Lecture IV: Ergodic Theory of DDSs & RDSs and the Statistics of Models & Observations Michael Ghil Ecole Normale Supérieure (ENS), Paris, & University of California at Los Angeles (UCLA) Please consult these sites for more info.: https://dept.atmos.ucla.edu/tcd , http://www.environnement.ens.fr/ & https://www.researchgate.net/profile/Michael_Ghil CliMathParis2019: Course I, Math Methods in Climate & GFD IHP, 23 Sept. 2019
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Lecture IV: Ergodic Theory of DDSs & RDSs · The kinetic theory of gases (L. Boltzmann), statistical mecanics (J.W. !Gibbs), Einstein’s explanation (1905, annus mirabilis) of Brownian
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Lecture IV: Ergodic Theory of DDSs & RDSsand the Statistics of Models & Observations
Michael GhilEcole Normale Supérieure (ENS), Paris, &
University of California at Los Angeles (UCLA)
Please consult these sites for more info.:https://dept.atmos.ucla.edu/tcd, http://www.environnement.ens.fr/& https://www.researchgate.net/profile/Michael_Ghil
CliMathParis2019: Course I,Math Methods in Climate & GFD
IHP, 23 Sept. 2019
Composite spectrum of climate variability!Standard treatement of frequency bands:! 1. High frequencies – white noise (or ‘‘colored’’) ! 2. Low frequencies – slow evolution of parameters !
From Ghil (2001, EGEC), after Mitchell* (1976)!* ‘‘No known source of deterministic internal variability’’!** 27 years – Brier (1968, Rev. Geophys.)!
Advanced Spectral Methods, Nonlinear Dynamics and the Geosciences
MotivationØ Time series in the geosciences have typically broad peaks on top of a continuous, “warm-colored” background è MethodØ Connections to nonlinear dynamics è TheoryØ Need for stringent statistical significance tests è Toolkit (*)Ø Applications to analysis and prediction è Examples
• The 19th century made substantial advances in grasping meteorological phenomena in mid-latitudes, via land and sea observations.!
• At the same time, the physical formulation and the mathematical analysis of the partial differential equations (PDEs) governing fluid flows (Navier-Stokes, shallow-water) made great strides.!
• V. Bjerknes (1904) and L.F. Richardson (1922) were the first to formulate and solve the problem of weather prediction as an initial-value problem for a systeme of PDEs; v. also F. Exner (1904) and M. Margules (1904).!
• With the spectacular growth of the number of meteorological observations during and after WWII, routine numerical weather prediction (NWP) will start in the 1960s.!
!
A little history of weather and chaos – II
• During the 1940s and 1950s, J. von Neumann assembles around him a group of meteorologists for the first experimental weather forecasts (Charney, Fjørtoft & V. Neumann, 1950).!
• P.D. Thompson (1961) organizes the transition from these experimental forecasts at Princeton (Inst. Adv. Studies) to routine, operational NWP via the JNWPU in Washington, DC.!
• Operational NWP gets better & better, due to improvements in the observations (satellites & other obs. systems), of the PDE models being used (subgrid-scale “parametrizations,” etc.), of the numerical methods used to solve them, of the data assimilation methods applied to combine the data and the models, and so on. But, oh heck ($✪v⌘!): these routine forecasts are (still) far from perfect?!?!!
• At that point, E.N. Lorenz (1963) formulates the problem of loss of forecast skill in NWP as one of stability and predictability of nonlinear dynamical systems.!
The great surprise of deterministic chaos – a major discovery of the 20th century?
• The “classical” point of view !!!The kinetic theory of gases (L. Boltzmann), statistical mecanics (J.W. !Gibbs), Einstein’s explanation (1905, annus mirabilis) of Brownian motion !led one to believe that irregular behavior in a medium could only result !!from the interaction of an infinite (or at least very large) number of particles.!
• The surprise of the ’60s and ’70s !! Three ordinary differential equations (ODEs) were enough to generate !! irregular behavior!!!
• Why 3 and not just 2 variables?!! In the “phase space” of a flow in one dimension !! (1-D) there can be only fixed points (FPs, i.e. equilibria); !! in 2-D only FPs and limit cycles (LCs, i.e. periodic solutions)(*); !! one needs 3-D to “accommodate” a “strange attractor”! !
! (*) The Jordan curve theorem and the Poincaré-Bendixson theorem explain this: ! no self-intersections!!
The sources of deterministic chaos – a bit of bibliometry (*)
• Mathematics :!!Poincaré (1890), Hadamard (1898), Smale (1968), etc.!!Poincaré (1889, book) = 223, (1890, article) = 50 ;!!Smale (1968) = 1423 !!v. also Van der Pol, Cartwright & Littlewood, etc.!
• Physics :!!D. Ruelle and F. Takens, 1971: On the nature of turbulence. Commun. Math. Phys., 20, 167–192 = 1318.!
• Meteorology and prediction:!!Lorenz (1963a) = 4832.!
!(*) Just for laughs, of course, according to ! Thomson Reuters Web of Knowledge, ! except for Poincaré (Google Scholar) – 6 May 2009.!
A little history of climate & stochasticity • A. Einstein’s (1905) Brownian motion paper.!• K. Itō (prof. at Kyoto U., RIMS director) ! formulates Itō calculus in 1942, enables solution ! of stochastic differential equations (SDEs);! Itō’s lemma is the stochastic counterpart of! Leibniz’s chain rule for differentiation.!• K. Hasselmann (Tellus, 1976) describes climate! as Brownian motion, with weather ! the stochastic driver.!• In this view, the deterministic part! of the model is stable, and random! perturbations decay to the mean.!!!
Kiyoshi Itō!
0 1 2 3 4 5 6 7 8−10
−5
0
5
10
t
X (t, ! ; X0), with X0 varying
Auto-regressive (AR) decay !
Nature is not deterministic or stochastic:
It depends on what we can, need & want to know !— more or less detail, with greater or lesser accuracy —!
larger scales more accurately, !smaller scales less so!
But we need both, deterministic and stochastic descriptions.!Knowing how to combine them is necessary, as well as FUN!!
Probability space =
Probability space & Borel sets
Hilbert’s 6th problem, of which Kolmogorov (1933) solved the 1st part
Venn (1880) diagrams & Borel sets
Borel algebra ={countable unions and intersectionsof open sets, andcomplements}
Emile Borel, founder of the IHP, in academician’s uniformUnion A [B, intersection A \B,
and complement A.<latexit sha1_base64="Ij9YQyaKl4kjxLhzw+LlkXe6oTk=">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</latexit>
{(Topological space X),(Borel �-algebra F),(probability measure P)}
Two (open) sets A&B<latexit sha1_base64="hA5e+tNy82u/mBBjFw12BbnfuSs=">AAACBnicdVBNSwMxEM36bf2qehQhWBUFKZsVtL1VvXisYFXolpJNpxrMJkuSVUrpyYt/xYsHRbz6G7z5b8y2FVT0wcDjvRlm5kWJ4Mb6/oc3Mjo2PjE5NZ2bmZ2bX8gvLp0ZlWoGNaaE0hcRNSC4hJrlVsBFooHGkYDz6Poo889vQBuu5KntJNCI6aXkbc6odVIzv3p6q/CWSkBuYwPW4PWDcCfcxOEOPlzPNfMFv+j7PiEEZ4Ts7/mOlMulgJQwySyHAhqi2sy/hy3F0hikZYIaUyd+Yhtdqi1nAnq5MDWQUHZNL6HuqKQxmEa3/0YPbzilhdtKu5IW99XvE10aG9OJI9cZU3tlfnuZ+JdXT2271OhymaQWJBssaqcCW4WzTHCLa2BWdByhTHN3K2ZXVFNmXXJZCF+f4v/JWVAku8XgJChUgmEcU2gFraEtRNA+qqBjVEU1xNAdekBP6Nm79x69F+910DriDWeW0Q94b5+kIZX4</latexit>
Outline Ø Motivation: spectrum has trends, broad peaks, and continuous
background + connections to nonlinear dynamics.Ø Methodology – univariate: singular-spectrum analysis (SSA) – multivariate: M-SSA + varimax rotation – statistical significance tests: MC-SSA & Procrustes rotation – SSA-MTM Toolkit – gap fillingØ Applications to analysis and prediction – Southern Oscillation Index (SOI)(*)
– Nile River floods – GPS data for Alaskan volcanoØ Concluding remarks and bibliography
• SSA is good at isolating oscillatory behavior via paired eigenelements.• SSA tends to lump signals that are longer-term than the window into
– one or two trend components.
•12/28
SSA-MTM Toolkit, I – A Free Toolkit for Spectral Analysis
Ø Developed at UCLA, with collaborations on 3 continents, since 1994.Ø Used extensively at the ENS and in various summer schools for teaching spectral methods to various audiences.Ø GUI based, for Linux, Unix and MacOSX platforms.Ø Latest developments by A. Groth and D. Kondrashov (UCLA).Ø Hundreds of downloads at every new version.Ø Available at: https://dept.atmos.ucla.edu/tcd/ssa-mtm-toolkit
Ø Reduce the variance of the spectral estimate of a time series, based on the periodogram (MTM), correlogram (BT) or other spectral analysis method (e.g., SSA).!
Ø Estimate peak frequencies to “fingerprint” limit cycles of the underlying dynamical system.!
Ø Provide statistical significance tests when such behavior is blurred by “noise.”!
Ø Allow rapid, visual and numerical comparison between the results of different methods: BT, SSA, MEM, MTM.!!!!
SSA-MTM Toolkit, II – General Goals
BT – Blackman-Tukey version of FT (Fourier transform), !MEM – Maximum Entropy Method, MTM – Multi-Taper Method!
SSA-MTM Toolkit, III – Targeted audiences u Non-specialists in time
series analysis!u Reasonable default
options!u Reads ASCII files!
u Non-specialists in computer management!u Precompiled
binaries!u User-friendly
interface!u Non-specialists in
dynamical systems!u Better if you do
know.!u No problem if you
don’t.!
Outline Ø Motivation: spectrum has trends, broad peaks, and continuous
background + connections to nonlinear dynamics.Ø Methodology – univariate: singular-spectrum analysis (SSA) – multivariate: M-SSA + varimax rotation – statistical significance tests: MC-SSA & Procrustes rotation – SSA-MTM Toolkit – gap fillingØ Applications to analysis and prediction – Southern Oscillation Index (SOI)(*)
– Nile River floods – GPS data for Alaskan volcanoØ Concluding remarks and bibliography
What is ENSO? • The El Niño-Southern Oscillation (ENSO) is a coupledocean–atmosphere phenomenon originating inthe Tropical Pacific. • It dominates interannualclimate variability globally viateleconnections.• Intrinsic variability interactswith the seasonal forcing andleads to irregular occurrencesof large climate anomalies,warm (El Niños) and cold(La Niñas). After Nastrom & Gage (JAS, 1985)
Time series of atmospheric pressure and sea surface temperature (SST) indices
Data courtesy of NCEP’s Climate Prediction Center Neelin (2006) Climate Modeling and Climate Change
SOI = mean monthly values of Δps (Tahiti – Darwin) Results (“undigested”) from 1933–1988 time interval (*) 1. For 18 < M < 60 months, singular spectra show a clear break at 5 < S < 17 (= “deterministic” part; M – S = “noise”);
2. 3 pairs of EOFs stand out: EOFs 1 + 2 (27%), 3 + 4 (19.7%), and 9 + 10 (3%);
3. the associated periods are ~ 60 mos. (“ENSO”), 30 mos. (QBO”), and 5.5 mos. (?!)
(*) E. M. (“Gene”) Rasmusson, X. Wang, and C.F. Ropelewski, 1990: The biennial component of ENSO variability. J. Marine Syst., 1, 71–96.
Sampling interval – τs = 1 month
Window width Mτs : 18τs < τw < 60τs or 1.5 yr < τw < 5 yr.
Each principal component (PC) is Fourier analyzed separately; individual variance indicated as well.
PCs (1+2) – period = 60 months, low-frequency or “ENSO” or quasi-quadrennial (QQ) component;
PCs (3+4) – period = 30 months quasi-biennial (QB) component;
PCs (9+10) – period = 5.5 months
Singular Spectrum Analysis (SSA) and M-SSA (cont’d)
• Break in slope of SSA spectrum distinguishes “significant” from “noise” EOFs• Formal Monte-Carlo test (Allen and Smith, 1994) identifies 4-yr and 2-yr ENSO oscillatory modes.
A window size of M = 60 is enough to “resolve” these modes in a monthly SOI time series
•
13/28
SSA (prefilter) + (low-order) MEM
“Stack” spectrum
In good agreement with MTM peaks of Ghil & Vautard (1991, Nature) for the Jones et al. (1986) temperatures & stack spectra of Vautard et al. (1992, Physica D) for the IPCC “consensus” record (both global), to wit 26.3, 14.5, 9.6, 7.5 and 5.2 years.
Peaks at 27 & 14 years also in Koch sea-ice index off Iceland (Stocker & Mysak, 1992), etc. Plaut, Ghil & Vautard (1995, Science)
2.0
1.5
1.0
0.5
0.0
0.05
25.0 years
14.2 years
7.7 years
5.5 years
0.10 0.15 0.20
Frequency (year-1)
Po
we
r sp
ectr
a
Total PowerThermohaline modeCoupled O-A modeWind-driven mode
Interannual
Interdecadal
Mid-latitude
L-F ENSOmode
8/28
The Nile River Records Revisited:How good were Joseph's predictions?
Michael Ghil, ENS & UCLAYizhak Feliks, IIBR & UCLA,
Dmitri Kondrashov, UCLA
17/28
Why are there data missing?
Hard Work
• Byzantine-period mosaic from Zippori, the capital of Galilee (1st century B.C. to 4th century A.D.); photo by Yigal Feliks, with permission from the Israel Nature and Parks Protection Authority )
18/28
Historical records are full of “gaps”....
Annual maxima and minima of the water level at the nilometer on Rodah Island, Cairo.20/28
SSA (M-SSA) Gap Filling
Main idea: utilize both spatial and temporal correlations to iteratively compute self-consistent lag-covariance matrix; M-SSA with M = 1 is the same as the EOF reconstruction method of Beckers & Rixen (2003)
Goal: keep “signal” and truncate “noise” — usually a few leading EOFs correspond to the dominant oscillatory modes, while the rest is noise.
(1) for a given window width M: center the original data by computing the unbiased value of the mean and set the missing-data values to zero.
(2) start iteration with the first EOF, and replace the missing points with the reconstructed component (RC) of that EOF; repeat the SSA algorithm on the new time series, until convergence is achieved.
(3) repeat steps (1) and (2) with two leading EOFs, and so on.
(4) apply cross-validation to optimize the value of M and the number of dominant SSA (M-SSA) modes K to fill the gaps: a portion of available data (selected at random) is flagged as missing and the RMS error in the reconstruction is computed.
Concluding Remarks Ø M-SSA allows one to extract spatial+temporal features from time series
without any a priori hypothesis on the physical, biological or economical processes at work or on the data set’s stochastic characteristics.
Ø We illustrated successful applications to both climatic and economic time series, including ENSO and Nile River floods, US and EU macroeconomic indicators.
Ø Many other successes in the physical, economic and life sciences, including synchronization of chaotic oscillators, solid-earth geosciences, fish population data, and agent-based predator–prey models.
Ø Further methodological developments: varimax rotation, Procrustes target target rotation.Ø These developments allow application to large data sets in many areas,
without prior dimension reduction, e.g. to complete temperature fields.Ø Availability of freeware toolkit(*) + other teaching and research aids.
To check a spectral feature, e.g., an oscillatory pair: 1. Find pair for given data set {Xn: n = 1,2, …N} and window width M. 2. Apply statistical significance tests (MC-SSA, etc.). 3. Check robustness of pair by changing M, sampling interval τs, etc. 3. Apply additional methods (MTM, wavelets, etc.) and their tests to {Xn}. 4. Obtain additional time series pertinent to the same phenomenon {Ym}, etc. 5. Apply steps (1)–(3) to these data sets. 6. Use multi-channel SSA (M-SSA) and other multivariate methods to check mutual dependence between {Xn}, {Ym}, etc.
7. Based on steps (1)–(6), try to provide a physical explanation of the mode. 8. Use (7) to predict an as-yet-unobserved feature of the data sets. 9. If this new feature is found in new data, go on to next problem. 10. If not, go back to an earlier step of this list.
(*) Ghil, M., M. R. Allen, M. D. Dettinger, K. Ide, D. Kondrashov, M. E. Mann, A. W. Robertson, A. Saunders, Y. Tian, F. Varadi, and P. Yiou, 2002: Advanced spectral methods for climatic time series, Rev. Geophys., 40(1), pp. 3.1–3.41, doi: 10.1029/2000RG000092.
Classical, paper-basedAlessio, S. M., 2016: Digital Signal Processing and Spectral Analysis for Scientists,
Springer, Heidelberg, Germany & New York, NY.Blackman, R. B., & J. W. Tukey, 1958: The Measurement of Power Spectra,
Dover, Mineola, NY.Broomhead, D. S., King, G. P., 1986. Extracting qualitative dynamics from
experimental data. Physica D, 20, 217–236.Chatfield, C., 1984: The Analysis of Time Series: An Introduction, Chapman & Hall,
New York.Ghil, M., et al., 2002: Advanced spectral methods for climatic time series,
Rev. Geophys., 40, doi:10.1029/2001RG000092.Hannan, E. J., 1960: Time Series Analysis, Methuen, London/Barnes & Noble,
New York, NY, 152 pp.Loève, M., 1978: Probability Theory, Vol. II, 4th ed., Graduate Texts in
Mathematics, vol. 46, Springer-Verlag, ISBN 0-387-90262-7.Percival, D. B.,& A. T. Walden, Spectral Analysis for Physical Applications,
Ghil, M., and R. Vautard, 1991: Interdecadal oscillations and the warming trend in global temperature time series, Nature, 350, 324–327.
Ghil, M., M. R. Allen, M. D. Dettinger, K. Ide, D. Kondrashov, M. E. Mann, A. W. Robertson, A. Saunders, Y. Tian, F. Varadi, and P. Yiou, 2002: Advanced spectral methods for climatic time series, Rev. Geophys., 40(1), pp. 3.1–3.41, doi: 10.1029/2000RG000092.
Groth, A., and M. Ghil, 2011: Multivariate singular spectrum analysis and the road to phasesynchronization, Phys. Rev. E, 84, 036206 (10 pp.), doi:10.1103/PhysRevE.84.036206.
Groth, A., and M. Ghil, 2015: Monte Carlo singular spectrum analysis (SSA) revisited: Detecting oscillator clusters in multivariate data sets, J. Climate, 28, 7873–7893.
Groth, A., and M. Ghil, 2017: Synchronization of world economic activity, Chaos, 27, 127002 (18 pp.), http://dx.doi.org/10.1063/1.5001820 .
Kondrashov, D., Y. Feliks, and M. Ghil, 2005: Oscillatory modes of extended Nile River records (A.D. 622–1922), Geophys. Res. Lett., 32, L10702, doi:10.1029/2004GL022156.
Kondrashov, D., and M. Ghil, 2006: Spatio-temporal filling of missing points in geophysical data sets, Nonlin. Processes Geophys., 13, 151–159.
Plaut, G., M. Ghil and R. Vautard, 1995: Interannual and interdecadal variability in 335 years of Central England temperatures, Science, 268, 710–713.
Vautard, R., and M. Ghil, 1989: Singular spectrum analysis in nonlinear dynamics, withapplications to paleoclimatic time series, Physica D, 35, 395–424.
Some specific references
Warming slow-down
It was a wonderful encounter !with some leading physicists!and mathematicians, as well!as with GFD & climate!researchers, and with great!students and post-docs. !!It taught me, as Erice had done !in March 1981, how well !organized the SIF and Italians !in general can be.!
But most of all, Michèle & I found out we’d be parents soon
Reserve slides
Diapos de réserve
• Time series analysis – The “smooth” and “rough” part of a time series – Oscillations and nonlinear dynamics
• Singular spectral analysis (SSA) – Principal components in time and space – The SSA-MTM Toolkit
• The Nile River floods – Longest climate-related, instrumental time series – Gap filling in time series – NAO and SOI impacts on the Nile River
Table 1a: Significant oscillatory modes in short records (A.D. 622–1470)
Table 1b: Significant oscillatory modes in extended records (A.D. 622–1922)
Periods Low High High-Low40–100yr 64 (9.3%) 64 (6.9%) 64 (6.6%)
20–40yr [32]
10–20yr 12.2 (5.1%), 18.0 (6.7%)
12.2 (4.7%), 18.3 (5.0%)
5–10yr 6.2 (4.3%) 7.2 (4.4%) 7.3 (4.4%)
0–5yr 3.0 (2.9%), 2.2 (2.3%)
3.6 (3.6%),2.9 (3.4%), 2.3 (3.1%)
2.9 (4.2%),
26/28
Some references Broomhead, D. S., King, G. P., 1986a. Extracting qualitative dynamics from experimental data.
Physica D, 20, 217–236.
Ghil, M., and R. Vautard, 1991: Interdecadal oscillations and the warming trend in global temperature time series, Nature, 350, 324–327.
Ghil, M., M. R. Allen, M. D. Dettinger, K. Ide, D. Kondrashov, M. E. Mann, A. W. Robertson, A. Saunders, Y. Tian, F. Varadi, and P. Yiou, 2002: Advanced spectral methods for climatic time series, Rev. Geophys., 40(1), pp. 3.1–3.41, doi: 10.1029/2000RG000092.
Karhunen, K., 1946. Zur Spektraltheorie stochastischer Prozesse. Ann. Acad. Sci. Fenn. Ser. A1, Math. Phys., 34.
Loève, M., 1978. Probability Theory, Vol. II, 4th ed., Graduate Texts in Mathematics, vol. 46, Springer-Verlag.
Plaut, G., M. Ghil and R. Vautard, 1995: Interannual and interdecadal variability in 335 years of Central England temperatures, Science, 268, 710–713.
Vautard, R., and M. Ghil, 1989: Singular spectrum analysis in nonlinear dynamics, with applications to paleoclimatic time series, Physica D, 35, 395–424.
E2C2 WP2 meeting, Paris 1-2 Feb 2007 12
Type of noise used in the toolkit
• Red noise: – AR(1) random process: – Decreasing spectrum (due to inertia)
)()()1( tbtaXtX +=+
€
CX (τ) =σ 2a τ
1− a2
€
PX ( f ) = CX (0)1− a2
1− 2acos 2πf + a2
E2C2 WP2 meeting, Paris 1-2 Feb 2007 26
Monte Carlo SSA (Allen et Smith, J. Clim., 1995)"
Goal: Assess whether the SSA spectrum can reject the null hypothesis that the time series is red noise.
Procedure: • Estimate red noise parameters with same variance and auto-
covariance as the observed time series X(t) • Compare the pdf of the projection of the noise covariance onto
the data EOFs:
The null hypothesis is rejected using the pdf of ΛΒ.