The Effect of Gleissberg's "Secular Smoothing"

Guest Post by Willis Eschenbach

ABSTRACT: Slow Fourier Transform (SFT) periodograms reveal the strength of the cycles in the full sunspot dataset (n=314), in the sunspot cycle maxima data alone (n=28), and the sunspot cycle maxima after they have been “secularly smoothed” using the method of Gleissberg (n = 24). In all three datasets, there is no sign of the purported 80-year “Gleissberg Cycle”. In addition, the effect on the periodograms of missing data is investigated.

Continuing my investigations of the non-existence of the purported “Gleissberg Cycle”, at the suggestion of a commenter I’ve now done periodograms of the full sunspot dataset, the maxima only, and the “secular smoothed” maxima using Gleissberg’s method. I’ve also re-written the code for my “slow Fourier transform” so that it deals properly with irregularly spaced data. To get started, let’s look at the data itself, including the maxima (red line) and the minima (blue line):

Figure 1. SIDC sunspot data since 1700. Red line shows the maximum value of each cycle. Blue line shows the cycle maxima after smoothing with Gleissberg’s “secular smooth”, a 1-2-2-2-1 trapezoidal filter.

For one thing, this would serve as the first real-world test for how well my “slow Fourier transform” performs when using a dataset that is both greatly reduced, and also irregular in time. So without further introduction, here are the periodograms of the sunspot data itself, and of the irregularly-spaced cycle peaks.

Figure 2. Periodograms of the total sunspot dataset (gold) and of the cycle maxima (red).

To begin with, let me say that I am amazed at how much information is contained in just the cycle peaks alone. Remember, the red line represents a mere 9% of the data, 91% of the data has been removed.

Next, looking at the full three centuries of sunspot data (gold), there are three main peaks, at 11 years, 102 years, and 52 years. There is no sign of Gleissberg’s 80-year cycle.

So how does using just the cycle peaks affect the results? Well, everything but the size of the main 11-year cycle has seen an increase in the reported strength of the cycle. This is because there is less data to constrain the fitting of the various lengths of sine waves, so they almost invariably end up larger than the corresponding cycle strength of the full dataset.

Despite all of that, however, the correlation between the two (red and gold) is impressively high, at 0.88. And it suggests that I should be able to further improve the results … more on that later, once I actually try it …

In any case, for purposes of investigating long-term results, there is little difference between using the full dataset and just the cycle peaks. Both of them, for example, show that rather than there being any strong “Gleissberg Cycle”, in fact 80 years is near the bottom of a dip in the cycle strength … and both the cycle peaks and the full dataset put the peak in the long-term cycles at about 100 years …

Having seen the results for the full data and the cycle maxima, what happens when we do the same analysis of Gleissberg’s “secularly smoothed” cycle maxima data? Figure 3 shows that result …

Figure 3. Periodograms of the total sunspot dataset (gold), of the cycle maxima (red, grayed out), and of the “secularly smoothed” cycle maxima).

Like I said, I had no idea what the periodogram of the “secularly smoothed” data would look like. One real surprise was that it totally wiped out the peak that exists at around 55 years in both the full and cycle maxima periodograms. It has also knocked out almost all of the power in the cycles from about 15-50 years. I wouldn’t have guessed either of those.

Curiously, the part that the “secular averaging” didn’t affect are the cycles of 70 years and longer. Well, it pushed the peak back to about 99 years instead of 102 years, but other than that all three tell the same tale.

And the tale they are all telling is that there is no such thing as an 80-year “Gleissberg Cycle”. Doesn’t exist in the sunspot data, even using Gleissberg’s crazy method.

Now, I’m sure people will jump up and down and say “but, but, but there are 80-year cycles in the Nile river data” or some other dataset … but so what? There is no 80-year cycle in the sunspot data, so if anything, your 80-year cycles in the Nile river data show that the sunspot cycles don’t affect the Nile river levels.

That’s what I started out to do regarding the lack of the Gleissberg Cycle, so I’ll leave the story there …

However, having seen how well my slow Fourier transform (SFT) performs when using the cycle maxima data, I’ve got to try randomly knocking out parts of the sunspot data to see how well the SFT performs … hang on while I go do that. … OK, here’s what happens when I randomly knock out 10% of the sunspot data.

Figure 4. Periodogram of the sunspot data, along with 30 instances of periodograms of the sunspot data with 10% of the data removed.

As can be seen, the loss of 10% of the data makes little practical difference to the results. This is quite encouraging. Next, here’s the same situation but with 50% of the data removed instead of 10% …

Figure 5. Periodogram of the sunspot data, along with 30 instances of periodograms of the sunspot data with 50% of the data removed.

Obviously, there’s much more variation with half of the data being missing, it’s getting sketchier, but the results still might be useable.

My final conclusion is that my method deals quite well with missing data. My next project? Well, now that I’ve modified my code to not require regular dates for the time series, I want to take a look at the ice core records …

Onwards … always more to learn.

w.

Like I’ve Said Before: If you disagree with something I or someone else has said, please quote the exact words you disagree with. This avoids many misunderstandings.

Data: The adjusted SIDC data, along with the R slow Fourier transform functions to do the periodograms, are both available in a zipped folder here. In accordance with the advice of Leif Svalgaard, all sunspot values before 1947 have been increased by 20% to account for the change in sunspot counting methods. It makes little difference to this analysis. I believe the R code to be complete and turnkey. I’ve included an example with the functions.