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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.

Fourier transform of the fossil-diversity curve...Click to Enlarge Image

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.








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