A. Serroukh, A. T. Walden and D. B. Percival (2000), `Statistical Properties and Uses of the Wavelet Variance Estimator for the Scale Analysis of Time Series,' Journal of the American Statistical Association, 95, no. 449, pp. 184--196.
Many physical processes are an amalgam of components operating on different scales, and scientific questions about observed data are often inherently linked to understanding the behaviour at different scales. We explore time-scale properties of time series through the variance at different scales derived using wavelet methods. The great advantage of wavelet methods over ad hoc modifications of existing techniques is that wavelets provide exact scale-based decomposition results. We consider processes which are stationary, nonstationary but with stationary dth order differences, and nonstationary processes which are locally stationary. We study an estimator of the wavelet variance based on the maximal-overlap (undecimated) discrete wavelet transform. The asymptotic distribution of this wavelet variance estimator is derived for a wide class of stochastic processes, not necessarily Gaussian or linear. The variance of this distribution is estimated using spectral methods. Simulations confirm the theoretical results. The utility of the methodology is demonstrated on two scientifically important series, the surface albedo of pack ice, a strongly non-Gaussian series, and ocean shear data, a non-stationary series.
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