Bayesian Wavelet Modeling with Applications in Turbulence

本研究将非线性小波收缩理论和贝叶斯统计小波建模的最新理论进展与随机现象的时间序列测量联系起来。研究人员研究了：(i)作为最佳小波基选择标准的成本度量，（ii）小波域的统计模型，范围从希尔伯特空间投影到相干贝叶斯模型，以及（iii）模型在小波域引起的收缩。结果表明，小波回归和密度估计是对湍流等复杂动态过程进行去噪和精简描述的有效工具。湍流测量以时域和频域的局部“爆发”为特征，为测试收缩方法、最佳基选择和小波建模提供了理想的介质。幂律的存在，与Kolmogorov的K41理论一致，被用来评估拟议的收缩方法。小波是各种科学领域中许多复杂现象的新颖构建模块和优秀描述符。本研究将小波应用于水文学和大气科学中一个重要而普遍存在的现象：湍流。统计建模和估计方面的最新理论进展与小波的几个内在特性相结合，产生了用于建模，去噪和分析复杂湍流测量的极好的工具。

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.