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Utilizing Topological Data Analysis to Detect Periodicity Elizabeth Munch University at Albany - SUNY :: Department of Mathematics & Statistics Oct 2, 2016 Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 1 / 30 Time series in biology

  1. Utilizing Topological Data Analysis to Detect Periodicity Elizabeth Munch University at Albany - SUNY :: Department of Mathematics & Statistics Oct 2, 2016 Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 1 / 30

  2. Time series in biology Mitosis Kredel et al. PLoS One 2009 Yeast gene expression Deckard et al., Bioinformatics 2013 Neuron Spike Trains ECG Curto et al. PLoS One 2008 Goldberg et al. 2000 Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 2 / 30

  3. Our definition of time series Definition A time series is a function f : R ≥ 0 − → D for some topological space D . Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 3 / 30

  4. Our definition of time series Definition A time series is a function f : R ≥ 0 − → D for some topological space D . Choice for D R - Classical time series analysis R m × n - R -valued m × n matrices (movies) Pers - Persistence diagram valued time series (vineyards) Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 3 / 30

  5. Commonly used tools Birth Radius Time Death Radius Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 4 / 30

  6. Common questions Classification/Clustering ◮ Is this signal Type A or Type B? Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 5 / 30

  7. Common questions Classification/Clustering ◮ Is this signal Type A or Type B? Periodicity ◮ Is this signal exhibiting periodic behavior? Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 5 / 30

  8. Common questions Classification/Clustering ◮ Is this signal Type A or Type B? Periodicity ◮ Is this signal exhibiting periodic behavior? Forecasting ◮ Given this previous signal, what do we expect to have happen next? Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 5 / 30

  9. Common questions Classification/Clustering ◮ Is this signal Type A or Type B? Periodicity ◮ Is this signal exhibiting periodic behavior? Forecasting ◮ Given this previous signal, what do we expect to have happen next? Segmentation ◮ Which pieces of this signal come from similar systems? Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 5 / 30

  10. Common questions Classification/Clustering ◮ Is this signal Type A or Type B? Periodicity ◮ Is this signal exhibiting periodic behavior? Forecasting ◮ Given this previous signal, what do we expect to have happen next? Segmentation ◮ Which pieces of this signal come from similar systems? Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 5 / 30

  11. Idea: Persistent homology and other TDA tools can be used to improve time series anaysis. Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 6 / 30

  12. Idea: Persistent homology and other TDA tools can be used to improve time series anaysis. This talk: Mechanical engineering ◮ Firas Khasawneh ◮ Jose Perea Atmospheric science ◮ Bill Dong ◮ Kristen Corbosiero ◮ Jason Dunion ◮ Ryan Torn Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 6 / 30

  13. Classification and Machining Dynamics 1 Periodicity and Hurricanes 2 Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 7 / 30

  14. Classification and Machining Dynamics 1 Periodicity and Hurricanes 2 Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 7 / 30

  15. Machining Dynamics Workpiece Stable Unstable feed Images courtesy Firas Khasawneh, SUNYIT; and Boeing. Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 8 / 30

  16. Left side: standard linear Deterministic model: oscillator y + 2 ζ ˙ ¨ y + y Right side: input based = K ρ α − 1 (1 + y ( t − τ ) − y ( t )) α on cutting forces 0.25 0.2 0.15 0.1 0.05 0 0.5 1 1.5 2 2.5 3 Khasawneh, F.A. & Mann, B. P. A spectral element approach for the stability of delay systems, International Journal for Numerical Methods in Engineering, 2011, 87, 566-592 Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 9 / 30

  17. Chatter Signal, [0.9, 0.07] Signal, [1.42, 0.05] Signal, [1.48, 0.25] 2.0 2.0 2.0 1.5 1.5 1.5 1.0 1.0 1.0 0.5 0.5 0.5 0.0 0.0 0.0 − 0.5 − 0.5 − 0.5 − 1.0 − 1.0 − 1.0 100 120 140 160 180 200 220 240 70 80 90 100 110 120 130 140 150 60 70 80 90 100 110 120 130 140 Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 10 / 30

  18. Takens embedding Definition Given a time series X ( t ), the Takens embedding is ψ m η : t �− → ( X ( t ) , X ( t + η ) , · · · , X ( t + ( m − 1) η )) . Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 11 / 30

  19. Persistent Homology of Point Cloud Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 12 / 30

  20. Noise resilience Original Signals 1.5 1.0 0.5 0.0 X 0.5 1.0 1.5 0 5 10 15 t Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 13 / 30

  21. Noise resilience Original Signals 1.5 1.0 0.5 0.0 X 0.5 1.0 1.5 0 5 10 15 t Persistence Diagrams Delay Embedding 1.8 1.5 1.6 1.0 1.4 1.2 0.5 32) 1.0 Death X ( t +1 . 0.0 0.8 0.6 0.5 0.4 1.0 0.2 1.5 0.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 X ( t ) Birth Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 13 / 30

  22. Comparing signals using persistence Signal, [0.9, 0.07] Signal, [1.42, 0.05] Signal, [1.48, 0.25] 2.0 2.0 2.0 1.5 1.5 1.5 1.0 1.0 1.0 0.5 0.5 0.5 0.0 0.0 0.0 − 0.5 − 0.5 − 0.5 − 1.0 − 1.0 − 1.0 100 120 140 160 180 200 220 240 70 80 90 100 110 120 130 140 150 60 70 80 90 100 110 120 130 140 Takens Embedding, [0.9, 0.07] Takens Embedding, [1.42, 0.05] Takens Embedding, [1.48, 0.25] 2.0 2.0 2.0 1.5 1.5 1.5 Y ( t + 2.13 ) Y ( t + 1.62 ) Y ( t + 1.56 ) 1.0 1.0 1.0 0.5 0.5 0.5 0.0 0.0 0.0 − 0.5 − 0.5 − 0.5 − 1.0 − 1.0 − 1.0 − 1.0 − 0.5 0.0 0.5 1.0 1.5 2.0 − 1.0 − 0.5 0.0 0.5 1.0 1.5 2.0 − 1.0 − 0.5 0.0 0.5 1.0 1.5 2.0 Y ( t ) Y ( t ) Y ( t ) Persistence Diagram, [0.9, 0.07] Persistence Diagram, [1.42, 0.05] Persistence Diagram, [1.48, 0.25] 1.8 1.8 1.8 1.6 1.6 1.6 Death Radius Death Radius Death Radius 1.4 1.4 1.4 1.2 1.2 1.2 1.0 1.0 1.0 0.8 0.8 0.8 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0.0 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 Birth Radius Birth Radius Birth Radius Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 14 / 30

  23. Comparing signals using persistence Signal, [0.9, 0.07] Signal, [1.42, 0.05] Signal, [1.48, 0.25] 2.0 2.0 2.0 1.5 1.5 1.5 1.0 1.0 1.0 0.5 0.5 0.5 0.0 0.0 0.0 − 0.5 − 0.5 − 0.5 − 1.0 − 1.0 − 1.0 100 120 140 160 180 200 220 240 70 80 90 100 110 120 130 140 150 60 70 80 90 100 110 120 130 140 Takens Embedding, [0.9, 0.07] Takens Embedding, [1.42, 0.05] Takens Embedding, [1.48, 0.25] 2.0 2.0 2.0 1.5 1.5 1.5 Y ( t + 2.13 ) Y ( t + 1.62 ) Y ( t + 1.56 ) 1.0 1.0 1.0 0.5 0.5 0.5 0.0 0.0 0.0 − 0.5 − 0.5 − 0.5 − 1.0 − 1.0 − 1.0 − 1.0 − 0.5 0.0 0.5 1.0 1.5 2.0 − 1.0 − 0.5 0.0 0.5 1.0 1.5 2.0 − 1.0 − 0.5 0.0 0.5 1.0 1.5 2.0 Y ( t ) Y ( t ) Y ( t ) Persistence Diagram, [0.9, 0.07] Persistence Diagram, [1.42, 0.05] Persistence Diagram, [1.48, 0.25] 1.8 1.8 1.8 1.6 1.6 1.6 Death Radius Death Radius Death Radius 1.4 1.4 1.4 1.2 1.2 1.2 1.0 1.0 1.0 0.8 0.8 0.8 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0.0 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 Birth Radius Birth Radius Birth Radius Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 14 / 30

  24. Overview Birth Radius Time Death Radius Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 15 / 30

  25. Overview Birth Radius Time Death Radius Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 15 / 30

  26. Differentiation by Max Persistence Signal, [0.9, 0.07] Signal, [1.42, 0.05] Signal, [1.48, 0.25] 2.0 2.0 2.0 1.5 1.5 1.5 1.0 1.0 1.0 0.5 0.5 0.5 0.0 0.0 0.0 − 0.5 − 0.5 − 0.5 − 1.0 − 1.0 − 1.0 100 120 140 160 180 200 220 240 70 80 90 100 110 120 130 140 150 60 70 80 90 100 110 120 130 140 Takens Embedding, [0.9, 0.07] Takens Embedding, [1.42, 0.05] Takens Embedding, [1.48, 0.25] 2.0 2.0 2.0 1.5 1.5 1.5 Y ( t + 2.13 ) Y ( t + 1.62 ) Y ( t + 1.56 ) 1.0 1.0 1.0 0.5 0.5 0.5 0.0 0.0 0.0 − 0.5 − 0.5 − 0.5 − 1.0 − 1.0 − 1.0 − 1.0 − 0.5 0.0 0.5 1.0 1.5 2.0 − 1.0 − 0.5 0.0 0.5 1.0 1.5 2.0 − 1.0 − 0.5 0.0 0.5 1.0 1.5 2.0 Y ( t ) Y ( t ) Y ( t ) Persistence Diagram, [0.9, 0.07] Persistence Diagram, [1.42, 0.05] Persistence Diagram, [1.48, 0.25] 1.8 1.8 1.8 1.6 1.6 1.6 Death Radius Death Radius Death Radius 1.4 1.4 1.4 1.2 1.2 1.2 1.0 1.0 1.0 0.8 0.8 0.8 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0.0 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 Birth Radius Birth Radius Birth Radius Liz Munch (UAlbany) TSA with TDA Oct 2 ACM-BCB 16 / 30

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