In this paper, a new method of state space reconstruction is proposed for the nonstationary time-series. The nonstationary time-series is first converted into its analytical form via the Hilbert transform, which retains both the nonstationarity and the nonlinear dynamics of the original time-series. The instantaneous phase angle θ is then extracted from the time-series. The first- and second-order derivatives θ˙, θ¨ of phase angle θ are calculated. It is mathematically proved that the vector field [θ,θ˙,θ¨] is the state space of the original time-series. The proposed method does not rely on the stationarity of the time-series, and it is available for both the stationary and nonstationary time-series. The simulation tests have been conducted on the stationary and nonstationary chaotic time-series, and a powerful tool, i.e., the scale-dependent Lyapunov exponent (SDLE), is introduced for the identification of nonstationarity and chaotic motion embedded in the time-series. The effectiveness of the proposed method is validated.

References

References
1.
Verdes
,
P. F.
,
Granitto
,
P. M.
,
Navone
,
H. D.
, and
Ceccatto
,
H. A.
,
2001
, “
Nonstationary Time-Series Analysis: Accurate Reconstruction of Driving Force
,”
Phys. Rev. Lett.
,
87
(
12
), p.
124101
.
2.
Cao
,
J. B.
,
2001
, “
Detecting Nonstationarity and State Transitions in a Time-Series
,”
Phys. Rev. E
,
63
(6), p.
066202
.
3.
Takens
,
F.
,
1981
, “
Detecting Strange Attractors in Turbulence
,”
Dynamical Systems and Turbulence
(Lecture Notes in Mathematics), Vol.
898
,
D. A.
Rand
and
L. S.
Young
, eds.,
Springer-Verlag
, Berlin, Germany, pp.
361
381
.
4.
Packard
,
N. H.
,
Crutchfield
,
J. P.
,
Farmer
,
J. D.
, and
Shaw
,
R. S.
,
1980
, “
Geometry From a Time-Series
,”
Phys. Rev. Lett.
,
45
(
1
), pp.
712
716
.
5.
Lai
,
Y. C.
,
2003
, “
Recent Developments in Time-Series Analysis
,”
Int. J. Bifurcation Chaos
,
13
(
6
), pp.
1383
1422
.
6.
Stark
,
J.
,
1999
, “
Delay Embeddings for Forced Systems. I. Deterministic Forcing
,”
J. Nonlinear Sci.
,
9
(
3
), pp.
255
332
.
7.
Figen
,
A. M.
,
Molkov
,
Y. I.
,
Mukhin
,
D. N.
, and
Loskutov
,
E. M.
,
2001
, “
Prognosis of Qualitative Behavior of a Dynamic System by the Observed Time-Series
,”
Radiophys. Quantum Electron.
,
44
(
5
), pp.
348
367
.
8.
Assireu
,
A. T.
,
Rosa
,
R. R.
,
Vijaykumar
,
N. L.
,
Lorenzzetti
,
J. A.
,
Rempel
,
E. L.
,
Ramos
,
F. M.
,
Abreu Sá
,
L. D.
,
Bolzan
,
M. J. A.
, and
Zanandrea
,
A.
,
2002
, “
Gradient Pattern Analysis of Short Nonstationary Time-Series: An Application to Lagrangian Data From Satellite Tracked Drifters
,”
Phys. D
,
168–169
, pp.
397
403
.
9.
Gribkov
,
D.
, and
Gribkova
,
V.
,
2000
, “
Learning Dynamics From Nonstationary Time-Series: Analysis of Electroencephalograms
,”
Phys. Rev. E
,
61
(
6
), pp.
6538
6545
.
10.
Chen
,
Y.
, and
Yang
,
H.
,
2012
, “
Multiscale Recurrence Analysis of Long-Term Nonlinear and Nonstationary Time Series
,”
Chaos, Solitons Fractals
,
45
(
7
), pp.
978
986
.
11.
Gao
,
J. B.
,
1999
, “
Recurrence Time Statistics for Chaotic Systems and Their Application
,”
Phys. Rev. Lett.
,
83
(
16
), pp.
3178
3181
.
12.
Gao
,
J. B.
, and
Cai
,
H. Q.
,
2000
, “
On the Structures and Quantification of Recurrence Plots
,”
Phys. Lett. A
,
270
(
1
), pp.
75
87
.
13.
Gao
,
J. B.
,
Cao
,
Y. H.
,
Gu
,
L. Y.
,
Harris
,
J. G.
, and
Principe
,
J. C.
,
2003
, “
Detection of Weak Transitions in Signal Dynamics Using Recurrence Time Statistics
,”
Phys. Lett. A
,
317
(
1
), pp.
64
72
.
14.
Gao
,
J. B.
,
Hu
,
J.
,
Mao
,
X.
, and
Perc
,
M.
,
2012
, “
Culturomics Meets Random Fractal Theory: Insights Into Long-Range Correlations of Social and Natural Phenomena Over the Past Two Centuries
,”
J. R. Soc., Interface
,
9
(
73
), pp.
1956
1964
.
15.
Gao
,
J. B.
,
Cao
,
Y. H.
,
Tung
,
W. W.
, and
Hu
,
J.
,
2007
,
Multiscale Analysis of Complex Time Series: Integration of Chaos and Random Fractal Theory and Beyond
,
Wiley
, Hoboken, NJ.
16.
Salisbury
,
J. I.
, and
Sun
,
Y.
,
2004
, “
Assessment of Chaotic Parameters in Nonstationary Electrocardiograms by Use of Empirical Mode Decomposition
,”
Ann. Biomed. Eng.
,
32
(
10
), pp.
1348
1354
.
17.
Huang
,
N. E.
,
Shen
,
Z.
,
Long
,
S. R.
,
Wu
,
M. C.
,
Shih
,
H. H.
,
Zheng
,
Q.
,
Yen
,
N. C.
,
Tung
,
C. C.
, and
Liu
,
H. H.
,
1998
, “
The Empirical Mode Decomposition and the Hilbert Spectrum for Nonlinear and Nonstationary Time-Series
,”
Proc. R. Soc. London A
,
454
(1971), pp.
903
995
.
18.
Flikkéma
,
P. G.
,
2002
, “
Spread-Spectrum Techniques for Wireless Communication
,”
IEEE Signal Process. Mag.
,
14
(
3
), pp.
26
36
.
19.
Magill
,
D. G.
,
Natali
,
F. D.
, and
Edwards
,
G. P.
,
1994
, “
Spread-Spectrum Technology for Commercial Applications
,”
Proc. IEEE
,
82
(
4
), pp.
572
583
.
20.
Itoh
,
M.
,
1999
, “
Spread Spectrum Communication Via Chaos
,”
Int. J. Bifurcation Chaos
,
9
(
01
), pp.
155
213
.
21.
Frei
,
M. G.
, and
Osorio
,
I.
,
2007
, “
Intrinsic Time-Scale Decomposition: Time–Frequency–Energy Analysis and Real-Time Filtering of Non-Stationary Signals
,”
Proc. R. Soc. A
,
463
(
2078
), pp.
321
342
.
22.
Ma
,
H. G.
,
Jiang
,
Q. B.
,
Liu
,
Z. Q.
,
Liu
,
G.
, and
Ma
,
Z. Y.
,
2010
, “
A Novel Blind Source Separation Method for Single-Channel Signal
,”
Signal Process.
,
90
(
12
), pp.
3232
3241
.
23.
Xu
,
F. F.
,
Chen
,
D. Y.
,
Zhang
,
H.
, and
Wang
,
F. F.
,
2015
, “
Modeling and Stability Analysis of a Fractional-Order Francis Hydro-Turbine Governing System
,”
Chaos, Solitons Fractals
,
75
, pp.
50
61
.
24.
Chen
,
D. Y.
,
Zhang
,
R. F.
,
Sprott
,
J. C.
,
Chen, H. T.
, and
Ma, X. Y.
,
2012
, “
Synchronization Between Integer-Order Chaotic Systems and a Class of Fractional-Order Chaotic Systems Via Sliding Mode Control
,”
Chaos
,
22
(
2
), p.
023130
.
25.
Chen
,
D. Y.
,
Wu
,
C.
,
Liu
,
C. F.
,
Yu, J. Y.
, and
Zhang, R. F.
,
2012
, “
Synchronization and Circuit Simulation of a New Double-Wing Chaos
,”
Nonlinear Dyn.
,
67
(
2
), pp.
1481
1504
.
26.
Chen
,
D. Y.
, and
Han
,
W. T.
,
2013
, “
Prediction of Multivariate Chaotic Time Series Via Radial Basis Function Neural Network
,”
Complexity
,
18
(
4
), pp.
55
66
.
27.
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
.
28.
Gao
,
J. B.
,
Hu
,
J.
,
Tung
,
W. W.
, and
Blasch
,
E.
,
2012
, “
Multiscale Analysis of Physiological Data by Scale-Dependent Lyapunov Exponent
,”
Front. Fractal Physiol.
,
2
(110), p.
110
.
29.
Shen
,
L. X.
,
Tay
,
F. E. H.
,
Qu
,
L. S.
, and
Shen
,
Y. D.
,
2000
, “
Fault Diagnosis Using Rough Sets Theory
,”
Comput. Ind.
,
43
(
1
), pp.
61
72
.
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