Thongchai Thailand

Nonlinear Dynamics: Is Climate Chaotic?

Posted on: May 25, 2018

  1. The video above plots the same time series twice – in red and in blue. In the red line, the events in the time series evolve as a deterministic system; independently with no dependence or persistence. Such independence is an important assumption in OLS (ordinary least squares) regression. In the blue line the events in the time series evolve as a chaotic system with dependence and persistence. A small dependence has been inserted as a 1% tendency for persistence. Without persistence rising and falling tendencies are always equal with 50% probability at each event. The result of that is seen in the red line where the ups and downs are not visible because they are small.
  2. The 1% persistence in the blue line means that if the prior change was positive, the probabilities change from 50%/50% to 51%/49% favoring a positive change for the next event. If it is positive again in the next event, the probabilities are changed to 52%/48% but if it is negative they change back to 50%/50%. The probabilities keep changing to favor the direction of change in the prior event. This behavior is called persistence and it is very common in nature, particularly in surface temperature.
  3. When you play the video you will see the blue line take various shapes and trends both rising and falling mostly staying very close to the Gaussian red line. BUT, it is capable of sudden departures from the Gaussian to form what appears to be patterns such as rising and falling trends. These shapes do not imply cause and effect phenomena. They are the random behavior of a chaotic system.
  4. All of these shapes are representations of the same underlying phenomenon. The differences among these curves have no interpretation because they represent randomness. The human instinct to look for causes for unusual patterns derives from Darwinian survival but it leads us astray when we study chaotic systems.
  5. The shapes and trends the blue line forms may be found to be statistically significant if the system is assumed to be deterministic and the violations of OLS assumptions are ignored; but that statistical significance is meaningless. Yet this kind of analysis is common. The conclusions they imply in terms of the phenomena of nature are have no interpretation because they are the product of violations of assumptions.
  6. Surface temperature and other climate variables are known to be chaotic and therefore, not all observed patterns in climate data contain contain information about cause and effect phenomena.That nature is chaotic is well understood. These three books present that argument with data, examples, and case studies: EA Jackson, Exploring Nature’s Dynamics; C Letellier, Chaos in Nature; Cushing et al, Chaos in Ecology.
  7. References to chaos in nature are also found in the climate change literature. These two excellent papers by Timothy Palmer are a good introduction to the subject of chaos in climate data: “A nonlinear dynamical perspective on climate prediction” Journal of Climate 12.2 (1999): 575-591, and “Predicting uncertainty in forecasts of weather and climate” Reports on Progress in Physics 63.2 (2000): 71.
  8. A common method of detecting fractal/chaotic behavior in long time series of field data is to compute the so called Hurst exponent “H” of the time series as a way of detecting persistence in the data.  Persistence implies that the the data in the time series do not evolve independently but contain a dependence on prior values such that prior changes tend to persist into the next time slice. This method was first described by Harold Edwin Hurst in 1951 in the only paper he ever published which is still cited hundreds of times a year every year ( Hurst, H.E. (1951). “Long-term storage capacity of reservoirs”. Transactions of American Society of Civil Engineers).
  9. The Hurst exponent method of detecting non-linear dynamics in time series data is used in climate change research. This trend in climate science has been led by the very charismatic and controversial hydrologist Demetris Koutsoyiannis, Professor of Hydrology, National Technical University of Athens. Demetris finds that many hydrology time series behaviors that climate science ascribes to emissions can be explained in terms of non-linear dynamics.
  10. Here are some examples from the literature of  Hurst persistence analysis of climate data: Cohn, Timothy etal “Nature’s style: Naturally trendy.” Geophysical Research Letters 32.23 (2005) (by “naturally trendy” Tim means that things that look like trends are actually randomness); Weber, Rudolf etal. “Spectra and correlations of climate data from days to decades.” Journal of Geophysical Research: Atmospheres 106.D17 (2001): 20131-20144; Koutsoyiannis, Demetris. “Climate change, the Hurst phenomenon, and hydrological statistics.” Hydrological Sciences Journal 48.1 (2003): 3-24; Markonis, Yannis, “Climatic variability over time scales spanning nine orders of magnitude: Connecting Milankovitch cycles with Hurst–Kolmogorov dynamics.” Surveys in Geophysics 34.2 (2013): 181-207; Pelletier, Jon etal. “Long-range persistence in climatological and hydrological time series: analysis, modeling and application to drought hazard assessment.” Journal of Hydrology 203.1-4 (1997): 198-208; Rybski, Diego, et al. “Long‐term persistence in climate and the detection problem.” Geophysical Research Letters 33.6 (2006).
  11. Much of the empirical work in climate science is presented in terms of time series of field data (field data means data made by nature over which the researcher has no control – as opposed to data collected in controlled experiments). Information about climate contained in these data is usually gathered by researchers in terms of OLS (ordinary least squares) regression analysis. Surprisingly, even at the highest levels of climate research little or no attention is paid to the assumptions of OLS analysis which include for example the assumption that the data in the time series evolve as I.I.D, (independent identically distributed). The stationarity assumption further enforces the requirement that the distribution must not change as the time series evolves.
  12. More information on regression analysis may be found in a companion post on SPURIOUS CORRELATIONS. A good reference for regression analysis of time series data is “Time Series Analysis: Forecasting and Control, Wiley 2015, by George Box & Gwilym Jenkins. Another is Time Series Analysis and Its Applications with Examples, Springer 2017, by Robert Shumway. These authors have shown that when time series analysis goes awry you can bet it has to do with violations of OLS assumptions.
  13. In my work with temperature, sea level rise, precipitation, solar activity, and ozone depletion, I tested the time series for OLS violations by computing the Hurst exponent H. The theoretical neutral value with no serial dependence is H=0.5 but it has been shown that the neutral value for comparison in empirical research needs to be adjusted for the specific sub-sampling strategy used in the estimation of H. In all my work, this estimation is carried out twice – once with the data and again with a Monte Carlo simulation of the data that generates a corresponding IID/stationary series.
  14. The two values of H are then compared. If the difference between the two values of H is not statistically significant, we conclude that there is no evidence of Hurst behavior in the data and OLS regression results may therefore be interpreted in terms of the phenomena under study. However if the value of H in the data is greater than the value of H in the IID simulation, then we can conclude that the data contain Fractal/Chaotic behavior by virtue of Hurst persistence and that therefore OLS results may not be interpreted strictly in terms of the phenomena under study. Nonlinear dynamics must be considered. A list of these studies appears below with links to the freely downloadable full text.
  1. Ozone Depletion Latitudinally Weighted Mean Global Ozone 1979-2015
  2. Warming Seasonality and Dependence in Daily Mean USCRN Temperature 
  3. Precipitation The Hurst Exponent of Precipitation
  4. Warming A Robust Test for OLS Trends in Daily Temperature Data
  5. Solar Activity The Hurst Exponent of Sunspot Counts
  6. Warming The OLS Warming Trend at Nuuk, Greenland
  7. Warming The Hurst Exponent of Surface Temperature
  8. Warming OLS Trend Analysis of CET Daily Mean Temperatures 1772-2016
  9. Precipitation The Hurst Exponent of Precipitation: England and Wales 1766-2016
  10. Warming  Hurst Persistence in UAH Zonal Mean Lower Troposphere Temperatures
  11. The abstract for the UAH paper: Zonal monthly means of UAH satellite based measurements of lower troposphere temperature anomalies 12/1978 to 4/2017 are examined for violations of the independence assumption of OLS linear regression. The Hurst Exponent (H), computed at a monthly time scale, is used as the measure of dependence. It is found that the independence assumption of OLS linear regression is violated in the UAH temperature anomaly time series. The apparent multi-decadal non-linearity found in a prior work may be related to non-linear dynamics implied by the Hurst phenomenon. The interpretation of OLS linear regression trend lines for these time series in terms of anthropogenic global warming should be made with caution, particularly so in view of the short time span of the data, because the data do not comply with OLS assumptions.

 

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1 Response to "Nonlinear Dynamics: Is Climate Chaotic?"

[…] Yet another contentious issue in event attribution with climate models is the known chaotic behavior of climate that is not contained in climate models. Non-linear dynamics and chaos is discussed in a related post: IS CLIMATE CHAOTIC? […]

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