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Colorado State University
1.
Teli, Mohammad Nayeem.
Dimensionality reduction and classification of time embedded EEG signals.
Degree: MS(M.S.), Computer Science, 2007, Colorado State University
URL: http://hdl.handle.net/10217/28637 
► Electroencephalogram (EEG) is the measurement of the electrical activity of the brain measured by placing electrodes on the scalp. These EEG signals give the micro-voltage…
(more)
▼ Electroencephalogram (EEG) is the measurement of the electrical activity of the brain measured by placing electrodes on the scalp. These EEG signals give the micro-voltage difference between different parts of the brain in a non-invasive manner. The brain activity measured in this way is being currently analyzed for a possible diagnosis of physiological and psychiatric diseases. These signals have also found a way into cognitive research. At
Colorado State University we are trying to investigate the use of EEG as computer input. In this particular research our goal is to classify two mental tasks. A subject is asked to think about a mental task and the EEG signals are measured using six electrodes on his scalp. In order to differentiate between two different tasks, the EEG signals produced by each task need to be classified. We hypothesize that a bottleneck neural network would help us to classify EEG data much better than classification techniques like Linear Discriminant Analysis(LDA), Quadratic Discriminant Analysis (QDA), and Support Vector Machines. A five layer bottleneck neural network is trained using a fast convergence algorithm (variation of Levenberg-Marquardt algorithm) and Scaled Conjugate Gradient (SCG). Classification is compared between a neural network, LDA, QDA and SVM for both raw EEG data as well as bottleneck layer output. Results indicate that QDA and SVM do better classification of raw EEG data without a bottleneck network. QDA and SVM always achieved higher classification accuracy than the neural network with a bottleneck layer in all our experiments. Neural network was able to achieve its best classification accuracy of 92% of test samples correctly classified, whereas QDA achieved 100% accuracy in classifying the test data.
Advisors/Committee Members: Anderson, Charles W. (advisor), McConnell, Ross (committee member), Kirby, Michael, 1961- (committee member).
Subjects/Keywords: SVM; electroencephalogram; QDA; support vector machines; bottleneck neural network; linear discriminant analysis; LDA; quadratic discriminant analysis; Brain-computer interfaces; Electroencephalography; EEG
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APA (6th Edition):
Teli, M. N. (2007). Dimensionality reduction and classification of time embedded EEG signals. (Masters Thesis). Colorado State University. Retrieved from http://hdl.handle.net/10217/28637
Chicago Manual of Style (16th Edition):
Teli, Mohammad Nayeem. “Dimensionality reduction and classification of time embedded EEG signals.” 2007. Masters Thesis, Colorado State University. Accessed March 01, 2021.
http://hdl.handle.net/10217/28637.
MLA Handbook (7th Edition):
Teli, Mohammad Nayeem. “Dimensionality reduction and classification of time embedded EEG signals.” 2007. Web. 01 Mar 2021.
Vancouver:
Teli MN. Dimensionality reduction and classification of time embedded EEG signals. [Internet] [Masters thesis]. Colorado State University; 2007. [cited 2021 Mar 01].
Available from: http://hdl.handle.net/10217/28637.
Council of Science Editors:
Teli MN. Dimensionality reduction and classification of time embedded EEG signals. [Masters Thesis]. Colorado State University; 2007. Available from: http://hdl.handle.net/10217/28637
Colorado State University
2.
Rutherford, Blake.
Lagrangian mixing and transport in hurricanes.
Degree: PhD, Mathematics, 2010, Colorado State University
URL: http://hdl.handle.net/10217/39048
► This study examines the role of transport and mixing in the dynamics of tropical cyclones from a mathematical viewpoint and their implications for intensity. While…
(more)
▼ This study examines the role of transport and mixing in the dynamics of tropical cyclones from a mathematical viewpoint and their implications for intensity. While this topic has seen extensive study, much of it has lacked the mathematical rigor allowed by a new class of Lagrangian techniques, which allow the study of particle transport through time-dependent flows. Lagrangian coherent structures (LCS's) are time-dependent boundaries which partition the flow into distinct regions, controlling the systematic transport of material between regions. In this study, the mathematics of Lagrangian transport is developed, and adapted to several tropical cyclone models. Three models are utilized to study mixing; the axisymmetric model of Rotunno and Emanuel (1987), the nondivergent barotropic 2D model of Schubert et al. (1999), and the 3D Penn
State-NCAR mesoscale model (MM5). For the study of mixing on the axisymmetric model, a new class of mixing rates is proposed which vary in initial time and integration time, and it is shown that mixing events precede changes in intensity. For the nondivergent barotropic model, orthogonal flow separation reveals coherent structures that are persistant through strong shear, and mixing is quantified through the shear during mesovortex interaction. The extension of the orthogonal separation methods to 3D provides a method for decomposing Lagrangian hyperbolicity from shear. The method is applied to the MM5 model to find the Lagrangian eye-eyewall interface (LEEI), which is responsible for dictating transport between the two regions. A new ridge extraction algorithm is used to extract the 2D manifolds of the 3D Lagrangian fields. By extending and automating this algorithm across varying initial time, a time-dependent and spatially smooth representation of the LEEI in terms of Fourier descriptors and radial basis functions is computed. The dynamics of the time-dependent LEEI indicate that the higher wavenumber asymmetries vanish, but the lower wavenumber asymmetries remain, quantifying the degree of axisymmetry in the storm from a transport perspective. The last study applies the new 3D techniques to an intensifying storm by studying the interaction of vortical hot towers (VHT's). VHT's are shown to not only be coherent structures, but to be associated with hyperbolic LCS's which play an important role in their interaction and in the formation of an eyewall. The length of the LCS's indicate that the VHT's have impact on a broad range that affects environmental flow into the primary vortex.
Advisors/Committee Members: Dangelmayr, G. (Gerhard), 1951- (advisor), Shipman, Patrick (committee member), Kirby, Michael, 1961- (committee member), Schubert, Wayne H. (committee member).
Subjects/Keywords: dynamics; hurricanes; Lagrangian; Hurricanes – Mathematical models; Vortex-motion – Mathematical models; Lagrangian functions; Transport theory – Mathematical models
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APA (6th Edition):
Rutherford, B. (2010). Lagrangian mixing and transport in hurricanes. (Doctoral Dissertation). Colorado State University. Retrieved from http://hdl.handle.net/10217/39048
Chicago Manual of Style (16th Edition):
Rutherford, Blake. “Lagrangian mixing and transport in hurricanes.” 2010. Doctoral Dissertation, Colorado State University. Accessed March 01, 2021.
http://hdl.handle.net/10217/39048.
MLA Handbook (7th Edition):
Rutherford, Blake. “Lagrangian mixing and transport in hurricanes.” 2010. Web. 01 Mar 2021.
Vancouver:
Rutherford B. Lagrangian mixing and transport in hurricanes. [Internet] [Doctoral dissertation]. Colorado State University; 2010. [cited 2021 Mar 01].
Available from: http://hdl.handle.net/10217/39048.
Council of Science Editors:
Rutherford B. Lagrangian mixing and transport in hurricanes. [Doctoral Dissertation]. Colorado State University; 2010. Available from: http://hdl.handle.net/10217/39048
Colorado State University
3.
Olson, Travis Andrew.
Hopf bifurcation in anisotropic reaction diffusion systems posed in large rectangles.
Degree: PhD, Mathematics, 2010, Colorado State University
URL: http://hdl.handle.net/10217/40474
► The oscillatory instability (Hopf bifurcation) for anisotropic reaction diffusion equations posed in large (but finite) rectangles is investigated. The work pursued in this dissertation extends…
(more)
▼ The oscillatory instability (Hopf bifurcation) for anisotropic reaction diffusion equations posed in large (but finite) rectangles is investigated. The work pursued in this dissertation extends previous studies for infinitely extended 2D systems to include finite-size effects. For the case considered, the solution of the reaction diffusion system is represented in terms of slowly modulated complex amplitudes of four wave-trains propagating in four oblique directions. While for the infinitely extended system the modulating amplitudes are independent dynamical variables, the finite size of the domain leads to relations between them induced by wave reflections at the boundaries. This leads to a single amplitude equation for a doubly periodic function that captures all four envelopes in different regions of its fundamental domain. The amplitude equation is derived by matching an asymptotic bulk solution to an asymptotic boundary layer solution. While for the corresponding infinitely extended system no further parameters generically remain in the amplitude (envelope) equations above the onset value of the control parameter, the finite-size amplitude equation retains a dependence on a rescaled version of this parameter. Numerical simulations show that the dynamics of the bounded system shows different behavior at onset in comparison to the unbounded system, and the complexity of the solutions significantly increases when the rescaled control parameter is increased. As an application of the technique developed, an anisotropic Activator-Inhibitor model with higher order diffusion is studied, and parameter values of the amplitude equations are calculated for several parameter sets of the model equations.
Advisors/Committee Members: Dangelmayr, G. (Gerhard), 1951- (advisor), Eykholt, Richard Eric, 1956- (committee member), Kirby, Michael, 1961- (committee member), Oprea, Iuliana (committee member).
Subjects/Keywords: Ginzburg; Landau; Hopf; Bifurcation theory; Differential equations; Rectangles; Boundary layer
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APA (6th Edition):
Olson, T. A. (2010). Hopf bifurcation in anisotropic reaction diffusion systems posed in large rectangles. (Doctoral Dissertation). Colorado State University. Retrieved from http://hdl.handle.net/10217/40474
Chicago Manual of Style (16th Edition):
Olson, Travis Andrew. “Hopf bifurcation in anisotropic reaction diffusion systems posed in large rectangles.” 2010. Doctoral Dissertation, Colorado State University. Accessed March 01, 2021.
http://hdl.handle.net/10217/40474.
MLA Handbook (7th Edition):
Olson, Travis Andrew. “Hopf bifurcation in anisotropic reaction diffusion systems posed in large rectangles.” 2010. Web. 01 Mar 2021.
Vancouver:
Olson TA. Hopf bifurcation in anisotropic reaction diffusion systems posed in large rectangles. [Internet] [Doctoral dissertation]. Colorado State University; 2010. [cited 2021 Mar 01].
Available from: http://hdl.handle.net/10217/40474.
Council of Science Editors:
Olson TA. Hopf bifurcation in anisotropic reaction diffusion systems posed in large rectangles. [Doctoral Dissertation]. Colorado State University; 2010. Available from: http://hdl.handle.net/10217/40474
Colorado State University
4.
Van Roekel, Luke P.
Influence of variations in penetrating solar radiation on the diurnal and intraseasonal structure of the oceanic boundary layer, The.
Degree: PhD, Atmospheric Science, 2010, Colorado State University
URL: http://hdl.handle.net/10217/40482
► The upper portion of the ocean is fairly well mixed and turbulent. The turbulence within the ocean boundary layer (OBL) is regulated by many mechanisms.…
(more)
▼ The upper portion of the ocean is fairly well mixed and turbulent. The turbulence within the ocean boundary layer (OBL) is regulated by many mechanisms. One process that is receiving a renewed interest is the effect of penetrating component of surface shortwave radiation on ocean dynamics. The influence of solar radiation has been parameterized in two ways. A limited set of models force all the incoming solar radiation to be absorbed in the top model layer. The second parameterization assumes that the irradiance (light) at a given level follows a multiple term exponential. Most commonly it is assumed that shortwave radiation is absorbed in two bands: visible and near infrared. The strength of the infrared absorption is assumed to be fixed. For the visible band, absorption depends on water clarity. Until recently, water clarity could take six different values (Jerlov water types). On climate scales, spatial and temporal variations in water clarity, based on surface chlorophyll, have a strong impact on the simulated ocean temperature, salinity, and momentum. For example, the sea surface temperature (SST) in the cold tongue is reduced. In addition, the strength of the Walker circulation is increased. However, this response is not consistent among different models and parameterizations. When chlorophyll is predicted, the influence of vertically variable water clarity on the thermodynamic and dynamic fields of the ocean can be examined. Studies that have incorporated an ecosystem model find minimal changes relative to using observed surface chlorophyll. Previous research has focused on longer climate time scales and most models do not consider vertical variations in water clarity. In this study the response of the ocean to diurnal and intraseasonal variations of water clarity is examined. The sensitivity to vertical variations in water clarity is also considered. To study the impact of variable solar radiation a model that accurately represents upper ocean physics is required. A new ocean mixing model is proposed that addresses some of the known deficiencies in previous models. The new model predicts entrainment based on turbulence at the OBL base, unlike other ocean models. An over prediction of the vertical heat flux in previous mixed layer models is avoided. The model framework discussed can be easily extended to any coordinate system. Further, this model can be coupled to an ocean biological model, which would determine the water clarity with depth, in a natural way. An evaluation of the new model against observations and a newly developed vector vorticity large eddy simulation (LES) model has shown that the new model preforms as well or better than previous OBL models in certain circumstances. This is especially with low vertical resolution. Since this version of the new model is local, it does not perform as well in pure convective simulations as OBL models with non-local forcing In this new model and K-Profile Parameterization (KPP), the temperature and velocity is very…
Advisors/Committee Members: Randall, David A. (David Allan), 1948- (advisor), Ito, Takamitsu (committee member), Denning, A. Scott (committee member), Kirby, Michael, 1961- (committee member).
Subjects/Keywords: chlorophyll; solar radiation; ocean; model; Solar radiation; Ocean-atmosphere interaction; Chlorophyll; Thermoclines (Oceanography)
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APA (6th Edition):
Van Roekel, L. P. (2010). Influence of variations in penetrating solar radiation on the diurnal and intraseasonal structure of the oceanic boundary layer, The. (Doctoral Dissertation). Colorado State University. Retrieved from http://hdl.handle.net/10217/40482
Chicago Manual of Style (16th Edition):
Van Roekel, Luke P. “Influence of variations in penetrating solar radiation on the diurnal and intraseasonal structure of the oceanic boundary layer, The.” 2010. Doctoral Dissertation, Colorado State University. Accessed March 01, 2021.
http://hdl.handle.net/10217/40482.
MLA Handbook (7th Edition):
Van Roekel, Luke P. “Influence of variations in penetrating solar radiation on the diurnal and intraseasonal structure of the oceanic boundary layer, The.” 2010. Web. 01 Mar 2021.
Vancouver:
Van Roekel LP. Influence of variations in penetrating solar radiation on the diurnal and intraseasonal structure of the oceanic boundary layer, The. [Internet] [Doctoral dissertation]. Colorado State University; 2010. [cited 2021 Mar 01].
Available from: http://hdl.handle.net/10217/40482.
Council of Science Editors:
Van Roekel LP. Influence of variations in penetrating solar radiation on the diurnal and intraseasonal structure of the oceanic boundary layer, The. [Doctoral Dissertation]. Colorado State University; 2010. Available from: http://hdl.handle.net/10217/40482
Colorado State University
5.
Lui, Yui Man.
Geometric methods on special manifolds for visual recognition.
Degree: PhD, Computer Science, 2010, Colorado State University
URL: http://hdl.handle.net/10217/39042
► Many computer vision methods assume that the underlying geometry of images is Euclidean. This assumption is generally not valid. Therefore, this dissertation introduces new nonlinear…
(more)
▼ Many computer vision methods assume that the underlying geometry of images is Euclidean. This assumption is generally not valid. Therefore, this dissertation introduces new nonlinear geometric frameworks based upon special manifolds, namely Graβmann and Stiefel manifolds, for visual recognition. The motivation for this thesis is driven by the intrinsic geometry of visual data in which the visual data can be either a still image or video. Visual data are represented as points in appropriately chosen parameter spaces. The idiosyncratic aspects of the data in these spaces are then exploited for pattern classification. Three major research results are presented in this dissertation: face recognition for illumination spaces on Stiefel manifolds, face recognition on Graβmann registration manifolds, and action classification on product manifolds. Previous work has shown that illumination cones are idiosyncratic for face recognition in illumination spaces. However, it has not been addressed how a single image relates to an illumination cone. In this dissertation, a Bayesian model is employed to relight a single image to a set of illuminated variants. The subspace formed by these illuminated variants is characterized on a Stiefel manifold. A new distance measure called Canonical Stiefel Quotient (CSQ) is introduced. CSQ performs two projections on a tangent space of a Stiefel manifold and uses the quotient for classification. The proposed method demonstrates that illumination cones can be synthesized by relighting a single image to a set of images, and the synthesized illumination cones are discriminative for face recognition. Experiments on the CMU-PIE and YaleB data sets reveal that CSQ not only achieves high recognition accuracies for generic faces but also is robust to the choice of training sets. Subspaces can be realized as points on Graβmann manifolds. Motivated by image perturbation and the geometry of Graβmann manifolds, we present a method called Graβmann Registration Manifolds (GRM) for face recognition. First, a tangent space is formed by a set of affine perturbed images where the tangent space admits a vector space structure. Second, the tangent spaces are embedded on a Graβmann manifold and chordal distance is used to compare subspaces. Experiments on the FERET database suggest that the proposed method yields excellent results using both holistic and local features. Specifically, on the FERET Dup2 data set, which is generally considered the most difficult data set on FERET, the proposed method achieves the highest rank one identification rate among all non-trained methods currently in the literature. Human actions compose a series of movements and can be described by a sequence of video frames. Since videos are multidimensional data, data tensors are the natural choice for data representation. In this dissertation, a data tensor is expressed as a point on a product manifold and classification is performed on this product space. First, we factorize a data tensor using a modified High Order Singular Value…
Advisors/Committee Members: Beveridge, J. Ross (advisor), Kirby, Michael, 1961- (committee member), Draper, Bruce A. (Bruce Austin), 1962- (committee member), Whitley, L. Darrell (committee member).
Subjects/Keywords: action classification; visual recognition; special manifolds; geometric methods; face recognition; Human face recognition (Computer science); Grassmann manifolds; Stiefel manifolds
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APA ·
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APA (6th Edition):
Lui, Y. M. (2010). Geometric methods on special manifolds for visual recognition. (Doctoral Dissertation). Colorado State University. Retrieved from http://hdl.handle.net/10217/39042
Chicago Manual of Style (16th Edition):
Lui, Yui Man. “Geometric methods on special manifolds for visual recognition.” 2010. Doctoral Dissertation, Colorado State University. Accessed March 01, 2021.
http://hdl.handle.net/10217/39042.
MLA Handbook (7th Edition):
Lui, Yui Man. “Geometric methods on special manifolds for visual recognition.” 2010. Web. 01 Mar 2021.
Vancouver:
Lui YM. Geometric methods on special manifolds for visual recognition. [Internet] [Doctoral dissertation]. Colorado State University; 2010. [cited 2021 Mar 01].
Available from: http://hdl.handle.net/10217/39042.
Council of Science Editors:
Lui YM. Geometric methods on special manifolds for visual recognition. [Doctoral Dissertation]. Colorado State University; 2010. Available from: http://hdl.handle.net/10217/39042
Colorado State University
6.
Bush, Keith A.
Echo state model of non-Markovian reinforcement learning, An.
Degree: PhD, Computer Science, 2008, Colorado State University
URL: http://hdl.handle.net/10217/28682
► There exists a growing need for intelligent, autonomous control strategies that operate in real-world domains. Theoretically the state-action space must exhibit the Markov property in…
(more)
▼ There exists a growing need for intelligent, autonomous control strategies that operate in real-world domains. Theoretically the
state-action space must exhibit the Markov property in order for reinforcement learning to be applicable. Empirical evidence, however, suggests that reinforcement learning also applies to domains where the
state-action space is approximately Markovian, a requirement for the overwhelming majority of real-world domains. These domains, termed non-Markovian reinforcement learning domains, raise a unique set of practical challenges. The reconstruction dimension required to approximate a Markovian
state-space is unknown a priori and can potentially be large. Further, spatial complexity of local function approximation of the reinforcement learning domain grows exponentially with the reconstruction dimension. Parameterized dynamic systems alleviate both embedding length and
state-space dimensionality concerns by reconstructing an approximate Markovian
state-space via a compact, recurrent representation. Yet this representation extracts a cost; modeling reinforcement learning domains via adaptive, parameterized dynamic systems is characterized by instability, slow-convergence, and high computational or spatial training complexity. The objectives of this research are to demonstrate a stable, convergent, accurate, and scalable model of non-Markovian reinforcement learning domains. These objectives are fulfilled via fixed point analysis of the dynamics underlying the reinforcement learning domain and the Echo
State Network, a class of parameterized dynamic system. Understanding models of non-Markovian reinforcement learning domains requires understanding the interactions between learning domains and their models. Fixed point analysis of the Mountain Car Problem reinforcement learning domain, for both local and nonlocal function approximations, suggests a close relationship between the locality of the approximation and the number and severity of bifurcations of the fixed point structure. This research suggests the likely cause of this relationship: reinforcement learning domains exist within a dynamic feature space in which trajectories are analogous to states. The fixed point structure maps dynamic space onto
state-space. This explanation suggests two testable hypotheses. Reinforcement learning is sensitive to
state-space locality because states cluster as trajectories in time rather than space. Second, models using trajectory-based features should exhibit good modeling performance and few changes in fixed point structure. Analysis of performance of lookup table, feedforward neural network, and Echo
State Network (ESN) on the Mountain Car Problem reinforcement learning domain confirm these hypotheses. The ESN is a large, sparse, randomly-generated, unadapted recurrent neural network, which adapts a linear projection of the target domain onto the hidden layer. ESN modeling results on reinforcement learning domains show it achieves performance comparable to lookup table and neural network architectures…
Advisors/Committee Members: Anderson, Charles W. (advisor), Draper, Bruce A. (Bruce Austin), 1962- (committee member), Kirby, Michael, 1961- (committee member), Young, Peter M. (committee member).
Subjects/Keywords: reinforcement learning (machine learning); mountain car problem; reinforcement learning; Markovian; echo state network; ESN; fixed point analysis; Hybrid systems
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APA ·
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APA (6th Edition):
Bush, K. A. (2008). Echo state model of non-Markovian reinforcement learning, An. (Doctoral Dissertation). Colorado State University. Retrieved from http://hdl.handle.net/10217/28682
Chicago Manual of Style (16th Edition):
Bush, Keith A. “Echo state model of non-Markovian reinforcement learning, An.” 2008. Doctoral Dissertation, Colorado State University. Accessed March 01, 2021.
http://hdl.handle.net/10217/28682.
MLA Handbook (7th Edition):
Bush, Keith A. “Echo state model of non-Markovian reinforcement learning, An.” 2008. Web. 01 Mar 2021.
Vancouver:
Bush KA. Echo state model of non-Markovian reinforcement learning, An. [Internet] [Doctoral dissertation]. Colorado State University; 2008. [cited 2021 Mar 01].
Available from: http://hdl.handle.net/10217/28682.
Council of Science Editors:
Bush KA. Echo state model of non-Markovian reinforcement learning, An. [Doctoral Dissertation]. Colorado State University; 2008. Available from: http://hdl.handle.net/10217/28682
Colorado State University
7.
Smith, Elin Rose.
Algorithms and geometric analysis of data sets that are invariant under a group action.
Degree: PhD, Mathematics, 2010, Colorado State University
URL: http://hdl.handle.net/10217/44768
► We apply and develop pattern analysis techniques in the setting of data sets that are invariant under a group action. We apply Principal Component Analysis…
(more)
▼ We apply and develop pattern analysis techniques in the setting of data sets that are invariant under a group action. We apply Principal Component Analysis to data sets of images of a rotating object in Chapter 5 as a means of obtaining visual and low-dimensional representations of data. In Chapter 6, we propose an algorithm for finding distributions of points in a base space that are (locally) optimal in the sense that subspaces in the associated data bundle are distributed with locally maximal distance between neighbors. In Chapter 7, we define a distortion function that measures the quality of an approximation of a vector bundle by a set of points. We then use this function to compare the behavior of four standard distance metrics and one non-metric. Finally, in Chapter 8, we develop an algorithm to find the approximate intersection of two data sets.
Advisors/Committee Members: Peterson, Christopher Scott, 1963- (advisor), Bates, Daniel J. (Daniel James), 1979- (committee member), Kirby, Michael, 1961- (committee member), McConnell, Ross M. (committee member).
Subjects/Keywords: principal component analysis; pattern analysis; minimal energy configuration; image analysis; group actions; data bundle; Geometric group theory; Geometric analysis; Invariant measures; Cluster analysis; Pattern perception
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APA ·
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APA (6th Edition):
Smith, E. R. (2010). Algorithms and geometric analysis of data sets that are invariant under a group action. (Doctoral Dissertation). Colorado State University. Retrieved from http://hdl.handle.net/10217/44768
Chicago Manual of Style (16th Edition):
Smith, Elin Rose. “Algorithms and geometric analysis of data sets that are invariant under a group action.” 2010. Doctoral Dissertation, Colorado State University. Accessed March 01, 2021.
http://hdl.handle.net/10217/44768.
MLA Handbook (7th Edition):
Smith, Elin Rose. “Algorithms and geometric analysis of data sets that are invariant under a group action.” 2010. Web. 01 Mar 2021.
Vancouver:
Smith ER. Algorithms and geometric analysis of data sets that are invariant under a group action. [Internet] [Doctoral dissertation]. Colorado State University; 2010. [cited 2021 Mar 01].
Available from: http://hdl.handle.net/10217/44768.
Council of Science Editors:
Smith ER. Algorithms and geometric analysis of data sets that are invariant under a group action. [Doctoral Dissertation]. Colorado State University; 2010. Available from: http://hdl.handle.net/10217/44768
. 
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