Browsing by Subject "Topological data analysis"

This work introduces geometric and topological data analysis (TDA) tools that can be used in conjunction with sliding window transformations, also known as delay-embeddings, for discovering structure in time series and dynamical systems in an unsupervised or supervised learning framework. For signals of unknown period, we introduce an intuitive topological method to discover the period, and we demonstrate its use in synthetic examples and real temperature data. Alternatively, for almost-periodic signals of known period, we introduce a metric called Geometric Complexity of an Almost Periodic signal (GCAP), based on a topological construction, which allows us to continuously measure the evolving variation of its periods. We apply this method to temperature data collected from over 200 weather stations in the United States and describe the novel patterns that we observe. Next, we show how geometric and TDA tools can be used in a supervised learning framework. Seizure-detection using electroencephalogram (EEG) data is formulated as a binary classification problem. We define new collections of geometric and topological features of multi-channel data, which utilizes temporal and spatial context of EEG, and show how it results in better overall performance of seizure detection than using the usual time-domain and frequency domain features. Finally, we introduce a novel method to sonify persistence diagrams, and more generally any planar point cloud, using a modified version of the harmonic table. This auditory display can be useful for finding patterns that visual analysis alone may miss.

This thesis provides a new take on problems in multimedia times series analysis by using a shape-based perspective to quantify patterns in time, which is complementary to more traditional analysis-based time series techniques. Inspired by the dynamical systems community, we turn time series into shapes via sliding window embeddings, which we refer to as ``time-ordered point clouds'' (TOPCs). This framework has traditionally been used on a single 1D observation function for deterministic systems, but we generalize the sliding window technique so that it not only applies to multivariate data (e.g. videos), but that it also applies to data which is not stationary (e.g. music).

The geometry of our time-ordered point clouds can be quite informative. For periodic signals, the point clouds fill out topological loops, which, depending on harmonic content, reside on various high dimensional tori. For quasiperiodic signals, the point clouds are dense on a torus. We use modern tools from topological data analysis (TDA) to quantify degrees of periodicity and quasiperiodicity by looking at these shapes, and we show that this can be used to detect anomalies in videos of vibrating vocal folds. In the case of videos, this has the advantage of substantially reducing the amount of preprocessing, as no motion tracking is needed, and the technique operates on raw pixels. This is also one of the first known uses of persistent H2 in a high dimensional setting.

Periodic processes represent only a sliver of possible dynamics, and we also show that sequences of arbitrary normalized sliding window point clouds are approximately isometric between ``cover songs,'' or different versions of the same song, possibly with radically different spectral content. Surprisingly, in this application, an incredibly simple geometric descriptor based on self-similarity matrices performs the best, and it also enables us to use MFCC features for this task, which was previously thought not to be possible due to significant timbral differences that can exist between versions. When combined with traditional pitch-based features using similarity metric fusion, we obtain state of the art results on automatic cover song identification.

In addition to being used as a geometric descriptor, self-similarity matrices provide a unifying description of phenomena in time-ordered point clouds throughout our work, and we use them to illustrate properties such as recurrence, mirror symmetry in time, and harmonics in periodic processes. They also provide the base representation for designing isometry blind time warping algorithms, which we use to synchronize time-ordered point clouds that are shifted versions of each other in space without ever having to do a spatial alignment. In particular, we devise an algorithm that lower bounds the 1-stress between two time-ordered point clouds, which is related to the Gromov-Hausdorff distance.

Overall, we show a proof-of-concept and promise of the nascent field of geometric signal processing, which is worthy of further study in applications of music structure, multimodal data analysis, and video analysis.

This dissertation is about building fundamental techniques for comparing data via a geometric and topological data analysis method called multiparameter persistent homology. The techniques used are largely algebraic. A new summary statistic, called the multirank function, is introduced as a measure of persistence output that detects relationships between important features of the data being analyzed. Also introduced is a technique for modifying existing metrics on the space of persistence outputs. Existing metrics can return infinite distances, which do not give as much information as a finite distance; the proposed modification gives fewer such situations. The final chapter of this dissertation details work in a long-term biology research project. Persistence is used to study the relationship between continuous morphological variation and rates of topologically abnormal morphologies in populations of fruit flies. Some preliminary computations showing proof of concept are included. Future plans involve using theoretical contributions from this dissertation for final analysis of the fly data.

The distance modification is joint work with Ezra Miller and the biology application is joint with Surabhi Beriwal, Ezra Miller, and biologists at the Houle Lab at Florida State University.