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Non-Negative Matrix Factorization (NMF) is a valuable matrix factorization technique which produces a "parts-based" decomposition of data sets. Wi-Fi user counts are a privacy-preserving indicator of population movements in smart and connected urban environments. In this paper, we apply NMF with a novel matrix embedding to Wi-Fi user count data from the University of Colorado at Boulder Campus for the purpose of automatically identifying patterns of human movement in a Smart and Connected infrastructure environment.

Growing use of network technology in Higher Education means that there has been increasing demand to adapt technology platforms and tools that transform student learning strategies, faculty teaching, research modalities, as well as general operations. Many of the new modalities are necessary for IHE business. In August 2019, we began collecting and analyzing data from the campus WIFI network. A goal of the research was to answer question like what passive sensing of the IHE WIFI might tell us about the dynamics of the WIFI weather in the IHE ecosystem and what does anonymized data tell us about the IHE ecosystem. The analogy with weather prediction seemed appropriate and a viable approach. Starting Fall 2019, data were collected in the observational phase. In the analysis phase, we applied Singular Spectrum Analysis decomposition, to deconstruct WIFI data from dorms, the central campus dining cafeteria, the recreation center, and other buildings on campus. That analysis led to the identification of clusters of buildings that behaved similarly. Just as in the case of models of the weather, a final component of this research was forecasting. We found that weekly forecast of WIFI behavior in the Fall 2019, were straight forward using SSA and seemed to present behavior of a low dimensional dynamical system. However, in Spring 2020, and the COVID perturbation, the campus ecosystem received a shock and data show that the campus changed very quickly. We found that as the campus moved to conduct remote learning, teaching, the closure of research labs, and the edict to work remotely, SSA forecasting techniques not trained on the Spring 2020, data after the shock, performed poorly. While SSA forecasting trained on a portion of the data did better., Comment: Contact E-mail: james.h.curry@colorado.edu, Applied Mathematics, University of Colorado, Boulder 80309-0526

Recent improvements in computing allow for the processing and analysis of very large datasets in a variety of fields. Often the analysis requires the creation of low-rank approximations to the datasets leading to efficient storage. This article presents and analyzes a novel approach for creating nonnegative, structured dictionaries using NMF applied to reordered pixels of single, natural images. We reorder the pixels based on patches and present our approach in general. We investigate our approach when using the Singular Value Decomposition (SVD) and Nonnegative Matrix Factorizations (NMF) as low-rank approximations. Peak Signal-to-Noise Ratio (PSNR) and Mean Structural Similarity Index (MSSIM) are used to evaluate the algorithm. We report that while the SVD provides the best reconstructions, its dictionary of vectors lose both the sign structure of the original image and details of localized image content. In contrast, the dictionaries produced using NMF preserves the sign structure of the original image matrix and offer a nonnegative, parts-based dictionary., Comment: 34 pages, 15 figures, 2 tables