Statistical Validation of

The Unified Cycle Theory

Purposes:1) Objectively determine wavelengths for

the Extra-Universal Wave Series cycles.2) Test the null hypothesis that random

fluctuations cause the EUWS cycles.

1If a question refers to a specific graph, please note the slide number →

Analytical Methods• Lomb-Scargle Periodogram – Spectral analysis method for determining the power of cycles detected in a time-series.

• Smoothed Periodogram – Estimates a wavelength when several peaks cluster together. Also used for determining confidence levels.

• Lagged Correlation Analysis – Used to compare time-series data to a model for all lags. This type of analysis assesses when the various EUWS cycles actually peaked and troughed.

• Monte Carlo Simulation – Random sequences generated to test their correlation to the model used in the Lagged Correlation Analysis.

• Probability Mass Function – Used for testing significance when only one phase of a cycle is certain, while other phases are unknown or less certain.

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Criteria for Validating EUWS CyclesCriteria A – For a time-series with minimal gaps in the data:

• Spectral Peak – The cycle must be among the top three peaks in the spectrum -- preferably, coming from the Lomb-Scargle spectrum.

• Wavelength – The cycle must fall within 3% of a theoretical EUWS period.

• Confidence Level – At the frequency being tested, the smoothed periodogram confidence bands must exceed the null continuum at the 95% level.

• Lagged Correlation – The lagged correlation analysis must produce a correlation coefficient with significance above the 90% confidence level.

• Myr and Gyr Cycles – The maximum correlation from the lagged correlation analysis must not deviate more than 20% from the cycle's theoretical phases.Note for Cycles below 1-Myr: For higher frequency cycles, delays following EUWS strikes often occur in global climate proxies because of natural response lags . In these cases, the lead-lag analysis becomes meaningless as a validation tool.

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Criteria for Validating EUWS Cycles

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Criteria B – Used when one phase of a cycle is better known than other phases:

• Probability Mass Function – Ideal for testing a discontinuous time-series.

• Exact Binomial Test – In the R-Statistical package, the Exact Binomial test performs the probability mass function calculation, along with hypothesis testing at a specified confidence level.

• Confirmation – An EUWS cycle is confirmed if the null hypothesis of randomness can be rejected at the 99% confidence level.

• Example of Usage – The times when major civilizations collapsed is well known, while the stages of ascent are less certain in many cases. The probability mass function can be used to test for a cyclical pattern in the collapses.

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Time-Series Information…• Star formation index• 120 observations to 12.8 Ga• [Hopkins & Beacom, 2006]

Test Results…• Age Errors: Up to 10%• Theoretical Period: 2.47-gyr• Periodogram Est.: 2.764-gyr• Lead/Lag Time: +0.084-gyr• Sm. Periodogram CL: 95%• Lag’d Correlation CL: 99%• No. of Cycles in Data: 4• Status: Failed

Comments…The test failed because the 2.764-gyr wavelength fell outside of the 3% tolerance. Large age-errors prevented accurate testing.

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Time-Series Information…• Volcanic activity proxy• Index of crustal formation• [McCulloch & Bennett, 1994]

Test Results…• Theoretical Period: 822-myr• Periodogram Est.: 830-myr• Lead/Lag Time: +9.1-myr• Sm. Periodogram CL: 95%• Lag’d Correlation CL: 95%• No. of Cycles in Data: 6• Status: Confirmed

Comments…Only 1 peak in the spectrum, which fell very close to the 822-myr theoretical period.

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Time-Series Information…• Macro-evolution index• Ages when new genes

appeared• [Ding et al., 2006]

Test Results…• Theoretical Period: 822-myr• Periodogram Est.: 812.8-myr• Lead/Lag Time: -119-myr• Sm. Periodogram CL: 99.9%• Lag’d Correlation CL: 99.9%• No. of Cycles in Data: 7• Status: Confirmed

Comments…Age-errors were unspecified; however, the data cross correlated with zircons and stromatolite mats. The slight lag behind volcanism suggests a chain reaction of volcanoes-environment-evolution.

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Time-Series Information…• 3-gyr climate proxy• Data from many sources• [Veizer, 2004]

Test Results…• Theoretical Period: 274-myr• Periodogram Est.: 273.3-

myr• Lead/Lag Time: -19.6-myr• Sm. Periodogram CL: 95%• Lag’d Correlation CL: 95%• No. of Cycles in Data: 10• Status: Confirmed

Comments…The period at 442-myr was notdetected in other time-series. Hence, the 274-myr cycle is the only valid one here.

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Time-Series Information…• Volcanic activity proxy• Histogram of igneous zircons• [Condie, 2009]

Test Results…• Theoretical Period: 274-myr• Periodogram Est.: 268.9-myr• Lead/Lag Time: -25.2-myr• Sm. Periodogram CL: 99.9%• Lag’d Correlation CL: 95%• No. of Cycles in Data: 14• Status: Confirmed

Comments…Once again, the 274-myr cycle is the only period that consistently appears in the spectra. Repetition of an estimate adds to its credibility.

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Time-Series Information…• Volcanic activity proxy• Histogram of zircons from

ancient sediments.• [Condie, 2009]

Test Results…• Theoretical Period: 274-myr• Periodogram Est.: 268.5-myr• Lead/Lag Time: +12.6-myr• Sm. Periodogram CL: 99%• Lag’d Correlation CL: 90%• No. of Cycles in Data: 7• Status: Confirmed

Comments…Once again, the spectrum shows the bands around 274-myr as a high probability area for a cycle.

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Time-Series Information…• Volcanic activity proxy• Histogram of zircons from

ancient sediments.• [Condie, 2009]

Test Results…• Theoretical Period: 91.3-myr• Periodogram Est.: 93.0-myr• Lead/Lag Time: -5.95-myr• Sm. Periodogram CL: 95%• Lag’d Correlation CL: 90%• No. of Cycles in Data: 11• Status: Confirmed

Comments…This test analyzed zircon distribution over the past 1 billion years, a span where age estimates contain the greatest accuracy.

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Time-Series Information…• Macro-evolution index• Ages when new genes

appeared• [Ding et al., 2006]

Test Results…• Theoretical Period: 91.3-myr• Periodogram Est.: 92.18-myr• Lead/Lag Time: -7.87-myr• Sm. Periodogram CL: 99%• Lag’d Correlation CL: 99%• No. of Cycles in Data: 30• Status: Confirmed

Comments…This evolutionary cycle is highly significant because it covers 30 cycles with a stable 91.3-myr oscillation through the entire span of the time-series.

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Time-Series Information…• Climate proxy• Data from many sources• [Veizer, 2004]

Test Results…• Theoretical Period: 30.4-myr• Periodogram Est.: 30.98-myr• Lead/Lag Time: +5.31-myr• Sm. Periodogram CL: 99%• Lag’d Correlation CL: 95%• No. of Cycles in Data: 17• Status: Confirmed

Comments…The time since the Cambrian contains the cleanest portion of Veizer’s data. That’s the portion used here to test the 30.4-myr EUWS cycle.

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Time-Series Information…• Volcanic activity proxy• Histogram of zircons from

modern river sediments.• [Condie, 2009]

Test Results…• Theoretical Period: 30.4-myr• Periodogram Est.: 30.33-

myr• Lead/Lag Time: +6.05-myr• Sm. Periodogram CL: 99%• Lag’d Correlation CL: 90%• No. of Cycles in Data: 13• Status: Confirmed

Comments…This tests covers the past 400-myr – a period where the age-errors for the zircons were the smallest.

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Time-Series Information…• Climate proxy• Derived from ocean

sediments• [Zachos, 2001]

Test Results…• Theoretical Period: 3.38-myr• Periodogram Est.: 3.36-myr• Lead/Lag Time: +0.01 myr• Sm. Periodogram CL: 99.9%• Lag’d Correlation CL: 95%• No. of Cycles in Data: 19• Status: Confirmed

Comments…The broad range of the peak around 3.38-myr, as well as the smoothed periodogram peak at 3.31-myr, indicates a powerful cycle in the vicinity of 3.38-myr.

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Time-Series Information…• Climate proxy• Derived from ocean sediments• [Zachos, 2001]

Test Results…• Theoretical Period: 1.13-myr• Periodogram Est.: 1.11-myr• Lead/Lag Time: -0.102-myr• Sm. Periodogram CL: 99.99%• Lag’d Correlation CL: 90%• No. of Cycles in Data: 19• Status: Confirmed

Comments…This test, covering the time from 44 Ma to 22 Ma, spans 19 oscillations. It estimates the wavelength at 1.11-myr, with a 90% confidence level derived from the Monte Carlo simulations.

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Time-Series Information…• Climate proxy • Derived from ocean sediments• [Zachos, 2001]

Test Results…• Theoretical Period: 1.13-myr• Periodogram Est.: 1.12-myr• Lead/Lag Time: -0.062-myr• Sm. Periodogram CL: 99.99%• Lag’d Correlation CL: 99%• No. of Cycles in Data: 19• Status: Confirmed

Comments…This test examined the time from 22 Ma to present – using the same time-series as the previous slide. Breaking the analysis in two tested stability in the cycle. A random process should not produce identical peaks in the spectra – as these tests did.

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Time-Series Information…• Climate proxy• Derived from ocean sediments• [Zachos, 2001]

Test Results…• Theoretical Period: 125-kyr• Periodogram Est.: 124.8-kyr• Lead/Lag Time: -14.6-kyr• Sm. Periodogram CL: 99%• Lag’d Correlation CL: 95%• No. of Cycles in Data: 58• Status: Confirmed

Comments…This periodogram confirmed two cycles. (1) The Eccentricity cycle came close to the theoretical value of 94.94-kyr. [Berger, 1978] (2) The 125-kyr EUWS cycle also appeared, significant at the 95% confidence level.

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Time-Series Information…• Geomagnetic proxy• 1.5-myr paleointensity index• [Channell, et al., 2009]

Test Results…• 2 x Theor. Period: 83.523-kyr • Periodogram Est.: 83.356-kyr• Lead/Lag Time: -11.4-kyr• Sm. Periodogram CL: 99.99%• Lag’d Correlation CL: not est.• No. of Cycles in Data: 17• Status: Confirmed

Comments…This EUWS period came within 0.2% of doubling the 41.7616-kyr cycle. The polarity visible in geomagnetism suggests other data only reflect the magnitude of EUWS oscillations.

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Time-Series Information…• Geomagnetic proxy• 2.2-myr paleointensity index• [Yamazaki & Oda, 2005]

Test Results…• Theoretical Period: 41.8-kyr• Periodogram Est.: 41.45-kyr• Lead/Lag Time: -2.16-kyr• Sm. Periodogram CL: 99%• Lag’d Correlation CL: 99.9%• No. of Cycles in Data: 53• Status: Confirmed

Comments…This geomagnetic index confirmed the 41.8-kyr cycle. In addition this test showed strong evidence of EUWS polarity by revealing periods at twice 13.9-kyr (95% CL) and twice 41.8-kyr (99.9% CL).

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Time-Series Information…• Climate proxy• Derived from ocean sediments• [Zachos, 2001]

Test Results…• Theoretical Period: 41.8-kyr• Periodogram Est.: 41.12-kyr• Lead/Lag Time: -15.9-kyr• Sm. Periodogram CL: 99.99%• Lag’d Correlation CL: 99.9%• No. of Cycles in Data: 25• Status: Confirmed, but

rejected

Comments…This EUWS confirmation was rejected because numerous tests produced an average of 41.0-kyr. The periodogram also showed a 23.72-kyr cycle. Both periods exactly match the Obliquity and Precession cycles. [Berger, 1978]

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Time-Series Information…• Atmospheric methane• Methane from Vostok ice-core• [Petit et al., 2001]

Test Results…• Theoretical Period: 41.8-kyr• Sm Periodogram Est: 41.7-

kyr• Lead/Lag Time: -8.2-kyr• Sm. Periodogram CL: 99%• Lag’d Correlation CL: 90%• No. of Cycles in Data: 10• Status: Confirmed

Comments…Methane cycles are theorized to result from gas released from melting ice sheets – or from bacterial breakdown of organic matter in wetlands. Alternatively, this cycle may indicate a 41.8-kyr pattern in volcanic discharge.

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Time-Series Information…• Volcanic activity proxy• Dust from Vostok ice-core• [Petit et al., 1990]

Test Results…• Theoretical Period: 41.8-kyr• Periodogram Est.: 42.4-kyr• Lead/Lag Time: +16.4-kyr• Sm. Periodogram CL: 75%• Lag’d Correlation CL: 99%• No. of Cycles in Data: 3.5• Status: Failed

Comments…This test only failed from a lack of cycles in the time-series (3.5) and a 75% confidence level from the smoothed periodogram. However, the broad peak around 41.8-kyr suggests of a significant volcanic cycle in the vicinity.

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Time-Series Information…• Volcanic activity proxy• Dust from Vostok ice-core• [Petit et al., 2001]

Test Results…• Theoretical Period: 41.8-kyr• Periodogram Est.: 39.76-kyr• Lead/Lag Time: +13.8-kyr• Sm. Periodogram CL: 99.9%• Lag’d Correlation CL: 99%• No. of Cycles in Data: 9• Status: Failed

Comments…Periodograms of volcanic activity consistently fail to reveal the Eccentricity & Precession cycles. This shows that volcanic activity results solely from EUWS cycles, and not from Milankovitch cycles.

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Time-Series Information…• Volcanic activity proxy• Dust from Vostok ice-core• [Petit et al., 1990]

Test Results…• Theoretical Period: 13.9-kyr• Periodogram Est.: 13.75-kyr• Lead/Lag Time: +1.45-kyr• Sm. Periodogram CL: 99.9%• Lag’d Correlation CL: 99.9%• No. of Cycles in Data: 12• Status: Confirmed

Comments…This test confirmed the 13.9-kyr cycle in volcanism. Once again, the Milankovitch cycles were not apparent in this volcanic time-series.

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Time-Series Information…• Climate proxy for Israel• Speleothem from Peqiin Cave• [Bar-Matthews et al., 2003]

Test Results…• Theoretical Period: 13.9-kyr• Periodogram Est.: 13.84-kyr• Lead/Lag Time: -4.48-kyr• Sm. Periodogram CL: 95%• Lag’d Correlation CL: 99%• No. of Cycles in Data: 14• Status: Confirmed

Comments…Getting back to a climate time-series, the 23.7-kyr Precession cycles reappeared. However, the 13.9-kyr EUWS cycle also made its presence felt – significant at the 95% confidence level.

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Time-Series Information…• Climate proxy for Israel• Speleothem from Soreq

Cave• [Bar-Matthews et al., 2003]

Test Results…• Theoretical Period: 13.9-kyr• Periodogram Est.: 14.15-

kyr• Lead/Lag Time: -5.11-kyr• Sm. Periodogram CL: 95%• Lag’d Correlation CL: 99%• No. of Cycles in Data: 12• Status: Confirmed

Comments…Another cave in Israel (Soreq) also showed the 13.9-kyr climate cycle. In this case, the 13.9-kyr cycle reached a power that rivaled the 23.7-kyr Precession cycle.

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Time-Series Information…• Climate proxy for Antarctica• From Dome Fuji ice-core• [Kawamura, et al., 2007]

Test Results…• Theoretical Period: 4.64-kyr• Periodogram Est.: 4.65-kyr• Lead/Lag Time: -1.80-kyr• Sm. Periodogram CL: 99.99%• Lag’d Correlation CL: 99%• No. of Cycles in Data: 37• Status: Confirmed

Comments…The newer portion of the Dome Fuji ice-core, from 172 Ka to present, strongly confirmed the 4.64-kyr EUWS cycle. This portion of the time-series contained the smallest age-errors.

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Time-Series Information…• Climate proxy for Turkey• Stalagmites from Sofular

Cave • [Fleitmann et al., 2009]

Test Results…• Theoretical Period: 4.64-kyr• Periodogram Est.: 4.674-kyr• Lead/Lag Time: -1.15-kyr• Sm. Periodogram CL: 95%• Lag’d Correlation CL: 99%• No. of Cycles in Data: 10• Status: Confirmed

Comments…And yet another climate proxy confirmed the 4.64-kyr EUWS cycle. However, this cycle was more powerful in the Antarctic climate time-series.

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Time-Series Information…• Climate proxy for Greenland• Greenland ice-core (GISP2) • [Nat’l Snow & Ice Data

Center]

Test Results…• Theoretical Period: 1.55-kyr• Sm Periodogram Est: 1.51-

kyr• Lead/Lag Time: +0.115-kyr• Sm. Periodogram CL:

99.99%• Lag’d Correlation CL: 90%• No. of Cycles in Data: 33• Status: Confirmed

Comments…The 1.55-kyr cycle is amazingly powerful in Greenland’s climate history. Spectral analysis showed a period slightly below the theoretical value – but close enough in the smoothed periodogram for confirmation.

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Time-Series Information…• Climate proxy for Turkey• Stalagmites from Sofular Cave • [Fleitmann et al., 2009]

Test Results…• Theoretical Period: 1.55-kyr• Periodogram Est.: 1.51-kyr• Lead/Lag Time: +0.58-kyr• Sm. Periodogram CL: 99.9%• Lag’d Correlation CL: 99%• No. of Cycles in Data: 16• Status: Confirmed

Comments…The Turkish time-series was split in two. Both tests revealed a powerful cycle in the vicinity of 1.55-kyr. Also, both tests showed a strong cycle at twice the 516-yr EUWS cycle (near frequency 1.0).

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Time-Series Information…• 11,400 year sunspot index• Derived from tree-rings• [Solanki et al., 2005]

Test Results…• Theoretical Period: 516-yr• Periodogram Est.: 521.5-yr• Lead/Lag Time: -33.9-yr• Sm. Periodogram CL: 99%• Lag’d Correlation CL: 99%• No. of Cycles in Data: 21• Status: Confirmed

Comments…The EUWS cycles match several significant sunspot and starspot cycles. However, every star contains its own internal magnetic cycles – independent of the EUWS cycles.

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Time-Series Information…• Volcanic activity proxy• Histogram of zircons• [Condie, 2009]

Lagged Correlation Analysis…The black lines in the graphs to the right show two zircon time-series after filtering.

The red lines show the EUWS models used to test correlation with the filtered data. Once the lagged correlation was found, it was compared with 36,000 Monte Carlo simulations.

This comparison determined how often random numbers produce equivalent correlations – thus providing an excellent estimate of statistical significance.

The Unified Cycle Theory Because of time limitations, this review of EUWS tests must

stop at 516-yr. This completes the statistical evaluation.

The hypothesis that random fluctuations cause the EUWS cycles has been rejected consistently for all but two of the cycles between 9.57-day and 822-gyr. To continue using a hypothesis of randomness, a proponent of choas must explain why a subjective judgment of randomness is preferred over the statistical assessment of EUWS periodicity.

Questions?

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