by Vaughan Pratt
Paul Clark, the developer of the immensely useful WoodForTrees website that plots climate data, has kindly joined the discussion at Climate Etc., to clarify the meanings of ‘From:’, ‘To:’, and ‘Isolate’ which had been giving some people trouble. In this post I’d like to focus on the third of these, ‘Isolate’, whose utility may not have been fully appreciated.
Those of you who’ve played around with the site Wood For Trees will have
found the moving-average function “Mean n” very useful, where n is the number of months. What “Mean n” accomplishes is to filter the signal with a so-called “box filter” or moving average of width n months. The effect is to completely kill any sinusoidal components of the signal in question whose periods are integer
submultiples of n months, namely periods of the form n/i months where i is an integer, otherwise called the i-th harmonic of that period.
(I’m assuming we’re all nerds here, and can follow this.)
For example “mean 60” will ensure that sine waves of period 60 months,
60/2 = 30 months, 60/3 = 20 months, and so on, will be completely
removed, not a single trace remaining. These removed components of your
signal are the harmonics of the fundamental frequency 158 picohertz, a
very slow oscillation compared to your 3 gigahertz CPU, or the million
gigahertz of ultraviolet light.
If you change “mean 60” to “mean 120” then the harmonics of 79 picohertz
are removed. These include all the harmonics that “mean 60” removed,
but they also include all the frequencies exactly midway between the
harmonics of “mean 60.” So exactly twice as many harmonics are removed
in this way.
In the digital filter business, “Mean n” is what is called a low-pass filter. The WoodForTrees function “Isolate n” is the corresponding high-pass filter that passes through exactly what “Mean n” takes out. The original signal (with ceiling(n/2) months deleted from each end) is equal to the sum of the Mean n and Isolate n signals.
If you want to be honest when filtering, exhibit both the mean and isolate signals together, along with their sum; that is, show all three signals. These are all shorter than the original, having had a total of 2*ceiling(n/2) months removed. State that the third signal is the sum of the other two (and hence is the original signal shortened). People can then judge for themselves whether they like what you’ve done. If you only show the low-passed portion they can’t judge whether they believe what you’ve removed is merely the high-passed stuff.
The downside of “Isolate” used in isolation is that the amplitude of the high frequency information (seasonal cycles in particular) tends to drown everything else out, including any low frequencies that mean n failed to pass through. For this reason it is often worthwhile to do one or two Mean n’s with smaller n as well.
A single use of Mean n convolves (technical term, google it, but basically it means filtering) with a box. Two uses is equivalent to convolving with a triangle instead of a box, which takes out most of the high frequencies that one box leaves in. That plus an Isolate 2n then gives the highest frequencies retained by the mean n’s.
Means and isolates all commute so you can give them in any order.
If instead of using Mean n twice you use Mean n and Mean 2n/3, the
latter flattens the lobe between 3n and 3n/2 (the 2nd harmonic of 3n)
even better but without taking out as much of the frequencies below
period 3n. For example you might use Mean 30 in conjunction with Mean
20, or Mean 54 with Mean 36. This kind of pairing makes for quite a
nice filter. More elaborate filters can be constructed along the same
lines with as many Mean’s as you want. The fact that they all commute
simplifies the space to be explored. It is highly unlikely that you’ll
ever see much benefit from using more than three Means; I’ve never had
occasion to explore beyond two.
With careful choices you can see all sorts of interesting things in the
various signals Paul has provided. For example the Joint Institute for
the Study of the Atmosphere and Ocean (JISAO)’s temperature record
of the Pacific Decadal Oscillation since 1900 can be smoothed with
this filter to remove essentially all seasonal variation. Note the
4.1 °C (7.4 °F) plunge in the 15 years between 1941 and 1956!
There is more to ocean variation than just the 0.1 °C amplitude
Atlantic Multidecadal Oscillation.
But why is the PDO called “decadal”? Is there a 10-year (120-month)
cycle? We could try to answer this using isolate 120 which passes
exactly 100% of any 120-month period, in combination say with the 2-3
punch of mean 120/2 mean 120/3. However this doesn’t give as good a
result as when you isolate 90 months (7.5 years) (with mean 30 mean 20),
which gives this beautiful plot. This alternates half-degree oscillations
(1910-1930, 1960-1985) with one-degree oscillations (1930-1960, 1985-2010).
What else is in the PDO data? Let’s try to isolate 240, which gives this
interesting plot. We see a 20-year-period oscillation roughly matching the Hale cycle. We’ll see (and explain) the Hale cycle again in the HADCRUT3 data below.
Anything else? Easy, just change “isolate” to “mean” as in this graph and we get a 60-year-period cycle with an impressive swing (2x amplitude) of more than a degree!
We can play around like this with other datasets than the PDO, for example HADCRUT3 (to get away from ocean oscillations for a bit). This plot picks out a cycle from the global land-sea temperature that correlates well with the solar cycle, sunspots, with the exception of a curious speed-up during 1920-1930 that is too fast to be sunspots. Put that on the stack of things to look into later.
In 1903 astronomer George Hale discovered that at every other sunspot
cycle the magnetic field of the Sun reversed, as though it contained a
magnet spinning with a period of 256 months (64/3 = 21.33 Earth years).
Two sunspot cycles are now called a Hale cycle in his honor. Pyramids and magnets are good for your health on Earth, but could a magnet spinning nearly a hundred million miles away have any perceptible impact on global climate Here’s a graph designed to test this hypothesis using isolate 250. You be the judge.
Incidentally I picked the first mean, mean 100, at random, with mean 67
following the above 2/3 rule to eliminate all high frequenices. Had the
second mean been 50 we’d have got this plot. Note the curious oscillation that suddenly pops up during 1920-1930. This consists of three cycles of a mysterious
67-month-period oscillation that snuck through the gap between mean 100
and mean 50.
Oh, but that was what we put on the stack four paragraphs ago. How very
interesting.
Obviously one can play this game forever, finding more and more spectral
curiosities in the various datasets at woodfortrees. Here for example is a single
60-year cycle in HADCRUT3. But you get the idea. Anyone can play.
Which raises the question, what exactly is the game here? Is it physics, metaphysics, philosophy, computer science, statistics, the latest weapons for skeptics and warmists to wage war, or what? My answer would be statistics. While I trust theory up to a point, I trust even more the evidence of my senses augmented by the powerful sensing instruments we’ve developed to date.
When there is a discrepancy between observation and theory, my first
thought is that the observational instrument is at fault. But when I
fail to find any fault, my attention turns to the theory. And sometimes
the theory is shown to be wrong, or at least incomplete. And this is
how science, our understanding of nature, has been progressing for as
long as anyone knows.
My high school French teacher, a white Russian who’d emigrated from Paris to Sydney, was in the habit of saying in class that if he’d taught even one of us to speak and write French properly he would have fulfilled his role here on
Earth. In that same spirit, if I’ve changed the mind of even one reader
of Judith’s blog, even by a little, I will feel that this post was not
entirely in vain.