Guest Post by Willis Eschenbach
Looking at a recent article over at Tallbloke’s Talkshop, I realized I’d never done a periodogram looking for possible cycles in the entire Central England Temperature (CET) series. I’d looked at part of it, but not all of it. The CET is one of the longest continuous temperature series, with monthly data starting in 1659. At the Talkshop, people are discussing the ~24 year cycle in the CET data, and trying (unsuccessfully but entertainingly in my opinion) to relate various features of Figure 1 to the ~22-year Hale solar magnetic cycle, or a 5:8 ratio with two times the length of the year on Jupiter, or half the length of the Jupiter-Saturn synodic cycle, or the ~11 year sunspot cycle. They link various peaks to most every possible cycle imaginable, except perhaps my momma’s motor-cycle … here’s their graphic:
First off, I have to say that their technique of removing a peak and voila, “finding” another peak is mondo sketchy on my planet. But setting that aside, I decided to investigate their claims. Let’s start at the natural starting point—by looking at the CET data itself.
Figure 2 shows the monthly CET data as absolute temperatures. Note that in the early years of the record, temperatures were only recorded to the nearest whole degree. Provided that the rounding is symmetrical, this should not affect the results.
Over the 350-year period covered by the data, the average temperature (red line) has gone up and down about a degree … and at present, central England is within a couple tenths of a degree of the long-term mean, which also happens to be the temperature when the record started … but I digress.
Figure 3 shows my periodogram of the CET data shown in Figure 2. My graphic is linear in period rather than linear in frequency as is their graphic shown in Figure 1.
Figure 3. Periodogram of the full CET record, for all periods from 10 months to 100 years. Color and size both show the p-value. Black dots show the cycles with p-values less than 0.05, which in this case is only the annual cycle (p=0.03). P-values are all adjusted for autocorrelation. The yellow line shows one-third the length of the ~350 year dataset. I consider this a practical limit for cycle detection. P-values for all but the one-year cycle are calculated after removal of the one-year cycle.
I show the periodogram in this manner to highlight once again the amazing stability of the climate system. One advantage of the slow Fourier transform I use is that the answers are in the same units as the input data (in this case °C). So we can see directly that the average annual peak-to-peak swing in the Central England temperature is about 13°C (23°F).
And we can also see directly that other than the 13°C annual swing, there is no other cycle of any length that swings even a half a degree. Not one.
So that is the first thing to keep in mind regarding the dispute over the existence of purported regular cycles in temperature. No matter what cycle you might think is important in the temperature record, whether it is twenty years long or sixty years or whatever, the amplitude of the cycle is very small, tenths of a degree. No matter if you’re talking about purported effects from the sunspot cycle, the Hale solar magnetism cycle, the synodic cycle of Saturn-Jupiter, the barycentric cycle of the sun, or any other planetasmagorica, they all share one characteristic. If they’re doing anything at all to the temperature, they’re not doing much. Bear in mind that without a couple hundred years of records and sophisticated math we couldn’t even show and wouldn’t even know such tiny cycles exist.
Moving on, often folks don’t like to be reminded about how tiny the temperature cycles actually are. So of course, the one-year cycle is not shown in a periodogram, too depressing. Figure 4 is the usual view, which shows the same data, except starting at 2 years.
Figure 4. Closeup of the same data as in Figure 3. Unlike in Figure 3, statistical significance calculations done after removal of the 1-year cycle. Unlike the previous figure, in this and succeeding figures the black dots show all cycles that are significant at a higher p-value, in all cases 0.10 instead of 0.05. This is because even after removing the annual signal, not one of these cycles is significant at the p-value of 0.05.
Now, the first thing I noticed in Figure 4 is that we see the exact same largest cycles in the periodogram that Tallbloke’s source identified in their Figure 1. I calculate those cycle lengths as 23 years 8 months, and 15 years 2 months. They say 23 years 10 months and 15 years 2 months. So our figures agree to within expectations, always a first step in moving forwards.
So … since we agree about the cycle lengths, are they right to try to find larger significance in the obvious, clear, large, and well-defined 24-year cycle? Can we use that 24-year cycle for forecasting? Is that 24-year cycle reflective of some underlying cyclical physical process?
Well, the first thing I do to answer that question is to split the data in two, an early and a late half, and compare the analyses of the two halves. I call it the bozo test, it’s the simplest of all possible tests, doesn’t require any further data or any special equipment. Figures 5a-b below show the periodograms of the early and late halves of the CET data.
I’m sure you can see the problem. Each half of the data is a hundred and seventy-five years long. The ~24-year cycle exists quite strongly in the first half of the data, It has a swing of over six tenths of a degree on average over that time, the largest seen in these CET analyses.
But then, in the second half of the data, the 24-year cycle is gone. Pouf.
Well, to be precise, the 24-year peak still exists in the second half … but it’s much smaller than it was in the first half. In the first half, it was the largest peak. In the second half, it’s like the twelfth largest peak or something …
And on the other hand, the ~15 year cycle wasn’t statistically significant at a p-value less than 0.10 in the first half of the data, and it was exactly 15 years long. But in the second half, it has lengthened almost a full year to nearly 16 years, and it’s the second largest cycle … and the second half, the largest cycle is 37 months.
Thirty-seven months? Who knew? Although I’m sure there are folks who will jump up and say it’s obviously 2/23rds of the rate of rotation of the nodes on the lunar excrescences or the like …
To me, this problem over-rides any and all attempts to correlate temperatures to planetary, lunar, or tidal cycles.
My conclusion? Looking for putative cycles in the temperature record is a waste of time, because the cycles appear and disappear on all time scales. I mean, if you can’t trust a 24-year cycle that lasts for one hundred seventy-five years, , then just who can you trust?
De Costumbre: If you object to something I wrote, please quote my words exactly. It avoids tons of misunderstandings.
Data and Code: I’ve actually cleaned up my R code and commented it and I think it’s turnkey. All of the code and data is in a 175k zip file called CET Periodograms.
Statistics: For the math inclined, I’ve used the method of Quenouille to account for autocorrelation in the calculation of the statistical significance of the amplitude of the various cycles. The method of Quenouille provides an “effective n” (n_eff), a reduced count of the number of datapoints to use in the various calculations of significance.
To use the effective n (n_eff) to determine if the amplitude of a given cycle is significant, I first need to calculate the t-statistic. This is the amplitude of the cycle divided by the error in the amplitude. However, that error in amplitude is proportional to
where n is the number of data points. As a result, using our effective N, the error in the amplitude is
where n_eff is the “effective N”.
From that, we can calculate the t-statistic, which is simply the amplitude of the cycle divided by the new error.
Finally, we use that new error to calculate the p-value, which is
p-value = t-distribution(t-statistic , degrees_freedom1 = 1 , degrees_freedom2 = n_eff)
At least that’s how I do it … but then I was born yesterday, plus I’ve never taken a statistics course in my life. Any corrections gladly considered.