Projects in Time Series Analysis: Chaos Detection with Persistent Homology & Process Monitoring
Abstract: Recently, new methods have emerged for systems with both known and unknown models to produce a definitive 0/1 diagnostic for chaos in time-series data. However, there still lacks a method which can reliably perform an evaluation for noisy time-series with no known model. In this work, we present a new chaos detection method which utilizes Persistent Homology, a tool from topological data analysis. Bi-variate density estimates of the randomly projected time-series in the p-q plane described in Gottwald and Melbourne's approach for 0/1 detection are used to generate a gray-scale image. We show that Wasserstein distances corresponding to the 1-D sub-level set persistence of the images can elucidate whether or not the underlying time series is chaotic. Case studies on the Lorenz and Rossler attractors are used to validate this claim. Similar to the original 0/1 test, our approach is unable to distinguish partially predicable chaotic and periodic behavior in the p-q space. However, although our method does not deliver a binary 0/1 result, we show that it is able to identify the shift points between chaotic and periodic dynamics in time-series even at high noise-levels.
Additionally, we present time-series analysis methods for signal classification with the application of in-situ monitoring of selective laser sintering (SLS). In-stiu monitoring of additively manufactured (AM) parts has become a topic of increasing interest to the manufacturing community, because defects in AM parts are generated as the part is built. Ensemble empirical mode decomposition (EEMD), singular spectrum analysis (SSA), and statistical measures of the time series partitions were used to extract feature vectors which correlate to pore formation. Sequential feature selection reduces the feature space and Support Vector Machine models classify the data successfully.
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