Mathematical Proof of the Synthetic Running Correlation Coefficient and Its Ability to Reflect Temporal Variations in Correlation

The running correlation coefficient(RCC) is useful for capturing temporal variations in correlations between two time series. The local running correlation coefficient(LRCC) is a widely used algorithm that directly applies the Pearson correlation to a time window. A new algorithm called synthetic running correlation coefficient(SRCC) was proposed in 2018 and proven to be reasonable and usable; however, this algorithm lacks a theoretical demonstration. In this paper, SRCC is proven theoretically. RCC is only meaningful when its values at different times can be compared. First, the global means are proven to be the unique standard quantities for comparison. SRCC is the only RCC that satisfies the comparability criterion. The relationship between LRCC and SRCC is derived using statistical methods, and SRCC is obtained by adding a constraint condition to the LRCC algorithm. Dividing the temporal fluctuations into high-and low-frequency signals reveals that LRCC only reflects the correlation of high-frequency signals; by contrast, SRCC reflects the correlations of high-and low-frequency signals simultaneously. Therefore, SRCC is the appropriate method for calculating RCCs.