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.

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