lecture iv: ergodic theory of ddss & rdss · the kinetic theory of gases (l. boltzmann),...

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

(*) SSA-MTM Toolkit, https://dept.atmos.ucla.edu/tcd/ssa-mtm-toolkit

A little history of weather and chaos – I

•  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)}

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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-MTM Toolkit, https://dept.atmos.ucla.edu/tcd/ssa-mtm-toolkit

Spectral Density (Math)/Power Spectrum (Science & Engng.)

Wiener-Khinchin (Bochner) Theorem Blackman-Tukey Method

i.e., the lag-autocorrelation function & the spectral density

are Fourier transforms of each other.

Continuous background + peaks

R(s) = limL!"

12L

L!

#L

x(t)x(t + s)dt

S(f) =12!

!!

"!

R(s)e"ifsds ! R(s)

S(f)

f

5/28

Power Law for Spectrum (cont’d)

Hypothesis: “Poles” correspond to the least unstable periodic orbits

Major clue to the physics

that underlies the dynamics

N.B. Limit cycle not necessarily elliptic, i.e. not

(a)“unstable limit cycles” “Poincaré section”

(x, y) = (afsin(ft), bfcos(ft))7/28

Singular

Spectrum

Analysis

10/28

Singular Spectrum Analysis (SSA)

Spatial EOFs SSA

x – space s – lag

k

λ

Statistical dimension

Pairs oscillations(nonlinear) sine + cosine pair

Colebrook (1978); Weare & Nasstrom (1982);Broomhead & King (1986: BK); Fraedrich (1986)

Vautard & Ghil (1989: VG) Physica, 35D, 395-424

BK+VG: Analogy between Mañe-Takens embedding and the Wiener-Khinchin theorem

C!(x,y)=E!(x,")!(y,")

=1T

Z T

o!(x, t)!(y, t)dt

CX(s)=EX(t + s,!)"(s,!)

=1T

Z T

oX(t)X(t + s)dt

11/28

ghil

Text Box

X(t+s)

Power Spectra & Reconstruction

A. Transform pair:

The ek’’s are adaptive filters,

the ak’’s are filtered time series.

B. Power spectra

C. Partial reconstruction

in particular:

ak(t) =M!

s=1

X(t + s)ek(s), ak(t)! PC

X(t + s) =M!

k=1

ak(t)ek(s), ek(s)! EOF

SX(f) =M∑

k=1

Sk(f); Sk(f) = Rk(s); Rk(s) ! 1T

∫ T

0ak(t)ak(t + s)dt

XK(t) =1M

!

k!K

M!

s=1

ak(t! s)ek(s);

9/28

K = {1, 2, ....., S} or K = {k} or K = {l, l + 1;!l ! !l+1}

ghil

Polygonal Line

Singular Spectrum Analysis (SSA)

SSA decomposes (geophysical & other) time series into

Temporal EOFs (T-EOFs) and Temporal Principal Components (T-PCs), based on the series’ lag-covariance matrix

Selected parts of the series can be reconstructed, via

Reconstructed Components (RCs)

Time series

RCs

T-EOFs

Selected References: Vautard & Ghil (1989, Physica D); Ghil et al. (2002, Rev. Geophys.)

• 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.!

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.

21/28

Nile River Records

• High level

• Low level

800 1000 1200 1400 1600 1800!4!2

024

a)Original records

Year (AD)

800 1000 1200 1400 1600 1800!4!2

024

Year (AD)

b)MSSA filled in

0 5 10 15 20 25 30100

102c)MSSA spectrum of filled Nile Records:M=100

No. of modes

24/28

Periods Low High High-Low

40–100yr 64 (13%) 85 (8.6%) 64 (8.2%)

20–40yr 23.2 (4.3%)

10–20yr [12], 19.7 (5.9%) 12.2 (4.3%), 18.3 (4.2%)

5–10yr [6.2] 7.3 (4.0%) 7.3 (4.1%)

0–5yr 3.0 (4%), 2.2 (3.3%)

4.2 (3.3%),2.9 (3.3%), 2.2 (2.9%)

[4.2], 2.9 (3.6%), 2.2 (2.6%)

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

Significant Oscillatory Modes

SSA reconstruction of the 7.2-yr mode in the extended Nile River records:

(a) high-water, and (b) difference.Normalized amplitude; reconstruction in the

large gaps in red.

Instantaneous frequencies of the oscillatorypairs in the low-frequency range (40–100 yr).

The plots are based on multi-scale SSA [Yiou et al., 2000]; local SSA performed in each

window of width W = 3M, with M = 85 yr.

27/28

How good were Joseph's predictions?

Pretty good!

28/28

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,

Cambridge Univ. Press, 1993. Web-basedhttps://dept.atmos.ucla.edu/tcd/ssa-mtm-toolkit,http://www.r-project.org

Some general references

https://dept.atmos.ucla.edu/tcd/ssa-mtm-toolkit

http://www.r-project.org

ghil

Highlight

ghil

Highlight

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

•  Concluding remarks – Cautionary remarks (“garde-fous”) — References

Motivation & Outline1..Data sets in the geosciences are often short and contain errors:: t this is both an obstacle and an incentive.

2. Phenomena in the geosciences often have both regular (“cycles”) and irregular (“noise”) aspects.

3. Different spatial and temporal scales: one person’s noise is another person’s signal..

4. Need both deterministic and stochastic modeling.

5. Regularities include (quasi-)periodicity spectral analysis via “classical” methods (see SSA-MTM Toolkit).

6. Irregularities include scaling and (multi-)fractality “spectral analysis” via Hurst exponents, dimensions, etc. (see Multi-Trend Analysis, MTA)

7. Does some combination of the two, + deterministic and stochastic modeling, provide a pathway to prediction?

For details and publications, please visit these two Web sites:

TCD — http://www.atmos.ucla.edu/tcd/ (key person – Dmitri Kondrashov!

E2-C2 — http://www.ipsl.jussieu.fr/~ypsce/py_E2C2.html 2/28

Time Series in Nonlinear Dynamics

The 1980s decade of greed & fast results

(LBOs, junk bonds, fractal dimension).

Packard et al. (1980), Roux et al. (1980);

Mañe (1981), Ruelle (1981), Takens (1981);

Method of delays:

Differentiation ill-posed ⇒ use differences instead!

1st Problem smoothness:

Whitney embedding lemma doesn’t apply to most attractors (e.g.,Lorenz)2nd Problem noise;3rd Problem sampling: long recurrence times.

Some rigorous results on convergence: Smith (1988, Phys. Lett. A), Hunt (1990, SIAM J. Appl. Math.)

x = F (x, x)!!

x = y,y = F (x, y)

xi = fi(x1, ....., xn)! x(n) = F (x(n!1), ....., x)

4/28

Power Law for Spectrum

S(f) ~ f – p + poles

i.e. linear in log-log coordinates

For a 1st-order Markov process or “red noise” p = 2

“Pink” noise, p = 1 (1/f , flicker noise)

“White” noise, p = 0

Low-order dynamical (deterministic ) systems

have exponential decay of S(f) (linear in log-linear coordinates)

e.g. for Smale horseshoe ∀k ∃2k unstable orbits of period k

N.B. Bhattacharaya, Ghil & Vulis (1982, J. Atmos. Sci.) showed a spectrum S ~ f – 2 for a nonlinear PDE with delay (doubly infinite-dimensional)

6/28

Synthetic I: Gaps in Oscillatory Signal

• Very good gap filling for smooth modulation; OK for sudden modulation.

0 50 100 150 200!5

0

5a)SSA gap filling in [20:40] range

0 2 4 6 8 10

0.2

0.4

No. of modes

CVL

Erro

r

Optimum window M and number of modes101520

0 50 100 150 200!5

0

5b)SSA gap filling in [120:140] range

0 2 4 6 8

0.2

0.4

No. of modesCV

L Er

ror

Optimum window M and number of modes

152025

0 0.1 0.2 0.3 0.4 0.5

100

SSA spectrum, M=10

Frequency0 0.1 0.2 0.3 0.4 0.5

100

SSA spectrum, M=20

Frequency

22/28

Synthetic II: Gaps in Oscillatory Signal + Noise

1 2 3 4 5 6 7 8 91

1.1

1.2

No. of modes

CVL

Erro

r

Optimum SSA window M and number of modesM=30M=40M=50M=60

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

100

SSA spectrum, M=50

Frequency

0 100 200 300 400 500 600!4

!2

0

2

Time

SSA filling of [80:120] gap

x(t) = sin( 2!300 t) ! cos( 2!

40 t + !2 sin 2!

120 t)23/28

Nile River Records

• High level

• Low level

800 1000 1200 1400 1600 1800!4!2

024

a)Original records

Year (AD)

800 1000 1200 1400 1600 1800!4!2

024

Year (AD)

b)MSSA filled in

0 5 10 15 20 25 30100

102c)MSSA spectrum of filled Nile Records:M=100

No. of modes

24/28

MC-SSA of Filled-in Records


40-100yr 64(9.3%) 64(6.9%) 64(6.6%)

20-40yr [32]

10-20yr 12.2 (5.1%), 18.3 (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%),


40-100yr 64(9.3%) 64(6.9%) 64(6.6%)

20-40yr [32]

10-20yr 12.2 (5.1%), 18.3 (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%),

0 0.1 0.2 0.3 0.4 0.510!1

100

101

71y24y 7.2y 4.2y 2.8y 2.2y

Freq (cycle/year)

High!water 622!1922, SSA M=75 years

0 0.1 0.2 0.3 0.4 0.5

10!1

100

101

71y12.2y 7.3y 2.9y18.9y 2.2y

Freq (year/cycle)

High!Low Water Difference, 622!1922, SSA M=75 years

SSA results for the extended Nile River records; arrows mark highly significant peaks (at 95%), in both SSA and MTM. 25/28


40–100yr 64 (13%) 85 (8.6%) 64 (8.2%)

20–40yr 23.2 (4.3%)

10–20yr [12], 19.7 (5.9%) 12.2 (4.3%), 18.3 (4.2%)

5–10yr [6.2] 7.3 (4.0%) 7.3 (4.1%)

0–5yr 3.0 (4%), 2.2 (3.3%)

4.2 (3.3%),2.9 (3.3%), 2.2 (2.9%)

[4.2], 2.9 (3.6%), 2.2 (2.6%)

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


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.

€

ΛB = RXt CR

Covar. bruit rouge{ RX

EOFs données}


Monte Carlo SSA: red noise test"

lecture iv: ergodic theory of ddss & rdss · the kinetic theory of gases (l. boltzmann),...

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