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:
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.
RA COOK replied to my comment
“TonyB (01:22:51) :
This is a very interesting and thought provoking article and Basil deserves our thanks for putting it together. So is global temperature rising sharply due to mans activities? It depends greatly on start point and methodolgy. The following comes from a longer article I wrote recently;”
Yes, I am absolutely aware of climate history since the lasrt ice age and that temperature has recopvereed naturally from the depths of the LIA. This is all in another part of the article but I did not want to divert peopple away from Basil’s article so curtyailed my own at the poiunt you see.
Personally I think we place far to much credence on many of the foundations of Agw w
Spence-UK .
.
Yes .
Nobody should write about the Hurst coefficient without reading D.Koutsoyianis papers first .
The purpose of the Hurst analysis is to identify (or better said to TRY to identify) power laws in probability density distributions .
If H = 0.5 then there is no power law and the process is iid (Gaussian , normal , white noise) .
Random walk is gaussian but the climatic parameters are clearly not gaussian therefore they can’t be random walk .
.
If H is not 0.5 then the process may be represented by a fractional gaussian noise (or a power law) .
May .
Or must not .
In any case it is not gaussian .
From the methodological point of view you are of course right to say that it is a heresy to apply the Hurst analysis to “smoothed” data or to moving averages because the Hurst analysis PRECISELY looks at autocorrelations at DIFFERENT scales from the smallest to the biggest .
By smoothing first , the small scales are destroyed and the computed “Hurst coefficient” looses any significance .
.
There is a third possibility too .
Dan Hughes applied the Hurst analysis to a known low dimensional chaotic process (Lorenz system) .
The Hurst analysis failed because the computed “Hurst coefficient” depended on scale what it should not .
However it stayed constant over large intervals of scales .
.
That allowed 2 conclusions :
– Deterministic chaos is not random (neither gaussian nor not gaussian) but that is something that people familiar with chaos theory already know .
– If you have not a huge amount of data (e.g a very long period of time) you might fall on a scale interval where there SEEMS to be a constant Hurst coefficient and therefore you’d think that you see a random process (gaussian or not gaussian) while in reality the system is chaotic . You only had bad luck that the Hurst coefficient didn’t vary at scales at which you looked .
Typically this applies to climate . Once we have some 10 000 years of daily temperature data , then we may be able to draw some conclusions from a Hurst analysis of this time series .
So only some 9900 years to wait 🙂
Sanchez, Politics aside, WUWT is website where open minds come to play… As a “Dr” of something or other, how is it that you don’t get that? Oh, wait…
In other news Athropogenic Continental Drift (h/t to Kate at SDA LOL!
Sorry my above comment ‘escaped’ before I finished and edited it!
As well as the nonsense of parsing historic Global temperatures to fractions of a degree as if they had been compiled in laboratory conditions, the manner in which ‘sea level rise since 1700’ has been put together also warrants an article in itself.
Tonyb
All I ever needed to learn about the climate I learned from my uncle Albert, the farmer. His words to describe the excursions of the temperature were: “one extreme always follows another”. So, when the summers were really hot he cut a lot of firewood.
For anyone interested in understanding the math & principles involved, an excellent related reference is: “The Misbehavior of Markets,” by Benoit Mandelbrot.
While the subject matter, stock market trends & crowd behavior, is very different than the focus of the analysis here, the underlying principles are basically the same. Such parallels between disciplines are very common…and reviewing related material in a diffent contexts is usually very helpful in increasing one’s understanding & appreciation of the basic analytical subject matter. But [more importantly] such breadth in exposure also helps one in both applying such tools, and [even more importantly still] is spotting the underlying patterns & opportunities for extracting new insights.
Besides, anyone enjoying the above “random walk” analysis will likely enjoy Mandelbrot’s book regardless.
The “random walk” jargon, by the way, hails to a classic book by Burton Malkiel, “A Random Walk Down Wall Street” 1st published in 1973.
I notice that the Danish Center for Snow and Ice shows the arctic temps dropping below freezing about two weeks before normal, and remarkably Arctic ROOS shows the ice area currently increasing (about two weeks before normal.) All the while UAH shows atmospheric temps substantially up. This just gets curiouser and curiouser.
It appears that our understanding of the earth’s climate and what drives it is still pretty primitive. One thing is obvious, though, the inexorable linear rise in Co2 that we are seeing doesn’t seem to correlate with ANYTHING (except perhaps increased agricultural yields.)
Sorry, Basil, but this must be grumpy day for me. Global temperatures — as we know them today — are not random, but crooked, fixed, cooked. Mosly lies. Until we locate those “thermometers” sited appropriately and read over a long period of time, we will never know. (See E.M. Smith and part of what I imagine Anthony’ssurface station projec will address.) The climate system may be chaotic, but the climate has changed over geological time — which involves the temperaure of land and ocean — and this truth is not random. I’m interested in the truths, not the lies of statistics from altered raw data.
More grumpy — the link to Orlin Grabbe on “Chaos and Fractals in Financial Markets”, you know, the one who is “an internationally recognized derivatives expert”. These are the guys who, in the guise of helping us to minimize risk, hide the fact that they have been dumping empy-of-any-value “equities” on unsuspecting (trusting) investors worldwide leading us to the disaster we are facing today. How about some substance, some truth in financial instruments, just like thermal energy instruments.
Until truth is determined in the HadCrut3 data ( global temperature anomalies), anything you do with it must fall under the GIGO principle. Good grief and bad grief! Physicists on an earlier thread cannot even agree on the definition of heat — a noun or a verb? thermal energy of a “substance” or transfer of energy? Help
Talking of obsessively following every new monthly value… About a month ago, when we’re were talking about surface-satellite ‘lag’ in
http://wattsupwiththat.com/2009/07/14/giss-for-june-way-out-there/
I said, “…I’m tempted to make a SWAG of UAH=+0.3 by September, and a crossover with GISS (adjusted for baselines) by the end of the year.”.
GISS for July is out at 0.6. Baseline-adjusted comparisons here:
http://www.woodfortrees.org/plot/wti/last:12/plot/hadcrut3vgl/last:12/offset:-0.15/plot/gistemp/last:12/offset:-0.24/plot/uah/last:12/plot/rss/last:12
OK, it’s a bit earlier than expected, but mine’s a pint of Doom Bar 🙂
This is Hansen’s very first climate projection from 1981.
Overall, includes some description of the random walks of the climate, temperature impact per doubling lines lower than later versions (basic math must have changed in the interim).
http://img23.imageshack.us/img23/7720/image002x.png
Quite often people confuse “random” with “arbitrary” , which results in unfortunate conclusions. When prior information is neglected ( Bayes theorem ) the result may bear little resemblance to reality.
But in this drunkard’s walk the drunkard gets further away from the lamp-post as the square-root of time as Leif has pointed out. Take a gaussian process (brownian motion, random walk, drunkard’s walk, whatever you wish to call it) and add an integrator to it. The integrator is important because other wise you’ll just bob around a long-term mean. The integrator produces something like 1/f noise. If the integrator as an infinite time constant your process will possess such a long memory about past events that you will never even be able to calculate a mean value–the central limit theorem does not apply to such a process and a mean value becomes meaningless.
The Earth probably does not have an integrator with infinite time constant but, there may be very long time constants, which means that no matter how long the time series one examines, there will always appear to be a linear trend.
By the way, the person who first pointed this out for geophysical processes was Benoit Mandelbrot, of fractals fame, back in 1968. He called it the Joseph Effect (seven fat years, etc). Mandelbrot looked at river discharge data, but the lesson has been there all along for people to apply to weather/climate. I have tried to point out on many occasions that negative feedback with sufficiently long characteristic time will make time series indistinguishable from the temprature series for any time duration one cares to examine, but the concept just doesn’t seem to click with anyone except my engineering students.
I’ve been interested in this notion for some time now. I’m no statistician, but I diddled around with Excel’s random number function to produce graphics that look amazingly like the average global temperature charts we’re all used to seeing.
Here’s a link to a web discussion group that I’ve posted some of this stuff on:
http://wc5.worldcrossing.com/webx?14@ur momisugly@.1de4fb6e/313
Some of the results I got with Excel:
http://i40.tinypic.com/wwb2hj.jpg
http://i41.tinypic.com/qq92ti.gif
http://i44.tinypic.com/11gj8t1.gif
Does it prove anything? I’m smart enough to know that trying to predict which way it’s all going to go is a fool’s errand. 100 years out? Ha ha ha ha ha ha ha ha ha!
But even a drunk wandering around a lamp post is moving around the sun, in a solar system circling the galaxy, in a galaxy moving across the cosmos, just as as the random walk of a snow flake takes place in the larger context of a moving storm system. What is the larger system that contains the temperature random walk? Just asking.
I’ve been interested in this notion for some time now. I’m no statistician, but I diddled around with Excel to create with random numbers graphics that look amazingly like the average global temperature charts we’re all used to seeing.
Here’s a link to a web discussion group that I’ve posted some of this stuff on:
http://wc5.worldcrossing.com/webx?14@ur momisugly@.1de4fb6e/313
Some of the results I got with Excel:
http://i40.tinypic.com/wwb2hj.jpg
http://i41.tinypic.com/qq92ti.gif
http://i44.tinypic.com/11gj8t1.gif
Does it prove anything? I smart enough to know that trying to predict which way it’s all going to go is a fool’s errand.
WoodforTrees
Looking at your chart I would say that someone has incorrectly calculated an algorithm, or adjusted in one go figures that were too ‘low’ in previous months.
Do you know of anyone that looks at the individual monthly figures used for global stations? If not separately perhaps as the total for the NHemisphere and the SH. THis is most useful as an actual temperature-not an anomaly. It would be interesting to see where this additional warmth is coming from. The fact that the US went trhe other way (I know it is just 2% of Earths land mass) suggests this is not global but regional.
Tonyb
Basil (05:44:03) :
If you want to chalk the apparent correspondence of the peaks in the cycles to solar cycle and lunar nodal cycle peaks as just fortuitous circumstance, what more can I say?
You can show a version of Figure 5 extended backwards to 1850 and to 1600 [the latter clearly without the observed data, but with the calculated values].
And, perhaps, comment on the fact that the average distance after N steps is not zero but SQRT(N), so no strong probability of reverting to the mean.
Spence_UK (01:46:17) :
I want to follow up here on the issue of computing a Hurst exponent from the raw monthly temperature analyses. The calculation is simple enough. For the HadCRUT data I am using, it is 0.967, close to the numbers you reference. Now this would ordinarily imply a very “strong trend” in the data. But before we can conclude that, we have to consider the impact of serial correlation. Here are two simple and straight forward estimates of the order of linear trend in the data:
(I’ll try to get this for format as “code” but without a preview function, I have no way of knowing it it will work.)
OLS estimates using the 1914 observations 1850:01-2009:06
Dependent variable: hc_g
HAC standard errors, bandwidth 9 (Bartlett kernel)
coefficient std. error t-ratio p-value
-----------------------------------------------------------
const -0.522158 0.0230692 -22.63 1.08E-100 ***
time 0.000367397 2.24001E-05 16.40 1.15E-056 ***
OLS estimates using the 1913 observations 1850:02-2009:06
Dependent variable: hc_g
HAC standard errors, bandwidth 9 (Bartlett kernel)
coefficient std. error t-ratio p-value
-----------------------------------------------------------
const -0.125384 0.0122823 -10.21 7.35E-024 ***
time 8.86101E-05 9.16300E-06 9.670 1.25E-021 ***
hc_g_1 0.759139 0.0235876 32.18 6.12E-182 ***
We’re interested here in the coefficients for the “time” variable. Since these are monthly data, we can multiply by 120 to derive an equivalent “decadal” trend rate. In the first case, which is the regression for the trend line represented in my original Figure 1, the decadal trend rate is 0.044°C. But there is a high degree of serial correlation (Durbin-Watson statistics = 0.48). In the second case, we control for serial correlation by adding a one month lag of the temperature as an explanatory variable (Durbin-Watson is now 2.40). The resulting trend estimate is now slashed by about 75%, to a decadal trend rate of 0.011°C.
So a high Hurst exponent, by itself, isn’t enough to indicate that we’ve accurately captured the real trend in the data.
And if anyone is curious, a similar trend analysis of the wavy blue line in Figure 3 yields this:
OLS estimates using the 1890 observations 1852:01-2009:06
Dependent variable: hpt_sd_hc_g
HAC standard errors, bandwidth 9 (Bartlett kernel)
coefficient std. error t-ratio p-value
----------------------------------------------------------------
const -0.00228909 0.00187566 -1.220 0.2225
time 2.83920E-06 1.68064E-06 1.689 0.0913 *
hpt_sd_hc__12 0.808243 0.0391915 20.62 2.21E-085 ***
Now, be careful, because here the annual trend is reflected in the constant term. It starts off negative, but has been becoming less negative over time, and at the end of the data the decadal equivalent is 0.031°C.
For those of you looking for a “drift” in the data over time that might be capturing AGW, it will be in the upward trend in the wavy blue line. Were this order of “drift” to continue, the decadal rate will have increased to about 0.065°C in another hundred years. That works out to about 0.43°C (interpolating) increase over the next century. If that’s all there is to the AGW impact, I think we can hold off on cap and trade for a while.
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.
Interesting statistical analysis!
But if you want to pursue publication you need to add [……..] “but these physical processes cannot explain global temperature increase in the industrial era.
The July MSU temp jump of .42c seems to mostly result from the readings in Antarctica, where they show a really big jump of 3.11c
The historical series for the South Pole land is volatile and it has spikes often, the 3.11 though is high and while there was a 3.3c reading in May, 2002, the 3.11 appears to be the second highest in the 1979 to present series
I would bet that August will show a much lower temp
woodfortrees (Paul Clark) (05:16:39) :
No doubt there is a major random component and each month’s data is rather irrelevant – however much partisan observers emphasise deviations which favour their point of view and ignore others! But it passes the time while the serious trends build up 🙂
However Basil’s posting seems to be shading dangerously close to trying to suggest that there is no underlying linear trend. Maybe that wasn’t the intention but some people seem to be taking it that way.
Paul,
While you were posting this, I was writing up my preceding reply in which I tried, somewhat unsuccessfully, to post up some formatted statistics relevant to your concern. You are right in thinking that it is not my intention of saying there is NO trend. But where I would look for it is in the blue wavy line of Figure 3. And it is there, though the significance level is marginal (90%), and is smaller than we get with a linear fit through the raw data.
Again, everyone, please, please, understand that what I’m seeing, and saying, is not that there is no trend, or that temperature is entirely a random walk. What I’m am saying is that I think the monthly anomalies “look” like a random walk, in part because of the high volatility of the stochastic component, and because the two components of the series have characteristics that seem to be offsetting, i.e. the true underlying “trend” (the wavy blue line in Figure 3) shows persistence, while the “random” or “stochastic” part (bottom pane of Figure 3) shows anti-persistence (mean reversion). With mean reversion working against the trend, we get something that looks like a random walk. But that is not to say that there are no physical processes at play. In the trending part, Anthony and I see a lunisolar influence in the oscillations, and there is an underlying “drift” that may, or may not, be due to AGW (and I say AGW, and not GHG, to allow for such impacts as UHI). In the random or stochastic part, we’re seeing the characteristic of a stable physical system (which climate has to be for life as we know it to exist) which responds to shocks by reverting back toward the mean.
Can I be any clearer?
Basil,
Thanks for an interesting post; I had not seen this type of analysis before.
1) It seems to me that by using the 1-year differences to transform the raw data (Figure 2) that you are taking something of a “first derivative” of the data, which ought to magnify the importance of short term variations, while reducing any longer term variation in the original data to small differences in the trend above or below the baseline. Since the overall trend for the raw data set is clearly positive, the average value for all the transformed data should be slightly above zero, with a value that depends on the slope of the least squares fit trend in the original data divided by the number of data points in the raw data set less 1. Do I understand the transform you have done correctly? Would a series of similar transforms with different time steps (2 yrs, 3 yrs, 4 yrs, etc.) not increasingly show the longer term variations in the raw data, while “filtering out” the short term variation? What is the reason for choosing 1 year changes as opposed to some other period?
2. If I understand correctly, you have assigned cause for the recent temperature history (the 20th century to now) to the “lunisolar” influence you described, and based on this, you project variation in average temperature around a flat trend for the next 20+ years (as shown in Figure 5). If this is true, then it seems to me that you are implicitly assigning a value of near zero for climate sensitivity to radiative forcing. Fair enough, it could be very low. But in this case, how can the climate have any measurable sensitivity to variation in TSI over the solar cycles? I think that Leif suggests an average solar cycle signature of about 0.075C in the historical data due to variation in TSI of about 1.4 watt/M^2 at the top of the atmosphere. This is consistent with a relatively low climate sensitivity, but not a near-zero sensitivity. If the sensitivity to radiative forcing is in fact near zero, then by what mechanism do you think the solar cycle shows up in the temperature data?
Well, what you would have here is a random walk for the “velocity” (i.e. the trend) around its average value. The fact that the Hurt coefficient is so close to 0.5 indicates a Gaussian distribution probability for it which, for large systems, is quite expected because of the central limit theorem. What do you think?
And I might add, with reference to all the concern about whether there is a trend, that there would be a trend component to the sinusoidal model that underlies Figure 5. But it may, for the next two decades, be “masked” by the behavior of the sinusoidal components of the model, leaving the anomaly series like we see it in Figure 5.
But do not read too much into this. Any extrapolation out for two decades, let alone for the next century (a la IPCC) involves what we call in my field “heroic assumptions.” And I’m quite well aware of that. What is useful in such analyses are exposing the assumptions that underlie them, so we can evaluate their plausibility.
E.M.Smith (02:46:34) :
Your analysis should be another talking point… a thread on it’s own merit. I would be interested to read a full discussion including “AGW convinced” rebuttals, on your thesis. You have given us much food for thought. Thx.