Topological Data Analysis for Clustering and Classifying Time Series

Topological Data Analysis (TDA), which refers to methods of utilizing topological features in data (such as connected components, tunnels, voids, etc.) has gained considerable momentum. More recently, TDA has been increasingly used for analyzing time series. The recent researches of using TDA in time series are mainly focused on unsupervised and supervised learning. In this thesis, a review of TDA will be provided, following by using TDA in time series for doing unsupervised and supervised learning for different applications. Topological Data Analysis for Clustering and Classifying Time Series

[1]  Jingyue Zhang,et al.  Novel Divide and Combine-Based Approach to Estimating Mixture Markov Model for Large Categorical Time Series Data: Application to Study of Clusters using Multiyear Travel Survey Data , 2019 .

[2]  Gunnar E. Carlsson,et al.  Topology and data , 2009 .

[3]  Peter Bubenik,et al.  Statistical topological data analysis using persistence landscapes , 2012, J. Mach. Learn. Res..

[4]  Mikael Vejdemo-Johansson,et al.  Automatic recognition and tagging of topologically different regimes in dynamical systems , 2013, ArXiv.

[5]  Mei-Po Kwan,et al.  Interactive geovisualization of activity-travel patterns using three-dimensional geographical information systems: a methodological exploration with a large data set , 2000 .

[6]  Konstadinos G. Goulias,et al.  LONGITUDINAL ANALYSIS OF ACTIVITY AND TRAVEL PATTERN DYNAMICS USING GENERALIZED MIXED MARKOV LATENT CLASS MODELS , 1999 .

[7]  Jason Lines,et al.  Time-Series Classification with COTE: The Collective of Transformation-Based Ensembles , 2015, IEEE Transactions on Knowledge and Data Engineering.

[8]  Firas A. Khasawneh,et al.  Chatter detection in turning using persistent homology , 2016 .

[9]  Steve Oudot,et al.  Persistence stability for geometric complexes , 2012, ArXiv.

[10]  M. Kubát.  An Introduction to Machine Learning , 2017, Springer International Publishing.

[11]  Leonidas J. Guibas,et al.  Persistence barcodes for shapes , 2004, SGP '04.

[12]  Kenji Fukumizu,et al.  Persistence weighted Gaussian kernel for topological data analysis , 2016, ICML.

[13]  G. Carlsson,et al.  Topology based data analysis identifies a subgroup of breast cancers with a unique mutational profile and excellent survival , 2011, Proceedings of the National Academy of Sciences.

[14]  Steve Oudot,et al.  Sliced Wasserstein Kernel for Persistence Diagrams , 2017, ICML.

[15]  Steve Oudot,et al.  Persistence-Based Pooling for Shape Pose Recognition , 2016, CTIC.

[16]  Eric R. Ziegel,et al.  The Elements of Statistical Learning , 2003, Technometrics.

[17]  Frédéric Chazal,et al.  Stochastic Convergence of Persistence Landscapes and Silhouettes , 2013, J. Comput. Geom..

[18]  Afra Zomorodian,et al.  The Theory of Multidimensional Persistence , 2007, SCG '07.

[19]  Carlo Ratti,et al.  Understanding individual mobility patterns from urban sensing data: A mobile phone trace example , 2013 .

[20]  Marian Gidea,et al.  Topology Data Analysis of Critical Transitions in Financial Networks , 2017, 1701.06081.

[21]  Sivaraman Balakrishnan,et al.  Confidence sets for persistence diagrams , 2013, The Annals of Statistics.

[22]  Elke A. Rundensteiner,et al.  Generalized Dynamic Time Warping: Unleashing the Warping Power Hidden in Point-Wise Distances , 2018, 2018 IEEE 34th International Conference on Data Engineering (ICDE).

[23]  Claire Brécheteau,et al.  A statistical test of isomorphism between metric-measure spaces using the distance-to-a-measure signature , 2019, Electronic Journal of Statistics.

[24]  Paul S. Bradley,et al.  Refining Initial Points for K-Means Clustering , 1998, ICML.

[25]  Patrick Truong.  An exploration of topological properties of high-frequency one-dimensional financial time series data using TDA , 2017 .

[26]  X. Liu,et al.  A fast algorithm for constructing topological structure in large data , 2012 .

[27]  Laura J. Grundy,et al.  A dictionary of behavioral motifs reveals clusters of genes affecting Caenorhabditis elegans locomotion , 2012, Proceedings of the National Academy of Sciences.

[28]  J. Climent,et al.  Applications of computational homology to the analysis of treatment response in breast cancer patients , 2010 .

[29]  R. Ho.  Algebraic Topology , 2022 .

[30]  P. Rousseeuw.  Silhouettes: a graphical aid to the interpretation and validation of cluster analysis , 1987 .

[31]  Herbert Edelsbrunner,et al.  Alpha, Betti and the Megaparsec Universe: On the Topology of the Cosmic Web , 2013, Trans. Comput. Sci..

[32]  David S. Stoffer,et al.  Spectral analysis for categorical time series: Scaling and the spectral envelope , 1993 .

[33]  Henry Adams,et al.  Evasion paths in mobile sensor networks , 2013, Int. J. Robotics Res..

[34]  Henry Adams,et al.  Persistence Images: A Stable Vector Representation of Persistent Homology , 2015, J. Mach. Learn. Res..

[35]  Chengyuan Wu,et al.  Weighted persistent homology , 2018, Rocky Mountain Journal of Mathematics.

[36]  G. Madey,et al.  Uncovering individual and collective human dynamics from mobile phone records , 2007, 0710.2939.

[37]  Clarke Wilson,et al.  Activity Patterns of Canadian Women: Application of ClustalG Sequence Alignment Software , 2001 .

[38]  Billur Barshan,et al.  Comparative study on classifying human activities with miniature inertial and magnetic sensors , 2010, Pattern Recognit..

[39]  Gunnar E. Carlsson,et al.  Topological pattern recognition for point cloud data* , 2014, Acta Numerica.

[40]  Jouni Helske,et al.  Mixture Hidden Markov Models for Sequence Data: The seqHMM Package in R , 2017, Journal of Statistical Software.

[41]  Nick S. Jones,et al.  Highly Comparative Feature-Based Time-Series Classification , 2014, IEEE Transactions on Knowledge and Data Engineering.

[42]  N. Huang,et al.  The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis , 1998, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[43]  Germain Forestier,et al.  Deep learning for time series classification: a review , 2018, Data Mining and Knowledge Discovery.

[44]  Jose A. Perea,et al.  Sliding Windows and Persistence: An Application of Topological Methods to Signal Analysis , 2013, Found. Comput. Math..

[45]  Robert H. Shumway,et al.  Time Series Analysis and Its Applications (Springer Texts in Statistics) , 2005 .

[46]  Ben D. Fulcher,et al.  Feature-based time-series analysis , 2017, ArXiv.

[47]  Steve Oudot,et al.  Eurographics Symposium on Geometry Processing 2015 Stable Topological Signatures for Points on 3d Shapes , 2022 .