Milankovic Theory and Time Series Analysis Mudelsee M Institute of Meteorology University of Leipzig...

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Milankovic Theory and

Time Series Analysis

Mudelsee M

Institute of MeteorologyUniversity of LeipzigGermany

Climate: Statistical analysis

Data (“sample”)

Climate system (“population”, “truth”, “theory”)

Climate: Statistical analysis

Data (“sample”) STATISTICS

Climate system (“population”, “truth”, “theory”)

Climate: Statistical analysis:

Time series analysis

Sample: t(i), x(i), i = 1, ..., n

Sample: t(i), x(i), i = 1, ..., nUNI-VARIATE TIME SERIES

Sample: t(i), x(i), y(i), i = 1, ..., nBI-VARIATE TIME SERIES

Sample: t(i), x(i), y(i), i = 1, ..., nTIME SERIES: DYNAMICS

[ t(i), x(i), y(i), z(i),..., i = 1 ]TIME SLICE: STATICS

Sample: t(i), x(i), y(i), i = 1, ..., nCLIMATE TIME SERIES

o uneven time spacing*

Low reso lu tion H igh resolu tionIce coreD irect observations,

A rch ive, sam plingD epth

Sedim ent core Sedim ent core

l( i+1)L ( i )

C lim ateAge, T

docum ents,c lim ate m odel

R ecent P ast

Top Bottom

Arch ive, sam plingEstim ated age, t

d ( i+1)D ( i )

A rch ive, tim e series, t( i )Estim ated age, t

D iffusion

Arch ive, tim e series, t'( i )"U psam pling", t'

"D ow nsam pling", t'

Strong in troduced dependence

N oW eak

D ' ( i )

Mudelsee M (in prep.) Statistical Analysis of Climate Time Series: A Bootstrap Approach. Kluwer.

UNEVEN TIME SPACING

0 2 00 4 00t ( i ) (k a)

d (i )(k a)

0 5 ,0 00 1 0 ,0 00t ( i ) (a B .P .)

d (i )(a)

2 ,0 00 6 ,0 00 1 0 ,0 00t ( i ) (a B .P .)

d (i )(a)

ICE CORE

(Vostok δD)

TREE RINGS

(atmospheric Δ14C)

STALAGMITE

(Qunf Cave δ18O)

UNEVEN TIME SPACING

o uneven time spacingo persistence*

PERSISTENCE

Low reso lu tion H igh resolu tionIce coreD irect observations,

A rch ive, sam plingD epth

Sedim ent core Sedim ent core

l( i+1)L ( i )

C lim ateAge, T

docum ents,c lim ate m odel

R ecent P ast

Top Bottom

Arch ive, sam plingEstim ated age, t

d ( i+1)D ( i )

A rch ive, tim e series, t( i )Estim ated age, t

D iffusion

Arch ive, tim e series, t'( i )"U psam pling", t'

"D ow nsam pling", t'

Strong in troduced dependence

N oW eak

D ' ( i )

ICE CORE

(Vostok δD)

TREE RINGS

(atmospheric Δ14C)

STALAGMITE

(Qunf cave δ18O)

PERSISTENCE

-2 0 0 2 0 4 0

D ( t( i)) [‰ ]

4 0D( t( i - 1 ))[‰ ]

-3 0 0 3 0

14C (t(i)) [‰ ]

3 014C( t( i - 1 ))[‰ ]

- 1 0 1

18O ( t( i)) [‰ ]

118O(t(i - 1))[‰ ]

o uneven time spacingo persistence

Milankovic theory

Theory: Orbital variations influenceEarth‘s climate.

Milankovic theory

Data: Climate time series

Milankovic theory

Data: Climate time seriesTIME SERIES ANALYSIS: TEST

Milankovic theory and

time series analysis

Part 1: Spectral analysis

Part 2: Milankovic & paleoclimate —back to the Pliocene

Acknowledgements

Berger A, Berger WH, Grootes P, Haug G, Mangini A, Raymo ME, Sarnthein M, Schulz M, Stattegger K, Tetzlaff G, Tong H, Yao Q, Wunsch C

Alert!

Mudelsee-bias

Sample: t(i), x(i), y(i), i = 1, ..., n

Simplification: uni-variate, only x(i),equidistance, t(i) = i

Sample: x(t) Time series

Population: X(t)

Population: X(t) Process

Part 1: Spectral analysis:

Process level

X(t)TIME DOMAIN

Process level

X(t)TIME DOMAIN

FOURIER TRANSFORMATION: FREQUENCY DOMAIN

Process level

X(t) +T

GT(f) = (2π)–1/2∫–T XT(t) e–2πift dt,

XT= X(t), –T ≤ t ≤ +T,0, otherwise.

TIME DOMAIN

FOURIER TRANSFORMATION: FREQUENCY DOMAIN

Process level

h(f) = limT→∞ [ E {|GT(f)|2 / (2T)} ]NON-NORMALIZED POWER SPECTRAL DENSITY FUNCTION,

“SPECTRUM”

Process level

h(f) = limT→∞ [ E {|GT(f)|2 / (2T)} ]NON-NORMALIZED POWER SPECTRAL DENSITY FUNCTION,

“SPECTRUM”

“ENERGY” (VARIATION) AT SOME FREQUENCY

Process level

Discrete spectrum

Harmonic process

Astronomy

0Fre q u e n cy, f

Process level

Discrete spectrum

Harmonic process

Astronomy

0Fre q u e n cy, f

Continuous spectrum

Random process

Climatic noise

Process level

Discrete spectrum

Harmonic process

Astronomy

0Fre q u e n cy, f

Continuous spectrum

Random process

Climatic noise

Mixed spectrum

Typical climatic

The task of spectral analysis is to estimate the spectrum.

There exist many estimation techniques.

Harmonic regression

X(t) = Σk [Ak cos(2πfk t) + Bk sin(2πfk t)] + ε(t)

HARMONIC PROCESS

Harmonic regression

If frequencies fk

known a priori:

Minimize Q = Σi {x(i) – Σk [Ak cos(2πfk t) + Bk sin(2πfk t)]}2

to obtain Ak and Bk.

HARMONIC PROCESS

Harmonic regression

If frequencies fk

known a priori:

Minimize Q = Σi {x(i) – Σk [Ak cos(2πfk t) + Bk sin(2πfk t)]}2

to obtain Ak and Bk.

HARMONIC PROCESS

LEAST SQUARES

Periodogram

If frequencies fk notknown a priori:

Take least-squaressolutions Ak and Bk, fk = 0, 1/n, 2/n, ..., 1/2,

to calculate P(fk) ~ (Ak)2 + (Bk)2.

Periodogram

to calculate P(fk) ~ (Ak)2 + (Bk)2. PERIODOGRAM

Periodogram

to calculate P(fk) ~ (Ak)2 + (Bk)2.

Where fk ≈ true f P(fk) has a peak.

PERIODOGRAM

Periodogram

P (fk)

Periodogram

Original paper:

Schuster A (1898) On the investigation of hidden periodicities with application to a supposed 26 day period ofmeteorological phenomena.Terrestrial Magnetism 3:13–41.

Periodogram

Hypothesis test (significance of periodogram peaks):

Fisher RA (1929) Tests of significance in harmonic analysis.Proceedings of the Royal Society of London, Series A, 125:54–59.

Periodogram

A wonderful textbook:

Priestley MB (1981) Spectral Analysis and Time Series.Academic Press, London, 890 pp.

Periodogram

Major problem with the periodogram as spectrum estimate:

Relative error of P(fk) = 200% for fk= 0, 1/2,

100% otherwise.

Periodogram

P (fk)

Periodogram

“More lives have been lost looking at the raw periodogram

than by any other action involving time series!”

Tukey JW (1980) Can we predict where ‘time series’ should go next? In: Directions in time series analysis (eds Brillinger DR, Tiao GC). Institute of Mathematical Statistics, Hayward, CA, 1–31.

Smoothing

0 t(i)

1stSegment

2ndSegment

3rdSegment

WELCH OVERLAPPEDSEGMENT AVERAGING(WOSA)

0 t(i)

1stSegment

2ndSegment

3rdSegment

WELCH OVERLAPPEDSEGMENT AVERAGING(WOSA)

ERROR REDUCTION <

Smoothing

Tapering: Weight time series

Spectral leakage reduced

(Hanning, Parzen,triangular windows, etc.)

Smoothing problem

Several segments averaged

Spectrum estimate more accurate :-)

Fewer (n‘ < n) data per segment

Lower frequency resolution :-(

Smoothing problem

Subjective judgement is unavoidable.

Play with parameters and be honest.

100-kyr problem

Δt = 1 fk = 0, 1/n, 2/n, ...

Δt = d fk = 0, 1/(n·d), 2/(n ·d), ...Δf = (n·d)–1

100-kyr problem

Δt = 1 fk = 0, 1/n, 2/n, ...

Δt = d fk = 0, 1/(n·d), 2/(n ·d), ...Δf = (n·d)–1

[ BW > (n·d)–1 SMOOTHING

100-kyr problem

n·d ≈ 650 kyr Δf = (650 kyr)–1*

100-kyr problem

n·d ≈ 650 kyr Δf = (650 kyr)–1

(100 kyr)–1 ± Δf = (118 kyr)–1 to(87 kyr)–1

100-kyr problem

n·d ≈ 650 kyr Δf = (650 kyr)–1

(100 kyr)–1 ± Δf = (118 kyr)–1 to(87 kyr)–1

[ ± BW wider SMOOTHING

100-kyr problem

The 100-kyr cycle existed not long enough to allow a precise enough frequency estimation.

Blackman–Tukey

h = Fourier transform of ACV

Blackman–Tukey

E [ X(t) · X(t + lag) ]

Blackman–Tukey

PROCESS LEVEL E [ X(t) · X(t + lag) ]

Blackman–Tukey

PROCESS LEVEL E [ X(t) · X(t + lag) ]

SAMPLE Σ [ x(t) · x(t + lag) ] / n

Blackman–Tukey

Fast Fourier Transform:

Cooley JW, Tukey JW (1965) An algorithm for the machine calculationof complex Fourier series.Mathematics of Computation 19:297–301.

Blackman–Tukey

Some paleoclimate papers:

Hays JD, Imbrie J, Shackleton NJ (1976) Variations in the Earth's orbit: Pacemaker of the ice ages. Science 194:1121–1132.

Imbrie J Hays JD, Martinson DG, McIntyre A, Mix AC, Morley JJ, PisiasNG, Prell WL, Shackleton NJ (1984) The orbital theory of Pleistocene climate: Support from a revised chronology of themarine δ18O record. In: Milankovitch and Climate (eds Berger A,Imbrie J, Hays J, Kukla G, Saltzman B), Reidel, Dordrecht,269–305.

Blackman–Tukey

Ruddiman WF, Raymo M, McIntyre A (1986) Matuyama 41,000-year cycles: North Atlantic Ocean and northern hemisphere ice sheets. Earth and Planetary Science Letters 80:117–129.

Tiedemann R, Sarnthein M, Shackleton NJ (1994) Astronomic timescale for the Pliocene Atlantic δ18O and dust flux records of Ocean Drilling Program Site 659. Paleoceanography 9:619–638.

Multitaper Method (MTM)

Spectral estimation with optimal tapering

Thomson DJ (1982) Spectrum estimation and harmonic analysis.Proceedings of the IEEE 70:1055–1096.

MINIMAL DEPENDENCE AMONG AVERAGED INDIVIDUAL SPECTRA

MINIMAL ESTIMATION ERROR

0 500 1000

Age t ( i ) [kyr]

2223242526O bliqu ity

x ( i ) [°]

-0 .08-0 .0400.040.08

Taper va lue

0 500 1000

Age t ( i ) [kyr]

-0 .08

0.08Tapered,detrendedx( i ) [°]

0 500 1000

Age t ( i ) [kyr]0 0.02 0.04

Frequency fk [kyr-1 ]

04080120160 M ultitaper

spectrum

Average(k = 0 , 1)

c d (41 kyr)-1

0 500 1000

Age t ( i ) [kyr]

2223242526O bliqu ity

x ( i ) [°]

-0 .08-0 .0400.040.08

Taper va lue

0 500 1000

Age t ( i ) [kyr]

-0 .08

0.08Tapered,detrendedx( i ) [°]

0 500 1000

Age t ( i ) [kyr]0 0.02 0.04

Frequency fk [kyr-1 ]

04080120160 M ultitaper

spectrum

Average(k = 0 , 1)

c d (41 kyr)-1

[ BETTER: DIRECTLY VIA ASTRONOMY EQS.]

Some paleoclimate papers:

Park J, Herbert TD (1987) Hunting for paleoclimatic periodicities in a geologic time series with an uncertain time scale. Journal ofGeophysical Research 92:14027–14040.

Thomson DJ (1990) Quadratic-inverse spectrum estimates: Applications to palaeoclimatology. Philosophical Transactions of the RoyalSociety of London, Series A 332:539–597.

Berger A, Melice JL, Hinnov L (1991) A strategy for frequency spectra ofQuaternary climate records. Climate Dynamics 5:227–240.

Further points

Uneven time spacing

Further points

Uneven time spacingUse X(t) = Σk [Ak cos(2πfk t) + Bk sin(2πfk t)] + ε(t)

Lomb NR (1976) Least-squares frequency analysis of unequallyspaced data. Astrophysics and Space Science 39:447–462.

Scargle JD (1982) Studies in astronomical time series analysis. II.Statistical aspects of spectral analysis of unevenly spaceddata. The Astrophysical Journal 263:835–853.

HARMONIC PROCESS

Further points

Red noise

0Fre q u e n cy, f

h (f) PERSISTENCE

Further points

Red noise

AR1 process for uneven spacing:

Robinson PM (1977) Estimation of a time series model from unequally spaced data. Stochastic Processes and their Applications 6:9–24.

0Fre q u e n cy, f

h (f) PERSISTENCE

Further points

Aliasing

0Fre qu e n cy, f

Further points

Aliasing

Safeguards: o uneven spacing (Priestley 1981)

o for marine records: bioturbationPestiaux P, Berger A (1984) In: Milankovitch

and Climate, 493–510.

0Fre qu e n cy, f

Further points

Running window Fourier Transform

0 t(i)

Priestley MB (1996) Wavelets and time-dependent spectral analysis.Journal of Time Series Analysis 17:85–103.

Further points

Detrending*

Further points

Errors in t(i): tuned dating,absolute dating,stratigraphy.

Errors in x(i): measurement error,proxy error,interpolation error.

Further points

Bi-variate spectral analysis

For example: x = marine δ18Oy = insolation

Further points

Higher-order spectra (bi-spectra, ...)

Further points

Etc., etc.

Part 2: Milankovic & paleoclimate

Less ice /w arm er

M ore ice /co lder

0 1 2 3 4A ge t (i ) [M yr]

1 18O [‰ ]benth ic

OD P 659

Less ice /w arm er

M ore ice /co lder

0 1 2 3 4A ge t (i ) [M yr]

OD P 659Northern Hemisphere Glaciation

Less ice /w arm er

M ore ice /co lder

0 1 2 3 4A ge t (i ) [M yr]

OD P 659Northern Hemisphere Glaciation

Mid-Pleistocene Transition

Climate transitions, trend

Age t ( i )

X fit(t) x2

x1, t < t1,Xfit(t) = x2, t > t2,

x1+ (t−t1) ·(x2−x1)/(t2−t1), t1 ≤ t ≤ t2.

x1, t < t1,Xfit(t) = x2, t > t2,

x1+ (t−t1) ·(x2−x1)/(t2−t1), t1 ≤ t ≤ t2.

LEAST SQUARES ESTIMATION

Less ice /w arm er

M ore ice /co lder

0 0.5 1 1.5A ge t (i ) [M yr]

OD P 659

M IS 23/24

Less ice /w arm er

M ore ice /co lder

0 0.5 1 1.5A ge t (i ) [M yr]

OD P 659

M IS 23/24100 kyr cycle

Mudelsee M, Schulz M (1997) Earth and Planetary Science Letters 151:117–123.

DSDP 552DSDP 607ODP 659ODP 677ODP 806

~ size of Barents/Kara Sea ice sheets

Database: 2–4 Myr, 45 marine δ18O records, 4 temperature records

benthicplanktonic

Mudelsee M, Raymo ME (submitted)

Results

2,000 3,000 4,000

2,000 3,000 4,000Age (kyr )

2 ,000 3,000 4,000

2,000 3,000 4,000Age (kyr )

Mudelsee & Raymo, Figure 1

606 b G .s.

606 b P .w .

1085 b

1143 b

1148 b

851 pG .sac.

96–100 M 2–M G 2 96–100 M 2–M G 2

High-resolution recordsMudelsee M, Raymo ME (submitted)

2,000 3,000 4,000

2,000 3,000 4,000Age (kyr )

2 ,000 3,000 4,000

2,000 3,000 4,000Age (kyr )

Mudelsee & Raymo, Figure 1

606 b G .s.

606 b P .w .

1085 b

1143 b

1148 b

851 pG .sac.

96–100 M 2–M G 2 96–100 M 2–M G 2

High-resolution records

Results

NHG was a slow global climate change (from ~3.6 to 2.4 Myr).

NHG ice volume signal: ~0.4 ‰.

Milankovic theory and

time series analysis: Conclusions

(1) Spectral analysis estimates thespectrum.

(2) Trend estimation is alsoimportant (climate transitions).

G O O D I E S

Climate transitions: error bars

t1, x1, t2, x2

Time series,size n

{t(i), x*(i)}

{t(i), x(i); i = 1,…, n } {t(i)}

Ramp estimation

t1*, x1*, t2*, x2*

Take standard deviation of simulated ramp

parameters

Simulated time series, x*(i) = ramp + noise

Simulated ramp parameters

Bootstrap errors

STD, PersistenceNoise estimation

Repeat 400 times

NHG amplitudes: temperature2,000 3,000 4,000

2,000 3,000 4,000Age (kyr )

D SDP 572S ST(via ostracoda)

D SDP 607B W T(via M g/C a)

O D P 806B W T(via M g/C a)

O D P 806S ST(via foram s)

96–100 M 2–M G 2

cooling (°C) in ~3,606−2,384 kyr

0.12 ± 0.47

0.62 ± 0.29

1.0 ± 0.5

−0.85 ± 0.17

NHG amplitudes: ice volume

Temperature calibration: 18OT/T = −0.234 ± 0.003 ‰/°C (Chen 1994; own error determination)

Salinity calibration: 18OS/T = 0.05 ‰/°C (Whitman and Berger 1992)

DSDP 572 p 18OT = 0.03 ± 0.12 ‰ 18OS = −0.01 ‰ 18OI = 0.34 ± 0.13 ‰

DSDP 607 b 18OT = 0.15 ± 0.07 ‰ 18OS = −0.03 ‰ 18OI = 0.41 ± 0.09 ‰

ODP 806 b 18OT = 0.24 ± 0.12 ‰ 18OS = − 0.05 ‰ 18OI = 0.25 ± 0.13 ‰

ODP 806 p 18OT = −0.20 ± 0.04 ‰ 18OS = 0.04 ‰ 18OI = 0.43 ± 0.06 ‰

(DSDP 1085 b cooling by 1 °C 18OI = 0.35 ‰)

Average 18OI = 0.39 ± 0.04 ‰

Milankovic Theory and Time Series Analysis Mudelsee M Institute of Meteorology University of Leipzig...

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