Bayesian Wavelet Modeling with Applications in Turbulence

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DMS 9626159 Vidakovic This research connects recent theoretical advances in both nonlinear wavelet shrinkage theory and Bayesian statistical wavelet modeling to time series measurements of stochastic phenomena. The researchers study: (i) cost measures that serve as criteria for the best wavelet basis selection, (ii) statistical models in the wavelet domain that range from Hilbert space projections to coherent Bayesian models, and (iii) the model induced shrinkage in the wavelet domain. It is demonstrated that wavelet regression and density estimation are excellent tools in denoising and parsimonious description of complex dynamic processes, such as turbulence. The turbulence measurements are characterized by local ``bursts'' in time and frequency domains are providing an ideal media for testing shrinkage methods, best basis choice, and wavelet modeling. The existence of power laws, consistent with Kolmogorov's K41 theory is used to assess the proposed shrinkage methods. Wavelets are novel building blocks and excellent descriptors of many complex phenomena in a variety of scientific fields. This research applies wavelets to an important and omnipresent phenomenon arising in hydrology and atmospheric science: the turbulence. Recent theoretical advances in statistical modeling and estimation are combined with the power of several intrinsic properties of wavelets to produce superb tools for modeling, denoising, and analyzing complex turbulence measurements.