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Applied Computational Topology for Point Clouds and Sparse Timeseries Data

Citation

Yeung, Melissa L. (2017) Applied Computational Topology for Point Clouds and Sparse Timeseries Data. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/Z9D798DF. http://resolver.caltech.edu/CaltechTHESIS:10262016-165619613

Abstract

The proliferation of sensors and advancement of technology has led to the production and collection of unprecedented amounts of data in recent years. The data are often noisy, non-linear, and high-dimensional, and the effectiveness of traditional tools may be limited. Thus, the technological advances that enable the ubiquitous collection of data from the cosmological scale to the subatomic scale also necessitate the development of complementary tools that address the new nature of the data.

Recently, there has been much interest in and success with developing topologically-motivated techniques for data analysis. These approaches are especially useful when a topological method is sensitive to large- and small-scale features that might not be detected by methods that require a level of geometric detail that is not provided by the data or by methods that may obscure geometric features, such as principal component analysis (PCA), multi–dimensional scaling (MDS), and cluster analysis.

Our work explores topological data analysis through two frameworks.

In the first part, we provide a tool for detecting material coherence from a set of spatially sparse particle trajectories via the study of a map induced on homology by the braid corresponding to the motion of particles. While the theory of coherent structures has received a great deal of attention and benefited from many advances in recent years, many of these techniques are limited when the data are sparse. We demonstrate through various examples that our work provides a practical and scalable tool for identifying coherent sets from a sparse set of particle trajectories using eigenanalysis.

In the second part, we formalize the local-to-global structure captured by topology in the setting of point clouds. We extend existing tools in topological data analysis and provide a theoretical framework for studying topological features of a point cloud over a range of resolutions, enabling the analysis of topological features using statistical methods. We apply our tools to the analysis of high-dimensional geospatial sensor data and provide a statistic for quantifying climate anomalies.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:mathematics ; topological data analysis
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Thesis Availability:Withheld
Research Advisor(s):
  • Desbrun, Mathieu
Thesis Committee:
  • Desbrun, Mathieu (chair)
  • Makarov, Nikolai G.
  • Marcolli, Matilde
  • Ni, Yi
Defense Date:8 June 2016
Funders:
Funding AgencyGrant Number
National Science Foundation Graduate Research FellowshipUNSPECIFIED
DOE CSGF (Department of Energy Computational Science Graduate Fellowship)DE-FG02-97ER25308
Caltech Department of Computing & Mathematical SciencesUNSPECIFIED
Caltech Department of MathematicsUNSPECIFIED
Lawrence Berkeley National LaboratoryUNSPECIFIED
Inria Sophia AntipolisUNSPECIFIED
Record Number:CaltechTHESIS:10262016-165619613
Persistent URL:http://resolver.caltech.edu/CaltechTHESIS:10262016-165619613
DOI:10.7907/Z9D798DF
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:9960
Collection:CaltechTHESIS
Deposited By: Melissa Yeung
Deposited On:09 Mar 2017 23:38
Last Modified:09 Mar 2017 23:38

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