Time sequence and spectral analysis - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 187

2.0

1.6

99%

1.2

95%
90%

0.8

0.4

0

4 6 8 10

Degrees of freedom

12 14 16

Figure 2.17 Length of the confidence interval, as a function of the number of
degrees of freedom, on logarithmic plots of spectral density, for confidence levels
of 90%, 95% and 99%.

peculiar to a particular observatory. To bring out features common to all obser-
vatories, the average product, or geometric mean, of the individual power spectra
has been developed (Smylie et al. , 1993). Since it is usual to plot the logarithm
of spectral density, the logarithm of the product spectrum is simply the arithmetic
mean of the logarithms of the individual spectra. The product spectrum may be
regarded as a generalisation of the cross spectrum used to find features common to
two records.
In order to establish confidence intervals for the product spectrum, we construct
the cumulative distribution function, F ( z ), for the random variable z that is the
product of n + 1 random variables x 1 , x 2 ,..., x n , x n + 1 . Thus,

z = x 1 x 2 ··· x n x n + 1 . (2.393)

Function F ( z ) can be found by integrating the joint probability density function
p ( x 1 , x 2 ,..., x n , x n + 1 ) over all of the realisation space below z . The integration then
includes all possible values of the first n − 1 random variables, but only those values

of the last two falling below the rectangular hyperbola

z ξ n − 1
x n

x n + 1 = (2.394)

in the ( x n , x n + 1 ) plane, with

1

=

ξ n − 1

x n − 1 .

(2.395)

x 1 x 2 ···