The astronomical theory and data compared - Ice Ages and Interglacials: Measurements, Interpretation, and Models - page 303

to establish the chronology of the time series, agreement between the frequency
spectra of solar and time series data might be to some extent a consequence of
circular reasoning. M&M placed great emphasis on spectral analysis. However, in
this topic I have relegated spectral analysis to a secondary role.
Consider some arbitrary function G ð t Þ . If the average value of G ð t Þ over all t
is h G i , we define deviation from the average as F ð t Þ¼ G ð t Þh G i .
Spectral analysis is based on the fact that almost any such function F ð t Þ can
be expressed as a Fourier transform in terms of an integral over cosine and sine
functions over all frequencies. Thus, if we consider an arbitrary function F ð t Þ
(measured from the average of G ð t Þ as specified above) which varies with the inde-
pendent variable t (in our case t is time and F ð t Þ may be temperature, ice volume,
or solar intensity) we may express F ð t Þ as:

ð þ1

Þ e 2 ift df F ð t Þ¼ H Tr ð f

1

e 2 ift ¼ cos ð 2 ft Þþ i sin ð 2 ft Þ

where f is frequency, and H Tr ð f Þ is the weighting function for various frequencies
that contribute to making up F ð t Þ . The subscript ''Tr'' is assigned to H to indicate
that this is the ''true'' mathematical distribution of frequencies that produce the
function F ð t Þ when integrated over all frequencies. The inverse of this integral is:

ð F ð t Þ e 2 ift dt

H Tr ð f Þ¼

and the integral is taken over all time.
In actual practice, one does not deal with a continuous function but, rather, a
set of discrete data points. Thus, one has a table of data representing a time series
such as:

t t 1 t 2 t 3 t 4 ...

F ð t Þ F 1 F 2 F 3 F 4 ...

The goal is to fit the data in this table to an expansion of cosines and sines to
find the frequency spectrum of F ð t Þ . If the function varies in a regular repeatable
way, the frequencies that contribute to the function may be narrowly peaked.
However,

if the function varies haphazardly and randomly, the frequency

spectrum may be very broad.
When a set of discrete data points are involved, we approximate the integral

as:

H ð f Þ¼ X F j ð cos ð 2

ft j Þþ i sin ð 2 ft j ÞÞ

where the sum is taken over all the data points ( j ¼ 1toN). Here, H ð f Þ is an

approximation to the true H Tr ð f Þ .