Below is an investigation of scale invariance or long term persistence (LTP) in time series including tree-ring proxies â€“ the recognition, quantification and implications for analysis â€“ drawn largely from Koutsoyiannis  (preprints available here). In researching this topic, I found a lot of misconceptions about LTP phenomena, such as LTP implying a long term memory process, and a lack of recognition of the implications of LTP. As to implications, the standard error of the mean of global temperatures at 30 data points is 4 times larger than the usual estimate for normal errors. Given that LTP is a fact of nature â€“ attributed by Koutsoyiannis to the maximum entropy (ME) principle â€“ this strongly suggests the H should be considered in all hypothesis testing.
There are a number of ways of modeling natural data, which leads to the question: What is an appropriate model? While basic features such as the mean, standard deviation and linear trends are usually the basis of analysis or data, little attention is paid to their autocorrelation properties, and how they scale. Most natural series have a very interesting character, and climate in particular seems to have long term internal correlations. These internal features have a bearing at a minimum on the information needed to predict, and the significance levels for testing of hypotheses. Understanding how these series are constructed helps to guide choice for modeling phenomena.
Here we construct a set of the basic series to simulate the global temperature series from CRU and use a number of tools to examine their properties. We then examine a set of proxy series provided at climateaudit.com as listed in Table 1.
An IID series is the simplest and most familiar series consisting of independent random numbers with a distribution such as the normal distribution. In order to predict future terms in the series, we could make use of the long term mean and variance of past data (hence identical), but the previous term is of no use in predicting the next term (hence independence). We first determine the parameters of the global temperature series to use for generating simulated data. The mean (or drift) in values is not of interest as all values are standardized, but the standard deviation (or dispersion) is estimated from the differenced values to eliminate the drift from the CRU values. A series of random numbers with a normal distribution and a standard deviation equal to CRU is then generated and appended to the data frame. The standard deviation is estimated as 0.15.
Moving averages are probably the next most familiar series, where the average of a limited set or window of values is calculated for every position in the series. In R this is done with the filter command, the filter being determined by a list of numbers to use as coefficients in a summation â€“ in this case 30 values of 1/30 provide a 30 year moving average for CRU. In a MA model the previous value of the series will help predict the next value. While a MA is often called a low frequency bandpass filter, suppressing high frequency fluctuations while passing the long frequency ones. It is also ‘non-stationary’ as the mean is not constant. We need to change the length of the filtered series to match the original series, as filtering reduces the length of the series.
> x <- filter(d$CRU, filter = rep(1, 30)/30)
> length(x) <- length(d$CRU)
> d <- cbind(d, list(CRU30 = x))
An auto-regression model is the next most familiar. Here each term in the series is determined by the previous terms plus some random error. In an AR(1) (or Markov) model only the previous term is useful in predicting the next term. An AR(1) series has the equation X(t) = e + aX(t – 1) where a is a coefficient and e is a random error term. A random walk is the simplest form where a = 1. It can be generated from the series of random numbers by taking the cumulative sum of the random series. We can estimate the value of a with the ar() function and generate an AR(1) model using the R facility arima.sim and the parameters from the CRU temperature data. The coefficient is a = 0.67 and standard deviation is sd = 0.15 for the AR(1) model of CRU.
The next and final series goes by many names: self similar, fractal, roughness, fractional Gaussian noise model (FGN), long term persistence (LTP), clustering or simple scaling series (SSS). Specifically they are characterized as having constantly scaling variance (or standard deviation) over all time or spatial scales. The well known proportionality of the variance of FGN series is:
V ar(k) k2H
where k is the scale, and H is a constant called the Hurst exponent. On a log-log plot of standard deviation this equation is a straight line from which H can be estimated.
log(StDev(k)) = c + Hlog(k)
Among other things, Koutsoyiannis shows a number of different methods of producing models that produce this scaling behavior . He also shows that FGN models of natural climatic phenomena satisfy the maximum entropy (ME) condition, which provides a satisfying link to thermodynamic principles . While there are a number of ways of approximating the FGN series one of the easiest is to generate a series with Gaussian errors and additional large and medium scale random fluctuations of the mean. The timing of the larger scale fluctuations is determined by a random variable with a logarithmic distribution in order to ensure no periodicity.
In Figure 1 the simulated series are plotted w.r.t. time. The AR(1) and the SSS series resemble quite closely the CRU natural series. However the IID series does not to capture the longer time scale fluctuations. In comparison, the random walk has very large long time scale fluctuations. While it can be seen by eye on Figure 1 that IID and random walk are not good models for the natural series more insightful methods are needed to distinguish the LTP properties. The FGN models are described unflatteringly as having ‘fat tails’. This refers to the way the distribution of less frequent difference values fades out into a thicker tail (power-type) rather than the exponential form of a normal distribution. When these distributions are plotted on Figure 1 it is hard to see which are Gaussian and which are not. We need more powerful ways to examine the data.
|Figure 1:||A: Plots of the global temperature (CRU), and the simulated series random, walk, ar(1), and sss. B: Probability distributions for the differenced variables.|
One of the main tools for examining the autocorrelation structure of data is the autocorrelation function or ACF. The ACF provides a set of correlations for each distance between numbers in the series, or lags. The autocorrelation decays in a characteristic fashion for each series as the lags get longer as shown in Figure 2. It can be seen that IID series decays very quickly (no long term correlation), the AR(1) model decays fairly quickly, the SSS next and the random walk most slowly. Another way to look at the autocorrelation properties is to plot the ACF at a constant lag, but at increasing aggregate time (or spatial scales). The aggregate is calculated as follows. For example, given a series of numbers X, the aggregated series X1, X2 and X3 is as follows.
This produces the plot on Figure 2B showing more clearly the decay in the autocorrelation of the series.
|Figure 2:||Autocorrelation function (ACF) of the simulated series for lags 1 to 40 with the 1/lag (solid line) for comparison. B: Lag 1 ACF of the aggregated processes at time scales 1 to 40.|
A plot of the logarithm of standard deviation of the simulated series against the logarithm of the level of aggregation or time scale on Figure 3 shows scale invariant series as straight lines. The plot of random numbers is a straight line of low slope (0.5). The random walk is also a straight line of higher slope (1) and hence with scale invariant standard deviation. The CRU and SSS are also a straight lines. The AR(1) model deviates from the initial slope of the SSS, towards the slope of the random line, showing decreasing standard deviation at larger scales and demonstrating the AR(1) model is not scale invariant.
The difference between the AR(1) model and the SSS model is seen even more clearly on a plot of the standard deviation of the mean against the lag-one autocorrelations against time scale (Figure 3). Here the implications of high self similarity or H value begin to be seen. Series such as the SSS, CRU and the random walk would maintain high standard deviations for very large numbers of data, while the s.d. of the IID series declines very quickly with more data This means that where a series has a high H, increasing numbers of data do not decrease our uncertainty in the mean very much. At the level of the CRU of H = 0.95 the uncertainty in a mean value of 30 points is almost as high as the uncertainty in a mean of a few points. It is this feature of LTP series that is of great concern where accurate estimates of confidence are needed.
|Figure 3:||A: Log-log plot of the s.d. of the aggregated simulated processes vs. time scale k. B. Log-log plot of the s.d. of the mean of the aggregated simulated processes vs. time scale k.|
We can calculate the H values for the simulated series from the slope of the regression lines on the log-log plot as listed in Table 2. The random series with no persistence has a Hurst exponent of about 0.5. The H of the AR(1) model is 0.67 while the SSS model we generated has an H of 0.83. The global temperature has a high H of 0.94 and the random walk is close to one.
What do the proxies look like? Figure 4 below is a log-log plot of the standard error of the mean of the proxies w.r.t scale. The lines indicate high levels of scale invariance although the slopes of the line differ slightly. In Figure 4 the lag-one ACF against scale for the proxies shows persistent correlation at long time scales. The Hurst exponents range from 0.82 to 0.97 as shown on Table 2.
|Figure 4:||A: Log-log plot of the s.d. of the proxy series vs time scale. B: Plot of the lag one ACF vs. time scale k.|
Which model should we use? The conclusions are clear.
1. Natural series are better modeled by SSS type models than AR(1) or IID models.
What are the consequences? The variance in an an SSS model is greater than IID models at all scales. Thus using IID variance estimates will lead to Type I errors, spurious significance, and the danger of asserting false claims. As variance of the natural LTP series is much higher than other series, the significance of trends in natural series is likely to be greatly underestimated by assuming IID errors and also underestimated by assuming AR(1) behaviour. The normal relationship for IID data for the standard error of the mean with number of data n
SE[IID] = /sqrt(n)
has been generalized to the following form in :
SE[LTP] = /n1–H
Where the errors are IID then H = 0.5 and the generalized from becomes identical to the upper IID form. How much larger is the standard deviation for non IID errors? By dividing and simplifying the equations above we get
SE[LTP]/SE[IID] = nH–0.5
This is plotted by n at a number of values of H below. It can be seen that at higher H values the SE[LTP] can be many times the SE[IID] Figure 5. When the 30 year running mean of temperature is plotted against the CRU temperatures it can be seen that the temperature increase from 1990 to 1950 is just outside the 95% confidence intervals for the FGN model (dotted line). The CIs for the IID model are very narrow however (dashed line),
|Figure 5:||A: Number of times by which the s.e. for FGN model exceeds s.e. for IID models at different H values. B: Confidence intervals for the 30 year mean temperature anomaly under IID assumptions (dashed line) and FGN assumptions (dotted lines).|
Even without this practical consequence, the relationship of FGN to maximum entropy provides a justification for their use. The self-similar property of natural series amounts to maximization of entropy simultaneously at all scales . This provides a link to the processes that may be responsible for the series consistent with the second law of thermodynamics, which states that overall entropy is never decreasing. While FGN models seem more complex than IID, they are justified simply because they are the ways things are. Other forms of model examined here do not necessarily maximize entropy. It is shown in  that the use of an annual AR(1) or similar model amounts to an assumption of a preferred (annual) time scale, but there is neither a justification nor evidence for a preferred time scale in natural series. Also, it is noted that the natural proxy series have only a slightly lower value for H than the CRU temperatures. While this provides additional support for the high values of the temperature series, it also may indicate a loss of some long term persistence in the temperature reconstructions by proxies.
 Demetris Koutsoyiannis. Uncertainty, entropy, scaling and hydrological stochastics, 2, time dependence of hydrological processes and time scaling. Hydrological Sciences Journal, 50(3):405â€“426, 2005.