Scaling Summaries in Multiscale Domains with Applications

在许多科学实验中，观察结果往往表现为噪音，使传统的数据分析技术不充分。提出的研究的主要目的是开发和探索统计模型，为这些噪声驱动的数据提供信息的缩放摘要，目的是推断，分类和预测。这是在最有效的多尺度工具之一小波的帮助下实现的。在过去的二十年里，从理论统计学到地球科学，多尺度方法在各个学科中都得到了发展。在分析大量、多维、嘈杂和看似混乱的数据集时，理解数据的缩放将带来重要的见解。拟议的研究将影响产生和利用高频数据和图像的各个科学领域：最显著的是卫生诊断和大气监测与预测领域。特别是，所提出的方法将应用于(1)通过筛选数字乳房x线照片和胸部x线图像的缩放特征来诊断乳腺癌和肺癌，以及(2)对湍流大气流动（如风速、温度和污染物浓度）进行地球科学分析，目的是建模和预测。该项目还将通过学生深入参与将新技术应用于各个科学领域的问题，为学生的教育和培训做出贡献。提出了一种评估数据中存在的尺度的替代框架，该框架利用了实和复杂尺度混合小波变换领域的统计建模。小波子空间的新尺度混合层次简洁地描述了数据中多元分量之间的“能量通量”。这样的描述符将以单分形和多重分形小波谱和共谱的形式提供额外的见解和信息摘要，以鲁棒的方式定义。该项目的三个科学目标包括：(i)分析不同分解（正交、非decimated和小波包）的尺度混合多维小波系数的性质，并研究它们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鲁棒度量的理论性质；（iii）将提议研究的理论进展转化为地球科学、生物信息学和医学诊断的应用。

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.