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音，使得传统的数据分析技术无法胜任。 拟议的研究的主要目标是开发和探索统计模型，为这些噪声驱动的数据产生信息丰富的缩放摘要，其目标是推断，分类和预测。 这是实现与小波的帮助下，最有效的多尺度工具之一。 在过去的二十年里，多尺度方法在从理论统计到地球科学的学科中得到了发展。 理解数据的缩放将在分析大量，多维，嘈杂和看似混乱的数据集时产生重要的见解。 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：最值得注意的是健康诊断和大气监测和预测领域。特别是，所提出的方法将应用于（1）乳腺癌和肺癌诊断，通过筛选数字乳房X线照片和胸部X线图像的缩放特征，以及（2）湍流大气流的地球科学分析，如风速，温度和污染物浓度，目标是建模和预测。该项目还将有助于教育和培训的学生，通过他们的新技术的应用，从不同的科学领域的问题的深入参与。提出的框架，目前在数据中的缩放的替代评估利用统计建模领域的真实的和复杂的尺度混合小波变换。小波子空间的新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重分形小波谱和共谱的形式提供更多的见解和信息摘要，以稳健的方式定义。该项目的三个科学目标包括：（i）分析不同分解的尺度混合多维小波系数的性质（正交，非抽取，小波包），并调查其相关性的缩放评估，（ii）建立理论属性的鲁棒措施，为规则和不规则的缩放在非标准的多尺度域，以及（iii）将拟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学的应用。

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.