Sliding Windows and Persistence: An Application of Topological Methods to Signal Analysis

Foundations of Computational Mathematics

Volumn 15, Issue 3, 2015, Pages 799-838

  Perea, Jose A ^a   Harer, John ^b  

^a DUKE UNIVERSITY   (United States)

^b Duke University   (United States)

Author keywords

Periodicity; Persistent homology; Time delay embeddings

Indexed keywords

COMMUNICATION CHANNELS (INFORMATION THEORY); DECODING; GENE EXPRESSION; TIME DELAY;

EMBEDDING DIMENSIONS; EMBEDDINGS; GENE EXPRESSION ANALYSIS; PERIODICITY; PERSISTENT HOMOLOGY; STATE-OF-THE-ART METHODS; THEORETICAL FRAMEWORK; TOPOLOGICAL PROPERTIES;

TOPOLOGY;

EID: 84929951586     PISSN: 16153375     EISSN: 16153383     Source Type: Journal    
DOI: 10.1007/s10208-014-9206-z     Document Type: Article

References (30)

1. A. Adcock, E. Carlsson and G. Carlsson, The Ring of Algebraic Functions on Persistence Bar Codes, Preprint http://comptop.stanford.edu/u/preprints/multitwo, 2012
2. A. J. Blumberg, I. Gal, M. A. Mandell and M. Pancia, Persistent homology for metric measure spaces, and robust statistics for hypothesis testing and confidence intervals, arXiv preprint http://arxiv.org/pdf/1206.4581, 2012
3. G. Carlsson, Topology and Data, Bulletin of the American Mathematical Society, vol 46(2), pp. 255–308, 2009.
4. G. Carlsson, T. Ishkhanov, V. de Silva and A. Zomorodian, On the local behavior of spaces of natural images, International Journal of Computer Vision, vol 7(1), pp. 1–12, 2008.
5. F. Chazal, D. Cohen-Steiner, M. Glisse, L. J. Guibas, and S. Y. Oudot, Proximity of persistence modules and their diagrams, In SCG, pp. 237–246, 2009.
6. D. Cohen-Steiner, H. Edelsbrunner and J. Harer, Stability of persistence diagrams, Discrete and Computational Geometry, vol 37(1), pp. 103–120, 2007.
7. D. Cohen-Steiner, H. Edelsbrunner, J. Harer and Y. Mileyko, Lipschitz Functions Have $$L^p$$Lp-Stable Persistence, Foundations of Computational Mathematics, vol 10(2), pp. 127–139, 2010.
8. D. Comaniciu and P. Meer, Mean shift: A robust approach toward feature space analysis, Pattern Analysis and Machine Intelligence, vol 24(5), pp. 603–619, 2002.
9. V. de Silva, D. Morozov and M. Vejdemo-Johansson, Persistent Cohomology and Circular Coordinates, Discrete & Computational Geometry, vol 45(4), pp. 737–759, 2011.
10. V. de Silva, P. Skraba and M. Vejdemo-Johansson, Topological Analysis of Recurrent Systems, Workshop on Algebraic Topology and Machine Learning, NIPS 2012, Preprint available at url: http://sites.google.com/site/nips2012topology/contributed-talks
11. A. Deckard, R. Analfi, D. Orlando, J. Hogenesch, S. Haase and J. Harer, Design and Analysis of Large-Scale Biological Rhythm Studies: A Comparison of Algorithms for Detecting Periodic Signals in Biological Data, Bioinformatics, btt541v1-btt541, 2013.
12. H. Edelsbrunner and J. Harer Computational Topology, an Introduction, American Mathematical Society, (2010) (241 pages).
13. E. F. Glynn, J. Chen and A. Mushegian, Detecting periodic patterns in unevenly spaced gene expression time series using Lomb-Scargle periodograms, Bioinformatics, vol 22(3), pp. 310–316, 2006.
14. A. Hatcher Algebraic Topology. Cambridge Univ. Press, England, 2002.
15. M. E. Hughes, J. B. Hogenesch and K. Kornacker, JTK-CYCLE: An Efficient Nonparametric Algorithm for Detecting Rhythmic Components in Genome-Scale Data Sets, Journal of Biological Rhythms, vol 25(372), pp. 372–380, 2010.
16. H. Kantz and T. Schreiber, Nonlinear Time Series Analysis, Cambridge University Press, 2003.
17. H. S. Kim, R. Eykholt and J. D. Salas, Nonlinear dynaimcs, delay times, and embedding windows, Physica D: Nonlinear Phenomena, vol 127(1), pp. 48–60, 1999.
18. N. R. Lomb, Least-squares frequency analysis of unequally spaced data, Astrophysics and Space Science, vol 39, pp. 447–462, 1976.
19. Y. Mileyko, S. Mukherjee, and J. Harer, Probability measures on the space of persistence diagrams, Inverse Problems, 27(12), p.p. 124007, 2011.
20. K. Mischaikow and V. Nanda, Morse Theory for Filtrations and Efficient Computation of Persistent Homology, To appear on Discrete and Computational Geometry, 2013.
21. J. R. Munkres Elements of Algebraic Topology. Addison-Wesley, Redwood City, California, 1984.
22. M. A. Pinsky, Introduction to Fourier Anlysis and Wavelets, The Brooks/Cole Series in Advanced Mathematics, USA, 2003.
23. J. A. Perea, A. Deckard, S. B. Haase and J. Harer, SW1PerS: Sliding Windows and 1-Persistence Scoring; Discovering Periodicity in Gene Expression Time Series Data, preprint (2013).
24. J. D. Scargle, Studies in astronomical time series analysis. II-Statistical aspects of spectral analysis of unevenly spaced data, Astrophysical Journal, vol 263, pp. 835–853, 1982.
25. C. J. Stam, Nonlinear dynamical analysis of EEG and MEG: Review of an emerging field, Clinical Neurophisiology, 116, pp. 2266–2301, 2005.
26. N. Hundewale, The application of methods of nonlinear dynamics for ECG in Normal Sinus Rythm, Int. J. of Computer Science, 9, pp. 458–467, 2012.
27. C. E. Shannon, Communication in the presence of noise, Proc. Inst. Radio Eng., vol. 37, no. 1, pp. 10–21, 1949.
28. F. Takens, Detecting strange attractors in turbulence. in D. A. Rand and L. -S. Young. Dynamical Systems and Turbulence, Lecture Notes in Mathematics, vol. 898. Springer-Verlag. pp. 366–381.
29. A. Tausz, M. Vejdemo-Johansson and H. Adams, JavaPlex: A research software package for persistent (co)homology, 2011, Software available at url: http://code.google.com/p/javaplex
30. A. Zomorodian and G. Carlsson, Computing Persistent Homology, Discrete & Computational Geometry, vol 33(2), pp. 249–274, 2005.