Wavelet analysis of covariance with application to atmospheric time series

Multiscale analysis of univariate time series has appeared in the literatur e at an ever increasing rate. Here we introduce the multiscale analysis of covariance between two time series using the discrete wavelet transform. Th e wavelet covariance and wavelet correlation are defined and applied to thi s problem as an alternative to traditional cross-spectrum analysis. The wav elet covariance is shown to decompose the covariance between two stationary processes on a scale by scale basis. Asymptotic normality is established f or estimators of the wavelet; covariance and correlation. Both quantities a re generalized into the wavelet cross covariance and cross correlation in o rder to investigate possible lead/lag relationships. A thorough analysis of interannual variability for the Madden-Julian oscillation is performed usi ng a 35+ year record of daily station pressure series. The time localizatio n of the discrete wavelet transform allows the subseries, which are associa ted with specific physical time scales, to be partitioned into both seasona l periods (such as summer and winter) and also according to El Nine-Souther n Oscillation (ENSO) activity, Differences in variance and correlation betw een these periods may then be firmly established through statistical hypoth esis testing. The daily station pressure series used here show clear eviden ce of increased variance and correlation in winter across Fourier periods o f 16-128 days, During warm episodes of ENSO activity, a reduced variance is observed across Fourier periods of 8-512 days for the station pressure ser ies from Truk Island and little or no correlation between station pressure series for the same periods.