University of Washington - Department of Statistics
Wavelets give rise to the concept of wavelet variance that decomposes the variance of a time series on a scale by scale basis and that has considerable appeal when physical phenomena are analyzed in terms of variations operating over a range of different scales. The wavelet variance has been applied to a variety of time series and is useful as an exploratory tool to identify important scales, to assess the exponent parameter of a power law process, to detect inhomogeneity and to estimate a time varying spectral density function. In this thesis, we propose a new method of obtaining estimates of the wavelet variance when the observed time series is â€œgappyâ€, i.e., is sampled at regular intervals, but certain observations are missing. We investigate properties of these estimates and discuss large sample inference. We then extend this results to the estimation of wavelet covariances for multivariate gappy time series. We apply our methodology to NOAA's tropical sea level barometric pressure data. Next we extend the concept of a wavelet variance for random fields based upon wavelet transform, and discuss estimation and large sample inference. We illustrate our methodology using images of clouds. Finally we develop theory for the robust estimation of wavelet variance when data are contaminated.