Scaling Summaries in Multiscale Domains with Applications

在许多科学实验中，观测结果往往表现为噪声，这使得传统的数据分析技术无法满足需要。拟议研究的主要目标是开发和探索统计模型，为这类噪声驱动的数据产生信息性的比例摘要，目的是推断、分类和预测。这是在小波的帮助下实现的，小波是最有效的多尺度工具之一。多尺度方法在过去二十年里在从理论统计到地球科学的多个学科中得到了发展。在分析海量、多维、噪声和看似混乱的数据集时，了解数据的缩放将带来重要的洞察力。拟议的研究将对产生和利用高频数据和图像的各个科学领域产生影响：最显著的是健康诊断和大气监测和预测领域。特别是，建议的方法将被应用于(1)乳腺癌和肺癌的诊断，通过筛选数字乳房X光照片和胸部X光图像的缩放特征，以及(2)对湍流大气流动的地球科学分析，如风速、温度和污染物浓度，目的是建模和预测。该项目还将通过学生深入参与新技术的应用来解决不同科学领域的问题，从而促进对他们的教育和培训。拟议的数据标度替代评估框架利用实数和复数尺度混合小波变换领域的统计建模。新的小波子空间尺度混合层次简洁地描述了数据中多变量分量之间的“能量通量”。这些描述符将以单分形和多分形子波谱和共谱的形式提供更多的洞察力和信息性摘要，并以稳健的方式定义。该项目的三个科学目标包括：(I)分析不同分解(正交、非抽取和小波包)的尺度混合多维小波系数的性质，并研究其与尺度评估的相关性；(Ii)建立非标准多尺度域中规则和不规则尺度的稳健度量的理论性质；以及(Iii)将拟议研究的理论进展转化为在地球科学、生物信息学和医学诊断中的应用。

In many scientific experiments, the observations often present themselves as noise, making traditional data analytic techniques inadequate. The main objective of the proposed research is to develop and explore statistical models that produce informative scaling summaries for such noise-driven data with the goal of inference, classification, and prediction. This is achieved with the help of wavelets, one of the most efficient multiscale tools. Multiscale methodologies have evolved in the last two decades in disciplines ranging from theoretical statistics to geosciences. Understanding the scaling in data will lead to significant insights when analyzing massive, multidimensional, noisy, and seemingly chaotic data sets. The proposed research will impact various scientific fields that produce and utilize high-frequency data and images: most notably the areas of health diagnostics and atmospheric monitoring and prediction. In particular, the proposed methods will be applied to (1) breast cancer and lung cancer diagnostics by screening for the scaling features of digital mammograms and chest x-ray images, and (2) geoscientific analysis of turbulent atmospheric flows such as wind velocities, temperatures, and pollutant concentrations with the goal of modeling and prediction. The project will also contribute to the education and training of students through their deep engagement in the applications of novel techniques to problems from various scientific fields.The proposed framework for an alternative assessment of scaling present in data utilizes statistical modeling in the domain of real and complex scale-mixing wavelet transforms. The novel scale-mixing hierarchies of wavelet subspaces succinctly describe "fluxes-in-energy" among the multivariate components in data. Such descriptors will provide added insights and informative summaries in the form of monofractal and multifractal wavelet spectra and co-spectra, defined in a robust manner. The three scientific aims of the project include: (i) analyzing the properties of scale-mixing multidimensional wavelet coefficients for different decompositions (orthogonal, non-decimated, and wavelet packets), and investigating their relevance to scaling assessment, (ii) establishing theoretical properties of robust measures for regular and irregular scaling in non-standard multiscale domains, and (iii) translating theoretical advances of the proposed research to applications in geosciences, bioinformatics, and medical diagnostics.