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.
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David Byrne’s lyrics seem to me to have a skeptical bend to them, though I sincerely doubt he himself is – this is an excerpt from “Cross eyed and Painless.” Seems to fit your recent analysis amazingly well…
____________________________
I’m ready to leave
I push the fact in front of me
Facts lost
Facts are never what they seem to be
Ah, nothing there
No information left of any kind
Lifting my head
Looking for the danger signs
And there was a line, there was a formula
Sharp as a knife, facts cut a hole in us
There was a line, there was a formula
Sharp as a knife, facts cut a hole in us
____________________________
getting interesting.
The red line is noisier since it has a lot less data, but the main peaks are confirmed.
“Curiously, the part that the “secular averaging” didn’t affect are the cycles of 70 years and longer. ”
Not curious. You passed a short period low-pass and it passed the low frequency components.
“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.”
You had data with roughly 10-11 year spacing and passed a slightly modified 5pt running mean. Not a real surprise either. By 60 years it’s starting to let some signal pass giving the false impression of a peak around 60.
What both the gold and the red plots show is a strong signal at 10 and 11 (on your course scaling) it may be worth a more detailed look at that portion with higher resolution, More detailed work shows periods are round numbers of years.
now if you add a your 10y period and your 11y period, it will produce an amplitude modulation of 10.5 modulated by 100 years (plus a residual of whichever short period is stronger).
The fact that you see both in the period analysis means that the process is non-linear. In fact this can be seen by eye that simple SSN counting looks like a “rectified” signal. The count is the amplitude of an underlying process where the sign matters.
In fact Ray Tomes has suggested that the noise level also indicates that it may be the square of whatever is the root cause.
” By 60 years it’s starting to let some signal pass giving the false impression of a peak around 60.”
Correction. On a closer look, it is the red line with less data points that is failing to resolve the 50 and 60 y periods in the full data, merging them in to an apparent 55 y peak. Having filtered out the 50y cmpt with the modified runny mean, the attenuated 60y comes through.
It may be better to regard this as a false peak with the resolution left after all the chopping , it is more a matter of chance that it lines up a real peak in the full data.
All in all, I would agree that there’s not much evidence of an 80 y cycle whatever way you dice it.
However, it is quite a good demonstration that there is a long period amplitude modulation of about 102 years, resulting from the combination of circa 10 and 11 year periods.
What I have never seen discussed in any mainstream work on SSN is what is needed to get to the root cause of this signal.
I was initially critical of Ray’s suggestion that the data should be square-rooted because it was an arbitrary choice ( typical econometrics way of doing things, not science ). But I think he is basically correct. It we be good to have a proper scientific reason, based on a physical mechanism to take the square root.
The ultimate need being for a way to ‘unfold’ the data and regain the signed time series. Once that is done the fourier analysis will become more meaningful and should tell us a lot more about the real underlying periodicities in the sun and that may finally start to give clues as to the cause.
henry@willis
as stated before, I don’t trust SSN as I found that there is a definitive trend upwards when measured against time. This is probably due to more stringent observation rules and better magnification technigues as time went by. Also, in addition, you even added data from years that Gleissberg never had, seeing as you took it to 2013. How do you justify that seeing as that I warned you before about the upward trend in SSN as time progresses?
Surely, you must see that this could affect your results?
The 88 year cycle is there, for a number of reasons
1) because I found it myself in the data on maximum temperatures
2) William Arnold found it and we both linked it to certain planetary configurations
this makes it very easy for me to predict the turning point in the cycle.
http://www.cyclesresearchinstitute.org/cycles-astronomy/arnold_theory_order.pdf
(note the cycle times given for the flooding of the Nile. He thought 1990 was the end of warming, I say it was 1995, when looking at energy in. Apparently he was also about 7 years out on the planetary configuration, compared to what actually happened on the sun.)
3)http://www.nonlin-processes-geophys.net/17/585/2010/npg-17-585-2010.html
shows both the Gleisberg (86.5) and De Vries cycles (208)
4)ersistence of the Gleissberg 88-year solar cycle over the last ˜12,000 years: Evidence from cosmogenic isotopes
Peristykh, Alexei N.; Damon, Paul E.
Journal of Geophysical Research (Space Physics), Volume 108, Issue A1, pp. SSH 1-1, CiteID 1003, DOI 10.1029/2002JA009390
Among other longer-than-22-year periods in Fourier spectra of various solar-terrestrial records, the 88-year cycle is unique, because it can be directly linked to the cyclic activity of sunspot formation. Variations of amplitude as well as of period of the Schwabe 11-year cycle of sunspot activity have actually been known for a long time and a ca. 80-year cycle was detected in those variations. Manifestations of such secular periodic processes were reported in a broad variety of solar, solar-terrestrial, and terrestrial climatic phenomena. Confirmation of the existence of the Gleissberg cycle in long solar-terrestrial records as well as the question of its stability is of great significance for solar dynamo theories. For that perspective, we examined the longest detailed cosmogenic isotope record—INTCAL98 calibration record of atmospheric 14C abundance. The most detailed precisely dated part of the record extends back to ˜11,854 years B.P. During this whole period, the Gleissberg cycle in 14C concentration has a period of 87.8 years and an average amplitude of ˜1‰ (in Δ14C units). Spectral analysis indicates in frequency domain by sidebands of the combination tones at periods of ≈91.5 ± 0.1 and ≈84.6 ± 0.1 years that the amplitude of the Gleissberg cycle appears to be modulated by other long-term quasiperiodic process of timescale ˜2000 years. This is confirmed directly in time domain by bandpass filtering and time-frequency analysis of the record. Also, there is additional evidence in the frequency domain for the modulation of the Gleissberg cycle by other millennial scale processes.
end quote
5) I have shown you about 30 proxies that have a recurring 80-100 year cycle. The reason why time differs in these proxies is because of the intricacy whereby earth stores energy and/or releases energy and/or causes obscurity due to volcanic eruptions.
Like I said before, for real scientists it is time to leave SSN alone and let it be…..
Rather try to figure out this graph here
http://ice-period.com/wp-content/uploads/2013/03/sun2013.png
The declining magnetic fields allows more energetic particles to escape from the sun. Note the binomial you can draw from the top (hyperbolic) and the bottom (parabolic) showing that we must come to the bottom of the field strength somewhere in 2016. I hope my planets will arrive in time, as otherwise I donot know where we will end up with this graph.
Have you thought about that? Come to think about it myself, if anything were to happen to the planets in our solar system so that they do not arrive in time (2016), we are all buggered here. We will all freeze to death……
.
“as stated before, I don’t trust SSN as I found that there is a definitive trend upwards when measured against time. This is probably due to more stringent observation rules and better magnification technigues as time went by.”
Just because a dataset has a finite “trend” we are supposed to “distrust” it? So we should only study data that are flat because everything else is nature is flat too , right?
DUH.
Also your comment about better magnification shows that you are totally ignorant of the subject. Considerable efforts are made to take this into account including using a small, antique telescope like used for early observations. This is used to ensure that counting methods from recent high quality observations are comparable to centuries old ones.
Greg says
Considerable efforts are made to take this into account including using a small, antique telescope like used for early observations
henry says
yet the upward trend is there
which should not be, if we have recurring cycles.
also there is no correlation as to why it should go up, so it was randomly going up.
If various methods say 86.5 to 88 (including my own method) then why would you trust SSN ( showing 100) more? Surely, SSN is a very subjective discretionary observation, depending, for example, even on the strength of somebody’s eyes?
“The wavelet power of the Gleissberg cycle is especially remarkable during 4750– 1400 BC, the power reaching peaks around 4215 BC, 3075 BC, 2755 BC, 2545 BC, 2075 BC and 1535 BC, with a statistical significance level higher than 95%. After 1400 BC, the wavelet power of the Gleissberg cycle becomes relatively weak.
Ogurtsov et al. (2002) proposed that the Gleissberg cycle has a wide frequency band with a double structure, i.e., 50–80 year and 90–140 year periodicities.”
http://ir.bao.ac.cn/bitstream/311011/930/2/3.pdf
Just a minor quibble: As I am sure you know well, the “sunspot number” is not the number of sunspots. Yet, a few places, notably the graph axis labels, are written as if SSN and sunspots are interchangeable.
Thanks Willis, for using the new posting format, without being prompted.
Yes, the abstract is a good idea. It makes it quick to see whether an article is of interest from the front page. Nice move.
Willis says
http://wattsupwiththat.com/2014/05/17/the-tip-of-the-gleissberg/#comment-1640546
henry says
I hope I did cover my position here
http://wattsupwiththat.com/2014/05/19/the-effect-of-gleissbergs-secular-smoothing/#comment-1640431
if not, before you trash Gleissberg completely, show us the (uncorrected) SSN data that he was working on?
Ulric says
http://wattsupwiththat.com/2014/05/19/the-effect-of-gleissbergs-secular-smoothing/#comment-1640464
henry says
thanks for that!
very interesting.
As I was saying, I would never trust building a whole theory on one data set of results.
I don’t think Gleiszberg did either. I am sure he also looked at other things besides SSN, like the level of lakes or rivers.
I am going on a holiday soon down the river Rhine. Perhaps I will pick up something here or there on the subject….;
“I was initially critical of Ray’s suggestion that the data should be square-rooted because it was an arbitrary choice ( typical econometrics way of doing things, not science ). But I think he is basically correct. It we be good to have a proper scientific reason, based on a physical mechanism to take the square root.”
sun spot count is not a proper physical unit. period. any equation you do with ‘spots’ will ever be dimensionally correct.
you can transform it however you like, its not a physical unit.
Now, if you want to switch to what matters.. like watts you’d be in a better position to understand physical systems.
Spots and spot counts are an abstraction. the climate doesnt ‘see’ spots. humans see spots and then they devised a methodology for counting them. does the number matter? yes its a proxy. what about the size?
if you drill down to the bottom and ask the simple question: what is a spot and how is it defined
you will see that the real physics you are after doesnt quantify over spots. spots are a crude tool or metric used to understand something else..
HenryP
I am convinced that we are entering a deep solar minimum which none of us remembers. The sharp decline in activity after the last maximum is a clear signal from Cosmos. The activity will now fall rapidly.
http://soho.nascom.nasa.gov/data/realtime/hmi_igr/512/latest.jpg
Steven Mosher says
http://wattsupwiththat.com/2014/05/19/the-effect-of-gleissbergs-secular-smoothing/#comment-1640655
Henry@steven
this must the first time ever that you post an argument that I absolute agree with….
100%
be blessed
henry@ren
don’t worry
my planets will come in time,
to switch the lever back to warming again (increasing solar magnetic polar strengths)
unless something happens to any one of them….
@ren
I do agree, that now, more than I can remember since I live here,
the sun is brighter/lighter than ever before
Amazing is it not? that a brighter/lighter/warmer sun causes a cooler earth…
it boggles the mind
does it not?
Greg Goodman says:
May 19, 2014 at 1:47 am
However, it is quite a good demonstration that there is a long period amplitude modulation of about 102 years, resulting from the combination of circa 10 and 11 year periods.
You have this backwards. There is a real 100-yr modulation of the amplitude of the cycle. This is a real physical effect: cycles vary in strength and we can understand why [or at least there are models that can ‘explain’ this]. The amplitude modulation then produces the side peaks at 10- and 12 years.
HenryP says:
May 19, 2014 at 5:49 am
Surely, SSN is a very subjective discretionary observation, depending, for example, even on the strength of somebody’s eyes?
The SSN shows 98% correlation with the microwave flux from the Sun and therefore can be trusted to give a very close measure of real solar activity. Experienced observers agree closely.
Greg says:
May 19, 2014 at 1:27 am
Thanks, Greg. Following Gleissberg, I used his 5-point filter. But in his method, the filter spans five sunspot cycle peaks, which is a varying period on the order of 44 years. Despite that, it has not eliminated the 11-year cycle much, and it has created a spurious 6-year cycle. So it is not acting like a normal low-pass filter, which would have left no cycles at all below 44 years or so.
I think that the oddness of those results comes from the fact that the data points that are being smoothed are all cycle peaks, but hey, I was born yesterday, and unlike many commenters, I’ve never taken a single class in any of these matters. So I’m always willing to learn.
A 5-point filter on data with 10-11 year spacing shouldn’t affect a 55-year cycle that much. That’s almost 25% longer than the filter width of 40-44 years, there should be little attenuation.
I’ve done that, and not found that “periods are round numbers of years”. They fall where they fall. And there is no reason they should be in round years, as solar cycles know nothing of puny Earthlings and their curious time unit, “years” … hang on, let me do that for you. To get higher resolution with my slow Fourier transform, you just use data with higher resolution. Here’s the periodogram of the monthly (rather than annual) sunspot data.
There are three prominent peaks, at ten years, eleven years, and eleven years ten months. There’s a much, much smaller cycle at twenty-one years one month. However, the fact that two of them are on even years is happenstance. Here’s the same analysis, starting 50 years later:
Note that the peaks are no longer on the exact 10- and 11-year boundaries.
There’s where we part company. There is no “your 10y period and your 11y period”. There is a single cyclic phenomenon which peaks at intervals ranging from 9 to 15 years. So you can’t just pick a couple frequencies and note that their synoptic period is kinda sorta the same length as some other observable cycle. A variable cycle that peaks at intervals between 9 and 15 years doesn’t automatically create an underlying century-long cycle, that’s just numerical handwaving.
Every rectified signal that I’ve ever seen started and ended every cycle at zero. That’s kinda the definition of “rectified” on my planet, that values below zero are converted to the same value above zero.
But the sunspot data doesn’t do that. What it resembles is something like the length of the line of people waiting for a bank teller. It has a daily peak, and it has low spots, just like the peaks and lows of sunspots. And sometimes it goes to zero, again like sunspots. But that doesn’t make either of them into rectified signals.
Thanks for that idea. It has led me to the concept of a curious expansion of the type of direct spectrum analysis I’ve called the “slow Fourier transform” (SFT). In the SFT, I’m using sine waves to probe the individual cycle lengths.
But your comment has led me to the idea that there is no reason to limit myself to sine waves. I could just as easily probe the data using log(sin), sin2, or other wave forms … always more adventure.
Regards,
w.
PS – One of the more interesting ideas I’ve had about using other waveforms is first determining the actual wave-form of the predominant 11-year cycle, and then using that idiosyncratic wave-form to probe the data … man, not enough hours in a day.
Willis Eschenbach says:
May 19, 2014 at 1:05 pm
There’s where we part company. There is no “your 10y period and your 11y period”. There is a single cyclic phenomenon
And Willis is correct here. The 102-yr real amplitude modulation creates the two side peaks at 10 and 12 years.
Damn! I’d just composed long reply and an accidental click zoomed you graphic. When I come back I’ve lost it.
OK, I’ll resume. Yes, abs(cos(x)) would be very interesting, you’ll get periods about twice what you see above.
Thanks for the high-res SFT, that is exactly what I was expecting. I wondered why there was not 11.8 in the original. 10y 1mo; 10y 11mo and 11y 10mo make up frequency side bands around the central period as Leif points out.
Such triplets , as I have frequently pointed out are the result of modulation. Lief was right to pick me up on how I wrote that above. The two ways of expressing the data are equivalent. I was not intending to say which was causing which physically.
Those numbers imply a modulation period of about 136 years, rather than 102, so maybe that something separate. It would be interesting to see what an abs(cos) SFT looks like. .
When I said “rectified” don’t take it to be like a power socket. It is well known that Schwabe cycle has a polarity change. That does not mean that whole of the sun has to suddenly stop moving while it switches over. Each of the circa 11y bumps are just half of a longer cycle of which SSN is measure of the magnitude ( or according to Ray Tomes, the square of the magnitude ).
You could try doing a sqrt(SSN) and fitting SFT .
I wanted to try and abs(cos()) fit to Arctic ice, did you post your SFT code anywhere?
Thanks.
Here’s what sqrt ( sun spot area ) looks like:
http://climategrog.wordpress.com/?attachment_id=941
That sloping profile looks like it has a component twice as fast as the basic 11y at about 30% amplitude. probably what is picked up around 5.5 in your latest plots.
replacing above sqrt(SSA) graph:
http://climategrog.wordpress.com/?attachment_id=942