A Wavelet Variance Primer

The wavelet variance is a decomposition of the variance of a time series. Because of its scale-based nature, the wavelet variance offers insight into various time series, particularly in the physical sciences. This primer is a basic introduction to the wavelet variance, starting with its definition in terms of the discrete wavelet transform, proceeding with a discussion of the large-sample statistical properties of its basic estimators, and then continuing with an examination of estimators appropriate for time series with either missing values or contamination by discordant values. The discussion then moves to two uses of the wavelet variance involving its across-scale patterns, namely, estimation of exponents of power-law processes and estimations of characteristic scales. The primer closes with examples of the wavelet variance applied to time series involving atomic clocks, sea-ice thickness, the albedo of Arctic ice, X-ray fluctuations from binary stars, and coherent structures in river flow.