Divergences or similarity measures between probability distributions have become a very useful tool for studying different aspects of statistical objects, such as time series, networks, and images. Notably, not every divergence provides identical results when applied to the same problem. Therefore, it seems convenient to have the widest possible set of divergences to be applied to the problems under study. Besides this choice, an essential step in the analysis of every statistical object is the mapping of each one of their representing values into an alphabet of symbols conveniently chosen. In this work, we choose the family of divergences known as the Burbea–Rao centroids (BRCs). For the mapping of the original time series into a symbolic sequence, we work with the ordinal pattern scheme. We apply our proposals to analyze simulated and real time series and to real textured images. The main conclusion of our work is that the best BRC, at least in the studied cases, is the Jensen–Shannon divergence, besides the fact that it verifies some interesting formal properties.
REFERENCES
1.
R. A.
Fisher
, “Theory of statistical estimation,” in Mathematical Proceedings of the Cambridge Philosophical Society (Cambridge University Press, 1925), Vol. 22, pp. 700–725.2.
T. M.
Cover
and J. A.
Thomas
, Elements of Information Theory
(John Wiley & Sons
, 2012
).3.
M.
Parry
and E.
Fischbach
, “Probability distribution of distance in a uniform ellipsoid: Theory and applications to physics
,” J. Math. Phys.
41
(4
), 2417
–2433
(2000
). 4.
S. N.
Ondimu
and H.
Murase
, “Effect of probability-distance based Markovian texture extraction on discrimination in biological imaging
,” Comput. Electron. Agric.
63
(1
), 2
–12
(2008
). 5.
A.
Kostin
, “Probability distribution of distance between pairs of nearest stations in wireless network
,” Electron. Lett.
46
(18
), 1299
–1300
(2010
). 6.
A. P.
Majtey
, P. W.
Lamberti
, and D. P.
Prato
, “Jensen-Shannon divergence as a measure of distinguishability between mixed quantum states
,” Phys. Rev. A
72
(5
), 052310
(2005
). 7.
G. E.
Crooks
and D. A.
Sivak
, “Measures of trajectory ensemble disparity in nonequilibrium statistical dynamics
,” J. Stat. Mech.: Theory Exp.
2011
(06
), P06003
. 8.
T. M.
Osán
, D. G.
Bussandri
, and P. W.
Lamberti
, “Monoparametric family of metrics derived from classical Jensen–Shannon divergence
,” Physica A
495
, 336
–344
(2018
). 9.
A. P.
Majtey
, P. W.
Lamberti
, and A.
Plastino
, “A monoparametric family of metrics for statistical mechanics
,” Physica A
344
(3-4
), 547
–553
(2004
). 10.
I.
Csiszár
, “Information measures: A critical survey,” in Transactions of the Seventh Prague Conference on Information Theory, Statistical Decision Functions, Random Processes (Hungarian Academy of Sciences, 1974), pp. 73–86.11.
J.
Burbea
and C. R.
Rao
, “Entropy differential metric, distance and divergence measures in probability spaces: A unified approach
,” J. Multivar. Anal.
12
(4
), 575
–596
(1982
). 12.
S. M.
Ali
and S. D.
Silvey
, “A general class of coefficients of divergence of one distribution from another
,” J. R. Stat. Soc.: Ser. B
28
(1
), 131
–142
(1966
). 13.
C.
Bandt
and B.
Pompe
, “Permutation entropy: A natural complexity measure for time series
,” Phys. Rev. Lett.
88
(17
), 174102
(2002
). 14.
M.
Zanin
and F.
Olivares
, “Ordinal patterns-based methodologies for distinguishing chaos from noise in discrete time series
,” Commun. Phys.
4
(1
), 190
(2021
). 15.
L.
Zunino
, F.
Olivares
, H. V.
Ribeiro
, and O. A.
Rosso
, “Permutation Jensen-Shannon distance: A versatile and fast symbolic tool for complex time-series analysis
,” Phys. Rev. E
105
(4
), 045310
(2022
). 16.
H. V.
Ribeiro
, L.
Zunino
, E. K.
Lenzi
, P. A.
Santoro
, and R. S.
Mendes
, “Complexity-entropy causality plane as a complexity measure for two-dimensional patterns
,” PLoS One
7
(8
), e40689
(2012
). 17.
L.
Zunino
and H. V.
Ribeiro
, “Discriminating image textures with the multiscale two-dimensional complexity-entropy causality plane
,” Chaos, Solitons Fractals
91
, 679
–688
(2016
). 18.
L.
Zunino
, M. C.
Soriano
, I.
Fischer
, O. A.
Rosso
, and C. R.
Mirasso
, “Permutation-information-theory approach to unveil delay dynamics from time-series analysis
,” Phys. Rev. E
82
(4
), 046212
(2010
). 19.
L.
Zunino
, M. C.
Soriano
, and O. A.
Rosso
, “Distinguishing chaotic and stochastic dynamics from time series by using a multiscale symbolic approach
,” Phys. Rev. E
86
(4
), 046210
(2012
). 20.
M.
Paluš
, V.
Komárek
, Z.
Hrnčíř
, and K.
Štěrbová
, “Synchronization as adjustment of information rates: Detection from bivariate time series
,” Phys. Rev. E
63
(4
), 046211
(2001
). 21.
A.
Krakovská
, J.
Jakubík
, H.
Budáčová
, and M.
Holecyová
, “Causality studied in reconstructed state space. Examples of uni-directionally connected chaotic systems,” arXiv:1511.00505 (2015).22.
J. F.
Restrepo
, D. M.
Mateos
, and G.
Schlotthauer
, “Transfer entropy rate through Lempel-Ziv complexity
,” Phys. Rev. E
101
(5
), 052117
(2020
). 23.
R. M.
May
, “Simple mathematical models with very complicated dynamics,” in The Theory of Chaotic Attractors (Springer, 2004), pp. 85–93.24.
Z.
Wan-zhen
, D.
Ming-zhou
, and L.
Jia-nan
, “Symbolic description of periodic windows in the antisymmetric cubic map
,” Chin. Phys. Lett.
2
(7
), 293
(1985
). 25.
R. M.
Benca
, M.
Okawa
, M.
Uchiyama
, S.
Ozaki
, T.
Nakajima
, K.
Shibui
, and W. H.
Obermeyer
, “Sleep and mood disorders
,” Sleep Med. Rev.
1
(1
), 45
–56
(1997
). 26.
R.
Stickgold
, “Sleep-dependent memory consolidation
,” Nature
437
(7063
), 1272
–1278
(2005
). 27.
J. J.
Pilcher
and A. S.
Walters
, “How sleep deprivation affects psychological variables related to college students’ cognitive performance
,” J. Am. Coll. Health
46
(3
), 121
–126
(1997
). 28.
R.
Boostani
, F.
Karimzadeh
, and M.
Nami
, “A comparative review on sleep stage classification methods in patients and healthy individuals
,” Comput. Methods Programs Biomed.
140
, 77
–91
(2017
). 29.
R. D.
Ogilvie
, “The process of falling asleep
,” Sleep Med. Rev.
5
(3
), 247
–270
(2001
). 30.
G.
Tononi
and C.
Cirelli
, “Sleep function and synaptic homeostasis
,” Sleep Med. Rev.
10
(1
), 49
–62
(2006
). 31.
A. L.
Goldberger
, L.
Amaral
, L.
Glass
, J. M.
Hausdorff
, P. C.
Ivanov
, R. G.
Mark
, J. E.
Mietus
, G. B.
Moody
, C. K.
Peng
, and H. E.
Stanley
, “PhysioBank, PhysioToolkit, and PhysioNet: Circulation
,” Discovery
101
(23
), 1
(1997
).32.
B.
Kemp
, A. H.
Zwinderman
, B.
Tuk
, H. A. C.
Kamphuisen
, and J. J. L.
Oberye
, “Analysis of a sleep-dependent neuronal feedback loop: The slow-wave microcontinuity of the EEG
,” IEEE Trans. Biomed. Eng.
47
(9
), 1185
–1194
(2000
). 33.
“Sleep-EDF database expanded,” PhysioNet database; see https://physionet.org/content/sleep-edfx/1.0.0/.
34.
N.
Nicolaou
and J.
Georgiou
, “The use of permutation entropy to characterize sleep electroencephalograms
,” Clin. EEG Neurosci.
42
(1
), 24
–28
(2011
). 35.
Q.
Noirhomme
, M.
Boly
, V.
Bonhomme
, P.
Boveroux
, C.
Phillips
, P.
Peigneux
et al., “Bispectral index correlate with regional cerebral blood flow during sleep in distinct cortical and subcortical structures in humans
,” Arch. Ital. Biol.
147
(1/2
), 51
–57
(2009
). 36.
D. M.
Mateos
, J.
Gómez-Ramírez
, and O. A.
Rosso
, “Using time causal quantifiers to characterize sleep stages
,” Chaos, Solitons Fractals
146
, 110798
(2021
). 37.
S.
Abdelmounaime
and H.
Dong-Chen
, “New Brodatz-based image databases for grayscale color and multiband texture analysis
,” Int. Sch. Res. Not.
2013
, 876386
. 38.
I.
Grosse
, P.
Bernaola-Galván
, P.
Carpena
, R.
Román-Roldán
, J.
Oliver
, and H. E.
Stanley
, “Analysis of symbolic sequences using the Jensen-Shannon divergence
,” Phys. Rev. E
65
(4
), 041905
(2002
). © 2023 Author(s). Published under an exclusive license by AIP Publishing.
2023
Author(s)
You do not currently have access to this content.