We investigate various estimators based on extreme value theory (EVT) for determining the local fractal dimension of chaotic dynamical systems. In the limit of an infinitely long time series of an ergodic system, the average of the local fractal dimension is the system’s global attractor dimension. The latter is an important quantity that relates to the number of effective degrees of freedom of the underlying dynamical system, and its estimation has been a central topic in the dynamical systems literature since the 1980s. In this work, we propose a framework that combines phase space recurrence analysis with EVT to estimate the local fractal dimension around a particular state of interest. While the EVT framework allows for the analysis of high-dimensional complex systems, such as the Earth’s climate, its effectiveness depends on robust statistical parameter estimation for the assumed extreme value distribution. In this study, we conduct a critical review of several EVT-based local fractal dimension estimators, analyzing and comparing their performance across a range of systems. Our results offer valuable insights for researchers employing the EVT-based estimates of the local fractal dimension, aiding in the selection of an appropriate estimator for their specific applications.
REFERENCES
1.
V.
Lucarini
, D.
Faranda
, A.
de Freitas
, J.
de Freitas
, M.
Holland
, T.
Kuna
, M.
Nicol
, M.
Todd
, and S.
Vaienti
, Extremes and Recurrence in Dynamical Systems, Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts (Wiley, 2016).2.
A. A.
Balkema
and L.
De Haan
, “Residual life time at great age
,” Ann. Probab.
2
, 792
–804
(1974
). 3.
J.
Pickands
, III, “Statistical inference using extreme order statistics
,” Ann. Stat.
3
, 119
–131
(1975
).4.
M.
Leadbetter
and H.
Rootzen
, “Extremal theory for stochastic processes
,” Ann. Probab.
16
, 431
–478
(1988
).5.
R. L.
Smith
and I.
Weissman
, “Estimating the extremal index
,” J. R. Stat. Soc., Series B
56
, 515
–528
(1994
).6.
D.
Faranda
, G.
Messori
, M. C.
Alvarez-Castro
, and P.
Yiou
, “Dynamical properties and extremes of northern hemisphere climate fields over the past 60 years
,” Nonlinear Processes Geophys.
24
, 713
–725
(2017
). 7.
S.
Buschow
and P.
Friederichs
, “Local dimension and recurrent circulation patterns in long-term climate simulations
,” Chaos
28
, 083124
(2018
). 8.
M.
Brunetti
, J.
Kasparian
, and C.
Vérard
, “Co-existing climate attractors in a coupled aquaplanet
,” Clim. Dyn.
53
, 6293
–6308
(2019
). 9.
D.
Faranda
, M. C.
Alvarez-Castro
, G.
Messori
, D.
Rodrigues
, and P.
Yiou
, “The hammam effect or how a warm ocean enhances large scale atmospheric predictability
,” Nat. Commun.
10
, 1316
(2019
). 10.
D.
Rodrigues
, M. C.
Alvarez-Castro
, G.
Messori
, P.
Yiou
, Y.
Robin
, and D.
Faranda
, “Dynamical properties of the North Atlantic atmospheric circulation in the past 150 years in CMIP5 models and the 20CRv2c reanalysis
,” J. Clim.
31
, 6097
–6111
(2018
). 11.
F.
Falasca
and A.
Bracco
, “Exploring the tropical pacific manifold in models and observations
,” Phys. Rev. X
12
, 021054
(2022
).12.
G.
Messori
, R.
Caballero
, and D.
Faranda
, “A dynamical systems approach to studying midlatitude weather extremes
,” Geophys. Res. Lett.
44
, 3346
–3354
, https://doi.org/10.1002/2017GL072879 (2017
). 13.
S.
Scher
and G.
Messori
, “Predicting weather forecast uncertainty with machine learning
,” Q. J. R. Meteorol. Soc.
144
, 2830
–2841
(2018
). 14.
A.
Hochman
, P.
Alpert
, T.
Harpaz
, H.
Saaroni
, and G.
Messori
, “A new dynamical systems perspective on atmospheric predictability: Eastern mediterranean weather regimes as a case study
,” Sci. Adv.
5
, eaau0936
(2019
). 15.
A.
Hochman
, P.
Alpert
, P.
Kunin
, D.
Rostkier-Edelstein
, T.
Harpaz
, H.
Saaroni
, and G.
Messori
, “The dynamics of cyclones in the twentyfirst century: The Eastern Mediterranean as an example
,” Clim. Dyn.
54
, 561
–574
(2019
).16.
A.
Hochman
, S.
Scher
, J.
Quinting
, J. G.
Pinto
, and G.
Messori
, “A new view of heat wave dynamics and predictability over the Eastern Mediterranean
,” Earth Syst. Dyn.
12
, 133
–149
(2021
). 17.
A.
Hochman
, S.
Scher
, J.
Quinting
, J. G.
Pinto
, and G.
Messori
, “Dynamics and predictability of cold spells over the Eastern Mediterranean
,” Clim. Dyn.
58
, 2047
–2064
(2022
). 18.
D.
Faranda
, G.
Messori
, and P.
Yiou
, “Dynamical proxies of North Atlantic predictability and extremes
,” Sci. Rep.
7
, 41278
(2017
). 19.
D.
Faranda
, Y.
Sato
, G.
Messori
, N. R.
Moloney
, and P.
Yiou
, “Minimal dynamical systems model of the northern hemisphere jet stream via embedding of climate data
,” Earth Syst. Dyn.
10
, 555
–567
(2019
). 20.
G.
Messori
, N.
Harnik
, E.
Madonna
, O.
Lachmy
, and D.
Faranda
, “A dynamical systems characterization of atmospheric jet regimes
,” Earth Syst. Dyn.
12
, 233
–251
(2021
). 21.
A.
Hochman
, G.
Messori
, J. F.
Quinting
, J. G.
Pinto
, and C. M.
Grams
, “Do Atlantic-European weather regimes physically exist?
,” Geophys. Res. Lett.
48
, e2021GL095574
, https://doi.org/10.1029/2021GL095574 (2021
). 22.
P.
De Luca
, G.
Messori
, F.
Pons
, and D.
Faranda
, “Dynamical systems theory sheds new light on compound climate extremes in Europe and Eastern North America
,” Q. J. R. Meteorol. Soc.
146
, 1636
(2020
).23.
K.
Giamalaki
, C.
Beaulieu
, S.
Henson
, A.
Martin
, H.
Kassem
, and D.
Faranda
, “Future intensification of extreme Aleutian low events and their climate impacts
,” Sci. Rep.
11
, 18395
(2021
). 24.
D.
Faranda
, G.
Messori
, P.
Yiou
, S.
Thao
, F.
Pons
, and B.
Dubrulle
, “Dynamical footprints of hurricanes in the tropical dynamics
,” Chaos
33
, 013101
(2023
). 25.
M.
Ginesta
, P.
Yiou
, G.
Messori
, and D.
Faranda
, “A methodology for attributing severe extratropical cyclones to climate change based on reanalysis data: The case study of storm Alex 2020
,” Clim. Dyn.
61
, 1
–25
(2022
).26.
G.
Messori
and D.
Faranda
, “Characterising and comparing different palaeoclimates with dynamical systems theory
,” Clim. Past
17
, 545
–563
(2021
). 27.
A.
Gualandi
, J.-P.
Avouac
, S.
Michel
, and D.
Faranda
, “The predictable chaos of slow earthquakes
,” Sci. Adv.
6
, eaaz5548
(2020
). 28.
F.
Hausdorff
, “Dimension und äußeres maß
,” Math. Ann.
79
, 157
–179
(1918
). 29.
C.
Nicolis
and G.
Nicolis
, “Is there a climatic attractor?
,” Nature
311
, 529
–532
(1984
). 30.
P.
Grassberger
, “Do climatic attractors exist?
,” Nature
323
, 609
(1986
). 31.
E. N.
Lorenz
, “Dimension of weather and climate attractors
,” Nature
353
, 241
–244
(1991
). 32.
F. M. E.
Pons
, G.
Messori
, M. C.
Alvarez-Castro
, and D.
Faranda
, “Sampling hyperspheres via extreme value theory: Implications for measuring attractor dimensions
,” J. Stat. Phys.
179
, 1698
–1717
(2020
). 33.
A. C. M.
Freitas
, J. M.
Freitas
, and M.
Todd
, “Hitting time statistics and extreme value theory
,” Probab. Theory Relat. Fields
147
, 675
–710
(2010
). 34.
D.
Faranda
, V.
Lucarini
, G.
Turchetti
, and S.
Vaienti
, “Numerical convergence of the block-maxima approach to the generalized extreme value distribution
,” J. Stat. Phys.
145
, 1156
–1180
(2011
). 35.
T.
Caby
, D.
Faranda
, G.
Mantica
, S.
Vaienti
, and P.
Yiou
, “Generalized dimensions, large deviations and the distribution of rare events,” arXiv:1812.00036 (2018).36.
T.
Carletti
and S.
Galatolo
, “Numerical estimates of local dimension by waiting time and quantitative recurrence
,” Physica A
364
, 120
–128
(2006
). 37.
R.
Vitolo
, M. P.
Holland
, and C. A.
Ferro
, “Robust extremes in chaotic deterministic systems
,” Chaos
19
, 043127
(2009
). 38.
P.
de Zea Bermudez
and S.
Kotz
, “Parameter estimation of the generalized Pareto distribution—Part I
,” J. Stat. Plann. Inf.
140
, 1353
–1373
(2010
). 39.
P.
de Zea Bermudez
and S.
Kotz
, “Parameter estimation of the generalized Pareto distribution—Part II
,” J. Stat. Plann. Inf.
140
, 1374
–1388
(2010
). 40.
41.
R. A.
Fisher
, “On an absolute criterion for fitting frequency curves,” in Messenger of Mathematics (Macmillan Publishers, 1912), Vol. 41, pp. 155–156. 42.
G.
Casella
and R. L.
Berger
, Statistical Inference
(Duxbury Pacific Grove
, CA
, 2002
), Vol. 2.43.
L.
Belzile
et al., mev: Modelling Extreme Values (2022), R package version 1.14.44.
S. D.
Grimshaw
, “Computing maximum likelihood estimates for the generalized Pareto distribution
,” Technometrics
35
, 185
–191
(1993
). 45.
J. A.
Villaseñor-Alva
and E.
González-Estrada
, “A bootstrap goodness of fit test for the generalized Pareto distribution,” in Computational Statistics and Data Analysis (Elsevier, 2009), pp. 3835–3841.46.
J. A.
Villaseñor-Alva
and E.
González-Estrada
, gPdtest: Bootstrap goodness-of-fit test for the generalized Pareto distribution (2012), R package version 0.4.47.
J. M.
Landwehr
, N.
Matalas
, and J.
Wallis
, “Probability weighted moments compared with some traditional techniques in estimating Gumbel Parameters and quantiles
,” Water Resources Res.
15
, 1055
–1064
, https://doi.org/10.1029/WR015i005p01055 (1979
). 48.
J. R. M.
Hosking
, J. R.
Wallis
, and E. F.
Wood
, “Estimation of the generalized extreme-value distribution by the method of probability-weighted moments
,” Technometrics
27
, 251
–261
(1985
). 49.
E. N.
Lorenz
, “Deterministic nonperiodic flow
,” J. Atmos. Sci.
20
, 130
–141
(1963
). 50.
H.
Hersbach
, B.
Bell
, P.
Berrisford
, S.
Hirahara
, A.
Horányi
, J.
Muñoz-Sabater
, J.
Nicolas
, C.
Peubey
, R.
Radu
, D.
Schepers
et al., “The ERA5 global reanalysis
,” Q. J. R. Meteorol. Soc.
146
, 1999
–2049
(2020
). 51.
P.
Grassberger
and I.
Procaccia
, “Characterization of strange attractors
,” Phys. Rev. Lett.
50
, 346
(1983
). 52.
D.
Viswanath
, “The fractal property of the Lorenz attractor
,” Phys. D
190
, 115
–128
(2004
). 53.
V. M.
Muggeo
, “Estimating regression models with unknown break-points
,” Stat. Med.
22
, 3055
–3071
(2003
). 54.
L.
Fery
, B.
Dubrulle
, B.
Podvin
, F.
Pons
, and D.
Faranda
, “Learning a weather dictionary of atmospheric patterns using latent dirichlet allocation
,” Geophys. Res. Lett.
49
, e2021GL096184
, https://doi.org/10.1029/2021GL096184 (2022
). 55.
F.
Pons
, G.
Messori
, and D.
Faranda
, “Stability of attractor local dimension estimates in non-Axiom A dynamical systems
,” Zenodo
(2023). © 2023 Author(s). Published under an exclusive license by AIP Publishing.
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