We describe the implementation of methods of nonlinear time series analysis which are based on the paradigm of deterministic chaos. A variety of algorithms for data representation, prediction, noise reduction, dimension and Lyapunov estimation, and nonlinearity testing are discussed with particular emphasis on issues of implementation and choice of parameters. Computer programs that implement the resulting strategies are publicly available as the TISEAN software package. The use of each algorithm will be illustrated with a typical application. As to the theoretical background, we will essentially give pointers to the literature.
REFERENCES
1.
The TISEAN software package is publicly available at The distribution includes an online documentation system.
2.
H. Kantz and T. Schreiber, Nonlinear Time Series Analysis (Cambridge University Press, Cambridge, 1997).
3.
T.
Schreiber
, “Interdisciplinary application of nonlinear time series methods
,” Phys. Rep.
308
, 1
(1998
).4.
D. Kaplan and L. Glass, Understanding Nonlinear Dynamics (Springer-Verlag, New York, 1995).
5.
E. Ott, Chaos in Dynamical Systems (Cambridge University Press, Cambridge, 1993).
6.
P. Bergé, Y. Pomeau, and C. Vidal, Order Within Chaos: Towards a Deterministic Approach to Turbulence (Wiley, New York, 1986).
7.
H.-G. Schuster, Deterministic Chaos: An Introduction (Physik-Verlag, Weinheim, 1988).
8.
A. Katok and B. Hasselblatt Introduction to the Modern Theory of Dynamical Systems (Cambridge University Press, Cambridge, 1996).
9.
E. Ott, T. Sauer, and J. A. Yorke, Coping with Chaos (Wiley, New York, 1994).
10.
H. D. I. Abarbanel, Analysis of Observed Chaotic Data (Springer-Verlag, New York, 1996).
11.
P.
Grassberger
, T.
Schreiber
, and C.
Schaffrath
, “Non-linear time sequence analysis
,” Int. J. Bifurcation Chaos Appl. Sci. Eng.
1
, 521
(1991
).12.
H. D. I.
Abarbanel
, R.
Brown
, J. J.
Sidorowich
, and L. Sh.
Tsimring
, “The analysis of observed chaotic data in physical systems
,” Rev. Mod. Phys.
65
, 1331
(1993
).13.
D.
Kugiumtzis
, B.
Lillekjendlie
, and N.
Christophersen
, “Chaotic time series I
,” Modeling Identification Control
15
, 205
(1994
).14.
D.
Kugiumtzis
, B.
Lillekjendlie
, and N.
Christophersen
, “Chaotic time series II
,” Modeling Identification Control
15
, 225
(1994
).15.
G. Mayer-Kress, in Dimensions and Entropies in Chaotic Systems (Springer-Verlag, Berlin, 1986).
16.
M. Casdagli and S. Eubank, in “Nonlinear modeling and forecasting,” Santa Fe Institute Studies in the Science of Complexity, Proceedings Vol. XII (Addison-Wesley, Reading, MA, 1992).
17.
A. S. Weigend and N. A. Gershenfeld, in “Time Series Prediction: Forecasting the future and understanding the past,” Santa Fe Institute Studies in the Science of Complexity, Proceedings Vol. XV (Addison-Wesley, Reading, MA, 1993).
18.
J. Bélair, L. Glass, U. an der Heiden, and J. Milton, in Dynamical Disease (AIP, Woodbury, NY, 1995).
19.
H. Kantz, J. Kurths, and G. Mayer-Kress, in Nonlinear Analysis of Physiological Data (Springer-Verlag, Berlin, 1998).
20.
T.
Schreiber
, “Efficient neighbor searching in nonlinear time series analysis
,” Int. J. Bifurcation Chaos Appl. Sci. Eng.
5
, 349
(1995
).21.
F. Takens, Detecting Strange Attractors in Turbulence, Lecture Notes in Mathematics Vol. 898 (Springer-Verlag, New York, 1981), Vol. 898.
22.
T.
Sauer
, J.
Yorke
, and M.
Casdagli
, “Embedology
,” J. Stat. Phys.
65
, 579
(1991
).23.
M.
Richter
and T.
Schreiber
, “Phase space embedding of electrocardiograms
,” Phys. Rev. E
58
, 6392
(1998
).24.
M.
Casdagli
, S.
Eubank
, J. D.
Farmer
, and J.
Gibson
, “State space reconstruction in the presence of noise
,” Physica D
51
, 52
(1991
).25.
A. M.
Fraser
and H. L.
Swinney
, “Independent coordinates for strange attractors from mutual information
,” Phys. Rev. A
33
, 1134
(1986
).26.
B.
Pompe
, “Measuring statistical dependences in a time series
,” J. Stat. Phys.
73
, 587
(1993
).27.
M.
Paluš
, “Testing for nonlinearity using redundancies: Quantitative and qualitative aspects
,” Physica D
80
, 186
(1995
).28.
M. B.
Kennel
, R.
Brown
, and H. D. I.
Abarbanel
, “Determining embedding dimension for phase-space reconstruction using a geometrical construction
,” Phys. Rev. A
45
, 3403
(1992
).29.
http://hpux.cs.utah.edu/hppd/hpux/Physics/embedding-26.May.93
30.
http://www.zweb.com/apnonlin/
31.
I. T. Jolliffe, Principal Component Analysis (Springer-Verlag, New York, 1986).
32.
D.
Broomhead
and G. P.
King
, “Extracting qualitative dynamics from experimental data
,” Physica D
20
, 217
(1986
).33.
W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes, 2nd ed. (Cambridge University Press, Cambridge, 1992).
34.
R.
Vautard
, P.
Yiou
, and M.
Ghil
, “Singular-spectrum analysis: a toolkit for short, noisy chaotic signals
,” Physica D
58
, 95
(1992
).35.
A.
Varone
, A.
Politi
, and M.
Ciofini
, “ laser with feedback
,” Phys. Rev. A
52
, 3176
(1995
).36.
R.
Hegger
and H.
Kantz
, “Embedding of sequences of time intervals
,” Europhys. Lett.
38
, 267
(1997
).37.
J. P.
Eckmann
, S.
Oliffson Kamphorst
, and D.
Ruelle
, “Recurrence plots of dynamical systems
,” Europhys. Lett.
4
, 973
(1987
).38.
M.
Casdagli
, “Recurrence plots revisited
,” Physica D
108
, 206
(1997
).39.
N. B.
Tufillaro
, P.
Wyckoff
, R.
Brown
, T.
Schreiber
, and T.
Molteno
, “Topological time series analysis of a string experiment and its synchronized model
,” Phys. Rev. E
51
, 164
(1995
).40.
41.
A.
Provenzale
, L. A.
Smith
, R.
Vio
, and G.
Murante
, “Distinguishing between low-dimensional dynamics and randomness in measured time series
,” Physica D
58
, 31
(1992
).42.
H. Tong, Threshold Models in Non-Linear Time Series Analysis, Lecture Notes in Statistics (Springer-Verlag, New York, 1983), Vol. 21.
43.
A.
Pikovsky
, “Discrete-time dynamic noise filtering
,” Sov. J. Commun. Technol. Electron.
31
, 81
(1986
).44.
G.
Sugihara
and R.
May
, “Nonlinear forecasting as a way of distinguishing chaos from measurement error in time series
,” Nature (London)
344
, 734
(1990
); reprinted in Ref. 9.45.
J.-P.
Eckmann
, S.
Oliffson Kamphorst
, D.
Ruelle
, and S.
Ciliberto
, “Lyapunov exponents from a time series
,” Phys. Rev. A
34
, 4971
(1986
); reprinted in Ref. 9.46.
J. D.
Farmer
and J.
Sidorowich
, “Predicting chaotic time series
,” Phys. Rev. Lett.
59
, 845
(1987
); reprinted in Ref. 9.47.
D.
Auerbach
, P.
Cvitanović
, J.-P.
Eckmann
, G.
Gunaratne
, and I.
Procaccia
, “Exploring chaotic motion through periodic orbits
,” Phys. Rev. Lett.
58
, 2387
(1987
).48.
O.
Biham
and W.
Wenzel
, “Characterization of unstable periodic orbits in chaotic attractors and repellers
,” Phys. Rev. Lett.
63
, 819
(1989
).49.
P.
So
, E.
Ott
, S. J.
Schiff
, D. T.
Kaplan
, T.
Sauer
, and C.
Grebogi
, “Detecting unstable periodic orbits in chaotic experimental data
,” Phys. Rev. Lett.
76
, 4705
(1996
).50.
P.
Schmelcher
and F. K.
Diakonos
, “A general approach to the finding of unstable periodic orbits in chaotic dynamical systems
,” Phys. Rev. E
57
, 2739
(1998
).51.
D.
Kugiumtzis
, O. C.
Lingjærde
, and N.
Christophersen
, “Regularized local linear prediction of chaotic time series
,” Physica D
112
, 344
(1998
).52.
L.
Jaeger
and H.
Kantz
, “Unbiased reconstruction of the dynamics underlying a noisy chaotic time series
,” Chaos
6
, 440
(1996
).53.
M.
Casdagli
, “Chaos and deterministic versus stochastic nonlinear modeling
,” J. R. Statist. Soc. B
54
, 303
(1991
).54.
D.
Broomhead
and D.
Lowe
, “Multivariable functional interpolation and adaptive networks
,” Complex Syst.
2
, 321
(1988
).55.
L. A.
Smith
, “Identification and prediction of low dimensional dynamics
,” Physica D
58
, 50
(1992
).56.
M.
Casdagli
, “Nonlinear prediction of chaotic time series
,” Physica D
35
, 335
(1989
);reprinted in Ref. 9.
57.
E. J.
Kostelich
and T.
Schreiber
, “Noise reduction in chaotic time series data: A survey of common methods
,” Phys. Rev. E
48
, 1752
(1993
).58.
M. E.
Davies
, “Noise reduction schemes for chaotic time series
,” Physica D
79
, 174
(1994
).59.
T.
Schreiber
, “Extremely simple nonlinear noise reduction method
,” Phys. Rev. E
47
, 2401
(1993
).60.
D. R. Rigney, A. L. Goldberger, W. Ocasio, Y. Ichimaru, G. B. Moody, and R. Mark, “Multi-channel physiological data: Description and analysis (Data set B),” in Ref. 17.
61.
P.
Grassberger
, R.
Hegger
, H.
Kantz
, C.
Schaffrath
, and T.
Schreiber
, “On noise reduction methods for chaotic data
,” Chaos
3
, 127
(1993
);reprinted in Ref. 9.
62.
H.
Kantz
, T.
Schreiber
, I.
Hoffmann
, T.
Buzug
, G.
Pfister
, L. G.
Flepp
, J.
Simonet
, R.
Badii
, and E.
Brun
, “Nonlinear noise reduction: a case study on experimental data
,” Phys. Rev. E
48
, 1529
(1993
).63.
M.
Finardi
, L.
Flepp
, J.
Parisi
, R.
Holzner
, R.
Badii
, and E.
Brun
, “Topological and metric analysis of heteroclinic crises in laser chaos
,” Phys. Rev. Lett.
68
, 2989
(1992
).64.
A. I.
Mees
and K.
Judd
, “Dangers of geometric filtering
,” Physica D
68
, 427
(1993
).65.
T. Schreiber and M. Richter, “Nonlinear projective filtering in a data stream,” Wuppertal preprint, 1998.
66.
M.
Richter
, T.
Schreiber
, and D. T.
Kaplan
, “Fetal ECG extraction with nonlinear phase space projections
,” IEEE Trans. Biomed. Eng.
45
, 133
(1998
).67.
J.-P.
Eckmann
and D.
Ruelle
, “Ergodic theory of chaos and strange attractors
,” Rev. Mod. Phys.
57
, 617
(1985
).68.
R.
Stoop
and J.
Parisi
, “Calculation of Lyapunov exponents avoiding spurious elements
,” Physica D
50
, 89
(1991
).69.
H.
Kantz
, “A robust method to estimate the maximal Lyapunov exponent of a time series
,” Phys. Lett. A
185
, 77
(1994
).70.
M. T.
Rosenstein
, J. J.
Collins
, and C. J.
De Luca
, “A practical method for calculating largest Lyapunov exponents from small data sets
,” Physica D
65
, 117
(1993
).71.
M.
Sano
and Y.
Sawada
, “Measurement of the Lyapunov spectrum from a chaotic time series
,” Phys. Rev. Lett.
55
, 1082
(1985
).72.
J. Kaplan and J. Yorke, “Chaotic behavior of multidimensional difference equations,” in Functional Differential Equations and Approximation of Fixed Points, edited by H. O. Peitgen and H. O. Walther (Springer-Verlag, New York, 1987).
73.
74.
T.
Sauer
and J.
Yorke
, “How many delay coordinates do you need?
,” Int. J. Bifurcation Chaos Appl. Sci. Eng.
3
, 737
(1993
).75.
76.
reprinted in Ref. 18.
77.
P.
Grassberger
, “Finite sample corrections to entropy and dimension estimates
,” Phys. Lett. A
128
, 369
(1988
).78.
F. Takens, in Dynamical Systems and Bifurcations, edited by B. L. J. Braaksma, H. W. Broer, and F. Takens, Lecture Notes in Mathematics (Springer-Verlag, Heidelberg, 1985), Vol. 1125.
79.
J.
Theiler
, “Lacunarity in a best estimator of fractal dimension
,” Phys. Lett. A
135
, 195
(1988
).80.
J. M.
Ghez
and S.
Vaienti
, “Integrated wavelets on fractal sets I: The correlation dimension
,” Nonlinearity
5
, 777
(1992
).81.
R.
Badii
and A.
Politi
, “Statistical description of chaotic attractors
,” J. Stat. Phys.
40
, 725
(1985
).82.
J.
Theiler
, S.
Eubank
, A.
Longtin
, B.
Galdrikian
, and J. D.
Farmer
, “Testing for nonlinearity in time series: The method of surrogate data
,” Physica D
58
, 77
(1992
);reprinted in Ref. 9.
83.
T.
Schreiber
and A.
Schmitz
, “Improved surrogate data for nonlinearity tests
,” Phys. Rev. Lett.
77
, 635
(1996
).84.
J.
Theiler
, P. S.
Linsay
, and D. M.
Rubin
, “Detecting nonlinearity in data with long coherence times,” in Ref. 17.85.
T.
Schreiber
, “Constrained randomization of time series data
,” Phys. Rev. Lett.
80
, 2105
(1998
).86.
T. Subba Rao and M. M. Gabr, An Introduction to Bispectral Analysis and Bilinear Time Series Models, Lecture Notes in Statistics (Springer-Verlag, New York, 1984), Vol. 24.
87.
C.
Diks
, J. C.
van Houwelingen
, F.
Takens
, and J.
DeGoede
, “Reversibility as a criterion for discriminating time series
,” Phys. Lett. A
201
, 221
(1995
).88.
T.
Schreiber
and A.
Schmitz
, “Discrimination power of measures for nonlinearity in a time series
,” Phys. Rev. E
55
, 5443
(1997
).89.
J.
Kadtke
, “Classification of highly noisy signals using global dynamical models
,” Phys. Lett. A
203
, 196
(1995
).90.
R.
Manuca
and R.
Savit
, “Stationarity and nonstationarity in time series analysis
,” Physica D
99
, 134
(1996
).91.
M. C.
Casdagli
, L. D.
Iasemidis
, R. S.
Savit
, R. L.
Gilmore
, S.
Roper
, and J. C.
Sackellares
, “Non-linearity in invasive EEG recordings from patients with temporal lobe epilepsy
,” Electroencephalogr. Clin. Neurophysiol.
102
, 98
(1997
).92.
T.
Schreiber
, “Detecting and analyzing nonstationarity in a time series using nonlinear cross predictions
,” Phys. Rev. Lett.
78
, 843
(1997
).
This content is only available via PDF.
© 1999 American Institute of Physics.
1999
American Institute of Physics
You do not currently have access to this content.