Abstract

Several pairs of algorithms were used to determine the phase space reconstruction parameters to analyze the dynamic characteristics of chaotic time series. The reconstructed phase trajectories were compared with the original phase trajectories of the Lorenz attractor, Rössler attractor, and Chen's attractor to obtain the optimum method for determining the phase space reconstruction parameters with high precision and efficiency. The research results show that the false nearest neighbor method and the complex autocorrelation method provided the best results. The saturated embedding dimension method based on the saturated correlation dimension method is proposed to calculate the time delay. Different time delays are obtained by changing the embedding dimension parameters of the complex autocorrelation method. The optimum time delay occurs at the point where the time delay is stable. The validity of the method is verified by combing the application of the correlation dimension, showing that the proposed method is suitable for the effective determination of the phase space reconstruction parameters.

References

1.
Xiong
,
Y.
, and
Zhao
,
H.
,
2019
, “
Chaotic Time Series Prediction Based on Long Short-Term Memory Neural Networks
,”
Chin. Sci.
,
49
(
12
), pp.
92
99
.10.1360/SSPMA-2019-0115
2.
Guo
,
X.
,
Sun
,
Y.
, and
Ren
,
J.
,
2020
, “
Low Dimensional Mid-Term Chaotic Time Series Prediction by Delay Parameterized Method
,”
Inf. Sci.
,
516
, pp.
1
19
.10.1016/j.ins.2019.12.021
3.
Farmer
,
J. D.
, and
Sidorowich
,
J. J.
,
1987
, “
Predicting Chaotic Time Series
,”
Phys. Rev. Lett
,
59
(
8
), pp.
845
848
.10.1103/PhysRevLett.59.845
4.
Packard
,
N. H.
,
Crutchfield
,
J. P.
,
Farmer
,
J. D.
, and
Shaw
,
R. S.
,
1980
, “
Geometry From a Time Series [J]
,”
Phys. Rev. Lett.
,
45
(
9
), pp.
712
716
.10.1103/PhysRevLett.45.712
5.
Duan
,
W.
,
Zhang
,
J.
,
Huang
,
W.
,
Shi
,
Y.
, and
Zhu
,
Y.
,
2001
, “
Study on the Derivative Reconstruction Method
,”
J. Sichuan Univ. (Eng. Sci. Ed.)
,
33
(
5
), pp.
102
106
.http://en.cnki.com.cn/Article_en/CJFDTOTAL-SCLH200105026.htm
6.
Chen
,
K.
, and
Han
,
B.
,
2005
, “
A Survey of State Space Reconstruction of Chaotic Time Series Analysis
,”
Comput. Sci.
,
4
, pp.
67
70
.10.1081/CEH-200044273
7.
Wang
,
Y.
, and
Xu
,
W.
,
2006
, “
The Methods and Performance of Phase Space Reconstruction for the Time Series in Lorenz System
,”
J. Vib. Eng.
,
9
(
2
), pp.
277
282
.10.1007/s11401-004-0494-5
8.
Camplani
,
M.
, and
Cannas
,
B.
,
2009
, “
The Role of the Embedding Dimension and Time Delay in Time Series Forecasting
,”
IFAC Proc. Vols.
,
42
(
7
), pp.
316
320
.10.3182/20090622-3-UK-3004.00059
9.
Takens
,
F.
,
1980
,
In Dynamical Systems and Turbulence
(Lecture Notes in Mathematics),
D. A.
Rand
and
L. S.
Young
, eds., Vol.
898
,
Springer-Verlag
,
New York
.
10.
Mane, R., 1981, “On the Dimension of the Compact Invariant Sets of Certain Nonlinear Maps,” Dynamical Systems and Turbulence, Warwick 1980, Lecture Notes in Mathematics, Vol. 898, Springer, New York, pp.
230
242
.
11.
Chen
,
N.
, and
Wei
,
J.
,
2011
, “
Application of Chaos Time Series Method in Groundwater Table Prediction
,”
J. Water Resour. Archit. Eng.
,
21
(
6
), pp.
1
4
.http://en.cnki.com.cn/Article_en/CJFDTOTAL-FSJS201106003.htm
12.
Liu
,
S.
,
Zhu
,
S.
, and
Yu
,
X.
,
2008
, “
Determinating the Embedding Dimension in Phase Space Reconstruction
,”
J. Harbin Eng. Univ.
,
29
(
4
), pp.
374
381
.http://en.cnki.com.cn/Article_en/CJFDTOTAL-HEBG200804012.htm
13.
Hu
,
Y.
, and
Chen
,
T.
,
2013
, “
Phase-Space Reconstruction Technology of Chaotic Attractor Based on C-C Method
,”
J. Electron. Meas. Instrum.
,
26
(
5
), pp.
425
430
.10.3724/SP.J.1187.2012.00425
14.
Tsonis
,
A.
, and
Elsner
,
J. B.
,
1988
, “
The Weather Attractor Over Very Short Timescales
,”
Nature
,
333
(
6173
), pp.
545
547
.10.1038/333545a0
15.
Lin
,
J.
,
Wang
,
Y.
,
Huang
,
Z.
, and
Shen
,
Z.
,
1999
, “
Selection of Proper Time-Delay in Phase Space Reconstruction of Speech Signals
,”
Signal Process.
,
15
(
3
), pp.
220
225
.http://en.cnki.com.cn/Article_en/CJFDTOTAL-XXCN199903004.htm
16.
Xu
,
Y.
,
Wang
,
B.
, and
Li
,
P.
,
2014
, “
Time-Delay Estimation for Phase Space Reconstruction Based on Detecting Nonlinear Correlation of a System
,”
Vib. Shock
,
33
(
8
), pp.
4
10
.10.13465/j.cnki.jvs.2014.08.002
17.
Ma
,
H.
, and
Han
,
C.
,
2006
, “
Selection of Embedding Dimension and Delay Time in Phase Space Reconstruction
,”
Front. Electr. Electron. Eng.
,
1
(
1
), pp.
111
114
.10.1007/s11460-005-0023-7
18.
Fraser
,
A. M.
, and
Swinney
,
H. L.
,
1986
, “
Independent Coordinates for Strange Attractors From Mutual Information
,”
Phys. Rev. A
,
33
(
2
), pp.
1134
1140
.10.1103/PhysRevA.33.1134
19.
Nichols
,
J. M.
, and
Nichols
,
J. D.
,
2001
, “
Attractor Reconstruction for Non-Linear Systems: A Methodological Note
,”
Math. Biosci.
,
171
(
1
), pp.
21
32
.10.1016/S0025-5564(01)00053-0
20.
Lei
,
M.
,
Wang
,
Z.
, and
Feng
,
Z.
,
2002
, “
A Method of Embedding Dimension Estimation Based on Symplectic Geometry
,”
Phys. Lett. A
,
303
(
2–3
), pp.
179
189
.10.1016/S0375-9601(02)01164-7
21.
Harikrishnan
,
K. P.
,
Jacob
,
R.
,
Misra
,
R.
, and
Ambika
,
G.
,
2017
, “
Determining the Minimum Embedding Dimension for State Space Reconstruction Through Recurrence Networks
,”
Ind. Acad. Sci. Conf. Ser. Bengaluru
,
1
(
1
), pp.
1
13
.10.29195/iascs.01.01.0004
22.
Xiu, C., Liu, X., and Zhang, Y., 2003. “Selection of Phase Space Reconstruction Delay Time and Embedding Dimension,”
J. Beijing University of Technology
, 23(2), pp. 219–224.10.3969/j.issn.1001-0645.2003.02.022
23.
Kennel
,
M. B.
, and
Abarbanel
,
H. D.
,
2002
, “
False Neighbors and False Strands: A Reliable Minimum Embedding Dimension Algorithm
,”
Phys. Rev. E
,
66
(
2
), p.
026209
.10.1103/PhysRevE.66.026209
24.
Maus
,
A.
, and
Sprott
,
J. C.
,
2011
, “
Neural Network Method for Determining Embedding Dimension of a Time Series
,”
Commun. Nonlinear Sci. Numer. Simulat.
,
16
(
8
), pp.
3294
3302
.10.1016/j.cnsns.2010.10.030
25.
Buzug
,
T.
, and
Pfister
,
G.
,
1992
, “
Optimal Delay Time and Embedding Dimension for Delay-Time Coordinates by Analysis of the Global Static and Local Dynamical Behavior of Strange Attractors
,”
Phys. Rev. A
,
45
(
10
), pp.
7073
7084
.10.1103/PhysRevA.45.7073
26.
Cao
,
L.
,
1997
, “
Practical Method for Determining the Minimum Embedding Dimension of a Scalar Time Series
,”
Phys. D
,
110
(
1–2
), pp.
43
50
.10.1016/S0167-2789(97)00118-8
27.
David
,
C.
,
2017
, “
Reliable Estimation of Minimum Embedding Dimension Through Statistical Analysis of Nearest Neighbors
,”
ASME J. Comput. Nonlinear Dyn.
,
12
(
5
), p.
051024
.10.1115/1.4036814
28.
Kennel
,
M. B.
,
Brown
,
R.
, and
Abarbanel
,
H. D.
,
1992
, “
Determining Embedding Dimension for Phase-Space Reconstruction Using a Geometrical Construction
,”
Phys. Rev. A
,
45
(
6
), pp.
3403
3411
.10.1103/PhysRevA.45.3403
29.
Cellucci
,
C. J.
,
Albano
,
A. M.
, and
Rapp
,
P. E.
,
2003
, “
Comparative Study of Embedding Methods
,”
Phys. Rev. E
,
67
(
6
), p.
066210
.10.1103/PhysRevE.67.066210
30.
Gao
,
J.
, and
Zheng
,
Z.
,
1993
, “
Local Exponential Divergence Plot and Optimal Embedding of a Chaotic Time Series
,”
Phys. Lett. A
,
181
(
2
), pp.
153
158
.10.1016/0375-9601(93)90913-K
31.
Goudarzi
,
S.
,
Anisi
,
M. H.
,
Kama
,
N.
,
Doctor
,
F.
,
Soleymani
,
S. A.
, and
Sangaiah
,
A. K.
,
2019
, “
Predictive Modelling of Building Energy Consumption Based on a Hybrid Nature-Inspired Optimization Algorithm
,”
Energy Build.
,
196
, pp.
83
93
.10.1016/j.enbuild.2019.05.031
32.
Yuan
,
Y.
,
Li
,
Y.
, and
Mandic
,
D. P.
,
2008
, “
A Comparison Analysis of Embedding Dimensions Between Normal and Epileptic EEG Time Series
,”
J. Physiol. Sci.
,
58
(
4
), pp.
239
247
.10.2170/physiolsci.RP004708
33.
Cecen
,
A. A.
, and
Erkal
,
C.
,
2008
, “
Effects of Trend and Periodicity on the Correlation Dimension and the Lyapunov Exponents
,”
Int. J. Bifurcation Chaos
,
18
(
12
), pp.
3679
3687
.10.1142/S0218127408022640
34.
Liu
,
S.
,
Ma
,
R.
,
Cong
,
R.
,
Wang
,
H.
, and
Zhao
,
H.
,
2012
, “
A New Approach for Embedding Dimension Determination Based on Empirical Mode Decomposition
,”
Kybernetes
,
41
(
9
), pp.
1176
1184
.10.1108/03684921211275180
35.
Rosenstein
,
M. T.
,
Collins
,
J. J.
, and
De Luca
,
C. J.
,
1993
, “
A Practical Method for Calculating Largest Lyapunov Exponents From Small Data Sets
,”
Phys. D
,
65
(
1–2
), pp.
117
134
.10.1016/0167-2789(93)90009-P
36.
Kim
,
H. S.
,
Eykholt
,
R.
, and
Salas
,
J. D.
,
1999
, “
Nonlinear Dynamics, Delay Times, and Embedding Windows
,”
Phys. D
,
127
(
1–2
), pp.
48
60
.10.1016/S0167-2789(98)00240-1
37.
Lu
,
Z.
,
Cai
,
Z.
, and
Jiang
,
K.
,
2007
, “
Determination of Embedding Parameters for Phase Space Reconstruction Based on Improved C-C Method
,”
J. Syst. Simul.
,
19
(
11
), pp.
2527
2529,
 2538.10.1360/jos182955
38.
Perinelli
,
A.
, and
Ricci
,
L.
,
2018
, “
Identification of Suitable Embedding Dimensions and Lags for Time Series Generated by Chaotic, Finite-Dimensional Systems
,”
Phys. Rev. E
,
98
(
5
), pp.
1
15
.10.1103/PhysRevE.98.052226
39.
Xie
,
Z.
,
2009
, “
Selection of Embedding Parameters in Phase Space Reconstruction
,”
China Science Technol. Inf.
,
4
, pp.
637
640
.10.1109/ICICTA.2009.868
40.
Ricci
,
L.
,
Perinelli
,
A.
, and
Franchi
,
M.
,
2020
, “
Asymptotic Behavior of the Time-Dependent Divergence Exponent
,”
Phys. Rev. E
,
101
(
4
), pp.
1
9
.10.1103/PhysRevE.101.042211
41.
Jiang
,
X.
, and
Adeli
,
H.
,
2003
, “
Fuzzy Clustering Approach for Accurate Embedding Dimension Identification in Chaotic Time Series
,”
Integr. Comput. Aided Eng.
,
10
(
3
), pp.
287
302
.10.3233/ICA-2003-10305
42.
Wolf
,
A.
, †
Swift
,
J. B.
,
Swinney
,
H. L.
, and
Vastano
,
J. A.
,
1985
, “
Determining Lyapunov Exponents From a Time Series
,”
Phys. D
,
16
(
3
), pp.
285
317
.10.1016/0167-2789(85)90011-9
43.
Takens
,
F.
,
1981
, “
Detecting Strange Attractors in Fluid Turbulence
,”
Lecture Notes in Mathematics
, 898(1), pp. 366–381.10.1007/BFb0091924
44.
Theiler
,
J.
,
1986
, “
Spurious Dimension From Correlation Algorithms Applied to Limited Time-Series Data
,”
Phys. Rev. A
,
34
(
3
), pp.
2427
2432
.10.1103/PhysRevA.34.2427
45.
Liebert
,
W.
,
Pawelzik
,
K.
, and
Schuster
,
H. G.
,
1991
, “
Optimal Embeddings of Chaotic Attractors From Topological Considerations
,”
EPL
,
14
(
6
), pp.
521
526
.10.1209/0295-5075/14/6/004
46.
Meng
,
L.
, and
Bi
,
Y.
,
2017
, “
Visual Analysis of Literature Review of Phase Space Reconstruction
,”
J. Syst. Simul.
,
29
(
12
), pp.
3167
3175
.10.16182/j.issn1004731x.joss.201712030
47.
Hong
,
M.
,
Wang
,
D.
,
Wang
,
Y.
,
Zeng
,
X.
,
Ge
,
S.
,
Yan
,
H.
, and
Singh
,
V. P.
,
2016
, “
Mid-and Long-Term Runoff Predictions by an Improved Phase-Space Reconstruction Model
,”
Environ. Res.
,
148
, pp.
560
573
.10.1016/j.envres.2015.11.024
48.
Rosenstein
,
M. T.
,
Collins
,
J. J.
, and
Luca
,
C. J. D.
,
1994
, “
Reconstruction Expansion as a Geometry-Based Framework for Choosing Proper Delay Times
,”
Phys. D Nonlinear Phenom.
,
73
(
1–2
), pp.
82
98
.10.1016/0167-2789(94)90226-7
49.
Yang
,
Z.
,
Wang
,
G.
, and
Chen
,
S.
,
1995
, “
Determination of Delay Time by Calculating Mutual Information With Equally Distance Space Cells
,”
Comput. Phys.
, (
4
), pp.
442
448
.
50.
Grassberger
,
P.
, and
Procaccia
,
I.
,
1983
, “
Characterization of Strange Attractors
,”
Phys. Rev. Lett.
,
50
(
5
), pp.
346
349
.10.1103/PhysRevLett.50.346
51.
Ding
,
C.
,
Zhu
,
H.
,
Sun
,
G. D.
,
Zhou
,
Y. K.
, and
Zuo
,
X.
,
2018
, “
Chaotic Characteristics and Attractor Evolution of Friction Noise During Friction Process
,”
Friction
,
6
(
1
), pp.
47
61
.
52.
Abarbanel
,
H. D. I.
,
Brown
,
R.
,
Sidorowich
,
J. J.
, and
Tsimring
,
S. L.
,
1993
, “
The Analysis of Observed Chaotic Data in Physical Systems
,”
Rev. Mod. Phys.
,
65
(
4
), pp.
1331
1392
.10.1103/RevModPhys.65.1331
53.
Chlouverakis
,
K. E.
, and
Sprott
,
J. C.
,
2005
, “
A Comparison of Correlation and Lyapunov Dimensions
,”
Phys. D Nonlinear Phenom.
,
200
(
1–2
), pp.
156
164
.10.1016/j.physd.2004.10.006
54.
Michalak
,
P. K.
,
2014
, “
How to Estimate the Correlation Dimension of High-Dimensional Signals
,”
Chaos Interdiscip. J. Nonlinear Sci.
,
24
(
3
), p.
033118
.10.1063/1.4891185
55.
Michalak
,
P. K.
,
2011
, “
Modifications of the Takens-Ellner Algorithm for Medium- and High-Dimensional Signals
,”
Phys. Rev. E
,
83
(
2
), p.
026206
.10.1103/PhysRevE.83.026206
56.
Zhang
,
Y.
, and
Ren
,
C.
,
2005
, “
The Methods to Confirm the Dimension of Reconstructed Phase Space
,”
J. Natl. Univ. Defense Sci. Technol.
, (
6
), pp.
101
105
.10.3969/j.issn.1001-2486.2005.06.022
57.
Li
,
T. F.
,
Yi
,
J.
,
Su
,
Y.
,
Hu
,
W.
, and
Gao
,
T.
,
2012
, “
Variable Selection for Nonlinear Modeling Based on False Nearest Neighbours in KPCA Subspace
,” ASME
J. Mech. Eng.
,
48
(
10
), pp.
192
220
.10.3901/JME.2012.10.192
58.
Liu
,
T.
,
Li
,
G.
,
Wei
,
H.
,
Mu
,
X.
, and
Xing
,
P. F.
,
2017
, “
Feature Parameter Extraction of a Friction Vibration Attractor Based on Singular Value Decomposition
,”
Vib. Shock
,
36
(
3
), pp.
172
175, 182
.http://en.cnki.com.cn/Article_en/CJFDTOTAL-ZDCJ201703027.htm
59.
Yuan
,
Z.
,
Duan
,
L.
, and
Wang
,
J.
,
2016
, “
Motor Rotor Imbalance Fault Recognition Based on Extreme Point SVD De-Noising and Correlation Dimension
,”
Pet. Sci. Bull.
,
1
(
3
), pp.
425
433
.http://en.cnki.com.cn/Article_en/CJFDTotal-SYKE201603012.htm
60.
Tang
,
L.
,
2019
, “
Application of Improved SVD Method for Dimension Reduction in High-Dimensional Dynamic System
,”
J. Northeast Pet. Univ.
,
43
(
2
), pp.
119
124 + 12
.http://en.cnki.com.cn/Article_en/CJFDTotal-DQSY201902013.htm
61.
Chen
,
Y.
,
Ma
,
J.
, and
Liu
,
Z.
,
1999
, “
The State Space Reconstruction Technology of Different Kinds of Chaotic Data Obtained From Dynamical System
,”
Acta Mech. Sin.
,
15
(
1
), pp.
82
92
.10.1007/BF02487904
62.
Abdi
,
H.
, and
Williams
,
L. J.
,
2010
, “
Principal Component Analysis
,”
Wiley Interdiscip. Rev.
,
2
(
4
), pp.
433
459
.10.1002/wics.101
63.
Ding
,
C.
,
Zhu
,
H.
,
Sun
,
G.
,
Zhou
,
Y.
, and
Zuo
,
X.
,
2018
, “
Chaotic Characteristics and Attractor Evolution of Friction Noise during Friction Process
,”
Friction
,
6
(
1
), pp.
47
61
.10.1007/s40544-017-0161-y
64.
Shirer
,
H. N.
,
Fosmire
,
C. J.
,
Wells
,
R.
, and
Suciu
,
L.
,
1997
, “
Estimating the Correlation Dimension of Atmospheric Time Series
,”
J. Atmos. Sci.
,
54
(
1
), pp.
211
230
.10.1175/1520-0469(1997)054<0211:ETCDOA>2.0.CO;2
65.
George
,
S. V.
,
Ambika
,
G.
, and
Misra
,
R.
,
2015
, “
Effect of Data Gaps on Correlation Dimension Computed From Light Curves of Variable Stars
,”
Astrophys. Space Sci.
,
360
(
1
), p.
5
.10.1007/s10509-015-2516-z
66.
Ji
,
C. C.
,
Zhu
,
H.
, and
Jiang
,
W.
,
2011
, “
A Novel Method to Identify the Scaling Region for Chaotic Time Series Correlation Dimension Calculation
,”
Chin. Sci. Bull.
,
56
(
9
), pp.
925
932
.10.1007/s11434-010-4180-6
67.
Qiao
,
M.
, and
Ma
,
X. A.
,
2012
, “
New Method on Solving Correlation Dimension of Chaotic Time-Series
,”
Intelligent Control & Automation
,
IEEE
, Beijing, China, Nov. 26.10.1109/WCICA.2012.6359391
68.
Zhao
,
L.
,
Zhou
,
Z.
,
Yin
,
Y.
,
Chen
,
R.
,
Liu
,
Q.
, and
Qin
,
W.
,
2014
, “
Feature Extraction of Rolling Bearing Fault Based on Ensemble Empirical Mode Decomposition and Correlation Dimension
,”
ASME
Paper No. MSEC2014-4070.10.1115/MSEC2014-4070
69.
J
,
L.
,
2002
, “
Chen's Chaotic Attractor and Its Characteristic Quantity
,”
Control Theory Appl.
,
19
(
2
), pp.
308
310
.http://en.cnki.com.cn/Article_en/CJFDTOTAL-KZLY200202035.htm
70.
Jiang
,
J. D.
,
Chen
,
J.
, and
Qu
,
L. S.
,
1999
, “
The Application of Correlation Dimension in Gearbox Condition Monitoring
,”
J. Sound Vib.
,
223
(
4
), pp.
529
541
.10.1006/jsvi.1998.2161
71.
Sprott
,
J. C.
, and
Rowlands
,
G.
,
2001
, “
Improved Correlation Dimension Calculation
,”
Int. J. Bifurcation Chaos
,
11
(
7
), pp.
1865
1880
.10.1142/S021812740100305X
You do not currently have access to this content.