Are there periodic booms and busts in the diversity of life on Earth? Hear a tale of fossils and Fourier transforms
Life has its ups and downs. Half a billion years ago, the sudden
proliferation of hard-bodied creatures in the Cambrian period was an
uptick; the mass extinction that wiped out the last of the dinosaurs
at the end of the Cretaceous was a major downer.
Several students of the history of life have suggested that these
peaks and valleys have a regularity to them—that they are not
just random fluctuations but periodic oscillations, possibly
synchronized to some external pacemaker. In the 1970s Keith Stewart
Thomson, then of Yale University (and now a columnist for this
magazine), noted surges in the diversity of various animal groups at
intervals of 62 million years. Then Alfred G. Fischer and Michael A.
Arthur of Princeton University suggested that extinctions come in
waves every 32 million years. Later, David M. Raup and J. John
Sepkoski, Jr., of the University of Chicago offered a revised
mass-extinction timetable with a period of 26 million years. Now
there's yet another sighting of cyclic tides in biodiversity, this
time with super-imposed wavelengths of 62 million years (again!) and
140 million years. The new report comes from Richard A. Muller, a
physicist at Lawrence Berkeley National Laboratory, and Robert A.
Rohde, a graduate student in physics at the University of
California, Berkeley. Their analysis was published in
Nature this past March.
To a naive observer, the sheer variety of these proposals invites a
certain skepticism. If there's a loud and steady drumbeat in the
history of life, shouldn't everyone hear the same rhythm? On the
other hand, if the signal is faint and has to be teased out of a
noisy background, could we be perceiving patterns in what is really
random noise? Just how do you go about detecting such an
oscillation, and how do you know whether or not it's real?
A reading of the various claims and counterclaims on periodicity in
the fossil record did not answer these questions for me. I felt an
urge to explore the data for myself, to see just how much teasing it
needs. Some years ago, such an undertaking would have been
unthinkable for anyone but insiders and experts—and I am
neither. But computational science is a great equalizer. The tools
and data are now widely available. The mathematics required is not
too daunting. Muller and Rohde have posted a detailed and very
helpful technical supplement—almost a how-to manual—on
the Nature Web site. If you're willing to write a few
programs, you too can create mass extinctions on your home computer.
Of course access to tools and data does not guarantee the skill to
use them well—as I shall demonstrate forthwith.
A Catalog of Life
The search for patterns in the history of life builds on the labor
of generations of paleontologists who went out in the field to dig
up fossils. It also owes a debt to one paleontologist who went into
the library to dig up thousands of records of fossil discoveries.
John Sepkoski began this research while he was still a student of
Stephen Jay Gould in the 1970s; by the time of Sepkoski's death in
1999 (at age 50), his database had grown to include more than 36,000
genera of marine organisms. The compendium was published in 2002,
both as a weighty tome and as a CD-ROM.
The Sepkoski database has a simple structure. For each genus, he
lists the oldest and the youngest geological layers in which at
least one member of the genus is reported to appear. For example,
the genus Tellinimera carries the notation "K (Camp-l)
- T (Dani), signifying that these bivalve molluscs are first
observed in the lower substage of the Campanian stage of the
Cretaceous period (which is abbreviated K to avoid confusion with
the Cambrian and the Carboniferous); the last appearances are in the
Danian stage of the Tertiary period. (Thus Tellinimera was
one of the lucky survivors of the K-T catastrophe, the extinction
that's famous for doing in the dinosaurs.)
Even though the Sepkoski compendium is available on CD-ROM, getting
it into a form suitable for further analysis is more than a routine
clerical chore. I did some preliminary reformatting with the
search-and-replace functions of a text editor, then wrote a small
program to do further processing, and finally imported the result
into a database manager. What's maddening about such a conversion
process is that even tiny typographical inconsistencies in the
text—a misplaced hyphen, an extra tab character—can
totally derail the operation. Other kinds of errors turn up, too.
For example, I found a few dozen entries where Sepkoski apparently
recorded the same genus twice. Such minor oversights are hardly a
surprise in a document that took decades to compile, and which the
author never had a chance to review and revise before publication.
In any case, for statistical purposes the database needn't be
perfect; random errors might blur a genuine periodic signal, but
they are unlikely to generate a spurious one.
The database gives the dates of fossils in terms of geologic
periods, epochs, stages and so on; for studies of periodicity, these
layers of the stratigraphic column have to be assigned dates and
durations in calendar years. As it happens, a new calibration of the
geologic sequence, assembled by the International Commission on
Stratigraphy, has just been published (in a tome even weightier than
the Sepkoski compendium). Based on radio-isotope measurements,
paleomagnetism and evidence of astronomical cycles, Geologic Time
Scale 2004, or GTS2004, gives dates for strata as far back as the
beginning of the Cambrian period—which according to GTS2004
was 542 million years ago.
The Sepkoski compendium mentions almost 300 geologic intervals, to
which Muller and Rohde assigned numerical dates based on the new
time scale. The task was not entirely straightforward because of
changes and variations in nomenclature. For example, Sepkoski refers
to a Wolf-campian epoch, which is not recognized in GTS2004; Muller
and Rohde defined it as the union of two stages.
How to Date a Fossil
Even after dates have been assigned to the stratigraphic layers, the
lifespans of the fossil organisms are still not quite pinned down.
Consider again the genus Tellinimera. Under the GTS2004
calibration, its first appearance in the lower Campanian could have
been at any time between 83.5 and 77.05 million years ago (mya), and
its last gasp in the Danian was somewhere between 65.5 and 60.2 mya.
Depending on how the dates of origination and extinction are chosen
within those intervals, Tellinimera could have lasted for
anywhere from 11 to 23 million years.
For genera whose dates are known with the greatest
precision—to the substage level of detail—Muller and
Rohde adopt a simple convention: If a genus first appears within a
substage, they set its date of origination to the beginning of that
substage. By this rule Tellinimera is assumed to arise at
83.5 mya. Likewise a last appearance within a substage is assigned
to the end of that substage. Where the data specify only a stage
rather than a substage, Muller and Rohde follow a more complicated
policy, allocating fractions of a genus to each possible
subdivision. Thus the extinction of Tellinimera is shared
equally between the two substages of the Danian stage; half of the
genus dies out at the end of the lower Danian (62.85 mya) and half
at the end of the upper Danian (60.2 mya). For genera dated only at
the epoch or period level, an even more elaborate algorithm comes
The net effect of this procedure is to divide geologic time into a
series of nonoverlapping units, with an average duration of roughly
three million years. Although the fractional allocations spread some
events over several of these units, it is still the case that all
originations and extinctions occur at the boundaries between units.
Nothing ever happens during a substage.
The decision to locate all changes at stratum boundaries has a
plausible argument in its favor. The boundaries were defined in the
first place because they mark distinctive shifts in fossil biota,
and so originations and extinctions ought to be clustered there.
Still, it can't be true that all taxa began and ended their
existence at those selected transition moments. So I decided to try
distributing the events more evenly, a decision made in the spirit
of idle experimentation, to see whether it would have any effect on
For each genus I assigned a date of origination by selecting a
moment at random from within the whole interval in which the
earliest fossil was reported. For Tellinimera all dates in
the lower Campanian, between 83.5 and 77.05 mya, would be equally
likely. Extinction dates are chosen in the same way, by picking a
number at random within the interval of last appearance. (Special
care is needed when a genus begins and ends in the same time unit:
It must not die before it is born.)
Under this plan, the average longevity of a genus is halfway between
the minimum and the maximum possible. Moreover, the scheme has the
attractive property that greater uncertainty in the dating of a
fossil automatically translates into greater variance in the
randomly assigned dates. If all we know about a genus is that it
arose sometime in the Permian, then the randomizing procedure can
assign it any date in the 48-million-year span of that period.
The major drawback of a randomized date assignment is that it makes
the analysis nondeterministic. Every run of the program gives a
slightly different result. But the law of large numbers protects us.
Although any particular genus may be assigned quite different dates
in successive runs, the outcome averaged over all 36,000 genera is
The final step in converting Sepkoski's database into a chronicle of
biological diversity is to construct a histogram giving the number
of extant genera as a function of time. My histograms have bins 1
million years wide, so 542 bins span the interval since the start of
the Cambrian. Once the bins are set up, a program scans through the
list of genera, placing each of the 36,000 origination and
extinction events in its proper bin. Then a pass through the bins
from earliest to latest increments the number of extant genera for
each origination and decrements it for each extinction. The result
is the graph at the top of this column.
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.
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.
For my own first experiments with the analysis of the diversity
curve, I decided to retain the entire set of 36,000 genera rather
than discard the doubtful cases; this proved not to be a problem.
But my attempts to get along without the cubic detrending curve were
unsuccessful. Fitting the data to linear or exponential curves left
large residuals, producing a massive low-frequency lump in the
spectrum that swamped all other signals. I tried piecing together
two linear trends, with a hinge point where the slope changes in the
Cretaceous, but that didn't help much. I had to concede the point:
If you want to examine midrange frequencies in this data set, you
need to remove lower frequencies first. A cubic curve seems to be
the best way to do that.
When I finally got a result, it was not what I expected. I would not
have been surprised to see a spectrum identical to that of Muller
and Rohde; after all, I was working from the same data and following
similar procedures. I would not have been astonished to see
something totally nonsensical, stemming from a bug in my program.
But in fact my graph was very similar to theirs, with peaks in the
same positions, yet there was also a conspicuous difference: The
spikes at 62 and 140 million years had swapped amplitudes. The
140-million-year peak was the higher one, looming over its
Tracking down the source of this discrepancy took more than a week.
My suspicion focused first on the decision to include all the
genera, even those of doubtful provenance. But when I reran the
analysis with only well-resolved genera, the outcome was very
similar, with energy still concentrated in the 140-million-year peak.
Next I considered the main visual difference between the histograms
that I generated and those published by Muller and Rohde.
Randomizing the dates of origination and extinction yields a
smoother contour, without the stairstep profile created when all
changes come at substage boundaries. Maybe the sharp corners of the
stairsteps somehow shift energy into the higher-frequency band?
Again the facts proved me wrong. I applied a smoothing filter to rub
the corners off the Muller-Rohde curve, and another algorithm to add
sharper local transitions to my own histogram. The spectra were
unchanged, continuing to disagree about the relative heights of the peaks.
In the course of my struggles with this issue, I tried altering my
methods in a number of ways, and eventually wound up with a
diversity curve that appeared to match the Muller-Rohde curve in all
but a few local details—and yet still the two spectra
disagreed. Could such tiny disparities have large consequences? The
puzzle was solved by Rohde, who guessed the source of the trouble as
soon as I sent him a copy of my graphs. Sepkoski had cataloged a
handful of genera from the Vendian period, which preceded the
Cambrian. Because the Vendian record was sparse and fragmentary,
Muller and Rohde had excluded it from their analysis. I knew of this
decision, but I had neglected to snip away the long tail of Upper
Vendian stragglers from my version of the database. (They were
included in the Fourier analysis, but were invisible in the graphs I
had been drawing.) Rohde correctly deduced that the presence of
those extra data points, spread out over an interval of 23 million
years, would cause just the distortion I was seeing, reinforcing the
140-million-year wave and damping the 62-million-year one. Once I
truncated my histograms at the start of the Cambrian, the spectra
produced by my program matched the ones published by Muller and Rohde.
Answers and More Questions
The question that launched me on this adventure was whether evidence
of periodicity is something blatant and robust and unmistakable, or
whether the procedures needed to detect it are subtle, temperamental
and subjective. My answers remain murky.
On the one hand, it was a relief to find that no careful selection
or heavy-handed mistreatment of the data were needed to bring forth
the two signals reported by Muller and Rohde. The peaks emerged
clearly from the entire data set or from many different subsets, and
the positions of the peaks along the frequency axis were quite
stable, unaffected by variations in analytic method. Even the
extraneous Vendian data altered only the heights of the peaks, not
their positions at 62 and 140 million years.
But my troubles with the heights of the peaks were chastening. In
the end the cause turned out to be a simple error; but, then again,
I knew that something was amiss only because I had the right answer
given to me. Under other circumstances, the decision to keep or to
discard the Vendian genera might be an open question. The choice
made about this seemingly minor component of the data—it
amounts to 0.2 percent—can have a visually conspicuous effect
on the outcome. (Whether the effect is also statistically
significant is a question I have not addressed.)
Mathematically, the Fourier transform is well-defined and
deterministic, with no more room for subjectivity than, say, the
conversion of rectangular to polar coordinates. The same input
always yields the same output. In practice, though, there are knobs
and dials to twiddle—choices to be made in preparing the input
and presenting the output. As with many other methods, it's these
niggling details—how to deal with outliers, how to correct for
systematic biases—that cause most of the trouble.
Perhaps it is foolish for an untrained amateur even to attempt using
such tools; certainly the tools are not to be blamed just because a
neophyte fails to get the right answer on the first try. But still I
cherish the notion that ordinary readers can assess a scientific
claim for themselves, by analyzing the evidence and working through
the steps of the argument, not by appeals to authority or consensus.
Apart from my methodological muddles, what should we make of the
oscillations in fossil diversity? A tall, sharp peak in a Fourier
spectrum implies that the underlying wave has a very steady
frequency and phase. Such long-term regularity is unusual in
biological systems, and so Muller and Rohde argue that there must be
some external driving force. Muller favors an astronomical
explanation, perhaps something related to the motion of the solar
system through the galaxy. Rohde is more partial to geological
causes, such as recurrent episodes of volcanism caused by periodic
events in the Earth's mantle.
Muller and Rohde have looked for correlations between the cycles in
fossil diversity and various geophysical phenomena, such as
indicators of past climate and sea level. They note a
135-million-year-cycle in glaciation, statistically
indistinguishable from the wavelength of their 140-million-year
cycle. There are several other possible matches as well, but none of
them is compelling enough for Muller and Rohde to endorse one
candidate cause among all the contenders.
© Brian Hayes