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Department of Mathematics

Permanent URI for this communityhttps://hdl.handle.net/10217/100468

These digital collections include faculty/student publications, theses, and dissertations from the Department of Mathematics.

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Browsing Department of Mathematics by Author "Anderson, Charles, committee member"

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    Anomaly detection in terrestrial hyperspectral video using variants of the RX algorithm
    (Colorado State University. Libraries, 2012) Schwickerath, Anthony N., author; Kirby, Michael, advisor; Peterson, Christopher, committee member; Anderson, Charles, committee member
    There is currently interest in detecting the use of chemical and biological weapons using hyperspectral sensors. Much of the research in this area assumes the spectral signature of the weapon is known in advance. Unfortunately, this may not always be the case. To obviate the reliance on a library of known target signatures, we instead view this as an anomaly detection problem. In this thesis, the RX algorithm, a benchmark anomaly detection algorithm for multi- and hyper-spectral data is reviewed, as are some standard extensions. This class of likelihood ratio test-based algorithms is generally applied to aerial imagery for the identification of man-made artifacts. As such, the model assumes that the scale is relatively consistent and that the targets (roads, cars) also have fixed sizes. We apply these methods to terrestrial video of biological and chemical aerosol plumes, where the background scale and target size both vary, and compare preliminary results. To explore the impact of parameter choice on algorithm performance, we also present an empirical study of the standard RX algorithm applied to synthetic targets of varying sizes over a range of settings.
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    Methods for network generation and spectral feature selection: especially on gene expression data
    (Colorado State University. Libraries, 2019) Mankovich, Nathan, author; Kirby, Michael, advisor; Anderson, Charles, committee member; Peterson, Chris, committee member
    Feature selection is an essential step in many data analysis pipelines due to its ability to remove unimportant data. We will describe how to realize a data set as a network using correlation, partial correlation, heat kernel and random edge generation methods. Then we lay out how to select features from these networks mainly leveraging the spectrum of the graph Laplacian, adjacency, and supra-adjacency matrices. We frame this work in the context of gene co-expression network analysis and proceed with a brief analysis of a small set of gene expression data for human subjects infected with the flu virus. We are able to distinguish two sets of 14-15 genes which produce two fold SSVM classification accuracies at certain times that are at least as high as classification accuracies done with more than 12,000 genes.
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    Schubert variety of best fit with applications and across domains sparse feature extraction
    (Colorado State University. Libraries, 2024) Karimov, Karim, author; Kirby, Michael, advisor; Peterson, Chris, committee member; Anderson, Charles, committee member; Adams, Henry, committee member
    This dissertation presents two novel approaches in applied mathematics for data analysis and feature selection, addressing challenges in both geometric data representation and multi-domain biological data interpretation. The first part introduces the Schubert Variety of Best Fit (SVBF) as a new geometric framework for analyzing sets of datasets. Leveraging the structure of Grassmann manifolds and Schubert varieties, we develop the SVBF-Node, a computational unit for solving related optimization problems. We demonstrate the efficacy of this approach through three classification algorithms and a new clustering method, SVBF-LBG. These techniques are evaluated on various datasets, including synthetic data, image sets, video sequences, and hyperspectral remote sensing data, showing improved performance over existing similar methods, particularly for complex, high-dimensional data. The second part proposes a multi-domain, multi-task (MDMT) architecture for feature selection in biological data. This method integrates multi-domain learning with masked feature selection, specifically applied to gene expression data from multiple tissues. We demonstrate its ability to identify novel biomarkers in host immune responses to infection, which are not detectable through single-domain analyses. The approach is validated using bulk RNA sequences from different tissues, revealing its potential to uncover cross-domain biological insights. Both contributions offer interpretable, mathematically grounded approaches to data analysis, providing new tools for researchers in applied mathematics, machine learning, and bioinformatics.
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    Subspace and network averaging for computer vision and bioinformatics
    (Colorado State University. Libraries, 2023) Mankovich, Nathan J., author; Kirby, Michael, advisor; Peterson, Chris, committee member; King, Emily, committee member; Anderson, Charles, committee member
    Finding a central prototype (a.k.a. average) from a cluster of points in high dimensional space has broad applications to complex problems like action clustering in computer vision or gene co-expression module representation in bioinformatics. A central prototype of a set of points may be cast as the solution to an optimization problem that either minimizes distance or maximizes similarity between the prototype and each point in the cluster. In this dissertation we offer four novel prototypes for a cluster of points: the flag median, maximally correlated flag, cluster expression vector and eigengene subspace. We will formalize the flag median and the maximally correlated flag using subspace representations for data, specifically the Grasmann and flag manifolds. In addition to introducing these prototypes, we will derive a novel algorithm which can be used to calculate subspace prototypes: FlagIRLS. The third and fourth prototypes, the cluster expression vector and eigengene subspace, are inspired by problems involving gene cluster (e.g., pathway or module) representations. The cluster expression vector leverages connections within networks of genes whereas the eigengene subspace is computed using Principal Component Analysis (PCA). In this work we will explore the theoretical under-pinnings of these prototypes, find algorithms to compute and them to computer vision and biological data sets.