Time Series Approaches to Understanding Earthquake Dynamics

1. Time Series Approaches to Understanding Earthquake Dynamics 新加坡南洋理工大学 CHEONG Siew Ann 张 寿 安 cheongsa@ntu.edu.sg http://www1.spms.ntu.edu.sg/~cheongsa/

2. Gutenberg-Richter Law Christensen et al., Universal scaling law for earthquakes, PNAS, 99, 2509, 2002 Power law scaling over >5 orders of magnitude Under-reporting? Frequent large earthquakes Why not like this? National Central University

3. Self Organization • Complex earth crust dynamics • Competing processes • Information/energy flow between different time/length scales • Transitions between dynamical phases • Earthquakes • Statistical Signatures • Critical slowing down • Dakos et al., PNAS 105, 14308 (2008) • Scheffer et al., Nature 461, 53 (2009) • Synchronization • Leung, Phys Rev E 58, 5704 (1998) • Osipov et al., Phys Rev Lett 91, 024101 (2003) • Goo et al., arXiv:0903.2099, 12 Mar 2009 National Central University

4. Macroscopic Physics • Macroscopic order parameters differentiate • Solid (S) • Liquid (L) • Gas (G) L S G National Central University

5. Microscopic Physics diffusive trajectories, δr2 increases with time • S, L, G time series distinguishable • S, L, G phase within single time series distinguishable δr fluctuates about 0, δr2 = αT time-independent ballistic trajectories, infrequent collisions National Central University

6. Macro ↔ Micro • Group statistically similar time series • Discover presence of different phases National Central University

7. Cross Correlations Between Time Series digital cross correlations Dow Jones US economic sector indices comovement digital cross correlations National Central University

8. Long-Time Correlation Matrix National Central University

9. No Separation of Scales National Central University

10. Complete Separation of Scales National Central University

11. Metallic Nanoclusters small Isothermal MD simulations Ag14, T = 300 K Prof S K Lai, NCU Physics intermediate Effective variables: {1321, 1551, 2331} {1541, 2211, 2321} {1211; 1431} gap large National Central University

12. Discover Effective Variables National Central University

13. Partial Separation of Scales Define effective variables National Central University

14. Partial Hierarchical Clustering • Seed cluster • Find Di*j* = maxijDij • Use c = {i*, j*} as seed cluster • Grow cluster • Add k* to cluster if • Dk,c = minl∈ cDkl • Dk*,c = maxkDk,c • Iterate • Cluster boundary • Plot correlation level Dk*,c against cluster size National Central University

15. Hierarchy of Effective Variables Singapore Stock Exchange (SGX): 2006-2007 financial molecule financial atom financial supermolecule National Central University

16. M > 6 New Zealand Earthquakes National Central University

17. The New Zealand GPS Network • 100+ stations • Oct 2006 to Mar 2008 • Daily time series data • r = (x, y, z) • Δr = (Δx, Δy, Δz) National Central University

18. Digital Vector Cross Correlations Δrj(t) within 30° cone of Δri(t): Cij → Cij + 1 IDEA:If the dynamics of two stations, i and j, are synchronized, then Δri(t) and Δrj(t) will very frequently be parallel or nearly parallel. Δrk(t) outside of 30° cone of Δri(t): no change to Cik National Central University

19. Synchronized Clusters Partial separation of scales First kink Stations on same tectonic plates Weaker kink Sharp kink Synchronized cluster Second kink Highest level of synchronization Synchronized cluster Next highest level of synchronization National Central University

20. Internal Dynamics of Synchronized Clusters Cross-sectional Across synchronized clusters Longitudinal Discover synchronized clusters loss-of-correlation features National Central University

21. Precursor Feature for SC2 precursor feature M > 6 5 < M < 6 National Central University

22. Spatio-Temporal Dynamics of Precursor Feature National Central University

23. Segmentation vs Clustering • Time Series Clustering • Discover effective mesoscopic variables in given time window • Discover slow time evolution of effective variables by sliding time window • Time Series Segmentation • Discover number/type of macroscopic phases • Discover lifetimes of macroscopic phases • Discover time scales of transitions between macroscopic phases National Central University

24. Modeling Nonstationary Time Series • Assume non-stationary time series • x = (x1, x2, …, xN) • M stationary segments • In segment m, data points drawn from (μm, σm2) Gaussian distribution • Recursive segmentation • One time series → two segments • Each segment → two subsegments • Iterate + optimize • Terminate National Central University

25. Jensen-Shannon Divergence • Single-segment likelihood for x = (x1, x2, …, xN) • Two-segment likelihood for x = (x1, …, xt, xt+1, …, xN) • ML estimates • Jensen-Shannon divergence National Central University

26. Recursive Segmentation National Central University

27. Segment Clustering National Central University

28. Temporal Distribution of Segments National Central University

29. Cross Section Study National Central University

30. Economic Recovery National Central University

31. Onset of Crisis National Central University

32. Response to Extrinsic Shocks National Central University

33. Conclusions • Potential for understanding macroscopic earthquake dynamics from microscopic time series • Time series clustering • Complete separation of scales • Metallic nanoclusters • Partial separation of scales • New Zealand 2007 earthquake precursor dynamics • Time series segmentation • US economy • Crisis and growth • Chemical picture of slow dynamics using MSTs National Central University

34. Acknowledgments • Time Series Clustering • Prof Lai S K • GOO Yik Wen • LIAN Tong Wei • ONG Wei Guang • CHOI Wen Ting • WANG Weihan • Time Series Segmentation • WONG Jian Cheng • Gladys LEE Hui Ting • ZHANG Yiting • Dr Manamohan PRUSTY National Central University

35. Thank You! National Central University