Is Global Temperature a Random Walk?

Guest Post by Basil Copeland

Figure 1

Each month, readers here at Watt’s Up With That, over at lucia’s The Blackboard, and elsewhere, anxiously await the latest global temperature estimates, as if just one more month of data will determine one way or the other the eternal destiny of AGW (the theory of “anthropogenic global warming”). For last month, July, the satellite estimates released by UAH and RSS were up sharply, with the UAH estimate up from essentially zero in June, to +0.41°C in July, while the RSS estimate was up from +0.081°C to +0.392°C. Does this sharp rise presage the resumption of global warming, after nearly a decade of relative cooling? Or is it just another in a series of meandering moves reminiscent of what statisticians know as a “random walk?”

I have not researched the literature exhaustively, but the possibility that global temperature follows a random walk was suggested at least as early as 1991 by A.H. Gordon in an article in The Journal of Climate entitled “Global Warming as a Manifestation of a Random Walk.” In 1995 Gordon’s work was extended by Olavi Kӓrner in a note in the Journal of Climate entitled “Global Temperature Deviations as a Random Walk.” Statistician William Briggs has written about climate behaving like a random walk on his blog.

Now even I will confess that the notion that global temperature, as a manifestation of climate processes, might be essentially random is difficult to accept. But I am coming around to that view, based on what I will present here, that monthly global temperature variations do, indeed, behave somewhat like a random walk. The qualifier is important, as I hope to show.

So, what is a “random walk” and why do some think that global temperature behaves, even if only somewhat, like a random walk? And what does it matter, anyhow?

While there are certainly more elegant definitions, a random walk in a time series posits that the direction of change at any point in time is essentially determined by a coin toss, i.e. by chance. As applied to global temperature, that is the same as saying that in any given month, it is just as likely to go up as it is to go down, and vice versa. Were global temperature a true random walk, there would be no underlying trend to the data, and any claimed evidence of a trend would be spurious. One of the best known “features” of a random walk is that in a time series it appears to “trend” up or down over extended periods of time, despite the underlying randomness of the direction of change at each point in time.

So why might we think global temperature follows a random walk? One reason is suggested by a close look at Figure 1. Figure 1 is the familiar HadCRUT3 time series of monthly global temperature anomalies since 1850, with a simple linear trend line fit through the data. When we look close, we see long periods, or “runs,” in which the data are above or below the trend line. If the data were truly generated by a linear process with random variations about the trend, we’d expect to see the deviations scattered approximately randomly above and below the trend line. We see nothing of the kind, suggesting that whatever is happening isn’t likely the result of a linear process.

On the other hand, when we perform what is a very simple transformation in time series analysis to the HadCRUT3 data, we get the result pictured in Figure 2.

Figure 2

A common transformation in time series to investigate the possibility of a random walk is to “difference” the data. Here, because we are using monthly data, a particularly useful type of differencing is seasonal differencing, i.e., comparing one month’s observation to the observation from 12 months preceding. Since 12 months have intervened in computing this difference, it is equivalent to an annual rate of change, or a one month “spot” estimate of the annual “trend” in the undifferenced, or original, series. When we look at Figure 2, it has the characteristic appearance of a random walk.

But we can do more than just look at the series. We can put a number to it: the Hurst exponent. Here’s a very understandable presentation of the Hurst exponent:

“The values of the Hurst Exponent range between 0 and 1.

• A Hurst Exponent value H close to 0.5 indicates a random walk (a Brownian time series). In a random walk there is no correlation between any element and a future element and there is a 50% probability that future return values will go either up or down. Series of this type are hard to predict.

• A Hurst Exponent value H between 0 and 0.5 exists for time series with “anti-persistent behaviour”. This means that an increase will tend to be followed by a decrease (or a decrease will be followed by an increase). This behaviour is sometimes called “mean reversion” which means future values will have a tendency to return to a longer term mean value. The strength of this mean reversion increases as H approaches 0.

• A Hurst Exponent value H between 0.5 and 1 indicates “persistent behavior”, that is the time series is trending. If there is an increase from time step [t-1] to [t] there will probably be an increase from [t] to [t+1]. The same is true of decreases, where a decrease will tend to follow a decrease. The larger the H value is, the stronger the trend. Series of this type are easier to predict than series falling in the other two categories.”

So what is the Hurst exponent for the series depicted in Figure 2? It is 0.475, which is very near the value of 0.5 which indicates a pure random walk. And when we exclude the data before 1880, which may be suspect because of a dearth of surface locations in computing the HadCRUT3 series, the Hurst exponent is 0.493, even closer to 0.5. So by all appearances, the global temperature series has the mark of a random walk. But appearances can be deceiving. In Figure 3 I fit a Hodrick-Prescott smooth to the data:

Figure 3

In the upper pane, the undulating blue line depicts the smoothed value derived using Hodrick-Prescott smoothing (lambda is 129,000, for the curious). In the lower panel are detrended seasonal differences, i.e., what is left after removing the smoothed series. Conceptually, the smoothed series can be taken to represent the “true” underlying “trend” in the time series, while the remainder in the bottom pane represents random variations about the trend. In other words, at times, the annual rate of change in temperature is consistently (or persistently, as we shall see) rising, while at other times it is consistently falling. That is, there are trends in the trend, or cycles, if you will. And while it is not obvious, because of the scaling involved, these are essentially the same cycles that Anthony and I have attributed to a lunisolar influence on global temperature trends. That should not be so surprising. In our paper, we smoothed the data first with Hodrick-Prescott smoothing, and then differenced it. Here we’re differencing it first, to show the random walk nature of the series, and then smoothing the differences. But either approach reveals the same pattern of cycles in global temperature trends over time.

Looking more closely at the smoothed series, and the random component (labeled “Cyclical component” in Figure 3), we have an interesting result when we compute the Hurst exponents for the two series. The Hurst exponent for the smoothed (blue line) series is 0.835, while the Hurst exponent for the detrended random component (bottom pane) is 0.383. The first is in the range associated with “persistent” behavior, while the second is in the range associated with “anti-persistent” behavior. Let’s discuss the latter first.

Anti-persistence is evidence of mean reversion or what is also sometimes called “regression toward the mean.” Simply put, when temperatures spike in one direction, there is a strong probability that they will subsequently revert back toward a mean value. Ignoring all other factors, this property would suggest that the dramatic rise in the temperature anomaly for July should lead to subsequent declines back toward some underlying mean or stable value. I think this is probably more what Gordon or Kӓrner had in mind for the physical processes at work when they proposed treating global temperatures as a random walk. I.e.,shocks to the underlying central tendency of the climate system from processes such as volcanism, ENSO events, and similar climate variations create deviations from the central tendency which are followed by reversions back toward the mean or central tendency. Carvalho et. al (2007), using rather complicated procedures, recently laid claim to having first identified the existence of anti-persistence in global temperatures. We’ve identified it here in a much simpler, and more straightforward, fashion. (I’m not trying to take away from the usefulness or significance of their work. Their procedures demonstrate the spatial-temporal nature of anti-persistence in global temperatures, especially on decadal time periods. I think WUWT readers would find their Figure 10 especially interesting, for while they do not use the term, it demonstrates “the great climate shift” of 1976 rather dramatically.)

With respect to the smoothed series, the Hurst exponent of 0.835 indicates persistent behavior, i.e. if the series is trending upward, it will have a tendency continue trending upwards, and vice versa. But that is to be expected from the cyclical undulations we observe in the smoothed series. As to the possible physical processes involved in generating these cycles, after Anthony and I posted our paper, comments by Leif Svalgaard prompted me to perform a “superposed epoch analysis” (also known as a “Chree analysis”) on these cycles:

Figure 4

While Leif contends that the analysis should be performed on the raw data, in this case I would beg to differ. As shown in Figure 3, the raw data is dominated by the essentially random character of the monthly changes, completely obscuring the underlying cycles in the data that emerge when we filter out (detrend) the raw data. Arguably, what we have in the blue line in Figure 3 is a “signal” that has been extracted from the “noise” depicted in the bottom pane. Now as such, the “signal” may mean something, or it may not. That is where Figure 4 comes in to play. The peaks in the cycles depicted by the blue line in Figure 3 show a strong correspondence to maximums in the lunar nodal cycle (the “luni” part of our suggestion of a “lunisolar” influence on global temperature trends). They also show a strong correspondence in solar maxima associated with odd numbered solar cycles, especially beginning with solar cycle 17. Are these correspondences mere coincidence? Anthony and I think not. While each may play an independent role in modulating global temperatures, since the 1920’s the solar and lunar influences appear to have been roughly in phase to strongly influence temperature trends on a bidecadal time frame. In other words, Figure 4 may be revealing the physical processes at work in explaining the persistence revealed by the Hurst exponent for the blue line in Figure 3.

Taken together, the two Hurst exponents – one for the true “signal” in the series, and the other for the “noise” in the series – essentially offset each other, leaving us with a Hurst exponent for the unsmoothed, raw, seasonal difference of ~0.5, i.e., essentially a random walk. And so on a monthly basis, the global temperature anomalies we await anxiously are essentially unpredictable. However, if the cycles in the smoothed series can be plausibly related to physical processes, as Anthony and I believe, that gives us a clue as the “general direction” of the monthly anomalies over time.

In our paper together, Anthony and I presented the following projection using a sinusoidal model based on the same cycles shown in the blue smooth in Figure 3:

Figure 5

The light purple line in Figure 5 is, essentially, a continuation, or projection, of the blue smooth in Figure 3. From this, we derived a projection for the HadCRUT3 anomaly (light blue in Figure 5) which has it essentially meandering between 0.3 and 0.5 for the foreseeable future (here, roughly, the next two decades).

But the monthly values will vary substantially around this basically flat trend, with individual monthly values saying little, if anything, about the long term direction of global temperature. In that sense, global temperature will be very much like a random walk.