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Application of Topological Data Analysis to Multi-Resolution Matching and Anomaly Detection
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Ulysses Data Analysis: Magnetic Topology of Heliospheric Structures
In this final technical report on research funded by a NASA grant, a project overview is given by way of summaries on nine published papers. Research has included: 1) Using suprathermal electron data to study heliospheric magnetic structures; 2) Analysis of magnetic clouds, coronal mass ejections (CME), and the heliospheric current sheet (HCS); 3) Analysis of the corotating interaction region (CIR) which develop from interactions between solar wind streams of different velocities; 4) Use of Ulysses data in the interpretation of heliospheric events and phenomena.
A Survey of Mathematical Structures for Lunar Networks
To sustain the current and increasing accessibility of space, a scalable communications infrastructure (i.e. the Solar System Internet, SSI) is necessary. The goal of this paper is to begin the discovery of the fundamental underlying mathematical structure of space networks to help the research community harness these structures for algorithm development and optimization. To ensure the applicability of the research, the approaches are considered through the lens of simulated scenarios inspired by the Artemis Back-to-the-Moon mission set for 2024. We note that any approach to an SSI must fit under the umbrella of Delay Tolerant Networking (DTN), due to celestial mobility, high link latencies, high variance in link latencies, disconnections, lack of end-to-end paths, and so on. These difficulties are exacerbated by the fact that the underlying structure of a space network is a time-evolving network and may experience multiple discontinuities in its topology. In this paper we propose several novel approaches to a mathematical foundation for Delay Tolerant Networking Theory that fall outside the traditional scope of temporal network theory. These techniques include methods from Topological Data Analysis, Dynamic Graph Analysis, Applied Algebraic Geometry, Probability Theory, and Game Theory. Some of these methods include tools adapted to the study of dynamic metric spaces, such as zigzag persistent homology and their higher parameter analogs. We find that several of these methods target desired engineering outcomes such as discovery and automatic sub-netting. While each approach is theoretical, they are also algorithmic in nature and offer immediate practical applications. The paper concludes with comparisons of the various methods along with suggestions for future work.
Dude Where's My Stars: A Novel Topologically Justified Approach to Star Tracking
In this paper, we consider two novel approaches to celestial navigation for spacecraft. Determining attitude without any prior knowledge using star tracking is known to be a difficult task, particularly given the computational complexity and the many potential sources of misinformation. We consider localization by optimizing matching parameters without explicit star identification in a computationally tractable manner. This is achieved using the mathematical tools of topological data analysis (TDA) and cellular sheaves to study the geometry and distribution of cataloged stars. A framework is gained that enhances the statistical approach to noise handling and false star detection, and heterogeneous sensor fusion. Finally, we discuss confidence bounds and minimum information requirements for successful operation.
Assuring and Securing Machine Learning
A short presentation highlighting using machine learning and topological data analysis to address the challenges of assuring and securing machine learning enabled systems.
Tuning the Performance of a Computational Persistent Homology Package
In recent years, persistent homology has become an attractive method for data analysis. It captures topological features, such as connected components, holes, and voids from point cloud data and summarizes the way in which these features appear and disappear in a filtration sequence. In this project, we focus on improving the performanceof Eirene, a computational package for persistent homology. Eirene is a 5000-line open-source software library implemented in the dynamic programming language Julia. We use the Julia profiling tools to identify performance bottlenecks and develop novel methods to manage them, including the parallelization of some time-consuming functions on multicore/manycore hardware. Empirical results show that performance can be greatly improved.
Performance Enhancement of a Computational Persistent Homology Package
In recent years, persistent homology has become an attractive method for data analysis. It captures topological features, such as connected components, holes, voids, etc., from a point cloud by finding out when these features appear and disappear in the filtration sequence. In this project, we focus on improving the performance of Eirene, a fancy computational persistent homology package. Eirene is a 5000-line opensource software implemented by using the dynamic programming language Julia. We use the Julia profiling tools to identify the performance bottlenecks and develop different methods to manage the bottlenecks, including the parallelization of some time-consuming functions on the multicore/manycore hardware. The empirical results show that the performance can be greatly improved.
Analysis and representation of complex structures in separated flows
We discuss our recent work on extraction and visualization of topological information in separated fluid flow data sets. As with scene analysis, an abstract representation of a large data set can greatly facilitate the understanding of complex, high-level structures. When studying flow topology, such a representation can be produced by locating and characterizing critical points in the velocity field and generating the associated stream surfaces. In 3D flows, the surface topology serves as the starting point. The 2D tangential velocity field near the surface of the body is examined for critical points. The tangential velocity field is integrated out along the principal directions of certain classes of critical points to produce curves depicting the topology of the flow near the body. The points and curves are linked to form a skeleton representing the 2D vector field topology. This skeleton provides a basis for analyzing the 3D structures associated with the flow separation. The points along the separation curves in the skeleton are used to start tangent curve integrations. Integration origins are successively refined to produce stream surfaces. The map of the global topology is completed by generating those stream surfaces associated with 3D critical points.
Topological grid structure - A data structure for earth science modeling
The automated analysis of land surface features is increasingly important to earth scientists. User-friendly algorithms for studying these features can be integrated into geographic information systems through the use of topological grid structure, which maintains the simplicity and transportability of standard grid structure while providing the essential capability to treat groups of contiguous, identically-classified pixels (corresponding to lakes, forests, fields, etc.) as distinct spatial entities.
Automated analysis for fluid flow topology
A new approach for visualizing vector data sets was developed by reducing the original vector field to a set of critical points and their connections, and was applied to fluid flow data sets. The critical point representation allows for considerable reduction in the data complexity. The representations are displayed as surfaces which are much simpler than the original data set, yet retain all the pertinent flow topology information. It is suggested that topological representations may be useful for database comparison.
Sneak circuit analysis of instrumentation and control systems
The evolvement and implementation is presented of a systematic means for detecting the sneak prone designs of an electrical system which could result in such unplanned modes so that they can be prevented prior to occurrence. Sneak circuit analysis is shown to be a formalized approach based upon topological techniques. The trees employed are produced from manufacturing detail data processed by a computer to orderly completion. The analysis techniques applied on a variety of NASA programs are shown to be particularly applicable to industrial instrumentation and control systems.
Coarse-Grain Bandwidth Estimation Scheme for Large-Scale Network
A large-scale network that supports a large number of users can have an aggregate data rate of hundreds of Mbps at any time. High-fidelity simulation of a large-scale network might be too complicated and memory-intensive for typical commercial-off-the-shelf (COTS) tools. Unlike a large commercial wide-area-network (WAN) that shares diverse network resources among diverse users and has a complex topology that requires routing mechanism and flow control, the ground communication links of a space network operate under the assumption of a guaranteed dedicated bandwidth allocation between specific sparse endpoints in a star-like topology. This work solved the network design problem of estimating the bandwidths of a ground network architecture option that offer different service classes to meet the latency requirements of different user data types. In this work, a top-down analysis and simulation approach was created to size the bandwidths of a store-and-forward network for a given network topology, a mission traffic scenario, and a set of data types with different latency requirements. These techniques were used to estimate the WAN bandwidths of the ground links for different architecture options of the proposed Integrated Space Communication and Navigation (SCaN) Network. A new analytical approach, called the "leveling scheme," was developed to model the store-and-forward mechanism of the network data flow. The term "leveling" refers to the spreading of data across a longer time horizon without violating the corresponding latency requirement of the data type. Two versions of the leveling scheme were developed: 1. A straightforward version that simply spreads the data of each data type across the time horizon and doesn't take into account the interactions among data types within a pass, or between data types across overlapping passes at a network node, and is inherently sub-optimal. 2. Two-state Markov leveling scheme that takes into account the second order behavior of the store-and-forward mechanism, and the interactions among data types within a pass. The novelty of this approach lies in the modeling of the store-and-forward mechanism of each network node. The term store-and-forward refers to the data traffic regulation technique in which data is sent to an intermediate network node where they are temporarily stored and sent at a later time to the destination node or to another intermediate node. Store-and-forward can be applied to both space-based networks that have intermittent connectivity, and ground-based networks with deterministic connectivity. For groundbased networks, the store-and-forward mechanism is used to regulate the network data flow and link resource utilization such that the user data types can be delivered to their destination nodes without violating their respective latency requirements.
Further analysis of a recent cosmic-ray antiproton experiment
Reference is made to the measurements of a cosmic ray antiproton flux at a few hundred MeV reported by Buffington et al. (1981), noting that one of the final background processes to be removed by the data analysis in that study was helium-induced events which satisfied the criteria for topology and timing. The response in the third scintillator S3 was used to identify and remove these events. For the top two scintillators S1 and S2, pulse size information was lost during the data-taking. A method is reported here for the partial retrieval of pulse size information for the scintillator S2. This is possible because a portion of this signal was subtracted from the Cerenkov response before trigger discrimination and data recording to remove scintillation from the Cerenkov response. For separating protons from more highly charged particles, the method is considered sufficient. It is pointed out that the sample of events identified as antiprotons, for which the method can be applied, has the expected unit charge in scintillator S2.
Visualization of second order tensor fields and matrix data
We present a study of the visualization of 3-D second order tensor fields and matrix data. The general problem of visualizing unsymmetric real or complex Hermitian second order tensor fields can be reduced to the simultaneous visualization of a real and symmetric second order tensor field and a real vector field. As opposed to the discrete iconic techniques commonly used in multivariate data visualization, the emphasis is on exploiting the mathematical properties of tensor fields in order to facilitate their visualization and to produce a continuous representation of the data. We focus on interactively sensing and exploring real and symmetric second order tensor data by generalizing the vector notion of streamline to the tensor concept of hyperstreamline. We stress the importance of a structural analysis of the data field analogous to the techniques of vector field topology extraction in order to obtain a unique and objective representation of second order tensor fields.
Surface representations of two- and three-dimensional fluid flow topology
We discuss our work using critical point analysis to generate representations of the vector field topology of numerical flow data sets. Critical points are located and characterized in a two-dimensional domain, which may be either a two-dimensional flow field or the tangential velocity field near a three-dimensional body. Tangent curves are then integrated out along the principal directions of certain classes of critical points. The points and curves are linked to form a skeleton representing the two-dimensional vector field topology. When generated from the tangential velocity field near a body in a three-dimensional flow, the skeleton includes the critical points and curves which provide a basis for analyzing the three-dimensional structure of the flow separation. The points along the separation curves in the skeleton are used to start tangent curve integrations to generate surfaces representing the topology of the associated flow separations.
Homology and the optimization of DNA sequence data
Three methods of nucleotide character analysis are discussed. Their implications for molecular sequence homology and phylogenetic analysis are compared. The criterion of inter-data set congruence, both character based and topological, are applied to two data sets to elucidate and potentially discriminate among these parsimony-based ideas. c2001 The Willi Hennig Society.
Topological analysis of chaotic orbits: Revisiting Hyperion
There is emerging interest in the possibility of chaotic evolution in astrophysical systems. To mention just one example, recent well-sampled ground-based observations of the Saturian satellite Hyperion strongly suggest that it is exhibiting chaotic behavior. We present a general technique, the method of close returns, for the analysis of data from astronomical objects believed to be exhibiting chaotic motion. The method is based on the extraction of pieces of the evolution that exhibit nearly periodic behavior-episodes during which the object stays near in phase space to some unstable periodic orbit. Such orbits generally act as skeletal features, tracing the topological organization of the manifold on which the chaotic dynamics takes place. This method does not require data sets as lengthy as other nonlinear analysis techniques do and is therefore well suited to many astronomical observing programs. Well sampled data covering between twenty and forty characteristic periods of the system have been found to be sufficient for the application of this technique. Additional strengths of this method are its robustness in the presence of noise and the ability for a user to clearly distinguish between periodic, random, and chaotic behavior by inspection of the resulting two-dimensional image. As an example of its power, we analyze close returns in a numerically generated data set, based on a model for Hyperion extensively studied in the literature, corresponding to nightly observations of the satellite. We show that with a small data set, embedded unstable periodic orbits can be extracted and that these orbits can be responsible for nearly periodic behavior lasting a substantial fraction of the observing run.