A novel two‐dimensional correlation coefficient for assessing associations in time series data

Abstract

ABSTRACTThe widely used Pearson's correlation coefficient calculated for assessing the linear relationship between two variables might produce misleading results especially in the comparison of periodic variables. A single correlation coefficient provides a measure of the overall dependence structure and generally might not be sufficient for assessing local differences between the variables (e.g. associations between each individual year might vary in hydrologic series). The reason for this deficiency is the consideration of the averages of the whole series while ignoring the variations of the local averages (e.g. annual averages or long year averages of months) throughout the observations. This study presents a two‐dimensional horizontal (row wise) and vertical (column wise) correlation calculation approach where the compared series are considered as two‐dimensional matrices in which each row represents a sub‐period (e.g. one calendar year of the precipitation data) of the investigated time series data. The method applies a normalization procedure by considering the averages of all rows (namely local averages) for calculating the horizontal correlation and the averages of all columns for calculating the vertical correlation instead of considering the averages of the whole matrices. This enables a separate determination of the degree of relationships between the rows and columns of the compared data matrices by using the horizontal and vertical variance and covariance values that constitute the base of the two‐dimensional correlation. The method is applied on 14 different linearly varying hypothetical matrices, 6 matrices for testing the influence of seasonal and inter‐annual variations and the monthly total precipitation records of 6 stations in southwest Turkey. The results have shown that the developed correlation approach assesses the two‐dimensional behaviour of time series data like precipitation and provides a measure which enables separate assessment of the contributions from the seasonal cycle vs. inter‐annual variability in the association between two time series.