Multiresolution wavelet analysis (MWA) is a powerful data processing tool that provides a characterization of complex signals over multiple time scales. Typically, the standard deviations of wavelet coefficients are computed depending on the resolution level and such quantities are used as measures for diagnosing different types of system behavior. To enhance the capabilities of this tool, we propose a combination of MWA with detrended fluctuation analysis (DFA) of detail wavelet coefficients. We find that such an MWA&DFA approach is capable of revealing the correlation features of wavelet coefficients in independent ranges of scales, which provide more information about the complex organization of datasets compared to variances or similar statistical measures of the standard MWA. Using this approach, we consider changes in the dynamics of coupled chaotic systems caused by transitions between different types of complex oscillations. We also demonstrate the potential of the MWA&DFA method for characterizing different physiological conditions by analyzing the electrical brain activity in mice.
REFERENCES
1.
S. G.
Mallat
, “A theory for multiresolution signal decomposition: The wavelet representation
,” IEEE Trans. Pattern Anal. Mach. Intell.
11
, 674
–693
(1989
). 2.
I.
Daubechies
, Ten Lectures on Wavelets
(Society for Industrial and Applied Mathematics
, Philadelphia, PA
, 1992
).3.
Y.
Meyer
, Wavelets-Algorithms and Applications
(Society for Industrial and Applied Mathematics
, Philadelphia, PA
, 1993
).4.
C.
Torrence
and G. P.
Compo
, “A practical guide to wavelet analysis
,” Bull. Am. Meteorol. Soc.
79
, 61
–78
(1998
). 5.
6.
G.
Beylkin
, R.
Coifman
, and V.
Rokhlin
, “Fast wavelet transforms and numerical algorithms I
,” Commun. Pure Appl. Math.
44
, 141
–183
(1991
). 7.
M. V.
Wickerhauser
, Adapted Wavelet Analysis: From Theory to Software
(A.K. Peters
, Wellesley, MA
, 1994
).8.
D. B.
Percival
and A. T.
Walden
, Wavelet Methods for Time Series Analysis
(Cambridge University Press
, Cambridge
, 2006
).9.
P. S.
Addison
, The Illustrated Wavelet Transform Handbook: Introductory Theory and Applications in Science, Engineering, Medicine and Finance
(CRC Press
, Boca Raton, FL
, 2017
).10.
W. J.
Staszewski
and D. M.
Wallace
, “Wavelet-based frequency response function for time-variant systems—An exploratory study
,” Mech. Syst. Signal Process.
47
, 35
–49
(2014
). 11.
B.
Li
and X.
Chen
, “Wavelet-based numerical analysis: A review and classification
,” Finite Elem. Anal. Des.
81
, 14
–31
(2014
). 12.
K. D.
Kompella
, V. G. R.
Mannam
, and S. R.
Rayapudi
, “DWT based bearing fault detection in induction motor using noise cancellation
,” J. Electr. Syst. Inf. Technol.
3
, 411
–427
(2016
). 13.
M. S.
Islam
, R.
Pears
, and B.
Bacic
, “A wavelet approach for precursor pattern detection in time series
,” J. Electr. Syst. Inf. Technol.
5
, 337
–348
(2018
). 14.
I. M.
Dremin
, V. I.
Furletov
, O. V.
Ivanov
, V. A.
Nechitailo
, and V. G.
Terziev
, “Precursors of stall and surge processes in gas turbines revealed by wavelet analysis
,” Control Eng. Pract.
10
(6
), 599
–604
(2002
). 15.
S.
Thurner
, M. C.
Feurstein
, and M. C.
Teich
, “Multiresolution wavelet analysis of heartbeat intervals discriminates healthy patients from those with cardiac pathology
,” Phys. Rev. Lett.
80
, 1544
–1547
(1998
). 16.
N. M.
Astafyeva
, I. M.
Dremin
, and K. A.
Kotelnikov
, “Pattern recognition in high multiplicity events
,” Modern Phys. Lett. A
12
(16
), 1185
–1191
(1997
). 17.
A. E.
Hramov
, A. A.
Koronovskii
, V. A.
Makarov
, A. N.
Pavlov
, and E.
Sitnikova
, Wavelets in Neuroscience
(Springer
, Berlin
, 2015
).18.
V. A.
Maksimenko
, A.
Pavlov
, A. E.
Runnova
, V.
Nedaivozov
, V.
Grubov
, A.
Koronovskii
, S. V.
Pchelintseva
, E.
Pitsik
, A. N.
Pisarchik
, and A. E.
Hramov
, “Nonlinear analysis of brain activity, associated with motor action and motor imaginary in untrained subjects
,” Nonlinear Dyn.
91
, 2803
–2817
(2018
). 19.
A. N.
Pavlov
, A. E.
Runnova
, V. A.
Maksimenko
, O. N.
Pavlova
, D. S.
Grishina
, and A. E.
Hramov
, “Detrended fluctuation analysis of EEG patterns associated with real and imaginary arm movements
,” Physica A
509
, 777
–782
(2018
). 20.
E.
Alickovic
, J.
Kevric
, and A.
Subasi
, “Performance evaluation of empirical mode decomposition, discrete wavelet transform, and wavelet packed decomposition for automated epileptic seizure detection and prediction
,” Biomed. Signal Process. Control
39
, 94
–102
(2018
). 21.
C.-K.
Peng
, S. V.
Buldyrev
, S.
Havlin
, M.
Simons
, H. E.
Stanley
, and A. L.
Goldberger
, “Mosaic organization of DNA nucleotides
,” Phys. Rev. E
49
, 1685
–1689
(1994
). 22.
C.-K.
Peng
, S.
Havlin
, H. E.
Stanley
, and A. L.
Goldberger
, “Quantification of scaling exponents and crossover phenomena in nonstationary heartbeat time series
,” Chaos
5
, 82
–87
(1995
). 23.
H. E.
Stanley
, L. A. N.
Amaral
, A. L.
Goldberger
, S.
Havlin
, P. C.
Ivanov
, and C.-K.
Peng
, “Statistical physics and physiology: Monofractal and multifractal approaches
,” Physica A
270
, 309
–324
(1999
). 24.
D. E.
Postnov
, T. E.
Vadivasova
, O. V.
Sosnovtseva
, A. G.
Balanov
, V. S.
Anishchenko
, and E.
Mosekilde
, “Role of multistability in the transition to chaotic phase synchronization
,” Chaos
9
, 227
(1999
). 25.
A. N.
Pavlov
, O. V.
Sosnovtseva
, and E.
Mosekilde
, “Scaling features of multimode motions in coupled chaotic oscillators
,” Chaos, Solitons Fractals
16
, 801
–810
(2003
). 26.
M.
Barfred
, E.
Mosekilde
, and N.-H.
Holstein-Rathlou
, “Bifurcation analysis of nephron pressure and flow regulation
,” Chaos
6
, 280
(1996
). 27.
E.
Mosekilde
, Topics in Nonlinear Dynamics: Applications to Physics, Biology and Economic Systems
(World Scientific
, Singapore
, 1996
).28.
W. H.
Press
, S. A.
Teukolsky
, W. T.
Vetterling
, and B. P.
Flannery
, Numerical Recipes, 3-rd Edition: The Art of Scientific Computing
(Cambridge University Press
, Cambridge
, 2007
).29.
A. N.
Pavlov
, O. V.
Semyachkina-Glushkovskaya
, Y.
Zhang
, O. A.
Bibikova
, O. N.
Pavlova
, Q.
Huang
, D.
Zhu
, P.
Li
, V. V.
Tuchin
, and Q.
Luo
, “Multiresolution analysis of pathological changes in cerebral venous dynamics in newborn mice with intracranial hemorrhage: Adrenorelated vasorelaxation
,” Phys. Meas.
35
, 1983
–1999
(2014
). 30.
A. N.
Pavlov
, D. S.
Grishina
, A. E.
Runnova
, V. A.
Maksimenko
, O. N.
Pavlova
, N. V.
Shchukovsky
, A. E.
Hramov
, and J.
Kurths
, “Recognition of electroencephalographic patterns related to human movements or mental intentions with multiresolution analysis
,” Chaos, Solitons Fractals
126
, 230
–235
(2019
). 31.
K.
Ivanova
and M.
Ausloos
, “Application of the detrended fluctuation analysis (DFA) method for describing cloud breaking
,” Physica A
274
, 349
–354
(1999
). 32.
P.
Talkner
and R. O.
Weber
, “Power spectrum and detrended fluctuation analysis: Application to daily temperatures
,” Phys. Rev. E
62
, 150
–160
(2000
). 33.
C.
Heneghan
and G.
McDarby
, “Establishing the relation between detrended fluctuation analysis and power spectral density analysis for stochastic processes
,” Phys. Rev. E
62
, 6103
–6110
(2000
). 34.
Q. D. Y.
Ma
, R. P.
Bartsch
, P.
Bernaola-Galván
, M.
Yoneyama
, and P. C.
Ivanov
, “Effect of extreme data loss on long-range correlated and anticorrelated signals quantified by detrended fluctuation analysis
,” Phys. Rev. E
81
, 031101
(2010
). 35.
O. N.
Pavlova
, A. S.
Abdurashitov
, M. V.
Ulanova
, N. A.
Shushunova
, and A. N.
Pavlov
, “Effects of missing data on characterization of complex dynamics from time series
,” Commun. Nonlinear Sci. Numer. Simulat.
66
, 31
–40
(2019
). 36.
N. S.
Frolov
, V. V.
Grubov
, V. A.
Maksimenko
, A.
Lüttjohann
, V. V.
Makarov
, A. N.
Pavlov
, E.
Sitnikova
, A. N.
Pisarchik
, J.
Kurths
, and A. E.
Hramov
, “Statistical properties and predictability of extreme epileptic events
,” Sci. Rep.
9
, 7243
(2019
). 37.
M. G.
Rosenblum
, A. S.
Pikovsky
, and J.
Kurths
, “Phase synchronization of chaotic oscillators
,” Phys. Rev. Lett.
76
, 1804
–1807
(1996
). 38.
M. G.
Rosenblum
, A. S.
Pikovsky
, and J.
Kurths
, “From phase to lag synchronization in coupled chaotic oscillators
,” Phys. Rev. Lett.
78
, 4193
–4196
(1997
). 39.
A. N.
Pavlov
, O. N.
Pavlova
, Y. K.
Mohammad
, and J.
Kurths
, “Characterization of the chaos–hyperchaos transition based on return times
,” Phys. Rev. E
91
, 022921
(2015
). 40.
O. V.
Sosnovtseva
, A. N.
Pavlov
, E.
Mosekilde
, and N.-H.
Holstein-Rathlou
, “Bimodal oscillations in nephron autoregulation
,” Phys. Rev. E
66
, 061909
(2002
). 41.
O. V.
Sosnovtseva
, A. N.
Pavlov
, E.
Mosekilde
, K.-P.
Yip
, N.-H.
Holstein-Rathlou
, and D. J.
Marsh
, “Synchronization among mechanisms of renal autoregulation is reduced in hypertensive rats
,” Am. J. Physiol. Renal. Physiol.
293
, F1545
–F1555
(2007
). 42.
D. E.
Postnov
, O. V.
Sosnovtseva
, E.
Mosekilde
, and N.-H.
Holstein-Rathlou
, “Cooperative phase dynamics in coupled nephrons
,” Int. J. Modern Phys. B
15
, 3079
–3098
(2001
). 43.
N. P.
Castellanos
and V. A.
Makarov
, “Recovering EEG brain signals: Artifact suppression with wavelet enhanced independent component analysis
,” J. Neurosci. Methods
158
, 300
–312
(2006
). 44.
A. N.
Pavlov
and V. S.
Anishchenko
, “Multifractal analysis of complex signals
,” Phys. Usp.
50
, 819
–834
(2007
). 45.
J. F.
Muzy
, E.
Bacry
, and A.
Arneodo
, “The multifractal formalism revisited with wavelets
,” Int. J. Bifurcation Chaos
4
, 245
–302
(1994
). 46.
G.
Medic
, M.
Wille
, and M. E.
Hemels
, “Short- and long-term health consequences of sleep disruption
,” Nat. Sci. Sleep
9
, 151
–161
(2017
). 47.
P.
Achermann
and A. A.
Borbély
, “Low-frequency ( Hz) oscillations in the human sleep electroencephalogram
,” Neuroscience
81
, 213
–222
(1997
). © 2021 Author(s).
2021
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