Saturday, January 26, 2008

Symmetries in the Global Temperature Record - Hadley CRUT3 Time Series

This post continues a series on the global temperature record, climate cycles and natural and human-caused temperature changes. I noted in my previous post that an update to the NOAA NCDC global temperature anomaly time series in January of this year muddled the intricate symmetries observed in the December 2007 NCDC release. This post will note that the December release of the Hadley CRUT3 global temperature anomaly time series contains similar numerical patterns.

I begin by noting that a zero month-to-month change in the global temperature anomaly will be treated as a "0" for the purpose of this analysis. The Hadley CRUT3 time series calculates the anomaly to three decimal places (rather than four for the NOAA NCDC time series); as a result, there are 14 zero monthly anomaly change values. The reader may question why I am making this assumption. Although I will give a more detailed explanation later on, the short answer is the assumption compensates for a likely anthropogenic effect (MUCH more on that later!).

The reader may recall from the previous post that the essential underlying pattern was of symmetries between the first and fourth periods and second and third periods respectively throughout the global temperature anomaly time series. For convenience, we repeat Table 10 from the previous post which sums across the corresponding 95 month periods for each of the four 380 month periods in the transformed NOAA NCDC time series:

Performing the same transformation using the December 2007 release for the Hadley CRUT3 global temperature anomaly time series gives the following result:

Although the symmetries in Table 10C are not perfect (as they were in Table 10), the numerical patterns are essentially the same. That is, the first and fourth and second and third periods are, respectively, symmetrical.

We next look at the first and third quarter symmetries in the Hadley CRUT3 time series. The tables are combined to ease comparison:

Note how in both quarters the first and fourth periods are symmetric while the second and third periods are symmetric across quarters. That is, period Q1a is symmetric with Q1d and period Q3a is symmetric with period Q3d while period Q1c is symmetric with period Q3c and period Q1b is symmetric, to a lesser extent, with period Q3b. To clearly compare these Hadley CRUT3 results with the NOAA NCDC results, the below table combines Tables 4 and 5 from my previous post:

The second and fourth periods of the Hadley CRUT3 temperature series contain numerical patterns similar to those found in the NOAA NCDC time series. For comparison, I repeat the results from Table 6 in the previous post for the comparable NOAA NCDC time series:

While there are differences in the results across the two time series, the signs are the same. Finally, I show the results for the four quarter periods for the two time series beginning with Hadley CRUT:

The similarities between the two time series are more significant than their differences. And why the differences between the two time series? The Hadley CRUT3 first differences (that is, the month-to-month anomaly temperature changes) have more volatility, especially before about 1940, than do the NOAA NCDC first differences. Thus, it is not surprising that the Hadley CRUT3 results appear, from the standpoint of the NOAA NCDC results, less "sharp". From such a standpoint, one may view the NOAA NCDC results as "better". Whether this is true or not, I will choose to focus on the NOAA NCDC results since they are, afterall, more asthetically pleasing and, essentially, more improbable.

Friday, January 25, 2008

Symmetries in the Global Temperature Record Part II

Fascinating Figures - Part II

This post continues a series on the global temperature record, climate cycles and natural and human-caused temperature changes. In my previous post I took global temperature anomaly time series data NOAA NCDC release for December 2007, created a new time series of the month-to-month change in the anomaly, transformed that change into a binary 0 and 1 format and examined numerical patterns in the full transformed time series and for partitions of the time series into halves and quarters. I left you with a very interesting symmetrical pattern for the quarter's time series partition, which for convenience is repeated below:

The first and third quarters have an equal number of 1's and 0's while the second and fourth quarters are perfectly symmetric. While this result is interesting, it may not mean very much since the pattern could have been produced by a random process. However if symmetries persist after further division of the time series, we can hypothesize that something more fundamental than random chance may be at play here.

Our next step then will be to subdivide the above quarters. Since the first and third quarters have an equal number of 1's and 0's, these quarters seem to be the next logical place to look for any underlying symmetric pattern. Because a division of the first and third quarters into equal halves would necessarily result in a symmetric outcome, it seems more interesting to divide each of these 380 month periods into four equal 95 month subdivisions. The results are fascinating:

A beautiful symmetry emerges from the data. The first 95-month period, Q1a, is perfectly symmetrical with the fourth 95-month period Q1d while the same is true for periods Q1b and Q1c. Before one gets too excited, it is important to remember that this result could have been produced by a completely random process. However, if there is a numerical pattern even remotely similar to this pattern somewhere else in the time series, we may begin to think that something special might be here afterall. The logical place to look for such a pattern would be in the third quarter since this quarter also had an identical number of 1's and 0's. Incredibly, the same numerical pattern emerges:

The numbers are not exactly the same as in Table 4 but the pattern is identical. That is, the first and fourth periods and second and third periods are, respectively, symmetrical just as in Table 4. If you think about, this is incredible and this is what I felt awhen I first stared at these figures for thon my computer screen on a cold, sunny March morning last year. A Dr. Seuss line comes to mind: "Fish in a tree, how can that be!" Indeed, how can that be?

But there's more. Let's next look at the second and fourth quarters. We begin by dividing each quarter into two equal halves of 190 month each and show the results for each quarter side by side for better comparison:

The corresponding halves in each quarter period are nearly symmetrical and the second half of each quarter period is particularly lopsided. I'll demonstrate in an upcoming post that the probability of such a one-sided outcome being generated by a random process is quite low. It is curious to note that in the most recent period, a time that saw rapidly rising global temperatures, saw a preponderence of negative month-to-month anomaly changes.

A further subdivision of results in Table 6 into 95-month periods appears appears at first to produce a jumbled, random pattern. However, a closer look at the numbers reveals an astonishing symmetric pattern lurking within:

Do you see the pattern? If you don't then the table below, which sums across the rows in Table 7, will make the pattern clear:

Isn't that lovely? More importantly, it's the same symmetrical pattern as in Tables 4 and 5. That is, the first and fourth periods and second and third periods, respectively, are symmetric. This becomes clear when we add up the rows across the periods in Tables 4 and 5 just as we did in Table 7:

Then, summing across the cells in Tables 8 and 9 gives the following result:

The symmetric pattern in Table 10 is the same pattern as that found in Tables 4, 5 and 8. That is, this symmetric pattern is found within both the first and third quarters, between the second and fourth quarters and across all quarters. Wow, what on earth is going on here?

If reading this makes you want to drop everything you are doing, run to the NOAA NCDC website, download the global temperature anomaly time series and derive these figures for yourself, be forewarned - the imprrrrrrrrrrrrrrrrrrrrroved time series update for January 2008 caused changes significant enough to completely muddle the symmerties between the first and third periods. The symmetries between the second and fourth quarters remain, but the beautiful patterns interwoven within and across the entire time series have, for the moment at least, been muddled into incomprehension. Perhaps the pond will clear again in time.

When I saw those updated figures, I felt as if I had been contemplating a beautiful reflection in a perfect pond on a perfect day when suddenly a rock thrown by a mischevious kid came crashing down to wreck this perfection. I had actually written an academic-type paper noting these symmerties and analyzing the probability of observing these symmertries at random. I was making final revisions to the paper and was a weekend or two away from sending the paper out for consideration when the revision was made.

For those readers who believe or suspect I am making all of this up, in my next posting I will show that something close to these symmetries are present in the Hadley CRU global temperature anomaly time series.

Thursday, January 24, 2008

Symmetries in the Global Temperature Record -Introduction and Part I


This is the first of a number of posts on the global temperature record, climate cycles and natural and human-caused temperature changes. In the first several posts of the series I will show you some fascinating numerical patterns lurking within the apparent chaos of the global temperature record. You will see that these patterns are, like snowflakes, beautifully symmetric and, even more amazing, the same symmetric pattern is found within and across portions of the temperature record as well as across the entire time series to be analyzed. For those readers needing the hard numbers of probabilities to be impressed by these symmetries, a post will detail how astonishingly unlikely these numerical patterns could be present within the temperature record by chance.

The series will then move to the most astonishing finding of all - the timing of these symmetries corresponds with major long-term temperature movements in the temperature record. The correspondence of intricate and improbable symmeties with major long-term movements in the temperature record suggests that the symmetries themselves are fingerprints of natural temperature cycles in the temperature record.

While the evidence of natural temperature cycles will undoubtably cheer skeptics of anthropogenic global warming theory, I will show that the temperature record in fact contains clear evidence of human-induced warming. I will demonstrate, making use of a very simple assumption, that in fact significant and measureable human-induced warming took place during the global cooling period of the 1940's to mid-1970's and that this human-induced warming accelerated from the mid-1970's to the present. However, while the temperature record contains clear evidence of human-induced global warming, I will argue that this record offers little evidence of major feedback effects from increasing concentrations of greenhouse gases in the atmosphere.

Finally, I will offer a global temperature forecast for the 21st century based on the above findings. I will suggest that recent episodes of unusually cold weather in many parts of the globe are in fact evidence the earth is just entering a new cooling phase, a cooling phase that will last for the next 30 years. We may expect however that human-induced warming will roughly offset this cooling phase resulting in very little temperature change in the world over the next 30 years.

Fascinating Figures - Part I

If you are like most people, your impression of the time series chart of global temperatures is probably quite vague. You might have seen this chart one of the reports put out by the UN's Intergovernmental Panel on Climate Change (IPCC) or perhaps in Al Gore's movie (or book) "An Inconvenient Truth". Chances are, all that may come to mind for most readers is a squiggly line that climbs up and to the right. Chances are any chart that you may have seen focused on the general temperature trend and deemphasized more short-term temperature movements.

That's because everyone knows that short-term movements are just "noise" in the data needing to be tamed so that we can see the forest of reality rather than get lost in the trees of detail. I've reproduced below such a "noisy" chart of monthly global temperatures, (expressed as anomalies, or departures from long term averages) for the period from 1880 to 2007:

It sure looks "noisy", especially if you look closely at a small piece of the chart, say a period covering a decade or so. Over such a short period it's hard to see any real trend and you really need to step back a bit and look at the entire chart to see that the long-term trend is definitely up. But you can easily imagine why people replace all that "noise" with squiggly, thick lines so that we can focus on the underlying trend. Afterall, the trend is what's really important, since knowing the past trend will help us to understand the future trend.

Of course, this assumes that the cause of the upward trend is understood. If we do understand the cause of the trend, then subtracting out the effect of that cause leaves us with random, directionless movement. In the case of global temperature movement most scientists studying the earth's climate believe that humans have caused this upward temperature trend, or at least have been the cause of this trend over the past 50 years or so. Most of these scientists would argue that without human intervention the global temperature movement, at least over the past 50 years or so, would have been random and directionless. This is essentially the argument made in the IPCC's most recent report on the earth's climate.

I am going to show you that this thinking is incorrect. I will show you that the general temperature trends in the above chart, over its entire period from start to finish, are ordered and not random. I will show you that the evidence for this order lies in a series of intricate and interwoven numerical patterns found within the data itself. These patterns can be found in a recent release of the NOAA NCDC global temperature anomaly time series (the figures in this post were calculated from the time series released in December of 2007). These numerical patterns are revealed by taking first differences (that is, the month to month changes) of the time series for the 1880 to 2006 period and transforming the first differences into a binary format. Successive divisions of the time series reveal these increasingly complex and interwoven numerical patterns.

The analysis to uncover these patterns will use monthly data for the January 1880 to September 2006 period, a period containing 1521 data points whose first difference produces 1520 data points. The analysis will use three simple steps:

1. Calculate first differences, that is, the one-period change in the global temperature anomaly.

2. Transform the resulting first differences into a binary format with values greater than zero being given a value of 1 and values less than or equal to zero being given values of 0 (note that reversing the value labels produces the same results) . For the record, a single first difference value in the transformed time series was equal to zero.

3. Examine the resulting numerical pattern of binary values for various partitions of the transformed time series.

We begin with the resulting distribution of binary values for the full time series:
There are an equal number of 0 values and 1 values in the full time series.

Since there are an equal number of 0 values and 1 values in the full time series, a partition of the time series into two equal halves produces a symmetrical numerical pattern:

We next partition the time series into four quarters of 380 data points each. Note that there is now no guarantee of a symmetric outcome for this partition; however, a very interesting pattern does, nonetheless, emerge:

The first and third quarters have an equal number of 1's and 0's while an excess of 1's in the second quarter are balanced by an excess of 0's in the fourth quarter. The patterns get even more interesting as we increase the number of partitions in the time series. More to come in my next post . . . .

New Blog Content

This blog, which previously contained postings on the Japanese economy and Japanese real estate market, will now be devoted to my various non-business related interests. In this regard, I have been studying the subject of climate change and global warming over the past year and this blog will, for some time at least, be focused on my own and others research and analysis of these subjects. I am an urban economist and statistician by academic training meaning that my postings on these subjects will have a distinctly economic and statistical bent. It is said that economists believe they can analyze virtually everything and some might conclude that this is yet another economist venturing into areas they know nothing about and about which should say even less. However, the important thing is offering insight and the reader can judge whether or not the content herein does so to their satisfaction.

As I am a contrarian by nature I tend to view consensus viewpoints as positions perturbed to a lesser or greater extent from reality; thus the title "Maintaining Equilibrium". I will contend that the consensus viewpoint on global warming and climate change is likely far enough removed from reality to render current consensus policy responses at best laughable and at worst damaging to both the economy and environment. I hope that readers of this content will be stimulated and challenged. I look forward to readers comments.