Long memory processes are those which exhibit substantial correlations between events separated by a long period of time. By analyzing a time series on a scale by scale basis, each scale corresponding to a range of frequencies, the ability to detect and locate a sudden change in the variance in the time series is introduced. Using this same procedure to detect a change in the long memory parameter is also investigated.
Applications involve the Nile River minimum water levels and vertical ocean shear measurements.
The concepts of wavelet covariance and wavelet correlation are introduced and applied to this problem as an alternative to cross-spectrum analysis. The wavelet covariance is shown to decompose the covariance between two stationary processes on a scale by scale basis. Asymptotic normality of estimators of the wavelet covariance and correlation is shown in order to construct approximate confidence intervals. The time-varying wavelet covariance is used to show the increase in positive correlation between the SOI and Truk Island station pressure in the first half of each year versus latter half.