In 1980 and 1981, two pioneering papers laid the foundation for what became known as nonlinear time-series analysis: the analysis of observed data—typically univariate—via dynamical systems theory. Based on the concept of state-space reconstruction, this set of methods allows us to compute characteristic quantities such as Lyapunov exponents and fractal dimensions, to predict the future course of the time series, and even to reconstruct the equations of motion in some cases. In practice, however, there are a number of issues that restrict the power of this approach: whether the signal accurately and thoroughly samples the dynamics, for instance, and whether it contains noise. Moreover, the numerical algorithms that we use to instantiate these ideas are not perfect; they involve approximations, scale parameters, and finite-precision arithmetic, among other things. Even so, nonlinear time-series analysis has been used to great advantage on thousands of real and synthetic data sets from a wide variety of systems ranging from roulette wheels to lasers to the human heart. Even in cases where the data do not meet the mathematical or algorithmic requirements to assure full topological conjugacy, the results of nonlinear time-series analysis can be helpful in understanding, characterizing, and predicting dynamical systems.
References
1.
G. E. P.
Box
and F. M.
Jenkins
, Time Series Analysis: Forecasting and Control
, 2nd ed. (Holden Day
, 1976
).2.
H.
Hurst
, “Long-term storage of water reservoirs
,” Trans. Am. Soc. Civ. Eng.
116
, 509
(1951
).3.
C.
Peng
, S.
Buldyrev
, S.
Havlin
, M.
Simons
, H.
Stanley
, and A.
Goldberger
, “Mosaic organization of DNA nucleotides
,” Phys. Rev. E
49
, 1685
(1994
).4.
5.
H.
Kantz
and T.
Schreiber
, Nonlinear Time Series Analysis
(Cambridge University Press
, 2004
).6.
J.
Crutchfield
, “Prediction and stability in classical mechanics
,” senior undergraduate thesis (University of California
, Santa Cruz, 1979
).7.
N.
Packard
, J.
Crutchfield
, J.
Farmer
, and R.
Shaw
, “Geometry from a time series
,” Phys. Rev. Lett.
45
, 712
(1980
).8.
F.
Takens
, “Detecting strange attractors in fluid turbulence
,” in Dynamical Systems and Turbulence
, edited by D.
Rand
and L.-S.
Young
(Springer
, Berlin
, 1981
), pp. 366
–381
.9.
T.
Sauer
, J.
Yorke
, and M.
Casdagli
, “Embedology
,” J. Stat. Phys.
65
, 579
–616
(1991
).10.
R.
Hegger
, H.
Kantz
, and T.
Schreiber
, “Practical implementation of nonlinear time series methods: The TISEAN package
,” Chaos
9
, 413
–435
(1999
).11.
A.
Fraser
and H.
Swinney
, “Independent coordinates for strange attractors from mutual information
,” Phys. Rev. A
33
, 1134
–1140
(1986
).12.
W.
Liebert
and H.
Schuster
, “Proper choice of the time delay for the analysis of chaotic time series
,” Phys. Lett. A
142
, 107
–111
(1989
).13.
P.
Grassberger
and I.
Procaccia
, “Measuring the strangeness of strange attractors
,” Physica D
9
, 189
–208
(1983
).14.
M.
Casdagli
, S.
Eubank
, J.
Farmer
, and J.
Gibson
, “State space reconstruction in the presence of noise
,” Physica D
51
, 52
–98
(1991
).15.
M. B.
Kennel
, R.
Brown
, and H. D. I.
Abarbanel
, “Determining minimum embedding dimension using a geometrical construction
,” Phys. Rev. A
45
, 3403
–3411
(1992
).16.
W.
Liebert
, K.
Pawelzik
, and H.
Schuster
, “Optimal embeddings of chaotic attractors from topological considerations
,” Europhys. Lett.
14
, 521
(1991
).17.
L.
Pecora
, L.
Moniz
, J.
Nichols
, and T.
Carroll
, “A unified approach to attractor reconstruction
,” Chaos
17
, 013110
(2007
).18.
P.
Grassberger
, T.
Schreiber
, and C.
Schaffrath
, “Nonlinear time sequence analysis
,” Int. J. Bifurcation Chaos
1
, 521
(1991
).19.
P.
Grassberger
and I.
Procaccia
, “Measuring the strangeness of strange attractors
,” Physica D
9
, 189
(1983
).20.
T.
Sauer
and J.
Yorke
, “How many delay coordinates do you need?
,” Int. J. Bifurcation Chaos
3
, 737
(1993
).21.
J.
Theiler
, “Spurious dimension from correlation algorithms applied to limited time series data
,” Phys. Rev. E
34
, 2427
(1986
).22.
P.
Grassberger
, “Finite sample corrections to entropy and dimension estimates
,” Phys. Lett. A
128
, 369
(1988
).23.
E.
Olbrich
and H.
Kantz
, “Inferring chaotic dynamics from time series: On which length scale determinism becomes visible
,” Phys. Lett. A
232
, 63
–69
(1997
).24.
L.
Smith
, “Intrinsic limits on dimension calculations
,” Phys. Lett. A
133
, 283
–288
(1988
).25.
J.
Eckmann
, S.
Oliffson-Kamphorst
, D.
Ruelle
, and S.
Ciliberto
, “Lyapunov exponents from time series
,” Phys. Rev. A
34
, 4971
(1986
).26.
A.
Wolf
, J.
Swift
, H.
Swinney
, and J.
Vastano
, “Determining Lyapunov exponents from time series
,” Physica D
16
, 285
(1985
).27.
M.
Sano
and Y.
Sawada
, “Measurement of the Lyapunov spectrum from a chaotic time series
,” Phys. Rev. Lett.
55
, 1082
(1985
).28.
T.
Sauer
, J.
Tempkin
, and J.
Yorke
, “Spurious Lyapunov exponents in attractor reconstruction
,” Phys. Rev. Lett.
81
, 4341
(1998
).29.
H.-L.
Yang
, G.
Radons
, and H.
Kantz
, “Covariant Lyapunov vectors from reconstructed dynamics: The geometry behind true and spurious Lyapunov exponents
,” Phys. Rev. Lett.
109
, 244101
(2012
).30.
Y.
Pesin
, “Characteristic Lyapunov exponents and smooth ergodic theory
,” Russ. Math. Sur.
32
, 55
(1977
).31.
32.
T.
Schreiber
and A.
Schmitz
, “Discriminating power of measures for nonlinearity in a time series
,” Phys. Rev. E
55
, 5443
(1997
).33.
J.
Theiler
, S.
Eubank
, A.
Longtin
, B.
Galdrikian
, and J.
Farmer
, “Testing for nonlinearity in time series: The method of surrogate data
,” Physica D
58
, 77
–94
(1992
).34.
T.
Schreiber
and A.
Schmitz
, “Improved surrogate data for nonlinearity tests
,” Phys. Rev. Lett.
77
, 635
(1996
).35.
T.
Schreiber
and A.
Schmitz
, “Surrogate time series
,” Physica D
142
, 346
–382
(2000
).36.
C. E.
Shannon
, “Prediction and entropy of printed English
,” Bell Syst. Tech. J.
30
, 50
–64
(1951
).37.
38.
D.
Lind
and B.
Marcus
, An Introduction to Symbolic Dynamics and Coding
(Cambridge University Press
, 1995
).39.
C.
Bandt
and B.
Pompe
, “Permutation entropy: A natural complexity measure for time series
,” Phys. Rev. Lett.
88
, 174102
(2002
).40.
J.
Amigó
, Permutation Complexity in Dynamical Systems: Ordinal Patterns, Permutation Entropy and All That
(Springer
, 2012
).41.
J.-P.
Eckmann
, S.
Kamphorst
, and D.
Ruelle
, “Recurrence plots of dynamical systems
,” Europhys. Lett.
4
, 973
–977
(1987
).42.
E.
Bradley
and R.
Mantilla
, “Recurrence plots and unstable periodic orbits
,” Chaos
12
, 596
–600
(2002
).43.
J.
Zbilut
and C.
Webber
, “Embeddings and delays as derived from recurrence quantification analysis
,” Phys. Lett. A
171
, 199
–203
(1992
).44.
C.
Webber
and J.
Zbilut
, “Dynamical assessment of physiological systems and states using recurrence plot strategies
,” J. Appl. Physiol.
76
, 965
–973
(1994
).45.
N.
Marwan
, M.
Romano
, M.
Thiel
, and J.
Kurths
, “Recurrence plots for the analysis of complex systems
,” Phys. Rep.
438
, 237
(2007
).46.
R.
Donner
, M.
Small
, J.
Donges
, N.
Marwan
, Y.
Zou
, R.
Xiang
, and J.
Kurths
, “Recurrence-based time series analysis by means of complex network methods
,” Int. J. Bifurcation Chaos
21
, 1019
–1046
(2011
).47.
J.-P.
Eckmann
and D.
Ruelle
, “Ergodic theory of chaos and strange attractors
,” Rev. Mod. Phys.
57
, 617
(1985
).48.
E.
Lorenz
, “Atmospheric predictability as revealed by naturally occurring analogues
,” J. Atmos. Sci.
26
, 636
–646
(1969
).49.
A.
Pikovsky
, “Noise filtering in the discrete time dynamical systems
,” Sov. J. Commun. Technol. Electron.
31
, 911
–914
(1986
).50.
51.
M.
Small
and C.
Tse
, “Predicting the outcome of roulette
,” Chaos
22
, 033150
(2012
).52.
Nonlinear Modeling and Forecasting
, edited by M.
Casdagli
and S.
Eubank
(Addison Wesley
, 1992
).53.
J.
Farmer
and J.
Sidorowich
, “Predicting chaotic time series
,” Phys. Rev. Lett.
59
, 845
–848
(1987
).54.
M.
Casdagli
, “Nonlinear prediction of chaotic time series
,” Physica D
35
, 335
–356
(1989
).55.
G.
Sugihara
and R.
May
, “Nonlinear forecasting as a way of distinguishing chaos from measurement error in time series
,” Nature
344
, 734
–741
(1990
).56.
Time Series Prediction: Forecasting the Future and Understanding the Past
, edited by A.
Weigend
and N.
Gershenfeld
(Santa Fe Institute Studies in the Sciences of Complexity
, Santa Fe, NM
, 1993
).57.
F.
Paparella
, A.
Provenzale
, L.
Smith
, C.
Taricco
, and R.
Vio
, “Local random analogue prediction of nonlinear processes
,” Phys. Lett. A
235
, 233
–240
(1997
).58.
M.
Ragwitz
and H.
Kantz
, “Markov models from data by simple nonlinear time series predictors in delay embedding spaces
,” Phys. Rev. E
65
, 056201
(2002
).59.
60.
M.
Cencini
, M.
Falcioni
, E.
Olbrich
, H.
Kantz
, and A.
Vulpiani
, “Chaos or noise: Difficulties of a distinction
,” Phys. Rev. E
62
, 427
(2000
).61.
J.
Theiler
and S.
Eubank
, “Don't bleach chaotic data
,” Chaos
3
, 771
–782
(1993
).62.
J.
Farmer
and J.
Sidorowich
, “Exploiting chaos to predict the future and reduce noise
,” in Evolution, Learning and Cognition
(World Scientific
, 1988
).63.
E.
Kostelich
and J.
Yorke
, “Noise reduction in dynamical systems
,” Phys. Rev. A
38
, 1649
–1652
(1988
).64.
P.
Grassberger
, R.
Hegger
, H.
Kantz
, C.
Schaffrath
, and T.
Schreiber
, “On noise reduction methods for chaotic data
,” Chaos
3
, 127
(1993
).65.
V.
Robins
, N.
Rooney
, and E.
Bradley
, “Topology-based signal separation
,” Chaos
14
, 305
–316
(2004
).66.
A.
Tsonis
, J.
Elsner
, and K.
Georgakakos
, “Estimating the dimension of weather and climate attractors: Important issues about the procedure and interpretation
,” J. Atmos. Sci.
50
, 2549
–2555
(1993
).67.
J.-P.
Eckmann
and D.
Ruelle
, “Fundamental limitations for estimating dimensions and Lyapunov exponents in dynamical systems
,” Physica D
56
, 185
–187
(1992
).68.
M.
Bär
, R.
Hegger
, and H.
Kantz
, “Fitting partial differential equations to space-time dynamics
,” Phys. Rev. E
59
, 337
(1999
).69.
U.
Parlitz
and C.
Merkwirth
, “Prediction of spatiotemporal time series based on reconstructed local states
,” Phys. Rev. Lett.
84
, 1890
(2000
).70.
T.
Sauer
, “Interspike interval embedding of chaotic signals
,” Chaos
5
, 127
(1995
).71.
R.
Hegger
and H.
Kantz
, “Embedding of sequences of time intervals
,” Europhys. Lett.
38
, 267
–272
(1997
).72.
D.
Holstein
and H.
Kantz
, “Optimal Markov approximations and generalized embeddings
,” Phys. Rev. E
79
, 056202
(2009
).73.
E.
Aurell
, G.
Boffetta
, A.
Crisanti
, G.
Paladin
, and A.
Vulpiani
, “Predictability in the large: An extension of the concept of Lyapunov exponent
,” J. Phys. A
30
, 1
(1997
).© 2015 AIP Publishing LLC.
2015
AIP Publishing LLC
You do not currently have access to this content.
