cite as: F. Bi. 2008. The empirical, the mode, and the decomposition. *Intl. J. Inact.*, 1:7–9

A while back the “global cooling” folks mentioned a paper[ZX07] by Zhen-Shan and Xian^{1} which uses an analysis method known as “empirical mode decomposition” (EMD). Being a bit of a signal processing geek, I decided to look up what EMD is all about.

The EMD, also known as the Hilbert-Huang transform, is a relatively new method for decomposing a (usually non-stationary) signal into a series of zero-mean, wave-like “intrinsic mode functions” (IMF), plus (optionally) a monotonic function. The original paper[HSL98] is pretty long (93 pages); it helps to know that the EMD method proper is described in pp. 917–923. (Actually, you can probably try implementing it in a program on your own computer — except for the little problem that it’s patented in the US, so *caveat haxor*.)

Anyway, one thing about the EMD algorithm is that the decomposition method works by using cubic splines and all that, which strikes me as … somewhat *ad hoc* (though the authors remind us that the output tend to be meaningful regardless). Indeed, Rilling et al.[RFG03] wrote that

the technique is faced with the difficulty of being essentially defined by an algorithm, and therefore of not admitting an analytical formulation which would allow for a theoretical analysis and performance evaluation.

Other potential problems also exist, such as the possibility of “mode mixing”[WH05]. Together, these mean that the output of EMD must be interpreted with great care.

Which leads me to the main problem with EMD, which isn’t with the algorithm itself, but *how* it’s used by Zhen-Shan and Xian. They try to extrapolate the IMFs in order to *predict* stuff. But here’s the breaks: **the IMFs are empirically derived functions which aren’t known to correspond to any neat formulae, so they can’t be extrapolated just like that** — at least, not without some more work.

Indeed, a close examination of Figure 1 in [ZX07] (shown on the right) will see what’s wrong with their attempts at interpretation and extrapolation:

- They claim that IMF3 “corresponds to 20-year periods”, but this is far from clear: there’s no maximum at 1900, and there are extra bumps at around 1970. Is this really indicative of a 20-year periodicity? Or is something else going on here?
- They claim that IMF4 “contains 60-year cycles”, based on just
*1 maximum*and*2 minima*. Sorry, but that’s not how to do things. - They claim that the residual (Res) “indicates larger timescale oscillation”. Why not a good old upward trend?

The best way to resolve these questions, of course, is to examine the underlying physics. That is, we’re back to the good old “climate models” which climate inactivists love to hate. 🙂

**Footnote**

- Or perhaps it should be Lin and Sun, and the authors wrote their names in the wrong order.

**References**

- [HSL98] N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. Zheng, N.-C. Yen, C. C. Tung, and H. H. Liu. 1998. The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis.
*Proceedings of the Royal Society of London A*, 454:903–995. - [RFG03] G. Rilling, P. Flandrin, and P. Gonçalvès. 2003. On empirical mode decomposition and its algorithms.
*Proceedings of IEEE-EURASIP Workshop on Nonlinear Signal and Image Processing*. - [WH05] Z. Wu and N. E. Huang. 2005. Ensemble empirical mode decomposition: A noise assisted data analysis method. Center for Ocean-Land-Atmosphere Studies Technical Report.
- [ZX07] L. Zhen-Shan and S. Xian. 2007. Multi-scale analysis of global temperature changes and trend of a drop in temperature in the next 20 years.
*Meteorology and Atmospheric Physics*, 95(1–2):115–121.

This got by the reviewers? Aren’t the empirically-derived waveforms completely arbitrary? What advantage does this method have over Fourier transformations? At least the sine waves derived from Fourier transforms are periodic.

Comment by Will TS — 2008/04/20 @ 04:42 |

Well, Rilling et al. also cited a paper by Souza Neto et al. which successfully used EMD to detect heart-rate oscillations at those moments when a person changed his posture (supine / seated / standing), oscillations which didn’t show up with good old short-time Fourier transforms (STFTs) — so EMD isn’t completely useless. 🙂 Still, Souza Neto et al. had to apply STFTs to the IMFs to see the oscillations.

Comment by frankbi — 2008/04/20 @ 05:42 |

How does this relate to wavelets (does it?)?

Comment by Gavin's Pussycat — 2008/04/21 @ 13:11 |

Well, my (very rudimentary) understanding is that wavelet analysis is like doing lots and lots of STFTs on the data with a smooth window set at various scales. While EMD is more like a method that simply tries to detect and subtract away high-level trends by repeated curve-fitting — it just gives you whatever oscillations it sees (rightly or wrongly) from the data, without being told what scale it should look at.

So I guess wavelet analysis and EMD are quite different beasts… and possibly complementary too. But that’s just my guess.

Comment by frankbi — 2008/04/21 @ 17:36 |