CEEMD vs Joe Fourier

Guest Post by Willis Eschenbach

In one of my late-night somnambulistic ramblings through the climate literature, I came across a 2021 study entitled “Evidence of solar 11-year cycle from Sea Surface Temperature (SST)” by Daniele Mazza and Enrico Canuto, hereinafter MC21. They claim that they can use Fourier analysis to show that there is a solar cycle signal in the 1948-2021 sea surface temperature in the tropical Pacific Nino 4 area (5°N to 5°S, 160°E – 150°W). They also make the same claim for a wider band in the same Nino 4 area of the tropics (10°N to 10°S, 160°E – 150°W).

In this analysis, I’ll show two very different reasons why their conclusions are not justified.

Let me start with the question, what is Fourier analysis when it’s at home? Well, back in 1807, a French genius named Joseph Fourier published his brilliant insight. He had realized that any signal, like say a time series of temperature observations, could be decomposed into a number of perfect sine waves. These sine waves, each with their own amplitude, period, and phase, all add together to exactly reconstruct the original signal.

Fourier analysis is a very powerful form of decomposing a signal. It has been immensely successful in analyzing, synthesizing, and understanding signals of all types, and it is used in a huge variety of analyses. However, it’s not the only game in town.

CEEMD stands for Complete Ensemble Empirical Mode Decomposition. Like Fourier analysis, it decomposes a signal into a number of simpler signals. However, Fourier decomposition breaks a signal into only pure, unvaryingly regular sine waves. CEEMD, on the other hand, breaks the signal into
“empirically determined” underlying signals. This means that the range of cycle lengths of the signals in each group is determined by the data itself. I discuss CEEMD in detail in my post entitled Noise Assisted Data Analysis.

What does this mean in practice? Well, let me apply CEEMD to the data that they have used in their analysis. First, here is the monthly sunspot data from 1948 to mid-2021.

Figure 1. Monthly sunspot counts, January 1948 — June 2021.

And here is the complete ensemble empirical mode decomposition (CEEMD) of the same sunspot signal.

Figure 2. CEEMD Decomposition of the monthly sunspot counts.

So what are we looking at here? The top panel shows the raw data. Panels C1 to C9 show the signal in each of the nine empirical modes. These, together with the residual trend in the bottom panel, will add together to perfectly reconstruct the original signal. Clearly, the majority of the sunspot signal is in empirical mode C6.

Below is another way to look at the CEEMD sunspot decomposition. This is to look at the periodograms of each of the empirical mode signals. Periodograms show the periods (cycle lengths) of the signals that make up that empirical mode. Figure 3 shows that result.

Figure 3. Periodograms of the CEEMD Decomposition of the monthly sunspot counts.

Here, we see that empirical mode C6 contains a strong sunspot cycle peaking at just under 11 years, with a bit more sunspot-related strength in empirical mode C7. Other than that, there’s little to be seen.

Now, we can recreate the long-period variations in the sunspot signal simply by adding together the empirical modes of ~ 11 years and longer shown in Figure 2. These are the modes C6 to C9 plus the residual trend. Figure 4 shows that result, superimposed on the underlying sunspot data.

Figure 4. Sunspot counts as in Figure 1, overlaid with the sum of the empirical modes of 11 years and longer.

So far, so good. You can see how well the CEEMD shows the variations in the sunspot data. Next, I’ll show the same kind of analysis for the wide NINO4 area used in the MC21 analysis. To start with, here’s the raw 1948-2021 data.

Figure 5. Monthly NINO4 10°N-10°S temperatures, January 1948 — June 2021.

Next, the complete ensemble empirical mode decomposition (CEEMD) of the same temperature signal.

Figure 6. CEEMD Decomposition of the monthly NINO4 temperatures.

Then we have the periodograms of each of the empirical mode signals.

Figure 7. Periodograms of the CEEMD Decomposition of the monthly NINO4 temperature.

Here, we can see that there is a strong cycle peaking at 12 years … and this is the reason that the authors of MC 21 claim a “solar signal” in the ocean temperatures. However, there is no 12-year cycle in the sunspot data. Look at Figure 3. It’s a few months shorter than an 11-year cycle.

Finally, as in Figure 4, we can reconstruct the underlying temperature cycle in the NINO4 10°N-10°S data by adding the 11-year and longer empirical modes plus the residual. Figure 8 shows that result.

Figure 8. Temperatures as in Figure 1, overlaid with the sum of the empirical modes of 11 years and longer.

As you can see, there’s a signal in there, and the CEEMD analysis gives a very good fit … but it’s very unlike the signal in the sunspots. To highlight the differences, let me show the sums of the eleven-year-plus CEEMD modes for the sunspots and the temperatures.

Figure 9. Comparison of underlying cycles in sunspots and NINO4 10°N-10°S temperatures. To allow direct comparison, the CEEMD residual trends have not been included in either result.

Here, the errors in their analysis become quite evident. They’ve claimed that a 12-year cycle in the temperature data is due to solar variations. But as you can see, although there is passable agreement up to 1980, even in that section the peaks and troughs of the temperature signal sometimes lead the solar signal by up to a year and a half. This would imply an impossibility, that the NINO4 temperature is causing the sunspot cycle … and at other times, the spots lead the temps by up to three and a half years.

Worse yet, post-1980 the NINO4 temperatures start shifting more and more to the right. This is a reflection of the difference between the 10.75-year cycle of the sunspot data over the Jan 1948 – Jun 2021 period, and the 12-year cycle of the NINO4 temperatures over the same period.

In short, while the cycles are close, they do not show any connection between the 11-year sunspot cycle and the 12-year temperature cycle over that period.

So why is there a similarity? They reveal the reason in their study, viz:

After having downloaded and analysed hundreds of temperature records of the earth surface, eventually, we found clear evidence for the sun’s 11-years cycle signature in some few cases, while for the vast majority of the others this wasn’t detectable, buried under other oscillations (seasonal or El-Nino related) or noise.

(In passing, I love their totally unsupported claim that the solar effect is everywhere but it just isn’t “detectable” because it’s “buried” under reasons … but I digress.)

The problem is that if you look in enough places you’ll eventually find a similar signal … but that’s probably not statistically significant. Here’s an example.

Suppose you have a random number generator that generates a new random number from one to one hundred each time it’s used. A man says “I can guess the range of the next number. It will be between one and five”. And sure enough, the next number is three.

Since the odds of him guessing it right by chance are only one in twenty (0.05), that result is said to be statistically significant at a “p-value” of 0.05, and perhaps the man is right that he can guess the number. Of course, with a p-value of 0.05, there’s still a 5% chance it was just dumb luck.

But suppose, on the other hand, that the next random number is thirty-two. The man is wrong. So he says “Let me try again” … and again he fails. So he tries again, and again, and of course, eventually he gets it right.

Is that result statistically significant? Does he get to claim success?

Well … no. As my dad used to say when I was a kid, “Even a blind hog will find an acorn once in a while”. (In my youth, I always misheard it as him saying “a blind hawk will find an acorn”, and so I spent years wondering what a hawk, blind or not, would do with an acorn anyhow … but again, I digress.)

No, it’s not significant and he can’t claim success, because if you make enough attempts, or in the current sunspot case if you look in enough places, you’ll eventually get a positive result.

To adjust for this, we use what is called the “Bonferroni Correction”. This was an extension of work by the Italian mathematician Carlo Emilio Bonferroni (1892-1960). The correction itself was developed by a woman named Olive Jean Dunn and published in 1961. She only mentioned Bonferroni once in her analysis, but she was a woman during the 1960’s so Bonferroni got the glory … go figure.

In any case, the Bonferroni/Dunn Correction says that if you are looking for statistical significance at some specified p-value of “α” (say 0.05 as in the example above, a value commonly used in climate science) and you look for it in “n” places, you need to adjust your p-value downwards as follows:

By their own description, the authors looked for the solar signal in “hundreds of temperature records” … so to find something statistically significant, it needs to be a very, very good match, with a Bonferroni-corrected p-value of

α of 0.05 / n of 100 = corrected p-value of 0.0005

This is a level of correspondence rarely seen in climate science … and in their analysis, they don’t even mention statistical significance.

So that’s my analysis of the MC21 study. And heck, if they’d just used plain old Fourier analysis and instead of just doing it on the NINO4 temperatures they’d compared it to the Fourier analysis of sunspots for the same time period, they’d have seen the problem right away:

Figure 10. Fourier periodograms of sunspots and NINO4 10°N-10°S temperatures.

As you can see, the periods are far from the same, which means that they will go into and out of phase with each other. This in turn means that, as shown above in Fig. 9, at times the changes in the NINO4 temperatures will lead the changes in the sunspots … and that means the sunspots cannot possibly be the cause of the NINO4 temperatures.

And another solar study bites the dust.

For reference, I started investigating this question of a sunspot-weather connection a couple of decades ago, and I was a true believer in the sunspot-weather connection. I thought it would be easy to find evidence that sunspots affected surface weather in some form.

But despite looking at a bunch of temperatures, rainfall, river levels, lake levels, ocean levels, and other phenomena which were claimed to contain a sunspot signal, I’ve never found one claim that stood up to close examination. See here for links to 24 of my sunspot analyses, all of which showed … nothing. Doesn’t mean there isn’t a connection between sunspots and surface phenomena—it just means that if it exists, I’ve been unable to find it.

---

Here, it’s a glorious spring day. I’m going outside. These are the redwood tree and the flowers in our front yard, basking in the abundant solar radiation and enjoying the warmth. Life is good.

Best to all,

w.

The Usual: When you comment, please quote the exact words you are discussing. It avoids heaps of misunderstandings.

A Sidenote For Those Interested: As has happened to me a couple of times with other discoveries, I independently derived the Bonferroni/Dunn correction from basic principles long before I ever heard about Bonferroni. I saw the problem and calculated the proper response.

However, the form I derived gives an exact answer, and the usual Bonferroni/Dunn correction is a greatly simplified approximation of that exact form.

If alpha is the desired p-value, and N is the number of tries, the exactly accurate form that I had independently derived is:

Corrected p-value = 1 – exp( log(1 – alpha) / N )

Now, the usual Bonferroni/Dunn correction is:

Corrected p-value = alpha / N

When I found out about the Bonferroni/Dunn correction a decade or so ago, I got to wondering how good an approximation it is. So I calculated the errors (actual minus approximation). Here are the errors for alpha = 0.05 and N from 2 to 10

 N  Error
 2 0.00032
 3 0.00029
 4 0.00024
 5 0.00021
 6 0.00018
 7 0.00016
 8 0.00014
 9 0.00013
10 0.00012

These errors are all definitely within tolerance. So I gave up using my own method and went for the approximation, much easier to remember and use.

Addendum: In addition to the Bonferroni Correction, I also independently derived the Koutsoyiannis method for determining effective N, and the Date-Compensated Discrete Fourier Transform, or DCDFT (Ferraz-Mello, S. 1981, Astron. J., 86, 619). A couple of people have asked me if it bothers me to find out that someone preceded me in deriving those methods, meaning that I was not the first one over the line.

Quite the opposite. I take it as evidence that I actually do understand the effective N statistics, the Bonferroni Correction, and the Fourier transform. I understand them well enough to derive them independently. And since I’m totally self-taught in these matters, never took even one statistics or signal analysis class, that’s an important validation for me.