COMPUTING SCIENCE
Life Cycles
Are there periodic booms and busts in the diversity of life on Earth? Hear a tale of fossils and Fourier transforms
Brian Hayes
Inside the Black Box
Does that wiggly line reveal a periodic oscillation? There are
certainly plenty of humps and dips, including deep valleys that
correspond to several mass extinctions. But are the ups and downs
periodic, with a fixed time scale? Or do they look more like the
meandering of a random walk? The eye is not a reliable judge in such
matters, sometimes inventing regularities that don't exist and
missing others that do.
A better tool for teasing out periodicity is Fourier analysis,
Joseph Fourier's mathematical trick for taking apart a curve with
arbitrarily intricate wiggles and reassembling it out of simple sine
waves. The Fourier transform identifies a set of component waves
that add up to a replica of any given signal. The result can be
presented as a power spectrum, which shows the amount of energy in
the signal at each frequency.
Fourier analysis is often treated as a black box. Put in any
time-domain signal, turn the crank, and out comes the
frequency-domain equivalent, with no need to worry about how the
process works. Muller has argued against this kind of mystification;
he is co-author (with Gordon J. MacDonald) of an excellent book on
spectral analysis that opens the lid of the box. Among other things,
Muller and MacDonald present a complete program for Fourier analysis
in seven lines of basic.
The black-box approach to Fourier transforms is not only unnecessary
but also misleading. It's simply not true that you can run any data
through a Fourier analysis and expect a meaningful result. On the
contrary, rather careful preprocessing is needed.
Here are the preliminaries Muller and Rohde went through with the
fossil-diversity data. First they selected only the
"well-resolved genera, those dated to the stage or substage
level; they also excluded all genera known only from a single
stratum. This refinement process discards fully half of the data
set. Next, they calculated the cubic polynomial that best fits the
data and subtracted this "detrending curve from the data. The
residual values left by the subtraction form a new curve in which
the largest-scale (or lowest-frequency) kinks have been straightened
out. This is the curve they finally submitted to Fourier analysis.
Muller and Rohde's result—or rather my reconstruction of
something like it—appears to the right. The spectrum has a
tall spike at a period of 62 million years and a lesser peak at 140
million years, indicating that these two periods account for most of
the energy in the signal.
» Post Comment