The formula to analyze correlation between original time series data, xi(i=1, ···, N)and time series data delayed by specified time interval (d), yi(xi+d ; i=1, ···N) is as follow;
This formula, called as Pearson’s Correlation or Linear Correlation Coefficient, is the most widely used method to obtain Autocorrelation Function. X bar, Y bar indicate mean values of xi(i=1, ···, N) and yi(xi+d ; i=1, ···N) respectively. Result from application of this formula while increasing the delayed time (d) from 0 to sampling time interval is Autocorrelation Function. Autocorrelation Function varying depending on signal is expressed as a value within -1 to 1. If the value is 1, it is said as “complete positive correlation.” Such case can be observed when xi and yi increase together. If Autocorrelation Function value (r) is -1, it is said as “complete negative correlation.” In this case, yi decreases while xi increases.The most important value in Autocorrelation Function is the delayed time when Autocorrelation Function value (r) firstly becomes 0 (zero). The point where Autocorrelation Function value (r) firstly becomes 0 (zero) is the time when correlation between xi and yi disappears, which is called as “Decorrelation time.”
Decorrelation time also means the time during which correlation continues. So, longer decorrelation time means longer time for forecasting of signal, while shorter decorrelation time means shorter time for forecasting. Further, shorter decorrelation time indicates shorter length of past data required for determination of the present data values. In short, former data close to decorrelation time have correlation with data at the present time point and data distant from decorrelation time do not have significant meaning for determination of data at the present time point.