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.

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.
H.
Abarbanel
,
Analysis of Observed Chaotic Data
(
Springer
,
1996
).
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.
H.
Schuster
and
W.
Just
,
Deterministic Chaos
(
Wiley
,
2005
).
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.
H.
Petersen
,
Ergodic Theory
(
Cambridge University Press
,
1989
).
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.
T.
Bass
,
The Eudaemonic Pie
(
Penguin
,
New York
,
1992
).
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.
J.
Garland
and
E.
Bradley
, “
Prediction in projection
,” preprint arXiv:1503.01678 (
2015
).
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
).
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