Biology Reference
In-Depth Information
consider the fraction of the variance that can be described by a sine or
cosine waveform of any particular period. The specific mathematical
expression is given by Eq. (9-5). The denominator of this fraction is the
weighted variance of the data minus the mean of the data. To compute
this variance, we first perform a weighted least-squares fit of the data
to a constant, as described in Chapter 8. Then, when the weighted least-
squares value of a is found, the variance of the residuals is calculated as
in Eq. (9-7). This gives the total variance of the data around a fixed
level a, where the value of a has been chosen to minimize the weighted
sum of squared residuals. The numerator of the fraction is the
corresponding weighted variance-of-fit when the data are fit to the
model given in Eq. (9-8)—a sum of a sine wave of period L, a cosine
wave with period L, and a constant a
0
. The estimates for a
0
, a
L
, and b
L
are
determined by a weighted linear least-squares procedure, and the
variance explained by the model is calculated for those values as in
Eq. (9-9).
0
500
1000
1500
Period
FIGURE 9-13.
The power spectrum of the GH data shown in
Figure 9-6.
Variance
Equation
ð
9
-
9
Þ
Variance
Equation
ð
9
-
7
Þ
Fraction of remaining variance
¼
(9-5)
Y
i
a
(9-6)
2
X
X
Y
i
a
SEM
i
R
i
Variance
¼
¼
(9-7)
i
i
a
0
2
þ
2
L
X
i
b
L
sin
2
L
X
i
Y
i
a
L
cos
þ
(9-8)
0
1
2
a
0
2
2
L
X
i
b
L
sin
2
L
X
i
Y
i
a
L
cos
X
X
@
A
R
i
:
Variance
¼
¼
SEM
i
i
i
(9-9)
The lower panel of Figure 9-14 presents a plot of the fraction of
remaining variance after the LH data have been fit to a Fourier
component, as in Eq. (9-8), as a function of the period L. In this example,
there is a dominant Fourier component with a period of about 230
minutes that explains 50.2% of the variance of the data. This component
is shown as a dashed line in the upper panel of Figure 9-14, along
with the original LH data from Figure 9-5. Although the model explains
half of the total variance, this Fourier component does not provide a
perfect description of the experimental data because the physiology
generates waveforms that are not a simple sum of a few sine/cosine
