Scaling Summaries in Multiscale Domains with Applications

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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.