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

Doubts about Detrending

When I first read about the practice of selecting and detrending the
data, it seemed highly manipulative: First you throw away half the
data, then you suppress the most conspicuous features in what
remains. The choice of a cubic polynomial for the detrending curve
was particularly troubling. Why a cubic, rather than, say, a linear
or an exponential trend line? The obvious answer is that the cubic
curve fits the data very well, and other curves don't, but that
seemed rather *ad hoc*. If you're allowed to invent any
detrending curve you please, then you can generate any result you want.

Muller and Rohde have persuaded me that these concerns were unfounded. The exclusion of genera with uncertain dates was not a way of enhancing the signal—which in fact is just as clear in the complete data set as it is in the well-resolved subset—but rather addressed concerns that poor-quality data might be seen as contaminating their result. And the detrending method has long been standard procedure in Fourier analysis. The polynomial curve is not meant to represent any meaningful trend in the data; it is simply a device for filtering out the lowest-frequency components of the signal, which would otherwise dominate the spectrum and obscure everything else. The long-period trends in diversity—especially the dramatic rise since the Cretaceous—may well be the most intriguing aspects of the fossil record, but they are not the subject of study here. The Fourier analysis is confined to a specific band of frequencies, corresponding to periods of roughly 200 down to 20 million years. The detrending process imposes the long-period limit, and a short-period cutoff comes from the finite resolution of the geologic time scale. Only features within this band are to be examined.

It is the nature of the Fourier transform to highlight the strongest
periodicities in any signal, whatever they might be. Because
*some* peaks are bound to emerge even in a spectrum made from
random data, a crucial question is whether the 62- and
140-million-year peaks climb far enough above the background level
to be considered statistically significant. Muller and Rohde address
this issue through Monte Carlo simulation, generating thousands of
random histograms and running them through their Fourier-analysis
mill. In essence they ask: If we had 10,000 planets like the Earth
and we could dig up fossils on all of them, how often would we see
spectral features as strong as those observed in the real fossil
record? They conclude that a peak as tall and narrow as the
62-million-year signal would turn up randomly no more than 1 percent
of the time; the case for the 140-million-year peak is less
compelling. I have not attempted to reproduce the Monte Carlo
analysis, although it is clearly key to evaluating claims of periodicity.

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