Theses and Dissertations at Montana State University (MSU)
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Item Directed graph descriptors and distances for analyzing multivariate time series data(Montana State University - Bozeman, College of Letters & Science, 2022) Belton, Robin Lynne; Chairperson, Graduate Committee: Tomas GedeonLocal maxima and minima, or extremal events, in experimental time series can be used as a coarse summary to characterize data. However, the discrete sampling in recording experimental measurements suggests uncertainty in the true timing of extrema during the experiment. This in turn gives uncertainty in the timing order of extrema within the time series. Motivated by applications in genomic time series and biological network analysis, we construct a weighted directed acyclic graph (DAG) called an extremal event DAG using techniques from persistent homology that is robust to measurement noise. Furthermore, we define a distance between extremal event DAGs based on the edit distance between strings. We prove several properties including local stability for the extremal event DAG distance with respect to pairwise L1 distances between functions in the time series data. Lastly, we provide algorithms, publicly free software, and implementations on extremal event DAG construction and comparison.Item DVM: a deep learning algorithm for minimizing functionals(Montana State University - Bozeman, College of Letters & Science, 2022) Bair, Dominic Robert; Chairperson, Graduate Committee: Dominique ZossoThe use of data-driven techniques to solve PDEs is a rapidly developing field. Current deep learning methods can find solutions to high-dimensional PDEs with great accuracy and efficiency. However, for certain classes of problems these techniques may be inefficient. We focus on PDEs with a so-called 'variational formulation'. Here the solution to the PDE is represented as a minimizer or maximizer to a functional. We propose a family of novel deep learning algorithms to find these minimizers with similar accuracy and greater efficiency than techniques using the PDE formulation. These algorithms can be also be used to minimize functionals which do not have an equivalent PDE formulation. We call these algorithms 'Deep Variational Methods' (DVM).Item Adapting archetypal analysis to scientific imaging applications(Montana State University - Bozeman, College of Letters & Science, 2022) Potts, Catherine Gabriel; Chairperson, Graduate Committee: Dominique ZossoScientific imaging applications create large sets of high-dimensional data, which may be difficult to process using traditional supervised machine learning representative models. First, many representative models generate computational elements that are difficult to interpret in terms of the scientific application and second, the high embedding dimension of the images often makes generating the models computationally inefficient. We propose using archetypal analysis (AA) as the representative model for these scientific imaging problems, since the computational elements, so called archetypes, resemble members of the original dataset. Specifically, the archetypes are generated as extreme points to an approximation of the convex hull of the data cloud, which means they maintain the structure of individual data points. To improve the computational task of generating the AA model, we propose a sketch-based AA method which projects the data to a lower embedding dimension before calculating the computational elements, lowering computation time for these high-dimensional problems, while at the same time retaining the geometric structure enough so that the computational elements closely match the results of AA. We also applied a primal-dual hybrid gradient (PDHG) solver to the AA algorithm structure attempting to speed up computation. To verify the significance of the interpretation of AA, we applied AA to transient fluorescent calcium images, recorded in the Kunze Neuroengineering lab as videos, in order to determine whether or not adding different nanoparticles changed the way the neurons in culture communicate. We also applied our sketch-based AA method to other sorts of imaging data sets, exploring the differences between our method and the standard AA method. Our experimentation shows the different ways that AA can be adapted to scientific imaging applications, providing a machine learning representation model that is interpretable in the context of the imaging problem and verifies the benefits of the sketch-based method in terms of computation time.Item Development and applications of particle swarm optimization for constructing optimal experimental designs(Montana State University - Bozeman, College of Letters & Science, 2021) Walsh, Stephen Joseph; Chairperson, Graduate Committee: John J. BorkowskiThe primary objective motivating this dissertation was to illustrate the efficacy of particle swarm optimization (PSO) as the engine of an algorithm to generate optimal design of experiments (DoE). PSO is a wildly popular and successful metaheuristic and machine learning algorithm which makes no assumptions regarding the behavior of the function being optimized. Optimal DoE, in part thanks to modern computing, has become the current dominant paradigm for practitioners to generate a DoE with some desirable property. We bring together these concepts first by extending the PSO to optimizing functions that take matrix inputs. Julia software was developed for this purpose and validated against published results. A detailed benchmarking study of three PSO variants was conducted and a preferred version of the algorithm was identified for further research and application. Next, we implemented the approach to generating G-optimal designs--a difficult mini-max optimization problem. New heretofore unknown G-optimal designs have been produced and the efficacy of PSO in generating efficiently (w.r.t. computing time) highly G-optimal designs is compared to current published results. Next, a new algorithm for generating optimal designs with specified replication structure, and thereby guaranteed a degrees-of-freedom for estimating the pure error variance, is proposed, illustrated, benchmarked and validated. Last, we propose a new algorithm for generating optimal mixture experiment designs which implements a PSO type search using a non-Euclidean geometry (specifically the Aitchison geometry). In this setting the space of candidate matrices is the Cartesian product of standard (K - 1)-simplices. The algorithm is extended to mixture experiments with lower and upper constraints on the mixture proportions; in this setting, the space of candidate matrices is the Cartesian product of high-dimensional irregular convex polytopes. The algorithm is validated against very recent published results. In total, the work presented in this dissertation speaks very favorably to PSO as a tool for generating optimal DoEs. We believe this approach should become part of the standard machine learning and statistical tool box for generating optimal experimental designs.Item Multi input minimax adaptive Antoulas-Anderson algorithm for rational approximation with stable poles(Montana State University - Bozeman, College of Letters & Science, 2021) Johns, William Richard; Chairperson, Graduate Committee: Lisa DavisThis thesis details the development of the 'symmetric stable multi-input multioutput Adaptive Antoulas Anderson' algorithm, we call this algorithm symmetric smiAAA. The symmetric smiAAA algorithm builds rational approximations, for multiple inputs. The approximations share a common set of parameters called the poles. The primary goal of this algorithm is to address shortcomings in multi-input multi-output rational approximation algorithms currently used in electro-magnetic transients programs. All state of the art algorithms currently follow a similar methodology: The user selects the number of poles to use and supplies an initial guess for their values. The algorithms optimize the shared poles and return the best approximation they found. The user is not guaranteed a specific accuracy in the approximations. If the results returned are not sufficiently accurate, the algorithm must be run again with additional poles. Symmetric smiAAA is designed with the goal of achieving user-defined accuracy, with no information about the number of poles. The user selects the desired accuracy of the approximations and the algorithm does the rest. Symmetric smiAAA returns approximations with the desired accuracy by finding the number of shared poles needed for the desired accuracy, and their values. This work introduces the following three features to the 'Adaptive Antoulas Anderson' algorithm. First, we extend the ideas from the single-input Adaptive Antoulas Anderson algorithm, to multi-input multi-output problems. Second, we introduce enforcement of constraints on the values of the poles. Lastly, we extend a single input post-processing optimization based upon the Lawson method, to multi-input multi-output problems. The symmetric smiAAA algorithm combines these three features with the symmetry enforcement introduced in the FastAAA algorithm. In order to test it against the current industry standards, we compare the symmetric smiAAA algorithm with Vector Fitting and the recently published RKFIT algorithms. These comparisons demonstrate that symmetric smiAAA produces approximations with similar accuracy and running time, while allowing the user to select only the desired accuracy. Symmetric smiAAA is a robust and powerful algorithm which provides the user full control over the final accuracy of the approximations.Item Space-filling designs for mixture/process variable experiments(Montana State University - Bozeman, College of Letters & Science, 2021) Obiri, Moses Yeboah; Chairperson, Graduate Committee: John J. BorkowskiThe ultimate objective of this dissertation was to present a statistical methodology and an algorithm for generating uniform designs for the combined mixture/process variable experiment. There are many methods available for constructing uniform designs and four of such methods have been used in this study. These are the Good Lattice Point (GLP) method, the cyclotomic field (CF) method, the square root sequence (SRS) method, and the power-of-a-prime (PP) method. A new hybrid algorithm is presented for generating uniform designs for mixture/process variable experiments. The algorithm uses the G function introduced by Fang and Yang (2000), and adopted by Borkowski and Piepel (2009) to map q-1 points from q + k - 1 points generated in the hypercube to the simplex. Two new criteria based on the Euclidean Minimum Spanning Tree (EMST) which are more computationally efficient for assessing uniformity of mixture designs and mixture/process variable designs are presented. The two criteria were found to be interchangeable and the geometric mean of the edge lengths (GMST) criterion is preferred to the average and standard deviation of edge lengths (adMST, sdMST) criterion. The GMST criterion uses only one statistic to quantify the uniformity properties of mixture and mixture/process variable designs. Tables of good uniform designs are provided for mixture experiments in the full simplex (S q) for q = 3; 4; 5 and practical design sizes, 9 _ n _ 30, using the four number theoretic methods in this study. A conditional approach based on the GMST criterion for generating good uniform mixture/process variable designs is also introduced and tables of good uniform designs are given for the combined q-mixture and k process variable experiments for q = 3; 4; 5, k = 1; 2 and practical numbers of runs, 9 _ n _ 30. A new algorithm is provided to augment existing mixture design points with space-filling points including designs with existing clustered design points. In this algorithm, new design points are chosen from a candidate set of points such that the resulting augmented design has good space-filling properties. The SRS method is found to produce the best augmented space-filling mixture designs.