Complex network approach
for recurrence analysis of time
series
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, Germany
2 Department of Physics, Humboldt University, Berlin, Germany
DSWC 09 Dresden, July 30th
Contact: [email protected], http://www.pik-potsdam.de/~donges
Outline
• Introduction
• Conceptual foundation
• Quantifying recurrence networks
• Application to climatological time series
• Conclusions & Outlook
Introduction
What do we start with?
1
kyr )
Time series x(t)
Terr Flux (g cm
2
4
2
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Time (Ma)
Marwan et al. (2009)
Recurrence plot
Marwan et al. (2009)
Complex network
Duality ?
Analogies
Recurrence plot
Complex network
Recurrence matrix
R
Adjacency matrix
A
RQA
Network statistics
Recurrence network
A=R-I
Conceptual foundation
Equivalence I
Phase space
Recurrence network
State x(i)
Vertex i
Recurrence
||x(i)-x(j)||< ε
Edge
(i,j)
Phase space (Lorenz)
Z
40
20
20
0
−20
0Y
X0
20 −20
Donner et al. (2009)
Equivalence II
Phase space
Recurrence network
ε - ball sequence
Path
Phase space (Rössler)
Z
20
10
0
10
Y
10
0
−10
−20 −10
0X
Donner et al. (2009)
Crucial properties
• Time ordering is lost - Arbitrary vertex
labeling
• Strong spacial constraints
• Recurrence network reflects attractor
geometry
Quantifying recurrence
networks
Degree centrality
40
Z
30
20
10
20
0
!20
X
0.001
0
0
Y
20 !20
0.011
0.021
0.031
Donner et al. (2009)
Clustering coefficient
1.00
0.83
40
30
0.65
20
Z
Z
30
40
20
0.48
10
10
20
0
! 20
0.31
0Y
X0
20 ! 20
0.13
20
0
! 20
0 Y
X0
20 ! 20
Donner et al. (2009)
Scalar measures
Recurrence network
Phase space
Edge density
Recurrence rate
Clustering
Higher order density
Average path length
Mean separation
Applications
Logistic map
Marwan et al. (2009)
4
Terr Flux (g cm
2
A
kyr 1)
Terrigenous dust flux
B
2
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0
0.5
1
1.5
2
2.5
Time (Ma)
3
3.5
4
4.5
8
L
6
4
2
C
C
0.8
0.6
0.4
ODP site 659
Marwan et al. (2009)
Lessons learned
• Clustering coefficient is sensitive to
periodicity (longer time scales)
• Average path length sensitive to transition
periods (shorter time scales)
Conclusions & Outlook
Advantages of
recurrence networks
• Method does NOT require equidistant time
scale
• Applicable to univariate and multivariate
time series
• Simple significance testing
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)
Related publications
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.
Thank you!