation Coefficient

Abstract

When one of two time series undergoes an obvious change in trend, the correlation coefficient between the two will be distorted. In the context of global warming, most meteorological time series have obvious linear trends, so how do variations in these trends affect the correlation coefficient? In this paper, the correlation between time series with trend changes is studied theoretically and numerically. Adopting the trend coefficient, which reflects the nature and size of the trend change, we derive a formula r = f(k, l) for the correlation coefficient of time series X and Y with respective trend coefficients k and l. Analysis of the function graph shows that the changes in correlation coefficient with respect to the trend coefficients produce a twisted saddle surface, and the saddle point coordinates are given by the trend coefficients of time series X and Y with the opposite signs. The curve f(k, l) = f(0, 0) divides the coordinate planes into regions where f(k, l) > f(0, 0) and f(k, l) < f(0, 0). When the trend coefficients k and l are very small and the correlation coefficient is also very small, then k > 0 and l > 0 (or k < 0 and l < 0) amplifies a positive correlation, whereas k > 0 and l < 0 (or k < 0 and l > 0) amplifies a negative correlation, as found in previous research. Finally, experiments using meteorological data verify the reliability and effectiveness of the theory.