This post is rather technical, and it is intended mainly for the historical completeness. So unless you are very, very interested in the tiny technical details of the HS saga, you can safely skip this.
As most readers are aware, and stated in my post few hours after CG broke out, Mike’s Nature trick was first uncovered by UC here. He was able to replicate (visually perfectly) the smooths of MBH9x thereby showing that the smooths involved padding with the instrumental data. The filter used by UC was the zero-phase Butterworth filter (an IIR filter), which has been Mann’s favourite since at least 2003. However, there was something else that I felt was odd: UC’s emulation required a very long (100 samples or so) additional zero padding. So about two years ago, I decided to take an additional look at the topic with UC.
Indeed, after digitalizing Mann’s smooths we discovered that UC’s emulation was very, very good but not perfect. After a long research, and countless hours of experimenting (I won’t bore you with the details), we managed to figure out the “filter” used by Mann before Mann (2004)-era. Mann had made his own version of the Hamming filter (windowing method, an FIR filter)! Instead of using any kind of usual estimate for the filter order, which is usually estimated from the transition bandwidth (see, e.g., Mitra: Digital Signal Processing) and has typically the length of a few dozen coefficients at maximum, he used the filter length equal to the length of the signal to be filtered! As Mann’s PCA was apparently just a “modern” convention, this must be a “modern” filter design. Anyhow, no digital signal processing expert I consulted about the matter had ever seen anything like that.
In order to see how absurd the “filter design” is, consider filtering a signal of length 999 samples. According to Mann, you should design a Hamming filter of the same length. One should always disregard half of the filter length amount (i.e., 499 in our example) of filtered values from both ends, so in Mann’s case one would end up with a single smoothed value! In Mann’s implementation, however, one ends up with a filtered series of the same length as the original signal.
Another way to think of Mann’s “filter” is to consider it as a normal filter with a huge (half the signal length) zero padding to the both ends of the signal. This interpretation also gives hints why UC’s emulation was so successful. One can also speculate, if the similarity of the results between zero-phase Butterworth and Mann’s original filters is the reason Mann chose the Butterworth filter in the first place.
If someone wants to explore this topic further, I’ve place my Octave/Matlab implementation of Mann’s smoother here. The code includes references to the original Mann’s code I uncovered. Finally, the exact parameters of the trick in MBH9x were as follows. MBH98 has 50-year smoothing with padding of 1981-1995 instrumental. Additionally, the smoothing is cut back 25 samples (half of the “filter length”) from both ends. MBH99 used 40-year filtering with 1981-1997 (not 1998!) instrumental padding. The smooth is cut back 20 samples from the end but not from the beginning.
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