Complete Ensemble Empirical Mode Decomposition

Guest Post by Willis Eschenbach

Back in 2015 in a post called “Noise Assisted Data Analysis“, I described a way to decompose a signal into its underlying components. It’s known by the acronym “CEEMD”, hence the title of this post. The most common kind of signal decomposition is called “Fourier Analysis”. However, it decomposes a signal into regular sine waves, constant signals with no changes in strength or frequency over time.

CEEMD, on the other hand, decomposes a signal into empirically-determined groups of time-varying underlying components. “Empirically determined” means that the division of the groups is based on the nature of the signal itself. This lets us see how each group of underlying components varies in strength and frequency over time.

I find CEEMD much more useful than Fourier Analysis because among other reasons it lets me investigate the relationship between different climate datasets. To demonstrate how this is done, let me begin with a look at the result of a CEEMD decomposition. Here’s the CEEMD analysis of the El Nino index called the “ONI”, the Oceanic Nino Index.

Figure 1. CEEMD analysis of the Oceanic Nino Index (ONI). Units are standard deviations of the underlying ONI signal.

From the top to the bottom, first you have the raw ONI data. Then you have the underlying signals, from the shortest period (highest frequency) signal group shown as empirical mode C1, to the longest period (lowest frequency) signal group shown as empirical mode C7. At the bottom you have the “Residual”, which is what’s left over after the removal of empirical mode signals C1 through C7. And if you add together all of those, C1 to C7 plus the residual … you reconstruct the original ONI signal exactly

As you can see, the strongest part of the signal is in empirical mode C5. This is the part of the signal with periods from about two to five years in length.

One of the most powerful uses of the CEEMD analysis is that it lets us see if two or more signals are closely related to each other. For example, Figure 2 below shows the empirical mode C5 for a variety of datasets. They include three temperature-based El Nino indices—the NINO34, ONI (Oceanic Nino Index), and MEI (Multivariate Enso Index).

Then there is one sea-level atmospheric pressure based El Nino index, the SOI (Southern Ocean Index). It is built around the difference in sea level pressure between Tahiti and Darwin, Australia.

There is one atmospheric temperature dataset, the UAH MSU satellite-based tropical lower troposphere temperature.

Finally, there are two total precipitable water (TPW) datasets. One is the tropical (23.5°N – 23.5°S) portion of the full ECMWF TPW dataset. The other is the RSS (Remote Sensing Systems) 20°N – 20°S ocean only TPW dataset. Here are the empirical mode 5 CEEMD results for all of them.

Figure 2. CEEMD empirical mode 5 results for a variety of datasets.

Interesting result, huh? You can see clearly that all of these datasets are moving in close harmony.

I got to thinking about this because in the comments to my last post, A Chain Of Effects, Matthew Sykes pointed out that the ECMWF total precipitable water (TPW) data was unlike the NVAP TPW data. To see who the outlier is, here are the CEEMD empirical mode 5 results for three different TPW results—the ECMWF, the RSS, and the NVAP total precipitable water datasets. The NVAP dataset starts in 1988, so I’ve started the comparison there.

Figure 3. CEEMD empirical mode 5 results for three total precipitable water datasets.

From this, it seems clear that the NVAP dataset is the clear outlier. It goes into and out of sync with the other two, while the other two agree well throughout.

That answers the question that I came in on, as well as demonstrating the usefulness of the CEEMD analysis. And further deponent sayeth not.


Meanwhile, here on our Northern California coastal hillside with a tiny view of the ocean, we’re supposed to get big rain all week starting this morning (Tuesday) … fingers crossed. They say it will rain 2.7 inches (6.9 cm) today alone, which ain’t no ordinary rain, it’s a frog-strangler. But since we’ve only had about 40% of normal rain so far, I can only wish for a tropical downpour.

Best of this wondrous world to everyone,

w.

PS—I am happy to discuss and defend what I’ve said. However, I cannot defend or discuss what you think I said. As a result, I ask that when you comment you quote the exact words you are discussing so that we can all be clear on both who and what you are referring to.

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Scissor
January 26, 2021 10:13 am

CEEMS like a good idea.

Rob_Dawg
January 26, 2021 11:04 am

My Mark 1 eyeball suggests the addition of a C8 1/2 sine periodicity c.22 years might even flatline the “residual.”

January 26, 2021 12:19 pm

Willis, Thanks so much for this explanation of the CMEED process. I’ve been a bit puzzled and interested in your frequent use of it…it obviously is very powerful tool but I’d like to be able to fully understand it.

However as a mathematical drongo I still don’t quite get it. For example how are the C1-C7 categories chosen. You say they are ordered from the highest (C1) to lowest frequencies (C7) but how does one determine that there should be 7 categories not 27? Will each data set have a different number of C categories? Is the frequency range for each C class pre set or does that pop out of the data set somehow.

Could you please reply as if you’re talking to a 6 year old. Thanks.

Reply to  Willis Eschenbach
January 26, 2021 12:57 pm

Thanks Willis, I’ll work through them and the comments tonight. If I still don’t get it (quite likely) I might get back to you if that’s OK.

Keep up the good work…we all appreciate it. Your posts add a lot of meat to WUWT rather than just political bickering and point scoring that some seem to love.

menace
Reply to  Willis Eschenbach
January 26, 2021 1:42 pm

From the Hilbert-Huange transform article in my other post (and I don’t pretend to fully understand this):

Mode mixing problemMode mixing problem happens during the EMD process. Straightforward implementation of sifting procedure produces mode mixing due to IMF mode rectification. Specific signal may not be separated into the same IMFs every time. This problem makes it hard to implement feature extraction, model training and pattern recognition since the feature is no longer fixed in one labeling index.

Ensemble empirical mode decomposition (EEMD)The proposed Ensemble Empirical Mode Decomposition is developed as follows:

  1. add a white noise series to the targeted data;
  2. decompose the data with added white noise into IMFs;
  3. repeat step 1 and step 2 again and again, but with different white noise series each time;and
  4. obtain the (ensemble) means of corresponding IMFs of the decompositions as the final result.

The effects of the decomposition using the EEMD are that the added white noise series cancel each other, and the mean IMFs stays within the natural dyadic filter windows, significantly reducing the chance of mode mixing and preserving the dyadic property.

menace
Reply to  Willis Eschenbach
January 26, 2021 1:26 pm

Looks like a form of Hilbert-Huang transform
https://en.wikipedia.org/wiki/Hilbert%E2%80%93Huang_transform#Definition

EMD is an iterative process to derive Intrinsic Mode Functions (IMFs) that are used to mathematically decompose non-stationary/non-linear data

EEMD somehow fixes the “mode mixing” problem of EMD by adding in white noise at the front end

However I’m not sure what the C=Complete for CEEMD means versus just plain EEMD

EEMD IMFs are by definition not generally mathematically complete (the E=empirical implies not complete, no?)

menace
Reply to  menace
January 26, 2021 1:49 pm

Maybe means completeness in this sense??…
https://en.wikipedia.org/wiki/Completeness_(order_theory)
whew way over my head though

January 26, 2021 1:15 pm

CEEMD on the ONI just produces noise plots from a noisy signal. Tell me when the NEXT big El Nino is from those any of those modes and I might think it means something. Predicting the past is meaningless.

dh-mtl
Reply to  Joel O'Bryan
January 26, 2021 2:58 pm

Looking at Figure 2, it seems to me that the primary mode for all of these datasets has a frequency of about 3.7 years. Interestingly, this is about 1/3 of a solar cycle.

Obviously, all of these data sets have the same primary driver, ocean temperatures. I would suspect that what controls, what seems to be a relatively stable frequency, is the mechanics of the circulation of ocean currents in the tropical Pacific.

When Paul Homewood posted an article on WUWT about Accumulated Cyclone Energy, on January 15, I found that there was a very good correlation between Global ACE and ENSO. I also found that the strongest peaks for Global ACE, like ENSO, occurred about 4 years after the solar cycle ramped up to its most active phase, i.e. what drives strong EL Ninos and active cyclone seasons is an interaction between ocean temperature cycles and the solar cycle.

So if you want to know when the next big El Nino will be, it is likely about 8 years off, ie. 2 1/2 ENSO cycles, and about 4 years or so after the current solar cycle ramps up to its most active phase.

stinkerp
Reply to  Joel O'Bryan
January 26, 2021 7:42 pm

CEEMD isn’t necessarily used to predict future cycles. The prior cycles may have random length or amplitude. CEEMD is extremely useful for comparing plots of different variables to see if “two or more signals are closely related to each other“, as Willis explained and demonstrated. The only time it could “predict the future” is if the cycles were of predictable length. It is an extremely useful technique when dealing with seemingly random or chaotic data.

Ben Vorlich
Reply to  stinkerp
January 27, 2021 1:32 am

Just a Teuchter’s view here. This method is good at eliminating what it isn’t and possibly what it is that causes something to happen. But until we have something predictable as the cause we can’t predict anything. But if more than one unpredictable is involved then it’s down to a guess. As all things in nature are inherently unpredictable to then we’re talking long range weather forecasting.

Rick Spielman
January 26, 2021 2:50 pm

Seems like a variant of the multi-resolution decomposition technique using wavelet basis functions (many different options of basis sets). One nice thing in this approach is the ability to decompose the data into the different 2^N levels and examine each level for the presence of stochastic noise. Then one can reassemble the data with the pure noise levels removed. This has been used for 20 year in oil field seismology. This is a great approach! Good work.

Old Cocky
January 26, 2021 4:02 pm

“They say it will rain 2.7 inches (6.9 cm) today alone, which ain’t no ordinary rain, it’s a frog-strangler.”
That’s a handy little scud, which I would have thought typical of a tropical thunderstorm.
I know this is way off the main topic, but the little asides such as “what counts as heavy rain in different areas” can be quite interesting as well.

Robert of Texas
January 26, 2021 4:24 pm

This is extremely interesting. Most of the plots in figure 2 are so in line that my “suspicion” radar turns on…when something looks too good to be true…Makes me wonder if the data does not rely on each other somehow when they “adjust” it. That or it’s just really related.

I note there are two large discrepancies. 1) SOI around 1985 is inverted from the other data sets – given it’s behavior after that it makes me wonder why. 2) TPW RSS appears really out of phase in the early 1990’s. Again, I wonder why that is so? Did they replacee a satellite sensor in or around 1996?

Wouldn’t it be cool if you have found a new way to check for faulty or badly calibrated data senors.

Ulric Lyons
Reply to  Robert of Texas
January 27, 2021 7:08 pm

Early 1990’s yes, but that is just before 1985, think volcanoes.

Pat from kerbob
January 26, 2021 6:48 pm

Looks like W gets his wish

stinkerp
January 26, 2021 7:46 pm

Thank you, Willis, for this excellent explanation and example of CEEMD. I’ve tried it myself on the UAH TEMP data set in R Studio to locate trends, but with very little understanding. This is extremely helpful, and the comparison with TPW is fascinating. Bookmarking this.

Last edited 8 months ago by stinkerp
David L Walker
January 27, 2021 7:40 am

Looks like Wavelet analysis under a different name.

Bear
January 27, 2021 10:16 am

Willis, sorry to bother you but I was wondering if you could help me with CEEMD. I wanted to use it on some other data but I thought I’d try to replicate what you did with the sunspot data just to make sure I could get reasonable answers. I tried replicating your sunspot graphs but I don’t see how you were generating the formatting. Was they part of the hht package, another package or did you roll your own with ggplot? The periodograms that you show are in the time domain(1/f) rather than frequency but all I managed to find in hht was frequency in one graph rather than stacked like yours.

Another question I had was what you were using for the white noise parameter noise.amp since there’s no default and the doc for hht doesn’t give any hints. My results “look” close to yours but there are some differences. I am using the latest data which has six more years than the one you did in 2014 so I don’t know if that has an effect either.

This is the code I’m using

cresults = CEEMD(sig=spnum.std,tt=ts(spyear),noise.amp = 1, trials=100);

Thanks for any help

Barry

Bear
Reply to  Willis Eschenbach
January 27, 2021 12:48 pm

Thank you so much.

Barry

peyelut
January 29, 2021 4:57 pm

“Deponement”? You made me look up a werd . . . Moncton makes me look up many more.

Last edited 8 months ago by peyelut
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