Extreme events capture the attention and imagination of the general public. Extreme events, especially meteorological and climatological extremes, cause significant economic damages and lead to a significant number of casualties each year. Thus, the prediction of extremes is of obvious importance. Here, I will survey the predictive skill and the predictability of extremes using dynamic-stochastic models. These dynamic-stochastic models combine deterministic nonlinear dynamics with a stochastic component, which consists potentially of both additive and multiplicative noise components. In these models, extremes are created by either the nonlinear dynamics, multiplicative noise, or additive heavy-tailed noises. These models naturally capture the observed clustering of extremes and can be used for the prediction of extremes.

1.
D. M.
Abramson
and
I.
Redlener
, “
Hurricane sandy: Lessons learned, again
,”
Disaster Med. Public Health Preparedness
6
,
328
329
(
2012
).
2.
P.
Embrechts
, “
Extremes in economics and the economics of extremes
,” in
Extreme Values in Finance, Telecommunications, and the Environment
(
Chapman & Hall, CRC Press
,
2003
), pp.
169
83
.
3.
Y.
Malevergne
and
D.
Sornette
,
Extreme Financial Risks: From Dependence to Risk Management
, 1st ed. (
Springer Science & Business Media
,
2006
).
4.
A. J.
McNeil
,
R.
Frey
, and
P.
Embrechts
,
Quantitative Risk Management: Concepts, Techniques and Tools
(
Princeton University Press
,
2015
).
5.
N. N.
Taleb
,
The Black Swan: The Impact of the Highly Improbable
(
Random House
,
2007
).
6.
G. A.
Meehl
,
K.
Thomas
,
D. R.
Easterling
,
S.
Changnon
 et al, “
An introduction to trends in extreme weather and climate events: observations, socioeconomic impacts, terrestrial ecological impacts, and model projections
,”
Bull. Am. Meteorol. Soc.
81
,
413
(
2000
).
7.
R. W.
Katz
,
G. S.
Brush
, and
M. B.
Parlange
, “
Statistics of extremes: Modeling ecological disturbances
,”
Ecology
86
,
1124
1134
(
2005
).
8.
E.
Castillo
, “
Extremes in engineering applications
,” in
Extreme Value Theory and Applications
(
Springer
,
1994
), pp.
15
42
.
9.
E.
Castillo
,
A. S.
Hadi
,
N.
Balakrishnan
, and
J.-M.
Sarabia
,
Extreme Value and Related Models with Applications in Engineering and Science
(
Wiley
,
Hoboken, NJ
,
2005
).
10.
S.
Coles
,
An Introduction to Statistical Modeling of Extreme Values
(
Springer
,
2001
), Vol.
208
.
11.
D.
Sornette
,
Critical Phenomena in Natural Sciences: Chaos, Fractals, Selforganization and Disorder: Concepts and Tools
, 2nd ed. (
Springer Science & Business Media
,
2006
).
12.
A. C.
Davison
and
R.
Huser
, “
Statistics of extremes
,”
Annu. Rev. Stat. Appl.
2
,
203
235
(
2015
).
13.
P.
Embrechts
,
C.
Klüppelberg
, and
T.
Mikosch
,
Modelling Extremal Events: For Insurance and Finance
(
Springer Science & Business Media
,
2013
), Vol.
33
.
14.
M.
Leadbetter
and
H.
Rootzen
, “
Extremal theory for stochastic processes
,”
Ann. Probab.
16
,
431
478
(
1988
).
15.
A.
Bunde
,
J. F.
Eichner
,
S.
Havlin
, and
J. W.
Kantelhardt
, “
The effect of long-term correlations on the return periods of rare events
,”
Physica A
330
,
1
7
(
2003
).
16.
A.
Bunde
,
J. F.
Eichner
,
J. W.
Kantelhardt
, and
S.
Havlin
, “
Long-term memory: A natural mechanism for the clustering of extreme events and anomalous residual times in climate records
,”
Phys. Rev. Lett.
94
,
048701
(
2005
).
17.
C. L. E.
Franzke
, “
Persistent regimes and extreme events of the north atlantic atmospheric circulation
,”
Philos. Trans. R. Soc. A
371
,
20110471
(
2013
).
18.
A. C. M.
Freitas
,
J. M.
Freitas
, and
M.
Todd
, “
Extreme value laws in dynamical systems for non-smooth observations
,”
J. Stat. Phys.
142
,
108
126
(
2011
).
19.
M. P.
Holland
,
R.
Vitolo
,
P.
Rabassa
,
A. E.
Sterk
, and
H. W.
Broer
, “
Extreme value laws in dynamical systems under physical observables
,”
Physica D
241
,
497
513
(
2012
).
20.
V.
Lucarini
,
D.
Faranda
, and
J.
Wouters
, “
Universal behaviour of extreme value statistics for selected observables of dynamical systems
,”
J. Stat. Phys.
147
,
63
73
(
2012
).
21.
V.
Lucarini
,
D.
Faranda
,
A. C.
Freitas
,
J. M.
Freitas
,
M.
Holland
,
T.
Kuna
,
M.
Nicol
,
M.
Todd
, and
S.
Vaienti
,
Extremes and Recurrence in Dynamical Systems
(
John Wiley & Sons
,
2016
).
22.
C. L. E.
Franzke
,
A. J.
Majda
, and
E.
Vanden-Eijnden
, “
Low-order stochastic mode reduction for a realistic barotropic model climate
,”
J. Atmos. Sci.
62
,
1722
1745
(
2005
).
23.
C. L. E.
Franzke
and
A. J.
Majda
, “
Low-order stochastic mode reduction for a prototype atmospheric GCM
,”
J. Atmos. Sci.
63
,
457
479
(
2006
).
24.
C. L. E.
Franzke
, “
Predictability of extreme events in a nonlinear stochastic-dynamical model
,”
Phys. Rev. E
85
,
031134
(
2012
).
25.
C. L. E.
Franzke
, “
Predictions of critical transitions with non-stationary reduced order models
,”
Physica D
262
,
35
47
(
2013
).
26.
S.
Kravtsov
,
D.
Kondrashov
, and
M.
Ghil
, “
Multilevel regression modeling of nonlinear processes: Derivation and applications to climatic variability
,”
J. Clim.
18
,
4404
4424
(
2005
).
27.
D.
Kondrashov
,
S.
Kravtsov
, and
M.
Ghil
, “
Empirical mode reduction in a model of extratropical low-frequency variability
,”
J. Atmos. Sci.
63
,
1859
1877
(
2006
).
28.
A. H.
Monahan
and
J.
Culina
, “
Stochastic averaging of idealized climate models
,”
J. Clim.
24
,
3068
3088
(
2011
).
29.
A.
Majda
,
C. L. E.
Franzke
, and
D.
Crommelin
, “
Normal forms for reduced stochastic climate models
,”
Proc. Natl. Acad. Sci. U. S. A.
106
,
3649
3653
(
2009
).
30.
P. D.
Sardeshmukh
,
G. P.
Compo
, and
C.
Penland
, “
Need for caution in interpreting extreme weather statistics
,”
J. Clim.
28
,
9166
9187
(
2015
).
31.
P. D.
Sardeshmukh
and
P.
Sura
, “
Reconciling non-gaussian climate statistics with linear dynamics
,”
J. Clim.
22
,
1193
1207
(
2009
).
32.
M.
Ghil
,
P.
Yiou
,
S.
Hallegatte
,
B.
Malamud
,
P.
Naveau
,
A.
Soloviev
,
P.
Friederichs
,
V.
Keilis-Borok
,
D.
Kondrashov
,
V.
Kossobokov
 et al, “
Extreme events: dynamics, statistics and prediction
,”
Nonlinear Processes Geophys.
18
,
295
350
(
2011
).
33.
P.
Sura
, “
A general perspective of extreme events in weather and climate
,”
Atmos. Res.
101
,
1
21
(
2011
).
34.
P.
Sura
and
M.
Perron
, “
Extreme events and the general circulation: Observations and stochastic model dynamics
,”
J. Atmos. Sci.
67
,
2785
2804
(
2010
).
35.
P.
Sura
, “
Stochastic models of climate extremes: Theory and observations
,” in
Extremes in a Changing Climate
(
Springer
,
2013
), pp.
181
222
.
36.
P.
Sura
and
A.
Hannachi
, “
Perspectives of non-gaussianity in atmospheric synoptic and low-frequency variability
,”
J. Clim.
28
,
5091
5114
(
2015
).
37.
G.
Woo
,
The Mathematics of Natural Catastrophes
(
Imperial College Press
,
London
,
1999
).
38.
G.
Woo
,
Calculating Catastrophe
(
Imperial College Press
,
London
,
2011
).
39.
Lloyd's Market Association
, “
Catastrophe modelling guidance for non-catastrophe modellers
,”
Technical Report
, Lloyd's Market Association, London,
2013
.
40.
R. N.
Mantegna
and
H. E.
Stanley
,
Introduction to Econophysics: Correlations and Complexity in Finance
(
Cambridge University Press
,
1999
).
41.
C. W.
Gardiner
,
Stochastic Methods: A Handbook for the Natural and Social Sciences
(
Springer
,
Berlin
,
2009
), Vol.
4
.
42.
M.
Wilczek
and
R.
Friedrich
, “
Dynamical origins for non-gaussian vorticity distributions in turbulent flows
,”
Phys. Rev. E
80
,
016316
(
2009
).
43.
P. D.
Ditlevsen
, “
Observation of alpha-stable noise induced millennial climate changes from an ice-core record
,”
Geophys. Res. Lett.
26
,
1441
1444
, doi: (
1999
).
44.
D.
Peavoy
and
C.
Franzke
, “
Bayesian analysis of rapid climate change during the last glacial using greenland δ 18O data
,”
Clim. Past
6
,
787
794
(
2010
).
45.
F.
Kwasniok
and
G.
Lohmann
, “
Deriving dynamical models from paleoclimatic records: Application to glacial millennial-scale climate variability
,”
Phys. Rev. E
80
,
066104
(
2009
).
46.
A. H.
Monahan
,
J.
Alexander
, and
A. J.
Weaver
, “
Stochastic models of the meridional overturning circulation: Time scales and patterns of variability
,”
Philos. Trans. R. Soc., A
366
,
2525
2542
(
2008
).
47.
S.
Kobayashi
,
Y.
Ota
,
Y.
Harada
,
A.
Ebita
,
M.
Moriya
,
H.
Onoda
,
K.
Onogi
,
H.
Kamahori
,
C.
Kobayashi
,
H.
Endo
 et al, “
The JRA-55 reanalysis: General specifications and basic characteristics
,”
J. Meteorol. Soc. Jpn.
93
,
5
48
(
2015
).
48.
G. A.
Gottwald
and
I.
Melbourne
, “
A huygens principle for diffusion and anomalous diffusion in spatially extended systems
,”
Proc. Natl. Acad. Sci.
110
,
8411
8416
(
2013
).
49.
A.
Clauset
,
C. R.
Shalizi
, and
M. E.
Newman
, “
Power-law distributions in empirical data
,”
SIAM Rev.
51
,
661
703
(
2009
).
50.
A. J.
Majda
,
I.
Timofeyev
, and
E. V.
Eijnden
, “
Models for stochastic climate prediction
,”
Proc. Natl. Acad. Sci. U. S. A.
96
,
14687
14691
(
1999
).
51.
A. J.
Majda
,
I.
Timofeyev
, and
E.
Vanden Eijnden
, “
A mathematical framework for stochastic climate models
,”
Commun. Pure Appl. Math.
54
,
891
974
(
2001
).
52.
A. J.
Majda
,
C. L. E.
Franzke
, and
B.
Khouider
, “
An applied mathematics perspective on stochastic modelling for climate
,”
Philos. Trans. R. Soc. A
366
,
2429
2455
(
2008
).
53.
D.
Peavoy
,
C. L.
Franzke
, and
G. O.
Roberts
, “
Systematic physics constrained parameter estimation of stochastic differential equations
,”
Comput. Stat. Data Anal.
83
,
182
199
(
2015
).
54.
C. L. E.
Franzke
,
T.
O'Kane
,
J.
Berner
,
P.
Williams
, and
V.
Lucarini
, “
Stochastic climate theory and modelling
,”
WIREs Clim. Change
6
,
63
78
(
2015
).
55.
C.
Penland
and
B. D.
Ewald
, “
On modelling physical systems with stochastic models: Diffusion versus lévy processes
,”
Philos. Trans. R. Soc. A
366
,
2455
2474
(
2008
).
56.
W. F.
Thompson
,
R. A.
Kuske
, and
A. H.
Monahan
, “
Stochastic averaging of dynamical systems with multiple time scales forced with α-stable noise
,”
Multiscale Model. Simul.
13
,
1194
1223
(
2015
).
57.
G. A.
Pavliotis
and
A.
Stuart
,
Multiscale Methods: Averaging and Homogenization
(
Springer Science & Business Media
,
2008
).
58.
T. G.
Kurtz
, “
A limit theorem for perturbed operator semigroups with applications to random evolutions
,”
J. Funct. Anal.
12
,
55
67
(
1973
).
59.
G. C.
Papanicolaou
, “
Some probabilistic problems and methods in singular perturbations
,”
Rocky Mt. J. Math.
6
,
653
(
1976
).
60.
K.
Hasselmann
, “
Stochastic climate models part I. theory
,”
Tellus
28
,
473
485
(
1976
).
61.
C.
Penland
, “
A stochastic model of indopacific sea surface temperature anomalies
,”
Physica D
98
,
534
558
(
1996
).
62.
C.
Penland
and
T.
Magorian
, “
Prediction of nino 3 sea surface temperatures using linear inverse modeling
,”
J. Clim.
6
,
1067
1076
(
1993
).
63.
G. E.
Box
,
G. M.
Jenkins
,
G. C.
Reinsel
, and
G. M.
Ljung
,
Time Series Analysis: Forecasting and Control
(
John Wiley & Sons
,
2015
).
64.
A. A.
Stanislavsky
,
K.
Burnecki
,
M.
Magdziarz
,
A.
Weron
, and
K.
Weron
, “
Farima modeling of solar flare activity from empirical time series of soft x-ray solar emission
,”
Astrophys. J.
693
,
1877
1882
(
2009
).
65.
C. L.
Franzke
,
T.
Graves
,
N. W.
Watkins
,
R. B.
Gramacy
, and
C.
Hughes
, “
Robustness of estimators of long-range dependence and self-similarity under non-gaussianity
,”
Philos. Trans. R. Soc., A
370
,
1250
1267
(
2012
).
66.
T.
Graves
,
C. L. E.
Franzke
,
N. N.
Watkins
, and
R.
Gramacy
, “
Systematic bayesian inference of the long-range dependence and heavy-tail distribution parameters
,”
Physica A
(submitted).
67.
T.
Mikosch
,
T.
Gadrich
,
C.
Kluppelberg
, and
R. J.
Adler
, “
Parameter estimation for arma models with infinite variance innovations
,”
Ann. Stat.
23
,
305
326
(
1995
).
68.
R. F.
Engle
, “
Autoregressive conditional heteroscedasticity with estimates of the variance of united kingdom inflation
,”
Econometrica
50
,
987
1007
(
1982
).
69.
C.
Brooks
,
Introductory Econometrics for Finance
(
Cambridge University Press
,
2014
).
70.
G.
Gottwald
,
D.
Crommelin
, and
C. L. E.
Franzke
, “
Stochastic climate theory
,” in
Nonlinear and Stochastic Climate Dynamics
, edited by
C. L. E.
Franzke
and
T.
O'Kane
(
Cambridge University Press
,
Cambridge
,
2016
).
71.
S.
Siegert
,
R.
Friedrich
, and
J.
Peinke
, “
Analysis of data sets of stochastic systems
,”
Phys. Lett. A
243
,
275
280
(
1998
).
72.
D.
Crommelin
and
E.
Vanden-Eijnden
 et al, “
Reconstruction of diffusions using spectral data from timeseries
,”
Commun. Math. Sci.
4
,
651
668
(
2006
).
73.
J.
Berner
, “
Linking nonlinearity and non-gaussianity of planetary wave behavior by the fokker-planck equation
,”
J. Atmos. Sci.
62
,
2098
2117
(
2005
).
74.
P.
Sura
,
M.
Newman
,
C.
Penland
, and
P.
Sardeshmukh
, “
Multiplicative noise and non-gaussianity: A paradigm for atmospheric regimes?
,”
J. Atmos. Sci.
62
,
1391
1409
(
2005
).
75.
A.
Majda
,
R. V.
Abramov
, and
M. J.
Grote
,
Information Theory and Stochastics for Multiscale Nonlinear Systems
(
American Mathematical Society
,
2005
), Vol.
25
.
76.
A.
Majda
,
I.
Timofeyev
, and
E.
Vanden-Eijnden
, “
A priori tests of a stochastic mode reduction strategy
,”
Physica D
170
,
206
252
(
2002
).
77.
P. D.
Sardeshmukh
and
C.
Penland
, “
Understanding the distinctively skewed and heavy tailed character of atmospheric and oceanic probability distributions
,”
Chaos
25
,
036410
(
2015
).
78.
C.
Penland
and
P. D.
Sardeshmukh
, “
Alternative interpretations of power-law distributions found in nature
,”
Chaos
22
,
023119
(
2012
).
79.
D.
Crommelin
and
E.
Vanden-Eijnden
, “
Fitting timeseries by continuous-time markov chains: A quadratic programming approach
,”
J. Comput. Phys.
217
,
782
805
(
2006
).
80.
D.
Crommelin
and
E.
Vanden-Eijnden
, “
Diffusion estimation from multiscale data by operator eigenpairs
,”
Multiscale Model. Simul.
9
,
1588
1623
(
2011
).
81.
D.
Crommelin
, “
Estimation of space-dependent diffusions and potential landscapes from non-equilibrium data
,”
J. Stat. Phys.
149
,
220
233
(
2012
).
82.
S.
Siegert
and
R.
Friedrich
, “
Modeling of nonlinear lévy processes by data analysis
,”
Phys. Rev. E
64
,
041107
(
2001
).
83.
P. J.
Mailier
,
D. B.
Stephenson
,
C. A.
Ferro
, and
K. I.
Hodges
, “
Serial clustering of extratropical cyclones
,”
Mon. Weather Rev.
134
,
2224
2240
(
2006
).
84.
R.
Vitolo
,
D. B.
Stephenson
,
I. M.
Cook
, and
K.
Mitchell-Wallace
, “
Serial clustering of intense european storms
,”
Meteorol. Z.
18
,
411
424
(
2009
).
85.
R.
Blender
,
C.
Raible
, and
F.
Lunkeit
, “
Non-exponential return time distributions for vorticity extremes explained by fractional poisson processes
,”
Q. J. R. Meteorol. Soc.
141
,
249
257
(
2015
).
86.
M. R.
Leadbetter
, “
Extremes and local dependence in stationary sequences
,”
Probab. Theory Relat. Fields
65
,
291
306
(
1983
).
87.
E.
Koscielny-Bunde
,
A.
Bunde
,
S.
Havlin
,
H. E.
Roman
,
Y.
Goldreich
, and
H.-J.
Schellnhuber
, “
Indication of a universal persistence law governing atmospheric variability
,”
Phys. Rev. Lett.
81
,
729
(
1998
).
88.
J. W.
Kantelhardt
,
E.
Koscielny-Bunde
,
H. H.
Rego
,
S.
Havlin
, and
A.
Bunde
, “
Detecting long-range correlations with detrended fluctuation analysis
,”
Physica A
295
,
441
454
(
2001
).
89.
C.
Franzke
, “
Nonlinear trends, long-range dependence, and climate noise properties of surface temperature
,”
J. Clim.
25
,
4172
4183
(
2012
).
90.
A. C.
Cameron
,
J. B.
Gelbach
, and
D. L.
Miller
, “
Bootstrap-based improvements for inference with clustered errors
,”
Rev. Econ. Stat.
90
,
414
427
(
2008
).
91.
H.
Kantz
,
E. G.
Altmann
,
S.
Hallerberg
,
D.
Holstein
, and
A.
Riegert
, “
Dynamical interpretation of extreme events: predictability and predictions
,” in
Extreme Events in Nature and Society
(
Springer
,
2006
) pp.
69
93
.
92.
S.
Hallerberg
,
J.
Bröcker
, and
H.
Kantz
, “
Prediction of extreme events
,” in
Nonlinear Time Series Analysis in the Geosciences
(
Springer
,
2008
) pp.
35
59
.
93.
G. W.
Brier
, “
Verification of forecasts expressed in terms of probability
,”
Mon. Weather Rev.
78
,
1
3
(
1950
).
94.
D.
Stephenson
,
B.
Casati
,
C.
Ferro
, and
C.
Wilson
, “
The extreme dependency score: A non-vanishing measure for forecasts of rare events
,”
Meteorol. Appl.
15
,
41
50
(
2008
).
95.
S.
Hallerberg
,
E. G.
Altmann
,
D.
Holstein
, and
H.
Kantz
, “
Precursors of extreme increments
,”
Phys. Rev. E
75
,
016706
(
2007
).
96.
S.
Hallerberg
and
H.
Kantz
, “
Influence of the event magnitude on the predictability of an extreme event
,”
Phys. Rev. E
77
,
011108
(
2008
).
97.
S.
Hallerberg
and
H.
Kantz
, “
How does the quality of a prediction depend on the magnitude of the events under study?
,”
Nonlinear Processes Geophys.
15
,
321
331
(
2008
).
98.
T.
Bódai
, “
Predictability of threshold exceedances in dynamical systems
,”
Physica D
313
,
37
50
(
2015
).
99.
A.
Sterk
,
M.
Holland
,
P.
Rabassa
,
H.
Broer
, and
R.
Vitolo
, “
Predictability of extreme values in geophysical models
,”
Nonlinear Processes Geophys.
19
,
529
539
(
2012
).
100.
T.
Bódai
and
C. L. E.
Franzke
, “
Predictability of extremes in heavy-tailed systems
,”
Extremes
(submitted).
101.
S.
Siegert
,
J.
Bröcker
, and
H.
Kantz
, “
Skill of data-based predictions versus dynamical models
,” in
Extreme Events: Observations, Modeling, and Economics
(
Wiley Online Library
,
2016
), pp.
35
50
.
102.
J. M.
Miotto
and
E. G.
Altmann
, “
Predictability of extreme events in social media
,”
PloS One
9
,
e111506
(
2014
).
103.
P.
Friederichs
and
A.
Hense
, “
Statistical downscaling of extreme precipitation events using censored quantile regression
,”
Mon. Weather Rev.
135
,
2365
2378
(
2007
).
104.
A.
Golightly
and
D. J.
Wilkinson
, “
Bayesian inference for nonlinear multivariate diffusion models observed with error
,”
Comput. Stat. Data Anal.
52
,
1674
1693
(
2008
).
105.
H.
Mori
, “
Transport, collective motion, and brownian motion
,”
Prog. Theor. Phys.
33
,
423
455
(
1965
).
106.
R.
Zwanzig
, “
Nonlinear generalized langevin equations
,”
J. Stat. Phys.
9
,
215
220
(
1973
).
107.
A. J.
Chorin
,
O. H.
Hald
, and
R.
Kupferman
, “
Optimal prediction and the Mori–Zwanzig representation of irreversible processes
,”
Proc. Natl. Acad. Sci.
97
,
2968
2973
(
2000
).
108.
J.
Wouters
and
V.
Lucarini
, “
Disentangling multi-level systems: Averaging, correlations and memory
,”
J. Stat. Mech.
2012
,
P03003
.
109.
J.
Wouters
and
V.
Lucarini
, “
Multi-level dynamical systems: Connecting the ruelle response theory and the Mori-Zwanzig approach
,”
J. Stat. Phys.
151
,
850
860
(
2013
).
110.
M. S.
Taqqu
, “
Fractional brownian motion and long-range dependence
,” in
Theory and Applications of Long-Range Dependence
(
Birkhauser
,
Boston, MA
,
2003
), pp.
5
38
.
111.
J.
Berner
,
G.
Shutts
,
M.
Leutbecher
, and
T.
Palmer
, “
A spectral stochastic kinetic energy backscatter scheme and its impact on flow-dependent predictability in the ecmwf ensemble prediction system
,”
J. Atmos. Sci.
66
,
603
626
(
2009
).
112.
F.
Tagle
,
J.
Berner
,
M. D.
Grigoriu
,
N. M.
Mahowald
, and
G.
Samorodnitsky
, “
Temperature extremes in the community atmosphere model with stochastic parameterizations
,”
J. Clim.
29
,
241
258
(
2016
).
113.
C.
Franzke
,
M.
Oliver
,
J.
Rademacher
, and
G.
Badin
, “
Systematic multi-scale methods for geophysical flows
,” in
Energy Transfers in Atmosphere and Ocean
, edited by
A.
Iske
and
C.
Eden
(
Springer
,
2017
).
You do not currently have access to this content.