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: donges@pik-potsdam.de, 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?

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)

Recurrence plot

Marwan et al. (2009)

Complex network

Duality ?

Analogies

Recurrence plot Complex network

Recurrence matrixR

Adjacency matrixA

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)

−200

20 −20

0

200

20

40

YX

Z

Donner et al. (2009)

Equivalence II

Phase space Recurrence network

ε - ball sequence Path

Phase space (Rössler)

−100

10

−20−10

0100

10

20

XY

Z

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

!20

0

20 !20

0

200

10

20

30

40

XY

Z

0.001 0.011 0.021 0.031

Donner et al. (2009)

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)

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)

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

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)

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.

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Thank you!