![Page 1: Complex network approach for recurrence analysis of time ...dswc09/CONTRIBUTIONS/donges_talk.pdf · Complex network approach for recurrence analysis of time series DSWC 09 Dresden,](https://reader035.vdocuments.mx/reader035/viewer/2022081615/5fd3ae191bf05f64db742851/html5/thumbnails/1.jpg)
Complex network approach for recurrence analysis of time
series
DSWC 09 Dresden, July 30th
J.F. Donges (1,2), N. Marwan (1), Y. Zou (1), R.V. Donner (1) and J. Kurths (1,2)
1 Potsdam Institute for Climate Impact Research, Germany2 Department of Physics, Humboldt University, Berlin, Germany
Contact: [email protected], http://www.pik-potsdam.de/~donges
![Page 2: Complex network approach for recurrence analysis of time ...dswc09/CONTRIBUTIONS/donges_talk.pdf · Complex network approach for recurrence analysis of time series DSWC 09 Dresden,](https://reader035.vdocuments.mx/reader035/viewer/2022081615/5fd3ae191bf05f64db742851/html5/thumbnails/2.jpg)
Outline
• Introduction
• Conceptual foundation
• Quantifying recurrence networks
• Application to climatological time series
• Conclusions & Outlook
![Page 3: Complex network approach for recurrence analysis of time ...dswc09/CONTRIBUTIONS/donges_talk.pdf · Complex network approach for recurrence analysis of time series DSWC 09 Dresden,](https://reader035.vdocuments.mx/reader035/viewer/2022081615/5fd3ae191bf05f64db742851/html5/thumbnails/3.jpg)
Introduction
![Page 4: Complex network approach for recurrence analysis of time ...dswc09/CONTRIBUTIONS/donges_talk.pdf · Complex network approach for recurrence analysis of time series DSWC 09 Dresden,](https://reader035.vdocuments.mx/reader035/viewer/2022081615/5fd3ae191bf05f64db742851/html5/thumbnails/4.jpg)
What do we start with?
Time series x(t)
Terr
Flux
(g c
m2 k
yr1 )
0 0.5 1 1.5 2 2.5 3 3.5 4 4.50
2
4
Time (Ma)
Marwan et al. (2009)
![Page 5: Complex network approach for recurrence analysis of time ...dswc09/CONTRIBUTIONS/donges_talk.pdf · Complex network approach for recurrence analysis of time series DSWC 09 Dresden,](https://reader035.vdocuments.mx/reader035/viewer/2022081615/5fd3ae191bf05f64db742851/html5/thumbnails/5.jpg)
Recurrence plot
Marwan et al. (2009)
![Page 6: Complex network approach for recurrence analysis of time ...dswc09/CONTRIBUTIONS/donges_talk.pdf · Complex network approach for recurrence analysis of time series DSWC 09 Dresden,](https://reader035.vdocuments.mx/reader035/viewer/2022081615/5fd3ae191bf05f64db742851/html5/thumbnails/6.jpg)
Complex network
![Page 7: Complex network approach for recurrence analysis of time ...dswc09/CONTRIBUTIONS/donges_talk.pdf · Complex network approach for recurrence analysis of time series DSWC 09 Dresden,](https://reader035.vdocuments.mx/reader035/viewer/2022081615/5fd3ae191bf05f64db742851/html5/thumbnails/7.jpg)
Duality ?
![Page 8: Complex network approach for recurrence analysis of time ...dswc09/CONTRIBUTIONS/donges_talk.pdf · Complex network approach for recurrence analysis of time series DSWC 09 Dresden,](https://reader035.vdocuments.mx/reader035/viewer/2022081615/5fd3ae191bf05f64db742851/html5/thumbnails/8.jpg)
Analogies
Recurrence plot Complex network
Recurrence matrixR
Adjacency matrixA
RQA Network statistics
![Page 9: Complex network approach for recurrence analysis of time ...dswc09/CONTRIBUTIONS/donges_talk.pdf · Complex network approach for recurrence analysis of time series DSWC 09 Dresden,](https://reader035.vdocuments.mx/reader035/viewer/2022081615/5fd3ae191bf05f64db742851/html5/thumbnails/9.jpg)
Recurrence network
A = R - I
![Page 10: Complex network approach for recurrence analysis of time ...dswc09/CONTRIBUTIONS/donges_talk.pdf · Complex network approach for recurrence analysis of time series DSWC 09 Dresden,](https://reader035.vdocuments.mx/reader035/viewer/2022081615/5fd3ae191bf05f64db742851/html5/thumbnails/10.jpg)
Conceptual foundation
![Page 11: Complex network approach for recurrence analysis of time ...dswc09/CONTRIBUTIONS/donges_talk.pdf · Complex network approach for recurrence analysis of time series DSWC 09 Dresden,](https://reader035.vdocuments.mx/reader035/viewer/2022081615/5fd3ae191bf05f64db742851/html5/thumbnails/11.jpg)
Equivalence I
Phase space Recurrence network
State x(i) Vertex i
Recurrence||x(i)-x(j)||< ε
Edge(i,j)
![Page 12: Complex network approach for recurrence analysis of time ...dswc09/CONTRIBUTIONS/donges_talk.pdf · Complex network approach for recurrence analysis of time series DSWC 09 Dresden,](https://reader035.vdocuments.mx/reader035/viewer/2022081615/5fd3ae191bf05f64db742851/html5/thumbnails/12.jpg)
Phase space (Lorenz)
−200
20 −20
0
200
20
40
YX
Z
Donner et al. (2009)
![Page 13: Complex network approach for recurrence analysis of time ...dswc09/CONTRIBUTIONS/donges_talk.pdf · Complex network approach for recurrence analysis of time series DSWC 09 Dresden,](https://reader035.vdocuments.mx/reader035/viewer/2022081615/5fd3ae191bf05f64db742851/html5/thumbnails/13.jpg)
Equivalence II
Phase space Recurrence network
ε - ball sequence Path
![Page 14: Complex network approach for recurrence analysis of time ...dswc09/CONTRIBUTIONS/donges_talk.pdf · Complex network approach for recurrence analysis of time series DSWC 09 Dresden,](https://reader035.vdocuments.mx/reader035/viewer/2022081615/5fd3ae191bf05f64db742851/html5/thumbnails/14.jpg)
Phase space (Rössler)
−100
10
−20−10
0100
10
20
XY
Z
Donner et al. (2009)
![Page 15: Complex network approach for recurrence analysis of time ...dswc09/CONTRIBUTIONS/donges_talk.pdf · Complex network approach for recurrence analysis of time series DSWC 09 Dresden,](https://reader035.vdocuments.mx/reader035/viewer/2022081615/5fd3ae191bf05f64db742851/html5/thumbnails/15.jpg)
Crucial properties
• Time ordering is lost - Arbitrary vertex labeling
• Strong spacial constraints
• Recurrence network reflects attractor geometry
![Page 16: Complex network approach for recurrence analysis of time ...dswc09/CONTRIBUTIONS/donges_talk.pdf · Complex network approach for recurrence analysis of time series DSWC 09 Dresden,](https://reader035.vdocuments.mx/reader035/viewer/2022081615/5fd3ae191bf05f64db742851/html5/thumbnails/16.jpg)
Quantifying recurrence networks
![Page 17: Complex network approach for recurrence analysis of time ...dswc09/CONTRIBUTIONS/donges_talk.pdf · Complex network approach for recurrence analysis of time series DSWC 09 Dresden,](https://reader035.vdocuments.mx/reader035/viewer/2022081615/5fd3ae191bf05f64db742851/html5/thumbnails/17.jpg)
Degree centrality
!20
0
20 !20
0
200
10
20
30
40
XY
Z
0.001 0.011 0.021 0.031
Donner et al. (2009)
![Page 18: Complex network approach for recurrence analysis of time ...dswc09/CONTRIBUTIONS/donges_talk.pdf · Complex network approach for recurrence analysis of time series DSWC 09 Dresden,](https://reader035.vdocuments.mx/reader035/viewer/2022081615/5fd3ae191bf05f64db742851/html5/thumbnails/18.jpg)
Clustering coefficient
! 20
0
20 ! 20
0
200
10
20
30
40
X Y
Z
0.13
0.31
0.48
0.65
0.83
1.00
! 20
0
20 ! 20
0
200
10
20
30
40
YX
Z
Donner et al. (2009)
![Page 19: Complex network approach for recurrence analysis of time ...dswc09/CONTRIBUTIONS/donges_talk.pdf · Complex network approach for recurrence analysis of time series DSWC 09 Dresden,](https://reader035.vdocuments.mx/reader035/viewer/2022081615/5fd3ae191bf05f64db742851/html5/thumbnails/19.jpg)
Scalar measures
Recurrence network Phase space
Edge density Recurrence rate
Clustering Higher order density
Average path length Mean separation
![Page 20: Complex network approach for recurrence analysis of time ...dswc09/CONTRIBUTIONS/donges_talk.pdf · Complex network approach for recurrence analysis of time series DSWC 09 Dresden,](https://reader035.vdocuments.mx/reader035/viewer/2022081615/5fd3ae191bf05f64db742851/html5/thumbnails/20.jpg)
Applications
![Page 21: Complex network approach for recurrence analysis of time ...dswc09/CONTRIBUTIONS/donges_talk.pdf · Complex network approach for recurrence analysis of time series DSWC 09 Dresden,](https://reader035.vdocuments.mx/reader035/viewer/2022081615/5fd3ae191bf05f64db742851/html5/thumbnails/21.jpg)
Logistic map
Marwan et al. (2009)
![Page 22: Complex network approach for recurrence analysis of time ...dswc09/CONTRIBUTIONS/donges_talk.pdf · Complex network approach for recurrence analysis of time series DSWC 09 Dresden,](https://reader035.vdocuments.mx/reader035/viewer/2022081615/5fd3ae191bf05f64db742851/html5/thumbnails/22.jpg)
Terrigenous dust fluxT
err
Flu
x (
g c
m2 k
yr
1)
A
0 0.5 1 1.5 2 2.5 3 3.5 4 4.50
2
4L
B
0 0.5 1 1.5 2 2.5 3 3.5 4 4.52
4
6
8
C
C
Time (Ma)0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
0.4
0.6
0.8
Marwan et al. (2009)ODP site 659
![Page 23: Complex network approach for recurrence analysis of time ...dswc09/CONTRIBUTIONS/donges_talk.pdf · Complex network approach for recurrence analysis of time series DSWC 09 Dresden,](https://reader035.vdocuments.mx/reader035/viewer/2022081615/5fd3ae191bf05f64db742851/html5/thumbnails/23.jpg)
Lessons learned
• Clustering coefficient is sensitive to periodicity (longer time scales)
• Average path length sensitive to transition periods (shorter time scales)
![Page 24: Complex network approach for recurrence analysis of time ...dswc09/CONTRIBUTIONS/donges_talk.pdf · Complex network approach for recurrence analysis of time series DSWC 09 Dresden,](https://reader035.vdocuments.mx/reader035/viewer/2022081615/5fd3ae191bf05f64db742851/html5/thumbnails/24.jpg)
Conclusions & Outlook
![Page 25: Complex network approach for recurrence analysis of time ...dswc09/CONTRIBUTIONS/donges_talk.pdf · Complex network approach for recurrence analysis of time series DSWC 09 Dresden,](https://reader035.vdocuments.mx/reader035/viewer/2022081615/5fd3ae191bf05f64db742851/html5/thumbnails/25.jpg)
Advantages of recurrence networks
• Method does NOT require equidistant time scale
• Applicable to univariate and multivariate time series
• Simple significance testing
![Page 26: Complex network approach for recurrence analysis of time ...dswc09/CONTRIBUTIONS/donges_talk.pdf · Complex network approach for recurrence analysis of time series DSWC 09 Dresden,](https://reader035.vdocuments.mx/reader035/viewer/2022081615/5fd3ae191bf05f64db742851/html5/thumbnails/26.jpg)
Take home messages
• Toolbox of complex network theory available to time series analysis
• RNs allow an intuitive and natural interpretation of results
• Complementary to established methods of time series analysis (linear, nonlinear, RQA)
![Page 27: Complex network approach for recurrence analysis of time ...dswc09/CONTRIBUTIONS/donges_talk.pdf · Complex network approach for recurrence analysis of time series DSWC 09 Dresden,](https://reader035.vdocuments.mx/reader035/viewer/2022081615/5fd3ae191bf05f64db742851/html5/thumbnails/27.jpg)
N. Marwan, J.F. Donges, Y. Zou, R.V. Donner and J. Kurths, Complex network approach for recurrence analysis of time series (2009), arXiv:0907.3368v1 [nlin.CD],
R.V. Donner, Y. Zou, J.F. Donges, N. Marwan and J. Kurths, Recurrence networks - A novel paradigm for nonlinear time series analysis (2009), in preparation.
Related publications
![Page 28: Complex network approach for recurrence analysis of time ...dswc09/CONTRIBUTIONS/donges_talk.pdf · Complex network approach for recurrence analysis of time series DSWC 09 Dresden,](https://reader035.vdocuments.mx/reader035/viewer/2022081615/5fd3ae191bf05f64db742851/html5/thumbnails/28.jpg)
Thank you!