Long-range Correlations and Extreme Events: Dynamics and … · pdf Solar Wind E. Reconstruction of...
Transcript of Long-range Correlations and Extreme Events: Dynamics and … · pdf Solar Wind E. Reconstruction of...
![Page 1: Long-range Correlations and Extreme Events: Dynamics and … · pdf Solar Wind E. Reconstruction of Dynamics - Time-delay embedding - Singular spectrum analysis - Input-output analysis](https://reader034.fdocuments.us/reader034/viewer/2022050213/5f5f842fa171f76c8a2d8b7e/html5/thumbnails/1.jpg)
Long-range Correlations and Extreme Events:Dynamics and Predictions
Surja SharmaUniversity of Maryland
ISSI, Bern16 Oct 2012
![Page 2: Long-range Correlations and Extreme Events: Dynamics and … · pdf Solar Wind E. Reconstruction of Dynamics - Time-delay embedding - Singular spectrum analysis - Input-output analysis](https://reader034.fdocuments.us/reader034/viewer/2022050213/5f5f842fa171f76c8a2d8b7e/html5/thumbnails/2.jpg)
Nonlinear Dynamics and Complexity
Geometry / Structure(Mandelbrot, 1977)
Real objects in nature (Trees, clouds, coastline, etc.)
Fractals, MultifractalsDynamics + Structure
Dynamics (Lorenz, 1963) Deterministic dynamicsChaosQuantitative resultsWeak connection with data
![Page 3: Long-range Correlations and Extreme Events: Dynamics and … · pdf Solar Wind E. Reconstruction of Dynamics - Time-delay embedding - Singular spectrum analysis - Input-output analysis](https://reader034.fdocuments.us/reader034/viewer/2022050213/5f5f842fa171f76c8a2d8b7e/html5/thumbnails/3.jpg)
Reconstruction of Dynamics“Geometry from a time series”
(Packard et al., PRL, 1980)
Embedding theorem (Takens, 1981)Time series data: x(t)Time-delay embedding:
xk(ti) = x(ti + (k-1)τ)Reconstructed space:
Xi = {x1(ti), x2(ti), x3(ti), ..} (Broomhead and King, Phys. A, 1986)
Actual
Reconstructed Time series data
Correlation Dimension
(Grassberger & Procaccia, PRL, 1983)
C(r) (r | Xi X j |) ~ r
![Page 4: Long-range Correlations and Extreme Events: Dynamics and … · pdf Solar Wind E. Reconstruction of Dynamics - Time-delay embedding - Singular spectrum analysis - Input-output analysis](https://reader034.fdocuments.us/reader034/viewer/2022050213/5f5f842fa171f76c8a2d8b7e/html5/thumbnails/4.jpg)
Earth’s Magnetosphere
Episodic nature of the magnetosphere:Storms and substorms driven by the solar wind
![Page 5: Long-range Correlations and Extreme Events: Dynamics and … · pdf Solar Wind E. Reconstruction of Dynamics - Time-delay embedding - Singular spectrum analysis - Input-output analysis](https://reader034.fdocuments.us/reader034/viewer/2022050213/5f5f842fa171f76c8a2d8b7e/html5/thumbnails/5.jpg)
Multiscale Magnetosphere
Global scaleMHD scaleHall-MHD scale Electron-MHD scale
![Page 6: Long-range Correlations and Extreme Events: Dynamics and … · pdf Solar Wind E. Reconstruction of Dynamics - Time-delay embedding - Singular spectrum analysis - Input-output analysis](https://reader034.fdocuments.us/reader034/viewer/2022050213/5f5f842fa171f76c8a2d8b7e/html5/thumbnails/6.jpg)
![Page 7: Long-range Correlations and Extreme Events: Dynamics and … · pdf Solar Wind E. Reconstruction of Dynamics - Time-delay embedding - Singular spectrum analysis - Input-output analysis](https://reader034.fdocuments.us/reader034/viewer/2022050213/5f5f842fa171f76c8a2d8b7e/html5/thumbnails/7.jpg)
Auroral Electrojet Indices
Envelopes:upper AUlower AL
![Page 8: Long-range Correlations and Extreme Events: Dynamics and … · pdf Solar Wind E. Reconstruction of Dynamics - Time-delay embedding - Singular spectrum analysis - Input-output analysis](https://reader034.fdocuments.us/reader034/viewer/2022050213/5f5f842fa171f76c8a2d8b7e/html5/thumbnails/8.jpg)
Reconstruction of Phase Space: Low Dimensional Dynamics
Correlation dimensionConvergence to a low value
Vassiliadis et al., GRL, 1990
Ortho-normal coordinates- Singular spectrum analysisSharma et al., GRL, 1993
Note: Uses Magnetospherric data alone:
Autonomous systemComplex Driven System:
Input-output model
![Page 9: Long-range Correlations and Extreme Events: Dynamics and … · pdf Solar Wind E. Reconstruction of Dynamics - Time-delay embedding - Singular spectrum analysis - Input-output analysis](https://reader034.fdocuments.us/reader034/viewer/2022050213/5f5f842fa171f76c8a2d8b7e/html5/thumbnails/9.jpg)
Strongly Driven Magnetosphere:Substorms during 81 intense storms in 2001
Solar wind data (ACE satellite)
Magnetospheric response (AL index)
Chen, Ph. D. Dissertation, 2007
1 min res.,0.5 x 106 data pts.
![Page 10: Long-range Correlations and Extreme Events: Dynamics and … · pdf Solar Wind E. Reconstruction of Dynamics - Time-delay embedding - Singular spectrum analysis - Input-output analysis](https://reader034.fdocuments.us/reader034/viewer/2022050213/5f5f842fa171f76c8a2d8b7e/html5/thumbnails/10.jpg)
Conditional Probability Distribution Functions
Multiscale phenomenonUkhorskiy et al., 2004Veeramani et al., 2007Sharma et al., 2008
Magnetospheric response
Solar Wind E
![Page 11: Long-range Correlations and Extreme Events: Dynamics and … · pdf Solar Wind E. Reconstruction of Dynamics - Time-delay embedding - Singular spectrum analysis - Input-output analysis](https://reader034.fdocuments.us/reader034/viewer/2022050213/5f5f842fa171f76c8a2d8b7e/html5/thumbnails/11.jpg)
Reconstruction of Dynamics- Time-delay embedding- Singular spectrum analysis- Input-output analysis
Input : solar wind induced E fieldOutput: auroral electrojet index AL
Leading components of the reconstructed space
![Page 12: Long-range Correlations and Extreme Events: Dynamics and … · pdf Solar Wind E. Reconstruction of Dynamics - Time-delay embedding - Singular spectrum analysis - Input-output analysis](https://reader034.fdocuments.us/reader034/viewer/2022050213/5f5f842fa171f76c8a2d8b7e/html5/thumbnails/12.jpg)
From vBs-AL Index DataSitnov et al., JGR, 2000;
Phys. Rev. E, 2001.
From vBs-Pseudo AL Index:Global MHD simulationsShao et al., JGR, 2003
Manifold of Magnetospheric Dynamics: - Phase Transition-like Behavior
![Page 13: Long-range Correlations and Extreme Events: Dynamics and … · pdf Solar Wind E. Reconstruction of Dynamics - Time-delay embedding - Singular spectrum analysis - Input-output analysis](https://reader034.fdocuments.us/reader034/viewer/2022050213/5f5f842fa171f76c8a2d8b7e/html5/thumbnails/13.jpg)
Mean-Field Model and Prediction
Mean state for t+1represented by
Predicted output:
Prediction procedure: search of mean response
cmttk
cmtt OO xxx ,)( 11 F
1tOnt+1
O(t)
cmtx
tx
cmt 1
~x
Ot11tx
Ukhorskiy et al., GRL, 2002
Nearest neighbors of current state xt
Center of mass - mean state for given input It
xtcm
xt1cm x2
x1
x3
![Page 14: Long-range Correlations and Extreme Events: Dynamics and … · pdf Solar Wind E. Reconstruction of Dynamics - Time-delay embedding - Singular spectrum analysis - Input-output analysis](https://reader034.fdocuments.us/reader034/viewer/2022050213/5f5f842fa171f76c8a2d8b7e/html5/thumbnails/14.jpg)
Ukhorskiy et al., GRL, 2004
Prediction with Mean-field Model
Distribution of past events- Conditional probabilities
Predicted and actual AL
Solar wind conditions
![Page 15: Long-range Correlations and Extreme Events: Dynamics and … · pdf Solar Wind E. Reconstruction of Dynamics - Time-delay embedding - Singular spectrum analysis - Input-output analysis](https://reader034.fdocuments.us/reader034/viewer/2022050213/5f5f842fa171f76c8a2d8b7e/html5/thumbnails/15.jpg)
Mean-Field & Weighted Mean-Field Models
Average over all nearest neighbors
Ukhorskiy et al., GRL, 2002, 2004
Closer nearest neighbors should contribute more
k
NN
kkn gX
NNO
11
1
NN
i ikk dd
g1
22
11
Mean-field model
Weighted mean-field model
Chen and Sharma, JGR, 2006,2007
dk – distance from center of mass
![Page 16: Long-range Correlations and Extreme Events: Dynamics and … · pdf Solar Wind E. Reconstruction of Dynamics - Time-delay embedding - Singular spectrum analysis - Input-output analysis](https://reader034.fdocuments.us/reader034/viewer/2022050213/5f5f842fa171f76c8a2d8b7e/html5/thumbnails/16.jpg)
Predictions for strongly driven magnetosphere
Chen et al., JGR, 2006Intense storms of 2003 (Oct 26 – Nov 3) and 2002 April
![Page 17: Long-range Correlations and Extreme Events: Dynamics and … · pdf Solar Wind E. Reconstruction of Dynamics - Time-delay embedding - Singular spectrum analysis - Input-output analysis](https://reader034.fdocuments.us/reader034/viewer/2022050213/5f5f842fa171f76c8a2d8b7e/html5/thumbnails/17.jpg)
Forecasting Space Weather
Near Real-time data of the solar wind from ACE satellite
Forecasts of AL and Dst using the weighted mean-field model
http://www.spp.astro.umd.edu/spaceweather
Pdf’s provide measures of predictability, and can used to estimate associated risk Forecast using the weighted mean-field model
![Page 18: Long-range Correlations and Extreme Events: Dynamics and … · pdf Solar Wind E. Reconstruction of Dynamics - Time-delay embedding - Singular spectrum analysis - Input-output analysis](https://reader034.fdocuments.us/reader034/viewer/2022050213/5f5f842fa171f76c8a2d8b7e/html5/thumbnails/18.jpg)
Spatio-Temporal Properties:Information Theoretic Approach
• Average mutual information (AMI):
• AMI represents the expectation of the average degree of independence incorporating all higher orders
• AMIs of 12 spatially distributed magnetometer time series
• Spread of Localized Integrated Mutual Information (SLIMI):
)},({}0,({)( iiiiI xIMinxIMax
K
i
K
j ji
ijij txptxp
txtxptxtxpxI
1 12 ))(())((
))(),((log))(),((),(
Edwards et al., AGU Spring Mtg., 2000Chen et al., JGR, 2008
Shannon, 1948, Wiener, 1949Jaynes, 1983
![Page 19: Long-range Correlations and Extreme Events: Dynamics and … · pdf Solar Wind E. Reconstruction of Dynamics - Time-delay embedding - Singular spectrum analysis - Input-output analysis](https://reader034.fdocuments.us/reader034/viewer/2022050213/5f5f842fa171f76c8a2d8b7e/html5/thumbnails/19.jpg)
• AMI represents the average degree of independence or correlation time• Spread among the AMIs of 12 spatially distributed magnetometers:
)},({}0,({)( iiiiI xIMinxIMax
AMl
Time delay τ
)(I
Average Mutual Information: Data from 12 Auroral stations
![Page 20: Long-range Correlations and Extreme Events: Dynamics and … · pdf Solar Wind E. Reconstruction of Dynamics - Time-delay embedding - Singular spectrum analysis - Input-output analysis](https://reader034.fdocuments.us/reader034/viewer/2022050213/5f5f842fa171f76c8a2d8b7e/html5/thumbnails/20.jpg)
Long-range Correlations:Detrended Fluctuation Analysis (DFA)
Detection of long-range correlations in time series data
Based on auto-correlation function C(s) Removes effects due to non-stationarities (?) Yields a scaling exponent:
F(s) ~ sα
α = 0.5 - uncorrelated (diffusion)> 0.5 - long-range correlation
![Page 21: Long-range Correlations and Extreme Events: Dynamics and … · pdf Solar Wind E. Reconstruction of Dynamics - Time-delay embedding - Singular spectrum analysis - Input-output analysis](https://reader034.fdocuments.us/reader034/viewer/2022050213/5f5f842fa171f76c8a2d8b7e/html5/thumbnails/21.jpg)
Why detrend data?
![Page 22: Long-range Correlations and Extreme Events: Dynamics and … · pdf Solar Wind E. Reconstruction of Dynamics - Time-delay embedding - Singular spectrum analysis - Input-output analysis](https://reader034.fdocuments.us/reader034/viewer/2022050213/5f5f842fa171f76c8a2d8b7e/html5/thumbnails/22.jpg)
Detrended Fluctuation Analysis
Kantelhardt et al. Physica A, 2001Bryce and Sprangue, Sci. Rep. 2:315, 2012
![Page 23: Long-range Correlations and Extreme Events: Dynamics and … · pdf Solar Wind E. Reconstruction of Dynamics - Time-delay embedding - Singular spectrum analysis - Input-output analysis](https://reader034.fdocuments.us/reader034/viewer/2022050213/5f5f842fa171f76c8a2d8b7e/html5/thumbnails/23.jpg)
Auto-correlation Function of AL index
![Page 24: Long-range Correlations and Extreme Events: Dynamics and … · pdf Solar Wind E. Reconstruction of Dynamics - Time-delay embedding - Singular spectrum analysis - Input-output analysis](https://reader034.fdocuments.us/reader034/viewer/2022050213/5f5f842fa171f76c8a2d8b7e/html5/thumbnails/24.jpg)
Mutual Information Function – AL index
![Page 25: Long-range Correlations and Extreme Events: Dynamics and … · pdf Solar Wind E. Reconstruction of Dynamics - Time-delay embedding - Singular spectrum analysis - Input-output analysis](https://reader034.fdocuments.us/reader034/viewer/2022050213/5f5f842fa171f76c8a2d8b7e/html5/thumbnails/25.jpg)
DFA analysis of AL and Solar Wind
- Uncorrelated beyond ~ 4-5 hrs
Veeramani et al., 2008
![Page 26: Long-range Correlations and Extreme Events: Dynamics and … · pdf Solar Wind E. Reconstruction of Dynamics - Time-delay embedding - Singular spectrum analysis - Input-output analysis](https://reader034.fdocuments.us/reader034/viewer/2022050213/5f5f842fa171f76c8a2d8b7e/html5/thumbnails/26.jpg)
Correlation Time of 4 – 5 hrs
Randomness from the definition of indicesArtificial stochasticity - Anthropogenic
Envelopes:upper AUlower AL
![Page 27: Long-range Correlations and Extreme Events: Dynamics and … · pdf Solar Wind E. Reconstruction of Dynamics - Time-delay embedding - Singular spectrum analysis - Input-output analysis](https://reader034.fdocuments.us/reader034/viewer/2022050213/5f5f842fa171f76c8a2d8b7e/html5/thumbnails/27.jpg)
DFA Analysis – Return Intervals
![Page 28: Long-range Correlations and Extreme Events: Dynamics and … · pdf Solar Wind E. Reconstruction of Dynamics - Time-delay embedding - Singular spectrum analysis - Input-output analysis](https://reader034.fdocuments.us/reader034/viewer/2022050213/5f5f842fa171f76c8a2d8b7e/html5/thumbnails/28.jpg)
Fractional Brownian motion with Hurst exponent of 0.8
![Page 29: Long-range Correlations and Extreme Events: Dynamics and … · pdf Solar Wind E. Reconstruction of Dynamics - Time-delay embedding - Singular spectrum analysis - Input-output analysis](https://reader034.fdocuments.us/reader034/viewer/2022050213/5f5f842fa171f76c8a2d8b7e/html5/thumbnails/29.jpg)
Fractional Gaussian noise with Hurst exponent of 0.8
![Page 30: Long-range Correlations and Extreme Events: Dynamics and … · pdf Solar Wind E. Reconstruction of Dynamics - Time-delay embedding - Singular spectrum analysis - Input-output analysis](https://reader034.fdocuments.us/reader034/viewer/2022050213/5f5f842fa171f76c8a2d8b7e/html5/thumbnails/30.jpg)
FGn data riding on FBm trend (FBm + σ.FGn)
![Page 31: Long-range Correlations and Extreme Events: Dynamics and … · pdf Solar Wind E. Reconstruction of Dynamics - Time-delay embedding - Singular spectrum analysis - Input-output analysis](https://reader034.fdocuments.us/reader034/viewer/2022050213/5f5f842fa171f76c8a2d8b7e/html5/thumbnails/31.jpg)
Detrended data (σ.FGn)
![Page 32: Long-range Correlations and Extreme Events: Dynamics and … · pdf Solar Wind E. Reconstruction of Dynamics - Time-delay embedding - Singular spectrum analysis - Input-output analysis](https://reader034.fdocuments.us/reader034/viewer/2022050213/5f5f842fa171f76c8a2d8b7e/html5/thumbnails/32.jpg)
Fluctuation analysis: Actual and Detrended data
![Page 33: Long-range Correlations and Extreme Events: Dynamics and … · pdf Solar Wind E. Reconstruction of Dynamics - Time-delay embedding - Singular spectrum analysis - Input-output analysis](https://reader034.fdocuments.us/reader034/viewer/2022050213/5f5f842fa171f76c8a2d8b7e/html5/thumbnails/33.jpg)
![Page 34: Long-range Correlations and Extreme Events: Dynamics and … · pdf Solar Wind E. Reconstruction of Dynamics - Time-delay embedding - Singular spectrum analysis - Input-output analysis](https://reader034.fdocuments.us/reader034/viewer/2022050213/5f5f842fa171f76c8a2d8b7e/html5/thumbnails/34.jpg)
Extreme Events
![Page 35: Long-range Correlations and Extreme Events: Dynamics and … · pdf Solar Wind E. Reconstruction of Dynamics - Time-delay embedding - Singular spectrum analysis - Input-output analysis](https://reader034.fdocuments.us/reader034/viewer/2022050213/5f5f842fa171f76c8a2d8b7e/html5/thumbnails/35.jpg)
Extreme Events: Data-enabled Modeling Framework
![Page 36: Long-range Correlations and Extreme Events: Dynamics and … · pdf Solar Wind E. Reconstruction of Dynamics - Time-delay embedding - Singular spectrum analysis - Input-output analysis](https://reader034.fdocuments.us/reader034/viewer/2022050213/5f5f842fa171f76c8a2d8b7e/html5/thumbnails/36.jpg)
Extreme Events: Data-enabled Modeling Framework
Information Theory- Long-range Corr.- Recurrence
Complexity Science- Dynamics- Predictability
Time Series Data- Natural phenomena- Anthropogenic
![Page 37: Long-range Correlations and Extreme Events: Dynamics and … · pdf Solar Wind E. Reconstruction of Dynamics - Time-delay embedding - Singular spectrum analysis - Input-output analysis](https://reader034.fdocuments.us/reader034/viewer/2022050213/5f5f842fa171f76c8a2d8b7e/html5/thumbnails/37.jpg)
“Simple Lessons from Complexity”Goldenfeld and Kadanoff, Science 1999
As science turns to complexity, one must realize that complexity demands attitudes quite different from those heretofore common in physics. Up to now, physicists looked for fundamental laws true for all times and all places. But each complex system is different; apparently there are no general laws of complexity. Instead, one must reach for “lessons” that might, with insight and understanding, be learned in one system and applied to another.
![Page 38: Long-range Correlations and Extreme Events: Dynamics and … · pdf Solar Wind E. Reconstruction of Dynamics - Time-delay embedding - Singular spectrum analysis - Input-output analysis](https://reader034.fdocuments.us/reader034/viewer/2022050213/5f5f842fa171f76c8a2d8b7e/html5/thumbnails/38.jpg)
Criticality and Long-range correlation
Unique relationship in Second order phase transition
Not all transitions in systems with long-range correlations are critical
“Self-organized complexity”Don Turcotte
Dynamics of systems with long-range correlations