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Statistical Validation of The Unified Cycle Theory. Purposes: Objectively determine wavelengths for the Extra-Universal Wave Series cycles. Test the null hypothesis that random fluctuations cause the EUWS cycles. If a question refers to a specific graph, please note the slide number →.

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statistical validation of the unified cycle theory

Statistical Validation ofThe Unified Cycle Theory

Purposes:

Objectively determine wavelengths for the Extra-Universal Wave Series cycles.

Test the null hypothesis that random fluctuations cause the EUWS cycles.

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

analytical methods

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.

criteria for validating euws cycles
Criteria for Validating EUWS Cycles

Criteria 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.

criteria for validating euws cycles1
Criteria for Validating EUWS Cycles

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.
slide5

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.
slide6

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.
slide7

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.
slide8

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 not
  • detected in other time-series. Hence, the 274-myr cycle is the only valid one here.
slide9

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.
slide10

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.
slide11

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.
slide12

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.
slide13

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.
slide14

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.
slide15

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.
slide16

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.
slide17

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.
slide18

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.
slide19

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.
slide20

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).
slide21

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]
slide22

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.
slide23

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.
slide24

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.
slide25

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.
slide26

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.
slide27

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.
slide28

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.
slide29

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.
slide30

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.
slide31

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).
slide32

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.
slide33

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
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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